The Illustrated Wavelet Transform Handbook
Introductory Theory and Applications in Science,
Engineering, Medicine and Finance


The Illustrated Wavelet
Transform Handbook
Introductory Theory and Applications in Science,
Engineering, Medicine and Finance
Paul S Addison
Napier University, Edinburgh, UK
loP
Institute of Physics Publishing
Bristol and Philadelphia

(Q lOP Publishing Ltd 2002
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For Hannah, Stephen, Anthony and Michael


Contents
Preface
.
Xl
1 Getting started
1.1 Introduction
1.2 The wavelet transform
1.3 Reading the book
1
1
2
3
2 The continuous wavelet transform 6
2.1 Introduction 6
2.2 The wavelet 6
2.3 Requirements for the wavelet 9
2.4 The energy spectrum of the wavelet 9
2.5 The wavelet transform 11
2.6 Identification of coherent structures 14
2.7 Edge detection 21
2.8 The inverse wavelet transform 25
2.9 The signal energy: wavelet-based energy and power spectra 28
2.10 The wavelet transform in terms of the Fourier transform 33
2.11 Complex wavelets: the Morlet wavelet 35
2.12 The wavelet transform, short time Fourier transform and Heisenberg
boxes 45
2.13 Adaptive transforms: matching pursuits 51
2.14 Wavelets in two or more dimensions 55
2.15 The CWT: computation, boundary effects and viewing 56
2.16 Endnotes 63
2.16.1 Chapter keywords and phrases 63
2.16.2 Further resources 63
3 The discrete wavelet transform 65
3.1 Introduction 65
3.2 Frames and orthogonal wavelet bases 65
3.2.1 Frames 65
. .
VB

Vl11 Contents
3.2.2 Dyadic grid scaling and orthonormal wavelet transforms 67
3.2.3 The scaling function and the multiresolution representation 69
3.2.4 The scaling equation, scaling coefficients and associated
wavelet equation 72
3.2.5 The Haar wavelet 73
3.2.6 Coefficients from coefficients: the fast wavelet transform 75
3.3 Discrete input signals of finite length 77
3.3.1 Approximations and details 77
3.3.2 The multiresolution algorithm-an example 81
3.3.3 Wavelet energy 83
3.3.4 Alternative indexing of dyadic grid coefficients 85
3.3.5 A simple worked example: the Haar wavelet transform 87
3.4 Everything discrete 91
3.4.1 Discrete experimental input signals 91
3.4.2 Smoothing, thresholding and denoising 96
3.5 Daubechies wavelets 104
3.5.1 Filtering 112
3.5.2 Symmlets and coiflets 115
3.6 Translation invariance 117
3.7 Biorthogonal wavelets 119
3.8 Two-dimensional wavelet transforms 121
3.9 Adaptive transforms: wavelet packets 133
3.10 Endnotes 141
3.10.1 Chapter keywords and phrases 141
3.10.2 Further resources 141
4 Fluids 144
4.1 Introduction 144
4.2 Statistical measures 145
4.2.1 Moments, energy and power spectra 145
4.2.2 Intermittency and correlation 152
4.2.3 Wavelet thresholding 153
4.2.4 Wavelet selection using entropy measures 159
4.3 Engineering flows 160
4.3.1 Jets, wakes, turbulence and coherent structures 160
4.3.2 Fluid-structure interaction 171
4.3.3 Two-dimensional flow fields 174
4.4 Geophysical flows 178
4.4.1 Atmospheric processes 178
4.4.2 Ocean processes 186
4.5 Other applications in fluids and further resources 187
5 Engineering testing, monitoring and characterization 189
5.1 Introduction 189
5.2 Machining processes: control, chatter, wear and breakage 189
5.3 Rotating machinery 195

Contents IX
5.3.1 Gears
5.3.2 Shafts, bearings and blades
5.4 Dynamics
5.5 Chaos
5.6 Non-destructive testing
5.7 Surface characterization
5.8 Other applications in engineering and further resources
5.8.1 Impacting
5.8.2 Data compression
5.8.3 Engines
5.8.4 Miscellaneous
195
199
202
208
211
221
224
224
225
228
229
6 Medicine 230
6.1 Introduction 230
6.2 The electrocardiogram 230
6.2.1 ECG timing, distortions and noise 231
6.2.2 Detection of abnormalities 234
6.2.3 Heart rate variability 236
6.2.4 Cardiac arrhythmias 239
6.2.5 ECG data compression 248
6.3 Neuroelectric waveforms 248
6.3.1 Evoked potentials and event-related potentials 249
6.3.2 Epileptic seizures and epileptogenic foci 252
6.3.3 Classification of the EEG using artificial neural networks 255
6.4 Pathological sounds, ultrasounds and vibrations 258
6.4.1 Blood flow sounds 259
6.4.2 Heart sounds and heart rates 260
6.4.3 Lung sounds 263
6.4.4 Acoustic response 264
6.5 Blood flow and blood pressure 267
6.6 Medical imaging 270
6.6.1 Ultrasonic images 270
6.6.2 Magnetic resonance imaging, computed tomography and
other radiographic images 270
6.6.3 Optical imaging 273
6.7 Other applications in medicine 275
6.7.1 Electromyographic signals 275
6.7.2 Sleep apnoea 275
6.7.3 DNA 276
6.7.4 Miscellaneous 276
6.7.5 Further resources 277
7 Fractals, finance, geophysics and other areas 278
7.1 Introduction 278
7.2 Fractals 278
7.2.1 Exactly self-similar fractals 279

x Contents
7.2.2 Stochastic fractals
7.2.3 Multifractals
7.3 Finance
7.4 Geophysics
7.4.1 Properties of subsurface media
7.4.2 Surface feature analysis
7.4.3 Climate, clouds, rainfall and river levels
7.5 Other areas
7.5.1 Astronomy
7.5.2 Chemistry and chemical engineering
7.5.3 Plasmas
7.5.4 Electrical systems
7.5.5 Sound and speech
7.5.6 Miscellaneous
282
292
294
298
299
305
307
309
309
310
311
311
312
313
Appendix Useful books, papers and websites
1 Useful books and papers
2 Useful websites
314
314
315
References
317
Index
351

Preface
Over the past decade or so wavelet transform analysis has emerged as a major new
time-frequency decomposition tool for data analysis. This book is intended to
provide the reader with an overview of the theory and practical application of wavelet
transform methods. It is designed specifically for the 'applied' reader, whether he or
she be a scientist, engineer, medic, financier or other.
The book is split into two parts: theory and application. After a brief first chapter
which introduces the main text, the book tackles the theory of the continuous wavelet
transform in chapter 2 and the discrete wavelet transform in chapter 3. The rest of the
book provides an overview of a variety of applications. Chapter 4 covers fluid flows.
Chapter 5 tackles engineering testing, monitoring and characterization. Chapter 6
deals with a wide variety of medical research topics. The final chapter, chapter 7,
covers a number of subject areas. In this chapter, three main topics are considered
first-fractals, finance and geophysics-and these are followed by a general discus-
sion which includes many of other areas not covered in the rest of the book.
The theory chapters (2 and 3) are written at an advanced undergraduate level. In
these chapters I have used italics for both mathematical symbols and key words and
phrases. The key words and phrases are listed at the end of each chapter and the
reader new to the subject might find it useful to jot down the meaning of each key
word or phrase to test his or her understanding of them. The applications chapters
(4 to 7) are at the same level, although a considerable amount of useful information
can be gained without an in-depth knowledge of the theory in chapters 2 and 3,
especially in providing an overview of the application of the theory.
It is envisaged that the book will be of use both to those new to the subject, who
want somewhere to begin learning about the topic, and also those who have been
working in a particular area for some time and would like to broaden their perspec-
tive. It can be used as a handbook, or 'handy book', which can be referred to when
appropriate for information. The book is very much 'figure driven' as I believe that
figures are extremely useful for illustrating the mathematics and conveying the
concepts. The application chapters of the book aim to make the reader aware of
the similarities that exist in the usage of wavelet transform analysis across disciplines.
In addition, and perhaps more importantly, it is intended to make the reader aware of
wavelet-based methods in use in unfamiliar disciplines which may be transferred to
Xl

xii Preface
his or her own area-thus promoting an interchange of ideas across discipline bound-
arIes.
The application chapters are essentially a whistle-stop tour of work by a large
number of researchers around the globe. Some examples of this work are discussed
in more detail than others and, in addition, a large number of illustrations have
been used which have been taken (with permission) from a variety of published
material. The examples and illustrations used have been chosen to provide an
appropriate range to best illustrate the wavelet-based work being carried out in
each subject area. It is not intended to delve deeply into each subject but rather
provide a brief overview. It is then left to the reader to follow up the relevant
references cited in the text for themselves in order to delve more deeply into each
particular topic as he or she requires.
I refer to over seven hundred scientific papers in this book which I have collected
and read over the past three or so years. I have made every effort to describe the work
of others as concisely and accurately as possible. However, if I have misquoted, mis-
represented, misinterpreted, or simply missed out something I apologise in advance.
Of course, all comments are welcome-e-mail address below.
The book stems from my own interest in wavelet transform analysis over the past
few years. This interest has led to a number of research projects concerning the wave-
let-based analysis of both engineering and medical signals including: non-destructive
testing signals, vortex shedding signals in turbulent fluid flows, digitized spatial
profiles of structural cracks, river bed sediment surface data sets, phonocardiographic
signals, pulse oximetry traces (photoplethysmograms) and the electrocardiogram
(ECG), the latter leading to patent applications and a university spin-off company,
Cardiodigital Ltd.
Quite a mixed bag, at first appearance, but with a common thread of wavelet
analysis running throughout. I have featured some of this work in the appropriate
chapters. However, I have tried not to swamp the application chapters with my
own work-although the temptation was high for a number of reasons including
knowledge of the work, ease of reproduction etc. I hope I have struck the correct
balance.
All books reflect, to some extent, the interests and opinions of the author and,
although I have tried to cover as broad a range of examples as possible, this one is
no exception. Coverage is weighted to those areas in which I have more interest:
fluids, engineering, medicine and fractal geometry. Geophysics and finance are
given less space and other areas (e.g. astronomy, chemistry, physics, non-medical
biology, power systems analysis) are detailed briefly in the final chapter.
There are some idiosyncrasies in the text which are worth pointing out. I am anf
person not an w person: I prefer Hertz to radians per second. I can tap my fingers at
approximately 5 Hz, or 1 Hz, I know what 50 Hz means (mains hum in the UK) and
so on: w, I have to convert. Hence the frequencies in the text are in the form of l/time
either in Hertz or non-dimensionalized. The small downside is that the mathematics,
in general, contain a few more terms-mostly 2s and 7rS. I have devoted a whole
chapter to the continuous wavelet transform. It is noticeable that many current
wavelet texts on the market deal only with the discrete wavelet transform, or give
the continuous wavelet transform a brief mention en route to the theory of the discrete

Preface Xl11
wavelet transform. I believe that the continuous wavelet transform has much to offer
a wide variety of data analysis tasks and I attempt, through this text, to redress the
balance somewhat. (Actually, the proportion of published papers which concern
the continuous wavelet transform as opposed to the discrete wavelet transform is
much higher than that represented by the currently available wavelet text books.)
The book is focused on the wavelet transform and makes only passing reference in
the application chapters to some of the other time-frequency methods now available.
However, I have added sections on the short time Fourier transform and matching
pursuits towards the end of chapter 2 and on wavelet packets at the end of chapter
3 respectively. Finally, note that I have developed the discrete wavelet transform
theory in chapter 3 in terms of scale rather than resolution, although the relationship
between the alternative notations is explained.
I would like to thank the following people for taking the time to comment on
various drafts of the manuscript: Andrew Chan of Birmingham University, Gareth
Clegg of Edinburgh University (formerly at The Royal Infirmary of Edinburgh),
Maria Haase of Stuttgart University and Alexander Droujinine of Heriot-Watt
University. I would like to thank Jamie Watson of CardioDigital Ltd for his
comments on the draft manuscript and for his close collaboration over the years
(and various bits of computer code!). I would also like to thank all those authors
and publishers who gave their consent to reproduce their figures within this text. I
am grateful to those funding bodies who have supported my research in wavelet
analysis and other areas over the years, including the Engineering and Physical
Science Research Council (EPSRC), the Medical Research Council (MRC) and the
Leverhulme Trust. And to those other colleagues and collaborators with whom my
wavelet research is conducted and who make it so interesting, thanks.
Special thanks to my wife, Stephanie, who has supported and encouraged me
during the writing of this book. Special thanks also to my parents for their support
and great interest in what I do.
Although it has been a long hard task, I have enjoyed putting this book together.
I have certainly got a lot out of it. I hope you find it useful.
Paul S Addison
January 2002
p.addison@napier.ac.uk

Chapter 1
Getting started
1.1 Introduction
The wavelet transform (WT) has been found to be particularly useful for analysing
signals which can best be described as aperiodic, noisy, intermittent, transient and
so on. Its ability to examine the signal simultaneously in both time and frequency
in a distinctly different way from the traditional short time Fourier transform
(STFT) has spawned a number of sophisticated wavelet-based methods for signal
manipulation and interrogation. Wavelet transform analysis has now been applied
in the investigation of a multitude of diverse physical phenomena, from climate
analysis to the analysis of financial indices, from heart monitoring to the condition
monitoring of rotating machinery, from seismic signal denoising to the denoising
of astronomical images, from crack surface characterization to the characterization
of turbulent intermittency, from video image compression to the compression of
medical signal records, and so on.
Many of the ideas behind wavelet transforms have been in existence for a
long time. However, wavelet transform analysis as we now know it really
began in the mid-1980s where they were developed to interrogate seismic signals.
Interest in wavelet analysis remained within a small, mainly mathematical
community during the rest of the 1980s with only a handful of scientific papers
coming out each year. The application of wavelet transform analysis in science
and engineering really began to take off at the beginning of the 1990s, with a
rapid growth in the numbers of researchers turning their attention to wavelet
analysis during that decade. The past few years have each seen the publication
of over one thousand refereed journal papers concerning the wavelet transform,
covering all numerate disciplines. Figure 1.1 shows this rapid increase in wave-
let-based scientific papers published in recent years. The wavelet transform is a
mathematical tool which is now common in many data analysts' toolboxes.
This book aims to provide the reader both with an introduction to the theory
of wavelet transforms and an overview of its use in practice. The two remaining
sections of this short chapter contain, respectively, a brief non-mathematical
description of the wavelet transform and a guide to subsequent chapters of the
book.
1

2
Getting started
1400
200
1200
1000
00
I-;
Q)
 800

o
I-;
Q)
"S 600
::s
s:::
400
o
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
year
Figure 1.1. Yearly count of scientific papers concerning wavelets 1990-2001. The plot contains the
number of papers with 'wavelet' or 'wavelets' in the title, keywords or abstract of refereed journal
papers. Source: Web of Science, http//wos.mimas.ac.uk/ (Note that a handful of these papers do
not concern wavelet transforms, but rather refer to physical phenomena where the term wavelet
has been used to describe a small localized wave.)
1.2 The wavelet transform
Wavelet transform analysis uses little wavelike functions known as wavelets. Actually,
'local' wavelike function is a more accurate description of a wavelet. Figure 1.2(a)
shows a few examples of wavelets commonly used in practice. Wavelets are used to
transform the signal under investigation into another representation which presents
the signal information in a more useful form. This transformation of the signal is
known as the wavelet transform. Mathematically speaking, the wavelet transform is
a convolution of the wavelet function with the signal and we will see exactly how
this is done in chapters 2 and 3. Here we stick to schematics.
The wavelet can be manipulated in two ways: it can be moved to various locations
on the signal (figure 1.2(b)) and it can be stretched or squeezed (figure 1.2(c)). Figure
1.3 shows a schematic of the wavelet transform which basically quantifies the local
matching of the wavelet with the signal. If the wavelet matches the shape of the
signal well at a specific scale and location, as it happens to do in the top plot of
figure 1.3, then a large transform value is obtained. If, however, the wavelet and
the signal do not correlate well, a low value of the transform is obtained. The trans-
form value is then located in the two-dimensional transform plane shown at the
bottom of figure 1.3 (indicated by the black dot). The transform is computed at vari-
ous locations of the signal and for various scales of the wavelet, thus filling up the
transform plane: this is done in a smooth continuous fashion for the continuous wave-
let transform (CWT) or in discrete steps for the discrete wavelet transform (DWT).

Reading the book
3
(a)
(b)
(c)

 
.
f\
 
.
Figure 1.2. The little wave. (a) Some wavelets. (b) Location. (c) Scale.
Plotting the wavelet transform allows a picture to be built up of the correlation
between the wavelet-at various scales and locations-and the signal. In subsequent
chapters we will cover the wavelet transform in more mathematical detail.
1.3 Reading the book
The purpose of the book is both to introduce the wavelet transform and to convey its
multidisciplinary nature. This is done in the subsequent chapters of the book by first
providing an elementary introduction to wavelet transform theory and then present-
ing a wide range of examples of its application. It will quickly become apparent that

4
Getting started
\
v
)
local matching of
wavelet and signal leads
to a large transform value
L1
Wavelet
Transform
L1
scale
current
wavelet -- --- --- --- --- --- --- --- -- --- -- ..
scale
/:::/::::::::::::::::::::::::::::::
/::::::// wavelet :::::::::::::::
:/:::/ transform :/::::/
 . 
/::/:: / plot /:://
/ /::::::::::::: // //:::::::/
....' ,.' ....' ".... ,.' ........'
....' ".'
location
current
wavelet
location
Figure 1.3. The wavelet, the signal and the transform.
very often the same wavelet methods are used to interrogate signals from very differ-
ent subject areas, where quite unrelated phenomena are under investigation.
The book is split into two distinct parts: the first part, chapters 2 and 3, deals
respectively with continuous wavelet transform theory and discrete transform
theory; while the second part, chapters 4, 5, 6 and 7, presents examples of their appli-
cation in science, engineering, medicine and finance. There are a number of ways to
read this book: from the linear (beginning to end) via the targeted (employing the
index) to the random (flicking through) approach. The reader unfamiliar with wavelet
theory should read chapters 2 and 3 before moving on to the sections of particular
relevance to his or her area of interest. The reader is also advised to look outside
his or her own area to see how wavelets are being employed elsewhere. Details of
further resources concerning the theory and applications of wavelet analysis are
provided at the end of each chapter. The appendix lists a selection of useful books,
papers and websites. The book contents are outlined in more detail as follows:
Chapter 2: This chapter presents the basic theory of the continuous wavelet trans-
form. It outlines what constitutes a wavelet and how it is used in the transformation of
a signal. In the latter part of the chapter the continuous wavelet transform is

Reading the book
5
compared both with the short time Fourier transform and the matching pursuit
method.
Chapter 3: The discrete wavelet transform is described in this chapter. Ortho-
normal discrete wavelet transforms are considered in detail, in particular those of
Haar and Daubechies. These wavelets fit into a multiresolution analysis framework
where a discrete input signal can be represented at successive approximations by
a combination of a smoothed signal component plus a sum of detailed wavelet
components. The chapter ends by looking briefly at wavelet packets, a generalization
of the discrete wavelet transform which allows for adaptive partitioning of the time-
frequency plane.
Chapter 4: This chapter deals with fluid mechanics, a subject that is always
hungry for new mathematical techniques. The time-frequency localization properties
of the wavelet transform have been employed extensively in the study of a wide variety
of fluid phenomena including the intermittent nature of fluid turbulence, the charac-
teristics of turbulent jets, the nature of fluid-structure interactions and the behaviour
of large scale geophysical flows. Chapter 4 also contains the mathematics for discrete
wavelet statistics and power spectra following on from some of the basic theory given
in chapter 3.
Chapter 5: In this chapter a close look is taken at the application of wavelet trans-
forms to a variety of pertinent problems in engineering. These applications include the
assessment of machine processes behaviour, condition monitoring of rotating
machinery, the analysis of nonlinear and transient oscillations, the characterization
of repeated impacting on structural elements, the interrogation of non-destructive
testing signals, and the characterization of rough surfaces.
Chapter 6: Medical applications of wavelet transform analysis are covered in this
chapter. Wavelet transform methods have been used to characterize a wide variety of
medical signals. Many of these are reviewed in this chapter, including the ECG, EEG,
EMG, pathological sounds (lung sounds, heart sounds and arterial sounds), blood
flows, blood pressures, DNA sequences and medical images (optical, x-ray, NMR,
ultrasound etc.).
Chapter 7: This final chapter covers a variety of areas of application. Most of
the chapter is devoted to three main subjects-fractal geometry, finance and geo-
physics-with a separate section devoted to each of them. The final part of the
chapter provides a brief account of the role wavelet transform analysis has played
in a number of other areas including astronomy, plasma physics, electrical power
systems, chemical analysis and more.
Appendix: The appendix contains a list of useful papers, books and websites
concerning wavelet transform theory and application. These have been chosen by
the author for their extensive content and/or clarity of presentation.

Chapter 2
The continuous wavelet transform
2.1 Introduction
This chapter covers the basic theory of the continuous wavelet transform (CWT).
We will first determine what constitutes a wavelet, how it is used in the transfor-
mation of a signal and what it can tell us about the signal. We then consider the
inverse wavelet transform and the reconstruction of the original signal. We will
look at the energy-preserving features of the wavelet transform and how it may be
used to produce wavelet power spectra. Finally, we will compare the wavelet
transform both with the short time Fourier transform (STFT) and matching pursuit
(MP) method.
2.2 The wavelet
The wavelet transform is a method of converting a function (or signal) into another
form which either makes certain features of the original signal more amenable to
study or enables the original data set to be described more succinctly. To perform
a wavelet transform we need a wavelet which, as the name suggests, is a localized
waveform. In fact, a wavelet is a function 1jJ( t) which satisfies certain mathematical
criteria. As we saw briefly in the previous chapter, these functions are manipulated
through a process of translation (i.e. movements along the time axis) and dilation
(i.e. spreading out of the wavelet) to transform the signal into another form which
'unfolds' it in time and scale. Note that in this chapter and the next we assume that
the signal to be analysed is a temporal signal, i.e. some function of time such as a
velocity trace from a fluid, vibration data from an engine casing, an ECG signal
and so on. However, many of the applications discussed later in the book concern
wavelet analysis of spatial signals, such as well logged geophysical data, crack surface
profiles, etc. In these cases, the independent variable is space rather than time;
however, the wavelet analysis is performed in exactly the same way.
A selection of wavelets commonly used in practice is shown in figure 2.1. We will
consider some of them in more detail as we proceed through the text. As we can see
from the figure they have the form of a small wave, localized on the time axis. There
6

The wavelet
7
1!J(t)
1!J(t)
(a) (b)
t t
1!J(t) 1!J(t)

-
(c)
(d)
t
t
Figure 2.1. Four wavelets. (a) Gaussian wave (first derivative of a Gaussian). (b) Mexican hat (second
derivative of a Gaussian). (c) Haar. (d) Morlet (real part).
are, in fact, a large number of wavelets to choose from for use in the analysis of our
data. The best one for a particular application depends on both the nature of the
signal and what we require from the analysis (i.e. what physical phenomena or process
we are looking to interrogate, or how we are trying to manipulate the signal). We will
begin this chapter by concentrating on a specific wavelet, the Mexican hat, which is
very good at illustrating many of the properties of continuous wavelet transform
analysis. The Mexican hat wavelet is shown in figure 2.1 (b) and in more detail in
figure 2.2( a). The Mexican hat wavelet is defined as
1jJ ( t) == (1 - t 2 ) e - t 2 / 2
(2.1)
The wavelet described by equation (2.1) is known as the mother wavelet or analysing
wavelet. This is the basic form of the wavelet from which dilated and translated
versions are derived and used in the wavelet transform. The Mexican hat is, in fact,
the second derivative of the Gaussian distribution function e- t2 / 2 : that is, with unit
variance but without the usual 1/ V21f normalization factor. The Mexican hat
normally used in practice, i.e. that given by equation (2.1) and shown in figure
2.2(a), is actually the negative of the second derivative of the Gaussian function.
All derivatives of the Gaussian function may be employed as a wavelet. Which is
the most appropriate one to use depends on the application. Both the first and
second derivatives of the Gaussian are shown in figures l(a) and l(b). These are
the two that are most often used in practice. Higher-order derivatives are less
commonplace.

8
The continuous wavelet transform
1
0.5
-..
......
 0
-7
-0.5
-1
-4 -3 -2 -1 0 1 2 3 4
 t
(a)
b
4
o
-0.4 -0.2
0.2 0.4 0.6 0.8 1.0
I
I p
Ie
.
.
a .
. .
.
.
I
S2

-1.0 -0.8 -0.6
(b)
V2
Peak frequency - !p = 2Jt
v5/2
Bandpass frequency - Ie = 2Jt
V2
and also - f, =--
P 2Jt
Figure 2.2. The Mexican hat mother wavelet and its associated energy spectrum. ( a) The Mexican hat
mother wavelet (named for an obvious reason!). Notice that, for the Mexican hat, the dilation
parameter a is the distance from the centre of the wavelet to where it crosses the horizontal axis.
(b) The energy spectrum of the Mexican hat shown in (a). Note that as it is a real wavelet its Fourier
spectrum is mirrored around the zero axis. (a is the standard deviation of the spectrum around the
vertical axis.)

The energy spectrum of the wavelet 9
2.3 Requirements for the wavelet
In order to be classified as a wavelet, a function must satisfy certain mathematical
cri teria. These are:
1. A wavelet must have finite energy:
E = ,[)(J I?j;(t) 1 2 dt < 00
(2.2)
where E is the energy of a function equal to the integral of its squared magnitude and
the vertical brackets II represent the modulus operator which gives the magnitude of
1jJ( t). If 1jJ( t) is a complex function the magnitude must be found using both its real
and complex parts.
2. If (f) is the Fourier transform of 1jJ(t), i.e.
(.f) = .eX) ?j;( t) e -i(27rf)t dt
(2.3)
then the following condition must hold:
A 2
C == .f oo 1 ?j;(.f) 1 df < 00
g 0 f
(2.4)
This implies that the wavelet has no zero frequency component, (O) == 0 or, to put it
another way, the wavelet 1jJ(t) must have a zero mean. Equation (2.4) is known as the
admissibility condition and C g is called the admissibility constant. The value of C g
depends on the chosen wavelet and is equal to 7r for the Mexican hat wavelet given
in equation (2.1).
3. An additional criterion that must hold for complex wavelets is that the Fourier
transform must both be real and vanish for negative frequencies. We shall consider
complex wavelets towards the end of this chapter when we will take a close look at
the Morlet wavelet.
2.4 The energy spectrum of the wavelet
Wavelets satisfying the admissibility condition (equation (2.4)) are in fact bandpass
filters. This means in simple terms that they let through only those signal components
within a finite range of frequencies (the passband) and in proportions characterized by
the energy spectrum of the wavelet. A plot of the squared magnitude of the Fourier
transform against frequency for the wavelet gives its energy spectrum. For example,
the Fourier energy spectrum of the Mexican hat wavelet is given by
EF(f) == I(f) 1 2 == 327r 5 f4 e-47r2f2
(2.5)
where the subscript F is used to denote the Fourier spectrum as distinct from the
wavelet-based spectrum defined later in section 2.9. A plot of the energy spectrum

10 The continuous wavelet transform
of the Mexican hat wavelet is shown in figure 2(b). Note that, as the Mexican hat
wavelet is a real function, its Fourier spectrum is symmetric about zero. We will see
later that complex wavelets do not have negative frequency components (requirement
3 above). The peak of the energy spectrum occurs at a dominant frequency of
/p == =:i=V2/27r. The second moment of area of the energy spectrum is used to define
the passband centre of the energy spectrum, fe' as follows:
Ie==
SOOO f21(f)12 df
Sooo I(f) 1 2 df
(2.6)
where Ie is simply the standard deviation of the energy spectrum about the vertical
axis. For the Mexican hat mother wavelet, fe is equal to J572/27r or 0.251 Hz. In
practice we require a characteristic frequency of the mother wavelet, such as /p, fe
or some other, in order to relate the frequency spectra obtained using Fourier
transforms to those obtained using wavelet transforms. Later we will see how these
characteristic frequencies change as the mother wavelet is stretched and squeezed
through its dilation parameter. When performing wavelet transform analysis it is
important that the energy spectrum of the wavelet is considered, as it indicates the
range and character of the frequencies making up the wavelet.
From equations (2.1) and (2.2) we see that the total energy of the Mexican hat
wavelet is finite and given by
.f 00 .f 00 2
E = -00 ?j;(t)2 dt = -00 [(1 - p) e- t /2f dt = h!7f
(2.7)
The energy of a function is also given by the area under its energy spectrum. For the
Mexican hat wavelet this gives us
.f 00 .f 00 2 2
E = -00 1(f)12 df = -00 32n 5 t e- 47f ! df =  y'7f
(2.8)
Hence
.Coo 1?j;(t)1 2 dt = .Coo 1(f)12df
(2.9)
This is a result we would expect for any function from Parseval's theorem.
Often, in practice, the wavelet function is normalized so that it has unit energy.
To do this for the Mexican hat we modify its definition given in equation (2.1).
From equation (2.7) we see that it is normalized to have unit energy by dividing it
by (3y'7f /4) 1/2. This gives
2 2 /
?j;(t) = yl3V7f (1 - p) e- t 2
(2.10)
Both equation (2.1) and equation (2.10) are commonplace in the literature. The only
alteration necessary when employing the normalized Mexican hat of equation (2.10)
rather than that defined in equation (2.1) is in the value of the admissibility constant
C g , which must be changed from 7r to 4y'7f/3. In the rest of this chapter we will stick to
our original definition of the Mexican hat given by equation (2.1).

The wavelet transform 11
2.5 The wavelet transform
Now we have chosen a mother wavelet, how do we put it to good use in a signal
analysis capacity? First we require our wavelet to be more flexible than that defined
earlier, i.e. 1jJ(t). We can perform two basic manipulations to make our wavelet
more flexible: we can stretch and squeeze it (dilation) or we can move it (translation).
Figure 2.3(a) shows the Mexican hat wavelet stretched and squeezed to respectively
1.0
0.5
-..

2, 0
? -------------',--""
'"
",
"
'''...,............__...tI
-0.5
-1.0
(a)
1.0
0.5
-..

I 0
......

-7
-0.5
-1.0
(b)
....----------
.
J'
.'
",'____J,.JJ
b
J
t
a2
a3
b l
t
b 2
b 3
Figure 2.3. Dilation and translation of a wavelet. (a) Stretching and squeezing a wavelet: dilation
(al = a2/ 2 ; a3 = a2 x 2). (b) Moving a wavelet: translation.

12 The continuous wavelet transform
double and half its original width on the time axis. This dilation and contraction of
the wavelet is governed by the dilation parameter a which, for the Mexican hat
wavelet, is (helpfully) the distance between the centre of the wavelet and its crossing
of the time axis. The movement of the wavelet along the time axis is governed by
the translation parameter b. Figure 2.3(b) shows the movement of a wavelet along
the time axis from b l via b 2 to b 3 . We can include the dilation parameter, a, and the
location parameter, b, within our definition of a wavelet given by equation (2.1).
These shifted and dilated versions of the mother wavelet are denoted 1jJ[(t - b)/a].
For example, in this form the Mexican hat wavelet becomes
?J; C  h ) = [1 - C a h YJ e-![(t-b)/af
(2.11 )
The original mother wavelet 1jJ(t) given by equation (2.1) simply had a == 1 and b == o.
In the form of equation (2.11) we can now transform a signal, x( t), using a range of a's
and b's. The wavelet transform of a continuous signal with respect to the wavelet func-
tion is defined as
J oo ( t - b )
T(a,h) = w(a) -00 x(t)?J;* ----;;- dt
(2.12 )
where w(a) is a weighting function. The asterisk indicates that the complex conjugate
of the wavelet function is used in the transform. We need not consider this when using
the Mexican hat wavelet as it is a real function, but we do need to take this into
account when using complex wavelets later in the chapter. The wavelet transform
can be thought of as the cross-correlation of a signal with a set of wavelets of
various 'widths'. Typically w( a) is set to 1/ va for reasons of energy conservation
(i.e. it ensures that the wavelets at every scale all have the same energy) and we
will use this value for the rest of this chapter. However, w(a) == l/a is sometimes
used and there is nothing to stop the user defining a function more appropriate
to the application, e.g. for the visual enhancement of the transform plot (see
section 2.15).
From here on we will use w(a) == l/va. Thus the wavelet transform is written
1 J oo ( t - b )
T(a,h) = va -00 x(t)?J;* a dt
(2.13 )
This is the continuous wavelet transform or CWT. Take a closer look at this equation.
It contains both the dilated and translated wavelet 1jJ((t - b)/a) and the signal x(t),
where x(t) could be a beating heart, an audio signal, gearbox vibration, a financial
index or perhaps even a spatial signal such as a crack profile or land surface heights.
In the equation, the product of the wavelet and the signal are integrated over the
signal range. In mathematical terms this is called a convolution. The normalized
wavelet function is often written more compactly as
1 ( t-b )
?J;a,b( t) = va ?J; ----;;-
(2.14 )

The wavelet transform 13
where the normalization is in the sense of wavelet energy. Hence, the transform
integral may be written as
T( a, b) = ,[)(J x( t)7/J:,b (t) dt
(2.15)
This is the nomenclature we will use in this chapter and we will refer to 1jJa,b(t) simply
as the wavelet. We can express the wavelet transform in even more compact form as
an inner product:
T(a, b) == (X,1jJa,b)
(2.16)
Figure 2.4 shows the effect that the dilation of a Mexican hat wavelet 1jJa,b(t) has on its
corresponding energy spectrum. As the wavelet expands, its corresponding energy
(a)
1.5
I'
, I
I I
I \
1.0
0.5
1!J a bet)
,
0 ---...... .....--
-0.5
-1.0
-10 -8 -6 -4 -2 0 2 4 6 8 10
t
(b)
10
EF(f) (\
5 / \
I
I
0
0 0.4 0.6 0.8 1.0
I
Figure 2.4. The Fourier energy spectra of dilating wavelets. (a) Three Mexican hat wavelets
'lj;a,b(t) = (I/VCi)'lj;[(t - b)/a] at three dilations, a = 0.5,1.0,2.0 and all located at b = o. (b) Energy
spectra corresponding to the wavelets in (a) all have identical energy and hence the same area under
their curves. Note that only the positive part of the energy spectrum is shown.

14 The continuous wavelet transform
spectrum contracts. This is an obvious consequence, as expansion in the time domain
must involve the lengthening of time periods and a corresponding lowering of asso-
ciated frequencies. The wavelet a scale is therefore inversely proportional to all its
characteristic frequencies, including its passband centre frequency, peak frequency,
central frequency (for complex wavelets) and so on. We will come back to this
relationship in more detail later in this chapter when we consider wavelet power
spectra in section 2.9.
The wavelet transform has been called a mathematical microscope, where b is the
location on the time series being 'viewed' and a is associated with the magnification at
location b. Now we have defined the wavelet and its transform, we are ready to see
how the transform is used as a signal analysis tool.
2.6 Identification of coherent structures
Figure 2.5(a) attempts to visualize the mechanics of the wavelet transform given by
equation (2.15). In the figure, a wavelet of scale a centred at location b on the time
axis is shown superimposed on top of an arbitrary signal. The time segments where
the wavelet and the signal are both positive result in a positive contribution to the
integral of equation (2.15), e.g. region A in the figure. Similarly, the time segments
where the wavelet and the signal are both negative result in a positive contribution
to the integral (region B). Regions where the signal and wavelet are of opposite
sign result in negative contributions to the integral, e.g. regions C, D and E in the
figure.
Figure 2. 5(b) shows a wavelet of fixed dilation at four locations on a signal. At
the first location (b l ), the wavelet is located on a segment of the signal in which the
positive and negative parts of the signal are reasonably coincidental with those of
the wavelet. This results in a relatively large positive value of T(a, b) given by
equation (2.15). At location b 2 , the positive and negative contributions to the integral
act to cancel each other out, resulting in a value near zero returned from equation
(2.15). At location b 3 , the signal and the wavelet are essentially out of phase which
results in a large negative value returned for T(a, b). At location b 4 the wavelet and
the signal are again out of phase, similar to location b 3 . This time, however, the
signal portion in the vicinity of the wavelet contains a large local mean component.
It is easy to see that the mean component contributes equal positive and negative
values to T(a, b). Thus, only the local signal feature is highlighted by the wavelet at
this location and the mean is disregarded. It is through this process that the wavelet
transform picks out 'coherent structures' in a time signal at various scales. By moving
the wavelet along the signal (increasing b) coherent structures relating to a specific a
scale in the signal are identified. This process is repeated over a range of a scales until
all the coherent structures within the signal, from the largest to the smallest, can be
dis tinguish ed.
Let us look at another simple example. Figure 2.6 shows a simple sinusoidal
waveform 'interrogated' at various locations by Mexican hat wavelets of various
dilations. The value of the transform convolution (equation (2.15)) depends upon
both the location and dilation of the wavelet. Figure 2.6(a) shows a wavelet of similar

Identification of coherent structures 15
positive
contribution
negative
contribution posItive
\ contribution
 negative
J / contribution
..
x(t)
'4Ja,b (t)
E
t
analysing
wavelet
a
(a)
b
local signal
mean at b 4
\
x(t)
'4Ja,b(t)
t
b I
b 2
b 3
b 4
(b)
Figure 2.5. The wavelet interrogation of the signal. (a) The wavelet of specific dilation and location on
the signal. The regions which give positive and negative contributions to the integral are delineated in
the sketch and marked with '+' and '-' respectively. (b) A wavelet of fixed dilation at four distinct
locations on the signal. A large positive value of T( a, b) is returned at location b l . A near-zero value
of T(a, b) is returned at b 2 and a large negative value of T(a, b) is returned at b 3 . At b 4 a local
minimum in the signal corresponds with the positive part of the wavelet and relatively higher
parts of the signal correspond with the negative parts of the wavelet. This combines to give a
large negative value of T (a, b).
'periodicity' to the signal waveform superimposed on the signal at location b, which
produces a reasonable local matching of the wavelet and signal. From the figure, it
can be seen that there is obviously a high correlation between the signal and wavelet
at this a scale and b location. The integral of the product of the signal with the wavelet
here produces a large positive value of T( a, b). Figure 2.6(b) shows the wavelet moved
to a new location where the wavelet and signal seem to be out of phase. In this case the
convolution produces a large negative value of T (a, b). In between these two
extremes, the value of the transform reduces from a maximum (figure 2.6(a)) to a

16 The continuous wavelet transform
sinusoidal waveform
p
; .. .. :
1 !
/ \, .... "- 1
! \
0.5 0.5
0 0
-0.5 \ -0.5
-1 -1
(a) b (d) b
1 1
0.5 0.5
0 0
-0.5 -0.5
-1 -1
(b) b (e) b
(c)
b
Figure 2.6. A wavelet interrogating a sinusoidal waveform. (a) The wavelet is in phase with the wave-
form giving good positive correlation. (b) The wavelet is now out of phase with the waveform giving
good negative correlation. (c) The wavelet is again out of phase with the waveform, this time giving
zero correlation. (d) The squeezed wavelet does not match the waveform locally. (e) The stretched
wavelet does not match the waveform locally.
minimum (figure 2.6(b)). Figure 2.6(c) shows the point at which the wavelet and signal
produce a zero value of T(a,b). Figures 2.6(a), (b) and (c) consider a wavelet which
matches locally the signal, i.e. it has approximately the same 'shape' and 'size' of the
signal in the vicinity of b. Figure 2.6( d) shows the effect that using a smaller a scale has
on the transform. From the plot we see that the positive and negative parts of the
wavelet are all convolved by roughly the same part of the signal, producing a value
of T (a, b) near zero. Hence, T (a, b) tends to zero as the dilation a tends to zero
width. T(a,b) also tends to zero as a becomes very large (figure 2.6(e)), as now the
wavelet covers many positive and negatively repeating parts of the signal, again
producing a near-zero value of T(a, b) in the integral of equation (2.15). Thus
when the wavelet function is either very small or very large compared with the
signal features, the transform gives near-zero values.
Continuous wavelet transforms are not usually computed at arbitrary dilations
and isolated locations but rather over a continuous range of a and b. A plot of
T(a, b) versus a and b for a sinusoidal signal is shown in figure 2.7, where the Mexican
hat wavelet has been used. This plot of T(a, b) against a and b is known as a wavelet
transform plot. Two methods are employed to present the resulting transformed

Identification of coherent structures 17
(a) I+-- P ----+I
r\/\/\/\/\]
(b)
0.05p
0.10p
0.25p
a
1.00p
2.50p
b
o
a
(c)
T(a,b)
Figure 2.7. Wavelet transform plots of a sinusoidal waveform. (a) Five cycles of a sinusoid of period p.
(b) Contour plot of T(a, b) for the sinusoid in (a). (Note the logarithmic scaling of the a axis. Note
also that a greyscale is used where white corresponds to transform maxima and black to minima. This
is the format used in all subsequent transform plots unless otherwise stated.) (c) Isometric surface
plot of T(a, b). Viewed with the smallest a scales to the fore.
signal in figure 2.7: as a contour plot (figure 2.7(b)) and as a surface plot (figure
2.7(c)). The contour plot is more commonly used in practice. The near-zero values
of T (a, b) are evident in the plot at both large and small values of a. However, at inter-
mediate values of a we can see large undulations in T (a, b) corresponding to the sinu-
soidal form of the signal. We can explain these large undulations by referring back to
figures 2.6(a)-(c), where wavelets of a 'size' comparable with the waveform move in

18 The continuous wavelet transform
and out of phase with the signal. For the Mexican hat wavelet, the a scale is required
to be roughly one quarter of the period, p, of the sine wave for this to occur. (This is
covered in more detail in section 2.9.) In figure 2.7(b) we can see that the maxima and
minima of the transform plot actually do occur at an a scale of approximately 0.25p,
indicating maximum correlation between the wavelet and signal at this scale. In the
figure, the a axis has logarithmic scaling. This is the most common form used in prac-
tice; however, note that linear scales are sometimes used. All wavelet transform plots
(a)
P2
..-
(b)
0.25p
a
0.25p
b
o
b
(c)
T(a,b)
Figure 2.8. Wavelet transform plots of two combined sinusoids. (a) A signal composed of a combina-
tion of two sinusoids of period PI and P2, where P2 = PI/5. (b) Contour plot of T(a, b) for the wave-
form in (a). (c) Isometric surface plot of T(a, b). Viewed with the smallest a scales to the fore.

Identification of coherent structures 19
in this chapter have logarithmic a scales, with the exception of figure 2.12. In addition,
in chapters 2 and 3 we stick to the convention of showing transform plots with the
smallest a scales at the top, although there is nothing to stop the reader plotting
them the other way up.
The signal shown in figure 2.8 is composed of two sinusoidal waveforms, one with
a period (PI) five times the other (P2). The transform plot below shows up very well
the two periodic waveforms in the signal at a scales of one quarter of each of the
periods. This figure clearly shows the ability of the transform to decompose the
signal into its separate components. The transform has unfolded the signal to show
its two constituent waveforms. Figure 2.9 contains a segment of a chirp signal
which has the form x( t) == sin x 2 . The chirp begins just into the signal window and
finishes just before the end of the window. The increase in frequency of the oscillation
can be seen in the signal. The transform plot of the transformed chirp is shown below
the signal. The transform plot shows the oscillation as peaks at decreasing a scales
from left to right.
Figure 2.1 O( a) contains a signal with a number of isolated features: four identical
wavegroups containing three sinusoidal oscillations of the same periodicity, a group
of three bumps, a small negative constant region (a block pulse) and a further sinu-
soidal wavegroup at a higher frequency. Three representations of the wavelet trans-
form are plotted directly below the signal in figures 2.1 O(b ), (c) and (d): these are,
small a
scales
a
large a
scales
b
Figure 2.9. Segment of a chirp signal with associated transform plot-Mexican hat wavelet.

20
The continuous wavelet transform
(a)
....-JP
scale
associated
with
wave group
I .
J
-r..:., . -:.. ..............
higher frequency
oscillation
bump signal
'1iA
C
II
(b)
(c)
periodicity of
wavegroup
 
(d)
spectral
energy
spectral
energy
(e)
,.. j 
frequency
frequency
Figure 2.10. The wavelet transform of an intermittent signal. (a) Intermittent signal. (b) Filled trans-
form plot. ( c) The unfilled transform plot using two contours. (d) The unfilled transform plot using
twelve contours. (The arrows point to the edges of the constant discontinuity.) (e) Fourier power
spectrum of signal. (Arbitrary axis units. Left-hand plot has both axes with linear scales, right-
hand plot has a linear horizontal scale and a logarithmic vertical scale.)

Edge detection 21
respectively, a filled plot, a contour plot using two equally spaced contours and a
contour plot using twelve equally spaced contours. The three different representations
are shown to illustrate their advantages and disadvantages as a method for viewing
the signal in the timescale plane ('wavelet domain' or 'wavelet space'). We can see
from all three transform plots that the four identical wavegroups all have the same
morphology in the wavelet domain. In addition, the periodicity of the wavegroups
can be easily differentiated from the periodicity of the waves within the group,
found from the associated wavelet scale as indicated in figure 2.1 O( c ). Wavelet trans-
forms are particularly good at picking out recognizable signal features in this way,
where the features occur intermittently. The wave group with the higher frequency
oscillation appears at a smaller a scale towards the top of the plot as we would
expect. The group of three bumps appears to have a similar form to the oscillations
in the filled plot, the difference being more apparent in the two-contour plot. The
edges of the block pulse are more apparent in the twelve-contour plot which points
to these discontinuities. These are located by the two arrows at the top of the plot.
Figure 2.1 O( e) contains the Fourier energy spectrum of the signal. The energy
spectrum on the left-hand side is plotted with linear scaling axes. The same energy
spectrum is plotted on the right-hand side with a logarithmic vertical axis which
shows up the rich structure in the Fourier domain. However, we can see that the
Fourier representation does not provide us with any useful information regarding
the coherent (obvious even) nature of the localized features within the signal.
2.7 Edge detection
Another useful property of the wavelet transform is its ability to identify abrupt
discontinuities ('edges') in the signal. A simple example of a discontinuity is shown
in figure 2.11(a), where a constant signal, x(t) == 1, suddenly drops to a constant
negative value, x( t) == -1. To see how the wavelet picks out such a discontinuity
we follow a wavelet of arbitrary dilation a as it traverses the signal discontinuity.
The effect of wavelet location, b, on the transform T(a, b) is discussed for each of
five locations on the signal, A, B, C, D and E.
Location A: At locations much earlier than the discontinuity, for example at loca-
tion A, the wavelet and the (constant) signal combine to give near-zero values of the
integral for T(a,b). As it is a localized function, the wavelet becomes approximately
zero at relatively short distances from its centre. Hence the wavelet transform
(equation (2.15)) effectively becomes a convolution of the wavelet with a constant
valued signal producing a zero value.
Location B: At this location the wavelet is just beginning to traverse the dis-
continuity. The left-hand lobe of the wavelet produces a negative contribution to
the integral, the right-hand lobe of the wavelet produces an equal positive contri-
bution, leaving the central bump of the wavelet to produce a significant positive
value for the integral at this location.
Location C: When the signal discontinuity coincides with the wavelet centre, b,
the right and left halves of the wavelet contribute to a zero value of the integral.

22 The continuous wavelet transform
x(t)
'4J«t-b)/a)
signal
discontinuity
/
(a)
A
BCD
E
t
(b)
B
C
D
T(a, b)
'width'
controlled
by a \
i local undulation
! V in T(a,b)
: :
: !
(c)
b
Figure 2.11. A schematic illustration of the wavelet interrogation of a signal discontinuity. ( a) A sche-
matic diagram of the wavelet interrogation of a signal discontinuity. The regions which give positive
and negative contributions to the wavelet transform integral are indicated. (b) A blow-up of the
wavelet as it traverses the discontinuity. (c) A plot of T(a, b) against location b at a specific a scale.
Note that, as the wavelet has zero mean by definition, we can see that the four regions
of the wavelet in the figure all have the same area.
Location D: This is similar to location B. As the wavelet traverses the discontinu-
ity further, the left-hand lobe of the signal produces a negative contribution to the
integral, the right-hand portion of the wavelet produces an equal positive contri-
bution, as with location B. This time, however, the central portion of the wavelet

Edge detection
23
( a) original signal
a=l
a=2
a=6
(b) a=16
a
a=l
(c)
a= 32
b
Figure 2.12. The wavelet decomposition of a signal discontinuity. (a) A signal with a step discontinuity.
(b) Plots of T( a, b) against b at four arbitrary a scales for the discontinuity: a = 1, 2, 6 and 16. (Total
window length = 64 units.) (c) The transform plot for the discontinuity in (a). Light greys correspond
to large positive values of T(a, b) and dark greys to large negative values of T(a, b). Note that here a
linear a scale axis is used. The plots in (b) represent vertical slices taken through the transform surface
T(a, b) plotted in (c).

24 The continuous wavelet transform
coincides with the negative constant signal and hence the integral produces a signifi-
cant negative value at this location.
Location E: At locations far greater than C, the wavelet and signal combine to
give near-zero values of the integral.
Hence, as the wavelet traverses the discontinuity there are first positive then negative
values returned by the transform integral. These values are localized in the vicinity of
the discontinuity. This is illustrated in figure 2.11(c), where a schematic diagram is
given of T (a, b) plotted against the b location on the time axis. From the figure, we
see an undulation in T(a, b) centred at the signal discontinuity. The width of this
ripple in T(a, b) is controlled by the width of the wavelet, a. In fact, it is directly propor-
tional to it. This is illustrated in figure 2.12, which plots the wavelet transform of a
signal discontinuity for various a scales. Below these the wavelet transform plot is
shown. Notice how the ripple in the transform plot becomes more localized as the dila-
tion parameter reduces. This has the effect of making the transform plot 'point' to the
location of discontinuities in the signal. Think for yourself what the result would be of
an antisymmetric wavelet (e.g. first derivative of the Gaussian shown in figure 2.1(a))
passing across the edge shown in figures 2.11 and 2.12. Can you see that T (a, b) plotted
against b would have the shape of a single bump with width proportional to the wavelet
a scale? Actually if we were using the first derivative of the Gaussian function as a wave-
let, the bump would be Gaussian in shape. Figure 2.13 contains another example of a
(a)
(b)
small a scales ----.
a
(log scale)
large a scales ----.
b
Figure 2.13. Pointing to an exponential discontinuity. (a) A sudden spike with an exponential tail. (b)
The transform plot for the discontinuity. (Note that in the transform plot a logarithmic a scale is used
and small a scales are located at the top of the plot. This is the format adopted for all the transform
plots in chapters 2 and 3 with the exception of figure 2.12.)

The inverse wavelet transform 25
signal discontinuity-a sudden spike in the signal halfway along its length followed by
a smooth exponential decay. As the transform plot has been oriented with the smallest a
scales at the top, it 'points' to the signal discontinuity in the signal above.
2.8 The inverse wavelet transform
As with its Fourier counterpart, there is an inverse wavelet transform, defined as
1 .f oo .f oo da db
x( t) == - c T( a, b )1)Ja,b (t) 2
g -00 0 a
(2.17)
This allows the original signal to be recovered from its wavelet transform by integrat-
ing over all scales and locations, a and b. Note that for the inverse transform, the
original wavelet function is used, rather than its conjugate which is used in the
forward transformation. If we limit the integration over a range of a scales rather
than all a scales, we can perform a basic filtering of the original signal. Figures
2.14 and 2.15 illustrate this on a segment of signal constructed from two sinusoidal
waveforms, one with a period one quarter of the other, plus a local burst of noise.
Figures 2. 14(a)-(c) show the three component waveforms which are added together
to make the composite signal shown in figure 2.14(d). The transform plot of the
composite signal (figure 2.14(e)) shows up the two constituent waveforms at scales
(a)
(d)
(b)
a l
a 2
(e)
(c)
Figure 2.14. A composite signal and its transform plot. (a) Sinusoidal waveform. (b) Sinusoidal
waveform with a period one quarter of that in (a). (c) A burst of high frequency noise. (d) Composite
signal obtained by combining (a), (b) and (c). (e) The transform plot of the composite signal (d).

26 The continuous wavelet transform
(a) (c)
(b)
(d)
Figure 2.15. Wavelet filtering of the composite signal. (a) Small scale (i.e. high frequency) components
above the line indicated are removed from the transform plot in figure 2.14( e ). (b) Reconstructed signal
using the transform plot in (a). (c) Small scale components above the line indicated are removed from
the transform plot in figure 2.14(e). (d) Reconstructed signal using the transform plot in (c).
al and a2. In addition, the high frequency (i.e. small a scale) burst of noise is shown up
as a patch within the top left-hand quadrant of the transform plot. Figure 2.15 shows
two reconstructions of the signal where the components in the transform plot, T (a, b),
are set to zero above the white line indicated. In effect we are reconstructing the signal
uSIng
1 .f oo .f oo da db
x(t) == - c * T(a, b )1)Ja,b(t) 2
g -00 a a
(2.18 )
i.e. over a range of scales a* < a < 00. The lower integral limit, a*, is the cut-off scale
indicated by the white lines in the figures. The reduction in the high frequency noise
components in the reconstructed signal is evident as the cut-off a scale value increases.
This simple noise reduction method is known as scale-dependent thresholding.
Figures 2.16(a) and (b) show, in a very simple fashion, the ability of the wavelet
transform to perform a manipulation of the signal which is localized in both time and
scale. Only those T (a, b) values in the region contained within the box in the trans-
form plot are set to zero. In this way the burst of noise can be dealt with locally in
the signal, thus the denoising does not affect other parts of the signal. Figures
2.16(c)-(e) show the effect of using a global cut-off at a much higher a scale. Here
the transform plot components are restricted to those associated mainly with the
low frequency waveform. The inverse transform (figure 2.16(d)) shows a sinusoidal-
like waveform. Figure 2.16( e) plots the reconstructed signal of figure 2.16( d) at a
greater vertical scale and compares it with the original waveform component. We
can see that a reasonably good match is obtained. However, as the spectral informa-
tion in the transform components is smeared across scales, perfect reconstruction of
the individual sinusoidal components is not achievable (it would be for the Fourier

The inverse wavelet transform 27
(a)
(c)
(b)
(d)
(e)
Figure 2.16. Further wavelet filtering. (a) Small scale (i.e. high frequency) components are removed
from the transform plot in figure 2.14( e) in the location within the box shown. (b) Reconstructed
signal using the transform plot in (a). (c) Components above the line indicated are removed from
the transform plot in figure 2.14(e). (d) Reconstructed signal using transform plot in (c). (e) Blow-
up of the reconstructed signal (solid line) in (d), together with the original low frequency sinusoidal
waveform (dotted line).
transform for the specific case of a signal composed of sinusoidal components). The
denoising strategies shown in figures 2.15 and 2.16 are very simple in nature and are
shown here as an illustration of the inverse transform. A better way to separate perti-
nent signal features from unwanted noise, or other larger scale artefacts, using the
continuous wavelet transform is by using a wavelet transform modulus maxima
method. Figure 2.17 shows a composite signal together with its transform plot and
corresponding modulus maxima lines. The modulus maxima lines are the loci of the
local maxima and minima of the transform plot, with respect to b, traced over wavelet
scales. Various signal features are identified within the modulus maxima plot. Modulus
maxima plots allow the salient information within the transform plot to be expressed in
a much more compact form. Following maxima lines down from large to small a scales
allows the high frequency information corresponding to large features within the signal
to be differentiated from high frequency noise components. This lends itself to novel
methods of filtering out noise from coherent signal features. We will come across
examples of the use of modulus maxima filtering methods as we proceed through
the rest of the book: in the context of the analysis of turbulent fluid measurements in

28 The continuous wavelet transform
U_
-r . ..- -r .
.,
...- -.....nr _-.-- .-, - 
(a)
irregular
maxima of "'1'........
. ...,
nOIse
(b)
regular maxima
of sinusoid
irregular
........- maxima of
noise extending
to larger scales
I
(c) discontinuity
;
end of
sinusoid
I
"'edge
effect
Figure 2.17. Modulus maxima of a composite signal. (a) A composite signal (bottom right)
constructed from the noise, curtailed sinusoid and exponential decay signals shown. (b) Wavelet
transform plot of the composite signal (Mexican hat wavelet). (c) Modulus maxima plot correspond-
ing to the transform plot in (b).
chapter 4; the filtering of non-destructive testing data in chapter 5, section 5.6 (see
figures 5.26-5.28); the filtering of ECG signals, Doppler ultrasound traces, DNA
sequences and medical images in chapter 6 (see figures 6.20 and 6.21) and the analysis
of multifractal signals in chapter 7.
2.9 The signal energy: wavelet-based energy and power spectra
The total energy contained in a signal, x(t), is defined as its integrated squared
magnitude
E = ,[)(J Ix(t)1 2 dt = Ilx(t)112
(2.19 )

The signal energy: wavelet-based energy and power spectra 29
For this equation to be useful the signal must contain finite energy. We have already
come across this expression in equation (2.7), where we found the energy in the
Mexican hat function (i.e. substitute x(t) for 1)J(t)). The relative contribution of the
signal energy contained at a specific a scale and b location is given by the two-
dimensional wavelet energy density function:
E(a,b) == IT(a,b)12
(2.20)
A plot of E(a,b) is known as a scalogram (analogous to the spectrogram, the energy
density surface of the short time Fourier transform-see section 2.12). In practice,
all functions which differ from 1 T(a, b) 1 2 by only a constant multiplicative factor
are also called scalograms, e.g. IT(a,b)1 2 /C g , IT(a,b)12/Cgfc, etc. The scalogram
can be integrated across a and b to recover the total energy in the signal using the
admissibility constant, C g , as follows:
E == C l J oo J oo 1 T ( a, b ) 1 2 d db ( 2.21 )
g -00 a a
Figures 2.18(a)-(c) show an experimental signal, x(t), with associated wavelet trans-
form plot, T(a,b), and scalogram, E(a,b). A Mexican hat wavelet was used in the
signal transformation. The scalogram (figure 2.18( c)) is very similar in form to
the wavelet transform plot. This is to be expected when using real wavelets as the
scalogram is simply the squared magnitude of the wavelet transform values. For
complex wavelets (see later) we can view the modulus, phase, real and complex
parts separately. The scalogram surface highlights the location and scale of dominant
energetic features within the signal.
The relative contribution to the total energy contained within the signal at a
specific a scale is given by the scale dependent energy distribution:
E(a) = J oo IT(a,b)12db
C g -00
Peaks in E ( a ) highlight the dominant energetic scales within the signal. Figure 2.18 ( d)
plots E (a) against a for the signal segment in figure 2.18( a). The plot shows that two
dominant scales exist within the signal which are linked to the dominant oscillatory
regime of the original experimental signal.
We may convert the scale dependent wavelet energy spectrum of the signal, E(a),
to a frequency dependent wavelet energy spectrum Ew(f) in order to compare
directly with the Fourier energy spectrum of the signal EF(f). To do this, we must
convert from the wavelet a scale (which can be interpreted as a representative
temporal, or spatial, period for physical data) to a characteristic frequency of the
wavelet. One of the most commonly used characteristic frequencies used in practice
is the passband centre of the wavelet's power spectrum. We will use this here, but
note that another representative frequency of the mother wavelet such as either the
spectral peak frequency, h, or the central frequency, fa, could be chosen and would
be equally valid in the following discussion. We saw in section 2.5, figure 2.4, that
the spectral components are inversely proportional to the dilation, i.e. f ex 1/ a, and
in section 2.4 we defined the passband centre frequency of the mother wavelet
(i.e. a == 1) as fc- Hence, using this passband frequency, the characteristic frequency
(2.22)

30 The continuous wavelet transform
0.3
0.2
0.1
o
o 2 4 6 8
E(a)
(a)
o
0.5
1.0
a
1.5
2.0
(d)
P w(f)
o
(e)
2
4
6
8
10
(b)
f
P F(f)
P w(f)
(arbitrary units)
0.01
(f)
0.1
1
f
10
100
(c)
Figure 2.18. Wavelet energy density and power spectra. (a) Experimental signal x(t): a velocity trace
taken within a vortex shedding regime in a fluid. (b) Transform plot T(a, b) using a Mexican hat
wavelet. (Negative values are shown dark grey to black. Positive values are shown light grey to
white.) (c) The wavelet scalogram 1 T(a, b) 1 2 . (Note all values are positive as the square of the modulus
is plotted. Hence all contours enclose peaks.) (d) Wavelet energy distribution E(a). (e) Wavelet power
spectral density Pw(f). The horizontal axis is related to that in (d) throughf = O.251ja (Mexican
hat wavelet). (f) Power spectral densities (logarithmic plot). Fourier spectrum PF(f), continuous
line. Wavelet spectrum Pw(f), circles. (Note that times are in seconds, frequencies are in Hertz
and the scale a = 1 corresponds to 1 second.)
associated with a wavelet of arbitrary a scale is given by
f= .fc
a
(2.23)
where the passband centre of the mother wavelet,h, becomes a scaling constant andf
is the representative or characteristic frequency for the wavelet at scale a. We saw in
section 2.4 thatfc is equal to J572/27r or 0.251 for the Mexican hat mother wavelet.

The signal energy: wavelet-based energy and power spectra 31
Hence, for this specific wavelet we have f == 0.251/ a and this is why the peaks in the
transform plot of the sinusoid in figure 2.7 occurred at around 0.25 of its period p
(== l/f). Using equation (2.23), we can now associate the scale dependent energy,
E(a), to the passband frequency of our wavelet. We can also see from equations
(2.21) and (2.22) that the total energy in the signal is given by
E == J oo E(a) da (2.24)
o a 2
We can rewrite this equation in terms of passband frequency by making the change
of variable f == fe/ a. The relationship between the derivatives is da/ a 2 == -df /fe
and, after modifying then swapping the integral limits to get rid of the negative
sign, we get
E = .r Ew(f) d[
where we define Ew(f) == E(a)/fe forf ==fe/a, and the subscript W corresponds to
'wavelet' to differentiate it from its Fourier counterpart. A plot of the wavelet
energy, Ew(f) against f (the wavelet energy spectrum) has an area underneath it
equal to the total signal energy and may be compared directly with the Fourier
energy spectrum EF(f) of the signal. (Remember that EF(f) is defined as the squared
magnitude of the Fourier transform of the signal. We have already come across it in
equation (2.5) where the energy spectrum of the Mexican hat wavelet function was
given.) From equation (2.22), we see that the total energy in the signal is given by
(2.25)
E == C 1 f J oo J oo 1 T(f, b) 1 2 d[ db
gJe -00 0
(2.26)
where we define T(f, b) == T(a, b) forf == fe/a. We can see also that the energy density
surface in the time-frequency plane, defined by E(f,b) == (IT(f,b)12)/(Cgfe),
contains a volume equal to the total energy of the signal, i.e.
E = .Coo .r E(f, b) d[ db
(2.27)
This energy density surface can be compared directly with the energy density surface
of the short time Fourier transform (the spectrogram). Note that the timescale repre-
sentations of the scalogram, E (a, b), and scale dependent energy distribution, E (a),
do not enclose, respectively, volumes and areas proportional to the energy of the
signal, whereas their time-frequency counterparts, E(f, b) and Ew(f), do. In fact,
the way we have defined E (f, b) and Ew (f) above means that they enclose a
volume and, respectively, an area exactly equal to the energy of the signal. However,
the peaks in E (a, b) and E (a) do correspond to the most energetic parts of the signal
as do the peaks in E(f,b) and E(f). We can, therefore, use both the scalogram and
the scale dependent energy distribution to determine the energy distribution relative
to wavelet scale. Scalograms are normally plotted with a logarithmic a scale axis. As
f == fe/ a and hence log(f) == logCf:) - log(a), the plot of 1 T(f, b) 1 2 using a logarith-
mic frequency scale is simply a shifted, inverted plot of 1 T(a, b) 1 2 using a logarithmic
a scale. For example, figure 2.18(c) containing IT(a,b)12 with logarithmically

32 The continuous wavelet transform
decreasing a scales towards the top of the plot can also be interpreted as a plot of
1 T(f, b) 1 2 with logarithmically increasing frequencies towards the top of the plot.
In the literature both representations are commonplace.
If the signal in figure 2.18(a) were infinitely long, we can see that its energy would
be infinitely large. However, in practice, experimental signals (see chapters 4 to 7) are
of finite length-usually long enough for the pertinent statistics of the signal to settle
down sufficiently for analysis. Hence, in practice, power spectra are more often used
to characterize experimental signals of finite length. The power spectrum is simply the
energy spectrum divided by the time period of the signal under investigation. Hence,
the area under the power spectrum gives the average energy per unit time (i.e. the
power) of the signal. For example, for a signal of length T, the Fourier and wavelet
power spectra are, respectively,
1
PF(f) == -EF(f)
T
PwU) = EwU) = :/c .I T ITU,b)12db
T T egO
(2.28)
(2.29)
Figure 2.18(e) plots the wavelet power spectrum, Pw(f), for the experimental signal
shown in figure 2.18( a). The wavelet power spectral density plot contains the same
two peaks as those of the scale energy distribution plot of figure 2.18(d), but in reverse
order as the horizontal frequency axis is the rescaled inverse of the scale axis. The area
underneath the Pw(f) plot is equal to the power of the signal. Figure 2.18(f) again
contains the wavelet power spectrum; this time logarithmic axes are used and the
corresponding Fourier power spectrum is also drawn for comparison. Such logarith-
mic power spectral plots are commonly used in practice (e.g. see the fluid turbulence
spectra of chapter 4) where, for example, some form of power-law scaling is expected
or when the pertinent spectral components span quite different orders of magnitude.
Due to the frequency distribution within each wavelet, the resulting wavelet power
spectrum is smeared compared with the Fourier spectrum. However, the wavelet
spectrum is more than simply a smeared version of the Fourier spectrum as the
shape of the wavelet itself is an important parameter in the analysis of the signal.
Some wavelets will correlate better with specific signal features than others, so accent-
uating these features in the resulting spectra. Note also that we are using the passband
centre of the wavelet as its representative frequency in our discussion; if we had used
another characteristic frequency of the wavelet then this would affect the resulting
wavelet power spectrum of the signal (and energy density plots), either squashing it
or stretching it while retaining the same overall shape and, of course, the same
power (respectively, energy).
Finally, it is worth noting that the wavelet variance, defined for the continuous
wavelet transform as
(,-2 ( a) =  .I T 1 T ( a, b ) 1 2 db
T 0
is often used in practice to determine dominant scales in the signal. Again we assume
that T is of sufficient length to gain a reasonable estimate of (J"2(a). We can see that
this expression is very similar to both the scale dependent energy distribution of
(2.30)

The wavelet transform in terms of the Fourier transform 33
equation (2.22) and the power spectral density function of equation (2.29), differing
from both equations only by constant multiplicative factors. We will come across
many examples of the use of wavelet and Fourier spectra (and wavelet variance) as
we proceed through the application chapters of this book.
2.10 The wavelet transform in terms of the Fourier transform
As we saw in equation (2.15), the wavelet transform is the convolution of the signal
with the wavelet function. Hence we can employ the convolution theorem to express
the wavelet transform in terms of products of the Fourier transforms of the signal,
x(f), and wavelet, a,b(f), as follows:
T(a, b) = ,CXJ x(f):,b(f) d[ (2.31)
where we note that the conjugate of the wavelet function is used. The Fourier trans-
form of the dilated and translated wavelet is
a,b(f) == J oo ?j; ( t - b ) e-i(27rf)t dt
-00 va a
Making the substitution t/ == (t - b) / a (hence dt == a dt') we obtain
 a,b (f) =  J oo ?j;( t ' ) e -i(27fJ) (at' + b) a dt'
y a -00
(2.32a)
(2.32b)
Separating out the constant part of the exponential function and dropping the prime
from t/ we get
aAf) = vae- i (27fJ)(b) .Coo ?j;(t) e- i (27faJ)(t) dt
(2.32c)
The integral expression in the above equation is simply the Fourier transform of the
wavelet at rescaled frequency af Hence we can write equation (2.32b) as
a,b(f) == va(af) e- i (27r f )b
(2.33)
The Fourier transform of the wavelet function conjugate is then simply
,b (f) == va* (af) e i (27r f )b
(2.34)
Hence equation (2.30) can be written in expanded form as
T(a, b) = va .Coo x(f)* (af) e i (27fJ)b d[
(2.35)
which we can see has the form of an inverse Fourier transform. This is a particularly
useful result when using discretized approximations of the continuous wavelet trans-
form in practice with large signal data sets, as the fast Fourier transform (FFT) algo-
rithm may be employed to facilitate rapid calculation of the wavelet transform and its
inverse. In addition, the Fourier transform of the wavelet function, a,b(f), is usually
known in analytic form and hence need not be computed using an FFT. Only an FFT

34
The continuous wavelet transform
of the original signal, x(f), is required. Then, to get T (a, b), we take the inverse FFT
of the product of the signal Fourier transform and the wavelet Fourier transform for
each required a scale and multiply the result by va. The equivalence between the time
convolution and Fourier integrals for determining T (a, b) is depicted in figure 2.19.
1!J a bet)
,
x(t)
x(t)
/
1!J a bet)
,
\
x (t)1!J a bet)
,
t
Near zero values of the
wavelet here lead to near
zero contributions to the
transform in regions far
from b
Features in the signal are
highlighted by the
wavelet in the vincinity of
b
(a)
x(f)ijJ (af)
 (af)
x(f)
x(f)
/
t
Frequencies in this
region are accentuated
by the filtering process
Frequencies in this
region are diminished by
the filtering process
(b)
Figure 2.19. Schematic representation of the wavelet transform in its time and frequency represen-
tations. (a) The convolution of the wavelet with the signal. (b) The convolution in (a) expressed
in the Fourier domain involves a product of the signal Fourier transform and the wavelet Fourier
transform.

Complex wavelets: the Morlet wavelet 35
The bandpass nature of the wavelet is evident from figure 2.19(b). The inverse trans-
form (equation (2.17)) can similarly be written in terms of an inverse Fourier function.
2.11 Complex wavelets: the Morlet wavelet
So far we have used the Mexican hat wavelet to illustrate many of the features of the
wavelet transform. In this section we consider wavelets which have both real and
imaginary parts. Complex or analytic wavelets have Fourier transforms which are
zero for negative frequencies (requirement 3 in section 2.3). By using such complex
wavelets we can separate the phase and amplitude components within the signal.
Actually, we can easily make a complex version of the Mexican hat wavelet by
taking its Fourier transform, setting the negative frequency components in the Four-
ier domain to zero and then performing an inverse Fourier transform to get the
complex wavelet. However, in this section, we focus on the most commonly used
complex wavelet, the Morlet wavelet, which is defined as
1jJ( t) == 7r -1/4 (ei27ffot _ e -(27ffo)2 /2) e _t 2 /2
(2.36)
wherefo is the central frequency of the mother wavelet. The second term in the brack-
ets is known as the correction term, as it corrects for the non-zero mean of the
complex sinusoid of the first term. In practice it becomes negligible for values of
fo » 0 and can be ignored, in which case, the Morlet wavelet can be written in a
simpler form as
'ljJ( t) =  ei27fJo/ e _/2/2
7r 1 / 4
/ r \
(2.37)
normalization complex Gaussian
factor sinusoid bell curve
This wavelet is simply a complex wave within a Gaussian envelope. We can see this by
looking at equation (2.37) in conjunction with figure 2.20(a). The complex sinusoidal
waveform is contained in the term ei27ffot (== cos(27rfot) + i sin(27rf o t)). The Gaussian
envelope e _t 2 /2 has unit standard deviation and 'confines' the complex sinusoidal
waveform. Figure 2.20(a) shows the real and imaginary parts of the Morlet wavelet
together with its confining Gaussian envelope. We can see that the real and imaginary
sinusoids differ in phase by a quarter period. The 7r 1 / 4 term is a normalization factor
which ensures that the wavelet has unit energy. Note that the function given by equa-
tion (2.37) is not really a wavelet as it has a non-zero mean, i.e. the zero frequency
term of its corresponding energy spectrum is non-zero and hence it is inadmissible
according to equation (2.4). However, it can be used in practice with fo » 0 with
minimal error.
The Fourier transform of the Morlet wavelet is given by
(f) == 7r 1 / 4 V2 e -!( 27f f - 27ffo)2
(2.38)

36 The continuous wavelet transform
real component
energy
density
Ig(f)1 2 4
Gaussian
envelope
2
1
o
-1 0
-6 -4 -2 0 2 4 6 0 0.5 1.0 1.5 2.0
(a) (b) frequency
fo f
1 1
,J  1'1(" "
" I \ \ -
0 0 - Il II n'._
. I i I r I ..'
- J,,. r r. "
-1 -1
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
(c) (d)
1 1
0
-.
-1 -1
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
(e) (f)
Figure 2.20. Morlet wavelets. (a) The Morlet wavelet a = 1 and fa = 0.894. (b) Energy spectrum of
the Morlet wavelet. (c)fo = 0.318 (a = 1). (d)fo = 1.909 (a = 1). (e) a = 0.5 (fa = 0.894). (f) a = 2
(fa = 0.894).
which has the form of a Gaussian function displaced along the frequency axis by fa.
Note that the central frequency of the Gaussian spectrum is generally chosen to be the
characteristic frequency of the analytic Morlet wavelet rather than the passband
frequency, which we used previously for the Mexican hat wavelet. The energy
spectrum (the squared magnitude of the Fourier transform) is given by
I(f) 1 2 == 27r 1 / 2 e-(27r f -27rfo)2
(2.39)
The integral of this gives the energy of the Morlet wavelet, which is equal to unity
according to our definition given by equation (2.37). The energy spectrum of the
Morlet wavelet is shown in figure 2.20(b). The central frequency fa is the frequency
of the complex sinusoid and its value determines the number of 'effective' or
'significant' sinusoidal waveforms contained within the envelop e, i.e. tho se which
are not very close to zero amplitude. The value of 0.849 (== V l / (21n 2)) is often
used in practice. It produces a decay where the magnitude of the two peaks in the
real waveform adjacent to central peak are half its amplitude. (In the literature the

Complex wavelets: the Morlet wavelet 37
angular frequency Wo == 27rfo of the wavelet is often quoted, hencefo == 0.849 becomes
Wo == 5.336 == 7r V 2/ln2. Values of Wo equal to 5 and 6 (and in between) are also
commonly used in practice. For values of Wo less than 5 (fo < 0.8) the full or
'complete' Morlet wavelet of equation (2.36) should be used as the simplified wavelet
function of equation (2.37) contains a significant non-zero mean.) Figures 2.20(c) and
(d) show Morlet wavelets with fo equal to 0.318 and 1.909 respectively. The figure
shows that the number of effective oscillations contained within the Gaussian
window increases withfo.
To construct the dilated and translated Morlet wavelet we replace t with (t - b) / a
as we did for the Mexican hat in equation (2.11). Its form is then
1jJ ( t - b ) =  ei27f!o[(t-b)/a] e-![(t-b)/a]2 (2.40)
a 7r 1 / 4
Figures 2.20(e) and (f) show Morlet wavelets with a scales of 0.5 and 2 respectively.
We can see directly from equation (2.40) that the standard deviation of the Gaussian
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.21. Morlet wavelet analysis of a two-component sinusoidal waveform. (a) Original signal.
(b) The real part of the wavelet transform Re( T ( a, b)) (posi ti ve maxima in w hi te, negative minima
in black). (c) The real part of the wavelet transform Re(T(a, b)) (same plot as (b) but with contours
added and a coarser shading used). (d) The imaginary part of the wavelet transform Im(T(a, b)).
(e) The phase of the wavelet transform. cjJ(a, b) = tan- 1 {[Im(T(a, b))]/[Re(T(a, b))} (-1r phase in
black, 1r pha se in white, zero phase in mid grey tone). (f) The modulus of the wavelet transform.
T(a, b) = yI [Re(T(a, b))]2 + [Im(T(a, b))]2 (maximum values in white, zero values in black).

38 The continuous wavelet transform
envelope on the time axis is, in fact, simply equal to a. Figures 2.20( e) and (f) show the
stretching and squeezing of the wavelet with a scale.
Figure 2.21 illustrates the use of the Morlet wavelet (fa == 0.849) in analysing a
two-component sinusoidal waveform. Performing the wavelet transform on the
signal using the complex Morlet wavelet results in complex transform values
T(a,b) which we may view in a number of ways. The real part ofT(a,b) is shown
in figure 2.21(b). As expected, the two wavelet components are picked up and
displayed as ripples in the transform plot at two distinct scales. Figure 2.21 (c) also
contains the real part of T (a, b). This time a coarser shading is used and contour
lines are added to enhance visualization of the periodic structure of the transform
plot. Figure 2.21(d) contains the imaginary part of T(a,b). Notice the similarity
between this transform plot and that of figure 2.21 (c). In fact, the imaginary plot is
a phase-shifted version of the real plot. The reason for this is apparent if we consider
the form of Morlet wavelet as shown in figure 2.20(a). In the figure we see that, for the
Morlet wavelet defined above, the imaginary part of the wavelet comprises a sinusoi-
dal waveform within a Gaussian envelope which leads the real part by one quarter of
a cycle. In other words it is phase shifted by one quarter of a cycle from the real part.
However, as we use the complex conjugate in the transform, the imaginary part is
(a)
(b)
(c)
Figure 2.22. Phase shifted sinusoid. (a) Original signal. (b) Phase. (c) Modulus.

Complex wavelets: the Morlet wavelet 39
inverted, leading to an imaginary waveform which lags behinds by one quarter of a
cycle from the real part. Hence, the imaginary part of the Morlet best matches one
quarter of a cycle later than the real part. Therefore, the maxima of the transform
plot for the imaginary part are phase shifted forward by one quarter of a cycle.
This we see when comparing figures 2.21(c) and (d).
As T(a, b) is a complex number, i.e. T(a, b) == Re(T(a, b)) + Im(T(A, b)), we can
write T (a, b) in terms of its phase cjJ( a, b) and modulus 1 T (a, b) I. The phase of the
Morlet transform plot is shown in figure 2.21 (e). The phase varies cyclically between
-7r and 7r over the duration of the component waveforms. Zero phase corresponds to
the real part of the Morlet wavelet centred at the maximum amplitude of the sinusoi-
dal waveforms. Hence, zero phase corresponds to the peaks on the real transform plot
of figure 2.21(c). A phase of 7r (and -7r) corresponds to the minima of the real trans-
form plot. Figure 2.21(f) contains the modulus of the transform plot where we can see
that the periodic sinusoidal waveforms manifest themselves as continuous bands
across the modulus plot.
Figures 2.22 to 2.28 show various simple signals together with their associated
Morlet wavelet transform plots. Figure 2.22 shows a sinusoidal waveform which is
(a)
-3 -2 -1 0 1 2 3
(b)
(c)
(d)
(d)
(f)
Figure 2.23. Morlet decomposition of signal with abrupt change in periodicity. ( a) Original signal.
(b) Morlet wavelet 10 = 0.849 (period = 1/10 = 1.117) at scale a = 1. (c) Phase of the transform.
(d) Real part of the transform. (e) Modulus of the transform. (f) Imaginary part of the transform.

40 The continuous wavelet transform
shifted by half a cycle in the middle. Hence, the resulting signal contains both a
discontinuity and a phase shift. Both transform plots indicated the location of the
discontinuity. The phase plot shows up the location of the phase shift in the wave-
form. Figure 2.23 shows the Morlet wavelet decomposition of a signal which contains
a change in periodicity midway along its length. The change in periodicity is clearly
shown in all four transform plots. Figure 2.24 shows the effect of interrogating the
same signal as figure 2.23 using different Morlet wavelets (i.e. different value of fa).
On the left-hand side of the figure, the signal is decomposed using a Morlet wavelet
with fa == 0.318. We see that the real part of this wavelet is more like a Mexican
hat in form. (As mentioned above, we should not really use such a low frequency
fa for a Morlet wavelet in practice, as its power spectrum is significantly non-zero
at the origin. Instead, its complete form, given by equation (2.36), should be
employed. An example of the use of the complete Morlet wavelet can be found in
section 5.6 and figure 5.29 of chapter 5.) Comparing the transform plots with those
of the previous figure we see some differences. The phase plot for the fa == 0.318 wave-
let is much smoother at the transition point. This is because this wavelet has essen-
(a)
(b)
Figure 2.24. The effect off 0 on the Morlet decomposition of signal with abrupt change in periodicity.
(a) Morlet wavelets fo = 0.318 (left) and Wo = 1.909 (right). (b) Phase plots corresponding to
fo = 0.318 (left) and Wo = 1.909 (right). (c) Modulus plots corresponding to fo = 0.318 (left) and
Wo = 1.909 (right).

Complex wavelets: the Morlet wavelet 41
power
(a)
frequency
power
#

#
  Fourier
#
wavelet
(b) /
frequency
power
#

L....... Fourier
#
wavelet
/
(c) frequency
I.
I.
I.
I.
Figure 2.25. Comparison of Morlet power spectra for three values of fOe (a) Power spectrum for
fa = 1.909 (arbitrary axis units). (b) Power spectrum for fa = 0.849 (arbitrary axis units). (c)
Power spectrum for fa = 0.318 (arbitrary axis units). (Note that the Fourier spikes are shown with
a finite height and width as they were calculated numerically using a discrete Fourier algorithm.
In theory they are Dirac delta functions.)
tially only a single positive bump which matches the signal smoothly over the transi-
tion region, whereas the fa == 0.849 wavelet has five distinct peaks within the window
which correspond to the five ridges converging at small scales in the phase plot in
figure 2.23( c). The right-hand side of figure 2.24 shows the Morlet wavelet decompo-
sition of the signal using fa == 1.909. The power spectra for the three Morlet wavelet
decompositions given in figures 2.23 and 2.24 are given in figure 2.25. We can see the
greater degree of frequency localization as fa increases (and hence the number of
cycles within the Gaussian envelope increases). This is also evident from the trans-
form plots in figures 2.23 and 2.24 where we can see that a narrower band in the
modulus transform plot is associated with higher values of fa. (Remember from
section 2.9 that we can interpret the logarithmic vertical axis of the transform plot
in terms of characteristic wavelet frequency rather than scale.) However, this greater
degree of frequency localization with increasing fa is associated with much poorer

42 The continuous wavelet transform
temporal localization, as can be seen when comparing both the modulus and phase
plots in figures 2.23 and 2.24.
Figure 2.26 shows two signals containing repeating features-bumps and spikes.
The transform plots are plotted below the signals. We can see that even for these
(a)
(d)
(b)
(e)
(t)
.... "-..
? '
! \
J 
(g)
(h)
Figure 2.26. Morlet decomposition of bump signals. (a) Single bump- positive part of sinusoid. (b)
Wavelet phase of signal (a). (c) Wavelet modulus of signal (a). (d) Bumps and spikes. (e) Wavelet
phase of signal (d). (f) Wavelet modulus of signal (d). (g) The Morlet wavelet at a periodicity
correlating best with the bump signal. This value of Morlet wavelet results in the dominant band
of (c). (h) The Morlet wavelet with half the period of that in (g). This value of Morlet wavelet results
in the second band in (c) at an a scale half that of (g).

Complex wavelets: the Morlet wavelet 43
relatively simple repeating signals the phase plots are already exhibiting a significant
degree of complexity. The modulus plot of the bump and spike signal points to the
discontinuous spikes in the signal (compare with the ECG signal shown in chapter
6, figure 6.12). Figure 2.26(g) shows a schematic of the wavelet at approximately
the same periodicity as the bump signal of figure 2.26(a). At this scale a maximum
correlation is produced which shows up as the dominant (white) band in the modulus
plot of figure 2.26(c). The next most dominant (grey) band in the modulus plot
is generated when the Morlet wavelet correlates with the signal as shown in
figure 2.26(h). Figure 2.27 contains the same chirp signal as that shown earlier in
(a)
(b)
(c)
(d)
Ridge
_............-
....... .................
a R ..... ......
,.....
(e)
-"
b R
Figure 2.27. Segment of a chirp signal with associated transform plots- Morlet wavelet. (a) Chirp
signal segment. (b) Real part of Morlet wavelet transform. ( c) Phase. (d) Modulus. (e) A schematic
of the ridge found from the maxima of the rescaled scalogram 1 T( a, b) 1 2 / a. The instantaneous
frequency at time b R can be found from aR . We can see the relation between maxima in the rescaled
scalogram and instantaneous frequency by substituting a complex sinusoid as the signal x(t) in the
wavelet transform integral given by equation (2.13), and using a Morlet wavelet. Then, using a
change of variable t' = (t - b) / a, it can be shown that maxima in the rescaled scalogram correspond
to the instantaneous frequencies through their associated scales.

44 The continuous wavelet transform
figure 2.9. The Morlet wavelet withfa == 0.849 was used to transform the signal. The
real part of the transformed signal is plotted in figure 2.27(b) and has similarities with
the Mexican hat transform plot in figure 2.9. The discontinuities at the beginning and
end of the chirp segment are picked up well in the phase plot of figure 2.27 (c). These
are located using arrows at the top of the phase plot. The continuous increase in
instantaneous frequency associated with the chirp is highlighted in the modulus
plot of figure 2.27( d). The instantaneous frequency associated with a signal can be
found from its wavelet transform ridges. These are the maxima found in the rescaled
wavelet transform scalogram, 1 T(a, b) 1 2 / a associated with the instantaneous
frequency of the signal. The ridge associated with the chirp signal is shown schema-
tically in figure 2.27(e), where the instantaneous scale aR at time b R can be used to find
the instantaneous frequency fR (== fa/ aR). The instantaneous amplitude and phase can
also be found from the ridge. Further, if we plot the rescaled scalogram in terms of a
characteristic wavelet frequency 1 T(f, b) 1 2 / a where f == fa/ a (refer back to equations
(2.26) and (2.27)), then the instantaneous frequency can be read directly off this plot.
We do not go into the derivation of ridges here, but rather refer the reader to section
2.16 at the end of the chapter.
A final test signal for the Morlet wavelet is shown in figure 2.28. It is composed of
a dominant sinusoid of unit amplitude which has three features superimposed on it.
(a)
(b)
(c)
(d)
(e)
Figure 2.28. Wavelet decomposition of a sinusoidal waveform with added small scale features. ( a)
Signal. (b) Phase. (c) Real part. (d) Modulus. (e) Unfilled contour plot of modulus.

The wavelet transform, short time Fourier transform and Heisenberg boxes 45
At the second peak a small spike has been added. The fourth oscillation has noise
added to it. A high frequency oscillatory component has been added to the sixth oscil-
lation of the dominant sinusoid. This has a frequency ten times that of the large wave-
form. The location of the spike can be easily identified in the phase plot, as are the
regions of random noise and the high frequency component. The real transform
plot locates the dominant peaks and troughs of the signal together with superimposed
signal features. The modulus is plotted as a filled plot in figure 2.28( d) and as a
contour plot in figure 2.28(e). The dominant waveform is obvious in both figures.
The high frequency oscillatory feature in the signal is made much more obvious in
the contour plot. In general, as the signal becomes more complex in nature the
phase and modulus information quickly becomes more difficult to interpret.
2.12 The wavelet transform, short time Fourier transform and
Heisenberg boxes
In this section we take a brief look at the time-frequency characteristics of the wavelet
transform and compare it with the short time Fourier transform (STFT). We will
consider the specific cases of the Morlet wavelet transform and the Gabor STFT,
both of which employ a Gaussian window.
As we will see below, the Morlet wavelet has a form very similar to the analysing
function used for the short time Fourier transform within a Gaussian window. The
important difference is that, for the Morlet wavelet transform, we scale the window
and enclosed sinusoid together, whereas for the STFT we keep the window length
constant and scale only the enclosed sinusoid. The wavelet can therefore localize
itself in time for short duration, i.e. high frequency, fluctuations. There is, however,
an associated spreading of the frequency distribution associated with wavelets of
short duration. Conversely, there is a spreading in temporal resolution at low
frequencies. This is illustrated in figure 2.29(a). The middle of the figure contains a
schematic of a Morlet wavelet (real part only) shown at three a scales. The energy
densities of the wavelets are plotted below in both the time and frequency domains,
i.e. respectively l1)Ja,b(t) 1 2 and Ia,b(f) 1 2 . We can see from the figure that, as the wavelet
contracts in time, it becomes composed of higher frequencies with a wider spread. The
spread of l1)Ja,b(t) 1 2 and Ia,b(f) 1 2 can be quantified using rJ"t and rJ"f respectively-the
standard deviations around their respective means. We can represent the spread of the
wavelets in the time-frequency plane by drawing boxes of side lengths 2rJ"t by 2rJ"f.
These are shown at the top of figure 2.29(a). These boxes are known as Heisenberg
boxes after the Heisenberg uncertainty principle, which tells us the minimum area
that these boxes can have. Specifically, the product rJ"trJ"f must be greater than or
equal to 1/ 47r, thus the area of the Heisenberg box is 1/ 7r. In fact, for the Gaussian
windowed functions used in the Morlet wavelet transform and STFT considered
here, rJ"trJ"f is exactly equal to 1/ 47r as the Gaussian distribution is the optimal
window shape. (You can verify this for the Morlet wavelet given in the time
domain by equation (2.37) and frequency domain by equation (2.38); integrating
their squared modulus leads to rJ"t == 1/0 and rJ"f == 0/47r.) The Heisenberg
uncertainty principle actually addresses the problem of the simultaneous resolution

46
The continuous wavelet transform
(a)
f
Heisenberg box in
the time-frequency
plane
h .................................................................................................................................................................................... ....
Of
h - ----------------------t-EB--
4 a t
It .... ........ I ...................... .....................  ........
t
.
1!J a,b(t)
(real part)
11!J a,b(t) 1 2
hI
h 2
h3
t
\
decrease in wavelet
width associated with
increase in spread of
spectral components
liP (f)1 2
a,b
/
It
h
h
f
Figure 2.29. Heisenberg boxes in the time-frequency plane. (a) Heisenberg boxes in the time-
frequency plane for a wavelet at various scales. Do not confuse O"j withi:, the passband frequency,
given earlier in section 2.4: Ie is the standard deviation of the spectrum around the origin; O"j is the
standard deviation of the spectrum around the mean spectral components 11,12 and 13 shown in the
figure.

The wavelet transform, short time Fourier transform and Heisenberg boxes 47
(b)
f
high to.................. I..m-!m../ ..
of I
memumto................t.. j..!.../ m
i . Ot
low tom.mm.mm.mm.mm.mm.mm.mm.mm. fm.mf.mm.f mm
.
t
1!J a,b(t) * low fo
(real part) 
mediumfo
t
.tw- high fo

-" ,,- t
Figure 2.29 (continued). (b) Heisenberg boxes in the time-frequency plane for a Morlet mother
wavelet with three different central frequencies set to a low, medium and high value. The confining
Gaussian windows are all of the same dimensions. Notice that altering the central frequency of the
mother wavelet simply shifts the associated 'mother' Heisenberg box up and down the time-
frequency plane without altering the box dimensions. This mother Heisenberg box then defines
the relative shapes of all the others in the time-frequency plane associated with each wavelet, i.e.
the pattern shown in (a) is simply shifted up or down the plane.
in time and frequency that can be attained when measuring a signal. To get a good
idea of the frequency composition we need to sample a long period of the signal. If
instead we pinpoint a small region of the signal to measure it with accuracy, then it
becomes very difficult to determine the frequency makeup of the signal in that
region. That is, the more accurate the temporal measurement (smaller (J"t) the less
accurate the spectral measurement (larger (J"f) and vice versa. Note that the Morlet
central frequency fa sets the location of the Heisenberg box in the time-frequency

48 The continuous wavelet transform
(c) f
 ... .-------------------------------------------------- ++--+
h............
Of
t
at
h --- +----1----+
t
hf,b(t)
(real part)
t
Ihf,b(t) 1 2
t
hI
b 2
b 3
A 2
Ihf,b(f) I
It
h
h
f
Figure 2.29 (continued). (c) Heisenberg boxes in the time-frequency plane for the STFT. Other
window lengths will produce longer and thinner or shorter and fatter boxes. However, once the
window length is fixed, all Heisenberg boxes in the time-frequency plane corresponding to the
STFT will have the same dimensions.
plane for the mother wavelet, and hence the relative locations for all dilated wavelets.
This is shown in figure 2.29(b). Thus, when comparing Heisenberg boxes centred at
the same location in the time-frequency plane, lower values of fo correspond to
Heisenberg boxes that are wider in frequency and narrower in time than boxes
corresponding to higher fos. Thus Morlet wavelets with lower fos correspond to
time-frequency decompositions that are 'more temporal than spectral' than their
higher central frequency counterparts. (See the example in chapter 5, figure 5.29.)
The Fourier transform of a signal x(t) is defined as
xU) = ,Cx> x( t) e -i(21fJ)t dt (2.41 )

The wavelet transform, short time Fourier transform and Heisenberg boxes 49
We can modify the Fourier transform to allow localized features in the signal to be
interrogated. This short-time Fourier transform (STFT) employs a window function
to localize the complex sinusoid. It is defined as
Fer, b) = .eX) x(t)h(t - b) e-i27fjt dt
(2.42)
where h(t - b) is the window function which confines the complex sinusoid e-i27rft.
The STFT is also commonly known as the windowed Fourier transform. There are
many shapes of window available, for example Hanning, Hamming, cosine, Kaiser
and Gaussian. We will consider the Gaussian windowed STFT, known as the
Gabor transform, which has the form
1 1 ( 2 / 2 )
h ( t) == e -2: t (J
yIo=7r I / 4
where a- is a fixed parameter (the standard deviation) which sets the width of the
Gaussian window on the time axis. The combined window plus complex sinusoid is
known as a windowed Fourier atom or more generally as a time-frequency atom,
denoted
(2.43a)
hf,b (t) == h( t - b) ei27rft
(2.43b)
Convolving the complex conjugate of this atom with the signal x(t) results in its time-
frequency decomposition: the short time Fourier transform. We can see that, when
the Gaussian window is employed within the STFT integral of equation (2.42), we
obtain
Fer, b) = ,[)(J x(t)hj,b(t) dt
== .f oo x( t) 1 e -![(t - b)2 j,?] e -i27fjt dt
-00 yIo=7r I / 4
This has a very similar form to the Morlet wavelet transform. We can see this by
combining equations (2.13) and (2.40) and rearranging the terms as follows:
( 2.44 )
T(a, b) = .[00 x(t)?j;,b(t) dt
== .f oo x(t) 1 e-![(t-b)2jt?] e- i27f (foja)(t-b) dt
-00 yl(i7r 1 / 4
The main difference between the Gabor STFT and the Morlet WT is now obvious: the
internal frequency, f, is allowed to vary within a Gaussian window of fixed width
(given by a-) in the former, whereas the latter employs an internal frequency
f (== fa/ a) which is linked to the window width (given by a). There is another less
significant difference in that the wavelet's complex sinusoid is centred at b on the
time axis whereas the complex sinusoid contained within the Gabor atom is 'centred'
at the origin (t == 0). To put it another way, in the STFT the sinusoid remains fixed
in relation to the origin and the window slides across it, whereas for the wavelet
(2.45)

50
The continuous wavelet transform
transform, the origin of the complex sinusoid is at b and hence the sinusoid and
window move together.
The atom used for the Gabor STFT is shown in figure 2.29(c) for three internal
frequencies within the fixed width window. This atom also has a constant width in the
+
(a)
>-.
u
s:::
Q)
::s
C"
Q)

- ----
time
(b)
>-.
u
s:::
Q)
6-
Q)

A.... 
time
(c)
>-.
u
s:::
Q)
::s
C"
Q)

time
>-.
u
s:::
Q)
::s
C"
Q)

time
Figure 2.30. STFT and WT time-frequency plots. (a) Sinusoid containing a spike at the location
indicated by the arrow. (b) Morlet wavelet transforms of the signal in (a). The wavelet transform
on the left has been generated using a Morlet wavelet with a central frequency half that used to
produce the transform plot on the right. (c) Gabor STFTs of the signal in (a). The left-hand plot
corresponds to a Gaussian window which is half the width of the one used to generate the right-
hand plot. Note that the horizontal axis is time and the vertical axis is frequency. In addition,
high to low energies correspond with white to black in the grey-scale used.

Adaptive transforms: matching pursuits 51
frequency domain. This leads to boxes in the time-frequency plane of equal shape,
regardless of the internal frequency. The dimensions of the Heisenberg boxes
shown in the time-frequency plane at the top of figure 2.29(c) are determined by
the preselected window width (J. Longer and thinner or shorter and fatter Heisenberg
boxes can be obtained by changing the window width. However, whatever length of
window is used, once fixed, the corresponding Heisenberg boxes associated with the
STFT all have exactly the same dimensions in the time-frequency plane. This is true
for all window shapes used in the STFT -not just Gaussians. As with the Morlet
wavelet transform, the Gaussian windowed complex sinusoids used in the Gabor
STFT have the smallest areas of Heisenberg boxes in the time-frequency plane-
that is, they have optimal time-frequency energy distributions. An example of the
difference between the WT and STFT is shown in figure 2.30 which contains a test
signal together with corresponding Morlet-based wavelet transforms and Gabor
STFTs. The signal comprises a sinusoid plus a single spike located by the arrow in
figure 2.30(a). We can see that the sinusoid manifests itself as a dominant horizontal
ridge in both the WT and STFT plots. The spike, however, is localized in time at high
frequencies by the wavelet transform, whereas it corresponds to the vertical ridge of
constant width in the STFT due to the constant width of its Heisenberg boxes. Hence,
the wavelet transform can discern individual high frequency features located close to
each other in the signal, whereas the STFT smears such high frequency information
occurring within its fixed width window (a good example of this effect is shown
in figure 6.17 of chapter 6). We can also see from figure 2.30(b) that the Morlet
wavelet transform corresponding to the lower central frequency (left-hand plot)
produces a sharper resolution in time but correspondingly poorer resolution in
frequency than the transform based on the higher central frequency (right-hand
plot). Similarly, comparing the plots in figure 2.30(c) we can see that the STFT
corresponding to the shorter window width produces a sharper resolution in time
but correspondingly poorer resolution in frequency than that generated using the
longer window width.
2.13 Adaptive transforms: matching pursuits
From the previous section we saw that the wavelet transform becomes more localized
as it interrogates smaller scales, whereas the STFT has a fixed window length for all
scales. We will now take a brief look at an adaptive transform called the matching
pursuit (MP) which offers an alternative, more flexible way of providing time-
frequency information. Although more flexible it does not provide a regular repeata-
ble or even full coverage of the time-frequency plane, but rather one that adapts to
each signal and is hence signal dependent.
The matching pursuit method involves the decomposition of the signal piece by
piece using a dictionary of analysing functions. At each stage in the decomposition an
analysing function is chosen which 'best' represents part or all of the remaining signal.
After a number of decompositions the original signal, x(t), can be represented to
some arbitrary resolution by a series of expansion coefficients, M i , i == 1,2, . . . , n,
where n is the number of iterations of the decomposition algorithm. The signal

52 The continuous wavelet transform
approximation reconstructed from these expansion coefficients is given by
n-l
Xn(t) == 2:: Mihi(t)
i=O
(2.46)
where hi(t) are the functions used in the decomposition. The essential differences
between wavelet transform analysis and matching pursuit analysis are in the way
the transform coefficients are selected and the flexibility in the choice of analysing
function used in the signal decomposition. The signal is first examined using each
of the analysing functions contained within a preselected dictionary of functions,
and the one which takes the most energy from the signal is chosen to decompose
the signal. The residual signal is constructed and then examined to find the next
function from the dictionary which takes the most energy from this new signal.
The process is repeated until the residual signal falls below some predetermined
cut-off. The procedure is illustrated in figure 2.31. In the figure we can see the original
signal with the first analysing function-a time-frequency atom-used to decompose
it shown directly below. The bottom plot in the figure contains the first residual
signal.
The example in figure 2.31 uses a Gabor atom, one of the most commonly
employed analysing functions in matching pursuit analysis. We have already come
across the Gabor atom used in the STFT. For the matching pursuit method the
atom is defined as
h ( t ) ==  h ( t - b ) e i (27r f )t
a,b,f 
ya a
(2.47)
where the scale, a, location, b, and frequency, f, can all be varied independently. Thus
it has an increased flexibility over the Gabor STFT atom. We will define the Gaussian
_ I<a)Si -  {::z:.l\]. ":
: f<b) at: -_ _' - :  .".  .10. 4 <. 10. 2f
-O . .  40 60 80 100 120 4Q iL lO 2 J
° l<c)re:._ <_  _  :
-1
o 20 40 60 80 100 120 140 160 180 200
time (samples)
Figure 2.31. The matching pursuit method. (a) A damped sinusoidal signal. (b) The first Gabor atom
chosen to represent the signal in (a). (c) The residual signal. (Note that this example comes from a
paper by Goodwin and Vetterli (1999), which goes on to show the difficulties in using symmetric
window functions in representing transient signals.) After Goodwin and Vetterli (1999) IEEE Trans-
actions on Signal Processing, 47(7) 1890-1902. (Q) IEEE 1999.

Adaptive transforms: matching pursuits 53
window as
h(t) == 2 1 / 4 e- 7rt2
(2.48)
Note that this is different in form from that used for the STFT in equation (2.43).
Both have unit energy and are equally valid, however, equation (2.48) is prevalent
in the literature for the matching pursuit method, hence we use it here. The time-
frequency atom defined by equations (2.47) and (2.48) has a similar form to both
those used in the Morlet wavelet transform and the short time Fourier transform
of equations (2.44) and (2.45). However, the Gabor atom used in the matching pursuit
method is more flexible in that its scale, location and internal frequency may all be
varied independently.
In order to get a decomposition with real expansion coefficients and real
residuals, which is often required in practice, real-only atoms are used of the form
2 1 / 4 2
h . b . f. ,/-.. ( t ) == K. - e -7r[(t - bi)/ad c os ( 27rl'.t + ,J.. )
a z , z,Jz,'Yz Z 1/1. J i If/z
yai
(2.49)
where ai and b i are the scale and location factors for the Gaussian envelope, fi and CPi
are respectively the frequency and phase of the real sinusoid within the Gaussian
envelope and K i is a normalization factor used to maintain unit energy for
hai,bi,fz,cpJ t). The subscript i relates to the specific set of parameters a, b, f and cp
used for the ith decomposition to get the Mi.
The expansion coefficients M i are determined in turn by examining the signal
with the analysing atom and selecting the parameter set (ai, bi,fi, cpi), which provides
the largest value of IM i l 2 where
M i ( ai, bi,.h, CPi) = .[x> x( t)h; (t) dt
(2.50)
and where we have used hi as a more compact representation of hai,bi,fz,CPi. In addition
to ai, b i , fi and cpi, we need to retain K i and, of course, M i at each iteration of the
matching pursuit method in order to perform the reconstruction. The matching
pursuit method is very flexible and the dictionary of analysing functions can include
functions other than the Gabor function hi, such as sinusoids within another shape of
window, continuous sinusoids (i.e. a Fourier function), wavelet functions, Dirac delta
functions, and so on. All these functions are kept within a preselected dictionary and
each are used in turn to determine which gives the maximum value IM i l 2 and, hence,
takes the most energy from the signal.
The matching pursuit method is an iterative method. At the first iteration the
signal is decomposed into two orthogonal components
x(t) == Xl(t) + Rlx(t)
(2.51)
which can also be written as
xo(t) == Moho(t) + R 1 x(t)
(2.52)
where R 1 x(t) is the residual vector after approximating the original signal xo(t) (now
subscripted with 0) in the direction of ho(t). R 1 x(t) and ho(t) are orthogonal to each

54 The continuous wavelet transform
other, hence the energy of the signal can be expressed as
Ilxo(t)112 == II M oho(t)112 + IIR 1 x(t)112
(2.53)
where IIxl1 2 == f Ixl 2 dx. In addition, as ho(t) has unit energy by definition and Mo is a
constant, then we can write
Ilxo(t)112 == IM o l 2 + IIR 1 x(t)112
(2.54 )
The first coefficient Mo is found by applying each of the dictionary functions to the
signal in turn and choosing the one which maximizes the value of Mo, or conversely,
the one which minimizes the energy of the residuaIIIR 1 x(t)112. This residual signal is
then examined in the same way to find the second coefficient M 1 and the new signal
residual R 2 x(t), and so on. At any stage in the decomposition, the original signal can
be partitioned into two components: the reconstructed part using all the expansion
coefficients (given by equation (2.46)), and the residual component after n iterations,
Rnx(t). This can be written as
n-I
xo(t) == 2:: Mihi(t) + Rnx(t) == xn(t) + Rnx(t)
i=O
(2.55)
where X n (t) is the signal approximation and R n x( t) is the residual signal left after the
nth iteration of the matching pursuit algorithm. In addition, the energy at the nth
decomposition is given by
n-I
IIXo(t) 11 2 == 2:: IM i l 2 + IIRnx(t) 11 2
i=O
(2.56)
In practice the matching pursuit algorithm is terminated either when the residual
energy is below a preset cut-off level c, defined by
IIRn x (t)112 < c 2 (xO(t)2)
(2.57)
or, alternatively, after a predetermined number of iterations, n.
Figure 2.32 shows an example of a synthetic signal decomposed using the
matching pursuit method selecting from a dictionary containing three basic elements:
Gabor atoms, Dirac delta functions (spikes in the time domain) and sinusoidal
Fourier functions (spikes in the frequency domain and continuous sinusoids in the
time domain). We can see from the figure that the MP algorithm leads to an
intermittent, patchy covering of the time-frequency plane as only those coefficients
obtained when the time-frequency atom is matched to the signal at locations of
maximum energy removal are retained. This contrasts both with the wavelet
transform and short time Fourier transform where the time-frequency plane is
covered evenly. The time-frequency plot used to represent the energy distribution
of the matching pursuit decomposition is based on a Wigner distribution. We do
not go into the details of this representation here, but instead the reader is referred
elsewhere at the end of this chapter. Modified versions of the basic matching pursuit
algorithm are now available and, in later chapters in this book, some examples are

Wavelets in two or more dimensions 55
>.
u
s:::
Q)
&
Q)

time
IV
III
II
I
Figure 2.32. The time-frequency decomposition of a test signal using the matching pursuit method. The
figure contains the time-frequency plane with the energy distributions associated with the MP
decomposition of the signal IV. Letters mark signal components and corresponding atoms or
groups of atoms: A, B = single transients; C, D = sinusoids modulated by a Gaussian function;
E = single sharp transient; F = sinusoid. Signal IV is a sum of I, II and III. The MP analysis
time-frequency plot for signal IV is shown above the signals. The darkness of the plot elements is
proportional to the logarithm of the energy distribution in the time-frequency plane according to
a Wigner- Ville distribution. With kind permission of P.l. Durka.
given of the use of matching pursuit methods in medicine, geophysics, finance and
. .
engIneerIng.
2.14 Wavelets in two or more dimensions
The two-dimensional Mexican hat specified on a tl, t2 coordinate plane is given by
1)J(t) == (2 - It1 2 ) e-ltI2/2
(2.58)
where t is the coordinate vector (t l ,t 2 ) and It I = Jti+t . Figure 2.33 shows a
couple of two-dimensional Mexican hats on the plane. The two-dimensional wavelet
transform is given by
1 J oo ( t b )
T(a, b) == - 1)J* x(t) dt
a -00 a
(2.59)
where b is the coordinate vector (b l , b 2 ). Note that the weighting function w(a) has
been set to 1/ a required to conserve energy across scales for the two-dimensional
wavelets. Note also that the coordinate vector t == (t l , t 2 ) is likely to specify two

56 The continuous wavelet transform
1!J(t)
t 1
I
/
2D Mexican hat on
the plane
side on view
reduced a scale
Figure 2.33. Two-dimensional Mexican hat.
spatial (rather than temporal) coordinates in practice, where the function x( t) could
for example be surface heights (e.g. fracture surfaces, topographic features) or image
greyscales (e.g. medical images, fluid visualization studies). The corresponding inverse
wavelet transform is
1 .f oo .f oo 1 ( t - b ) da
x(t) == - c - g T(a, b) 3 db
g -00 0 a a a
(2.60)
Wavelet transforms in higher dimensions, D, are also possible simply by extending
the length of the vectors t and b to D components. To preserve energy in the
D-dimensional transformation, the weighting function becomes 1/ a D / 2 . Hence, the
D-dimensional wavelet is defined as
1 ( t-b )
1)J a,b (t) == aD /2 1)J a
(2.61 )
The transform in D dimensions becomes
T(a, b) = ,CX) ?j;,b(t)X(t) dt
(2.62)
with inverse
1 .f oo .f oo da
x(t) == - c 1)Ja,b(t)T(a, b) D+l db
g -00 0 a
(2.63)
and the energy of the signal may be found from
1 .f oo .f 00 2 da
E == - c IT (a, b) I D+l db
g -00 0 a
(2.64 )
2.15 The CWT: computation, boundary effects and viewing
As with all mathematical tools used to investigate physical phenomena, a number of
practical issues must be taken into consideration. This is no less the case when using

The CWT: computation, boundary effects and viewing 57
the wavelet transform. The results obtained by the investigator must be viewed in
terms of the limitations in the data analysis method used. These limitations stem
from a number of sources, including the discrete nature of the data, the finite resolu-
tion of the data, the finite extent of the data, the wavelet used, the discretization and
numerical computation of the transform, and so on.
To compute the continuous wavelet transform we could simply perform a naive
discretization of the transform integral given by equation (2.13), replacing the
integral with a discrete summation involving the sampling interval of the measured
time series t::..t together with a suitable discretization for the a and b parameters-
usually logarithmic for a and linear steps of t::..t for b. This is a very cumbersome
way to compute the CWT integral and a much better approach is to use the FFT
method described in section 2.10 and given by equation (2.35). The FFT provides a
much faster algorithm for the computation of the transform integral. In fact, this
approach has been used in all the transform plots presented in this chapter. We
will consider again the discretization of the wavelet transform in the next chapter
when we tackle the discrete wavelet transform (DWT). However, note that the
DWT is fundamentally different from the discretized CWT (see chapter 3, section
3.2.2) and in the scientific literature the term 'continuous wavelet transform' generally
includes all discretizations using continuous wavelets such as the Mexican hat,
Morlet, etc.
In practice, experimental data sets are finite in extent and often the investigator
wants to analyse the whole of the available data. However, an obvious consequence
of wavelet analysis of a finite data set is that, as the wavelet gets closer to the edge of
the data, parts of it begin 'spill over' the edge as illustrated in figure 2.34(a). This
creates a boundary effect, where transform values close to the boundary of the
signal are tainted by the discontinuous nature of the signal edge. The affected
region increases in extent as the dilation of the analysing wavelet increases. An
example of boundary effects is shown in the transform plot of figure 2.34(c), which
is generated for the simple signal shown in figure 2.34(b) using a Mexican hat wavelet.
The transform plot points to the location of the signal discontinuity as expected, but
also treats the edges of the signal as additional discontinuities. Hence, large T(a, b)
values are realized close to the edge of the transform plot, which increase in extent
as a increases. This region affected by a discontinuity is known as the cone of
influence. The extent of the cone increases linearly with a, i.e. it is proportional to
the temporal support (or 'width') of the wavelet. (We have already seen the cone of
influence corresponding to a signal discontinuity in figure 2.12.) However, when
plotting T(a, b) using a logarithmic a scale (e.g. figure 2.34(c)) the cone contours
become curved. The cone boundaries at either ends of the signal define the region
which is significantly influenced by the signal edges. It is very much up to the
investigator where to take the boundary of the cone by choosing a limiting value of
the distance from the wavelet centre-a multiple of the wavelet a scale-at which
point pertinent information contained in the transform plot is not considered
masked by edge effects.
A number of methods have been developed to cope with the boundaries
of signals of finite extent. A variety of these are illustrated in figure 2.35 and
include: adding a line of zero values at either end of the signal (zero padding,

58 The continuous wavelet transform
x(t) ...
1!J«t-b)/a)
(a)
End of signal
Part of wavelet
not within signal
segment
(b)
Edge effects
(c)
Figure 2.34. The manifestation of boundary effects in the scalogram. ( a) Schematic diagram of the
wavelet encountering the signal boundary. (b) A simple signal with a discontinuity halfway along
its length. (c) The scalogram of the signal in (b).
figure 2.35(a)); adding a line of constant values at either end of the signal equal to the
last value of the signal (value padding, figure 2.35(b )); adding some form of decay to
zero for the last value at each end of the signal (decay padding, figure 2.35(c));
continuing the signal on from the last point back to the first point (periodization,
figure 2.35(d)); reflecting the signal at the edges (reflection, figure 2.35(e));
convolving the signal with a window function which reduces the edge values of the
signal to zero (windowing, figure 2.35(f)); using a polynomial extrapolation of the

The CWT: computation, boundary effects and viewing 59
x(t)
Signal set to zero here
\
Signal set to zero here
j

t
(a)
Beginning of
signal
segment
End of
signal
segment
x(t)
Signal set to initial Signal set to final
value here value here \

r t
Beginning of End of
(b) signal signal
segment segment
x(t)
Signal decays Signal decays
to zero to zero

t
(c)
Beginning of
signal
segment
End of
signal
segment
Figure 2.35. Schemes to deal with signal boundaries. (a) Zero padding. (b) Value padding. (c) Decay
padding.
signal at either end (polynomial fitting, figure 2.35(g)); and, if we are focusing on a
small segment of a much larger signal available to us, we may simply use the data
points outside the segment under consideration (signal following, figure 2.35(h)). In
addition, there is the wraparound method which uses only the length of signal
available but wraps the parts of the wavelet which fall off each end of the signal
back to the other end. Wraparound is similar to periodization but gives wavelet
transform values only within the extent of the signal. Wraparound is used
extensively in the next chapter. Whichever method is employed to cope with the
signal edges, we must be aware that features appearing near the edges of the
transform plot will contain information (synthetic or real) from outside the region
of the signal segment under consideration. In addition, this effect will increase in

60
The continuous wavelet transform
x(t)
Repeated segment
Repeated segment

(d)
Beginning of
signal
segment
End of
signal
segment
x(t)
Reflected segment
Reflected segment
I
(e)
Beginning of
signal
segment
End of
signal
segment
x(t)
Signal set to zero here
\
Smoothing 0 .. I
. ngina
wIndow . I
\ sIgna
///------------,.,>/ Modified signal
J : .
. .
Signal set to
zero here
i
1
t
(f)
Beginning of
signal
segment
End of
signal
segment
Figure 2.35 ( continued). (d) Periodization involves the repetition of the signal along the time axis.
(e) Reflection. (f) Smoothing window.
extent as the wavelet a scale increases and the wavelet extends further beyond the edge
of the signal when close to it.
Finally, when plotting the wavelet transform, we want to be able to discern
features associated with the signal at specific a scales. It is often the case that features

The CWT: computation, boundary effects and viewing 61
x(t)
r t
Beginning of End of
signal signal
(g) segment segment
Polynomial fit
x(t)
Signjutwith segment
Signal outwith segment
r
t
(h)
Beginning of
signal
segment
End of
signal
segment
Figure 2.35 (continued). (g) Polynomial fitting. (h) Signal following.
at certain scales dominate the transform plot, obscuring the detail at other scales. In
order to accentuate this hidden detail we can change the weighting parameter w(a) in
the transform from the usual 1/ va value to a value which is more suitable at
highlighting features at a specific a scale of interest. This is shown in figure 2.36 for
a sinusoidal signal containing a burst of noise. The transform plot with w( a) set to
1/ va shows up the sinusoidal waveform well, but this dominates and the noise is
not highlighted particularly well. By changing the weighting to w(a) == a-1. 5 we can
enhance the noise within the plot at lower a scales. When w( a) is set to a -2.5 we
can see that the noise now dominates the transform plot. It is sometimes useful to
vary w( a) in this way to accentuate features at different scales in the transform plot
which might otherwise have been missed. Another way to weight the transform
plot to show up small and large amplitude features is to use a logarithmic scale for
T(a, b). However, two problems arise with this approach. The first is that we
cannot take the logarithm of the negative values of T (a, b) and the second is that
taking the logarithm of near-zero values of T(a, b) produces very large negative

62 The continuous wavelet transform
(a)
(b)
(c)
(d)
(e)
Figure 2.36. Accentuating features in the wavelet transform plot. (a) Original signal. (b) Transform
plot with w(a) = l/va. (c) Transform plot with w(a) = a-1.5. (d) Transform plot with
w(a) = a- 2 . 5 . (e) Logarithmic plot of the modulus of the Mexican hat wavelet transform IT(a, b) I.
Near-zero values of I T( a, b) I produce large negative logarithmic values. These are omitted using a
minimum cut-off value, where all I T( a, b) I values less than the cut-off are set to this value before
computing logarithms.
numbers. We can get around these two problems, respectively, by using the modulus
of the transform I T(a, b) I to avoid negatives and setting a cut-off, or floor, value of
IT(a,b)1 in order to both avoid zero and near-zero values and limit the extent of
the logarithmic scale. This is illustrated in figure 2.36( e). Logarithmic plotting is
particularly good at highlighting simultaneously features in the signal occurring at
very different orders of magnitude, see for example the logarithmic plots of the
ECG signals in figures 6.12 to 6.15 of chapter 6.

2.16 Endnotes
Endnotes 63
2.16.1 Chapter keywords and phrases
(Y ou may find it helpful to jot down your understanding of each of them.)
wavelet
mother/analysing wavelet
Mexican hat wavelet
Fourier transform
admissibility condition
admissibility constant
energy spectrum
bandpass filter
2.16.2 Further resources
passband centre
dilation parameter
location parameter
wavelet transform
inverse wavelet transform
wavelet transform modulus
maxzma
scalogram
spectrogram
wavelet variance
Morlet wavelet
wavelet transform ridges
short time Fourier transform
windowed Fourier atom
time-frequency atom
matching pursuit
boundary effects
There are a number of sources of information at an introductory level concerning
the continuous wavelet transform. Gade and Gram-Hansen (1997) compare wavelet
transforms with short time Fourier transforms, presenting time-frequency plots of a
variety of signals including speech and engine vibrations. Sarkar and Su (1998) detail
the properties of the continuous wavelet transform from the perspective of electrical
engineers. The relationship between Fourier and wavelet spectra is described by
Perrier et al (1995). Wong and Chen (2001) provide a number of illustrative examples
of phase and modulus plots in their Morlet wavelet-based study of nonlinear struc-
tural oscillations. See also chapter 1 of the book by Holschneider (1995) which
contains many informative modulus and phase plots for a variety of wavelets and
various simple signal features. Holshneider gives details of many other wavelets
including Bessel, chirp, Cauchy, Poisson and Marr. Many web sites contain good
introductory material on the continuous wavelet transform. The appendix lists
some websites from which to begin a search.
Concise treatments of the continuous wavelet transform which provide a little
more of the mathematical background at a reasonable mathematical level are to be
found in chapter 3 of the book by Kaiser (1994), chapter 3 of the book by Blatter
(1998), chapter 5 of the book by Vetterli and Kovacevic (1995) and the paper
by Koornwinder in the book edited by the same author (Koornwinder, 1993).
Heisenberg's uncertainty principle is covered by many authors; see for example
Mallat (1998) or Kaiser (1994) for some extra information. Ridges have been used
by Staszewski (1997, 1998a) in a new procedure for nonlinear system identification.
(More details are given in chapter 5, section 5.4.) See also the early paper by Delprat
et al (1992) and the algorithms for ridge detection of noisy signals by Carmona et al
(1997, 1999). On a related topic, the relationship between standard wavelet scalo-
grams and Fourier wavelengths is discussed by Meyers et al (1993), who provide
brief mathematical detail in the appendix. See also Torrence and Compo (1998)
who provide the information on how to find the Fourier wavelengths for the
Morlet, Paul and all the derivatives of Gaussian wavelets (including the Mexican

64 The continuous wavelet transform
hat). More information on modulus maxima and signal reconstruction can be found
in the papers by Mallat and Hwang (1992) and Mallat and Zhong (1992). Examples of
the use of modulus maxima and ridge methods are to be found in chapters 4 to 7 of
this book (e.g. figures 5.27 and 5.28 in chapter 5 and figures 6.20 and 6.21 in chapter
6). Further information on the application of ridges, modulus maxima, the complete
Morlet wavelet and complex Mexican hat can also be found in Addison et al (2002a).
In this chapter we considered the two-dimensional Mexican hat wavelet. More
information concerning the two-dimensional continuous wavelet transform can be
found in the book by Antoine (1999). A two-dimensional Morlet wavelet is given
by Kumar and Foufoula-Georgiou (1994, p 29) and Peyrin and Zaim (1996).
Koornwinder (1993) provides a brief outline of the mathematics of multidimensional
wavelet transforms. See also the discussion by Daubechies (1992, pp 33-34)
concerning the continuous wavelet in higher dimensions where rotations can be
introduced into the definition of higher-dimensional wavelets.
Numerous, diverse applications of the continuous wavelet transform to real signals
are given in chapters 4 to 7 of this book, all of which require a discretized version of the
transform. We dealt briefly with this issue in section 2.15. For more information on the
discretization of the continuous wavelet transform see for example Jones and Baraniuk
(1991), Sadowsky (1994) and Jordan et al (1997). The latter authors discuss many of the
implementation issues that arise from the discretization itself. In particular they show
how to convert from the non-dimensional times and frequencies of the mother wavelet
to physical measures. They use the Morlet wavelet to illustrate their discussion, apply-
ing it to the velocity fluctuations measured in a subsonic wake undergoing transition to
turbulence. Note that they use angular frequency and a Morlet wavelet which is not
normalized to have unit energy. In this chapter we have not dealt directly with the
issue of the physical meaning of the a scale parameter. It is non-dimensional. However,
in practice it has to be linked to the timescales under investigation. The easiest thing to
do is relate a = 1 to an appropriate unit of time, e.g. 1 second, 1 day, 1 year, etc. This has
been done in this chapter when necessary: see for example figure 2.18 containing the
vortex shedding time series where a == 1 corresponded to 1 second. It can also be
linked to the sampling interval t::..t or some multiple of it. We do not even have to
use the a scale parameter to link to a temporal scale but can employ another measure
of spread such as the standard deviation of the wavelet energy density in time; this has
been done for the Morlet wavelet by Jordan et al (1997).
A good account of the matching pursuit method is given in the original paper by
Mallat and Zhang (1993). The method has been applied in many areas of science,
technology and finance. We will come across some of these applications in subsequent
chapters of this book. However, the paper by Zygierewicz et al (1999), which investi-
gates sleep patterns in the EEG, gives a concise account in its appendix of the time-
frequency (Wigner) representation of energy distribution used for the matching
pursuit (something we have not gone into detail with here). The matching pursuit
is a member of a larger family of adaptive approximation techniques. Jaggi et al
(1998) give a brief account of related methods and introduce a high resolution pursuit
which overcomes some of the shortcomings of the matching pursuit. Goodwin and
Vetterli (1999) used a matching pursuit method which incorporates damped sinusoids
as the analysing functions.

Chapter 3
The discrete wavelet transform
3.1 Introduction
In this chapter we consider the discrete wavelet transform (DWT). We will see that when
certain criteria are met it is possible to completely reconstruct the original signal using
infinite summations of discrete wavelet coefficients rather than continuous integrals (as
required for the CWT). This leads to a fast wavelet transform for the rapid computa-
tion of the discrete wavelet transform and its inverse. We will then see how to perform a
discrete wavelet transform on discrete input signals of finite length: the kind of signal we
might be presented with in practice. We will also consider briefly biorthogonal wavelets,
which come in pairs, and some space is devoted to two-dimensional discrete wavelet
transforms. The chapter ends with wavelet packets: a generalization of the discrete
wavelet transform which allows for adaptive partitioning of the time-frequency plane.
3.2 Frames and orthogonal wavelet bases
3.2.1 Frames
In chapter 2, the wavelet function was defined at scale a and location b as
1 ( t-b )
'lj; a,b ( t) = va 'lj; ----;;-
(3.1)
In this section the wavelet transform of a continuous time signal, x( t), is
considered where discrete values of the dilation and translation parameters, a and
b, are used.
A natural way to sample the parameters a and b is to use a logarithmic discretiza-
tion of the a scale and link this, in turn, to the size of steps taken between b locations.
To link b to a we move in discrete steps to each location b which are proportional to
the a scale. This kind of discretization of the wavelet has the form
'lj;m,n(t) =  'lj; ( t - nodO )
a o a o
(3.2)
65

66 The discrete wavelet transform
where the integers m and n control the wavelet dilation and translation respectively; ao
is a specified fixed dilation step parameter set at a value greater than 1, and b o is the
location parameter which must be greater than zero. The control parameters m and n
are contained in the set of all integers, both positive and negative. It can be seen from
the above equation that the size of the translation steps, b == boao, is directly
proportional to the wavelet scale, ao.
The wavelet transform of a continuous signal, x( t), using discrete wavelets of the
form of equation (3.2) is then
.f oo 1
T m,n == x( t) m/2 1)J( ao m t - nb o ) dt
-00 a
o
(3. 3a)
which can also be expressed as the inner product
T m,n == (x,1)Jm,n)
(3.3b)
where Tmn are the discrete wavelet transform values given on a scale-location grid of
,
index m,n. For the discrete wavelet transform, the values Tmn are known as wavelet
,
coefficients or detail coefficients. These two terms are used interchangeably in this
chapter as they are in the general wavelet literature. To determine how 'good' the
representation of the signal is in wavelet space using this decomposition, we can
resort to the theory of wavelet frames which provides a general framework for study-
ing the properties of discrete wavelets. Wavelet frames are constructed by discretely
sampling the time and scale parameters of a continuous wavelet transform as we
have done above. The family of wavelet functions that constitute a frame are such
that the energy of the resulting wavelet coefficients lies within a certain bounded
range of the energy of the original signal, i.e.
00
AE < 2::
00
2:: I T m,n1 2 < BE
(3.4)
m= -00 n=-oo
where T m n are the discrete wavelet coefficients, A and B are the frame bounds, and E
,
is the energy of the original signal given by equation (2.19) in chapter 2:
E == f 00 Ix(t) 1 2 dt == IIX( t) 11 2 , where our signal, x(t), is defined to have finite energy.
The values of the frame bounds A and B depend upon both the parameters ao and
b o chosen for the analysis and the wavelet function used. (For details of how to deter-
mine A and B see Daubechies (1992).) If A == B the frame is known as 'tight'. Such
tight frames have a simple reconstruction formula given by the infinite series
1 0000
x( t) = A 2:: 2:: Tm,n'l/;m,n(t)
m=-oo n=-oo
(3.5)
A tight frame with A (== B) > 1 is redundant, with A being a measure of the
redundancy. However, when A == B == 1 the wavelet family defined by the frame
forms an orthonormal basis. If A is not equal to B a reconstruction formula can
s till be written:
/ 2  
x(t)== A+B  
m=-oo n=-oo
T m,n 1)Jm,n (t)
(3.6)

Frames and orthogonal wavelet bases 67
0.5a
scale m
a = 6/Z)m
b
width
xyz
0.5a
b
Figure 3.1. The nearly tight Mexican hat wavelet frame with ao = 2 1 / 2 and ho = 0.5. Three consecutive
locations of the Mexican hat wavelet for scale indices m (top) and m + 1 (lower) and location indices
n, n + 1, n + 2. That is, a = 2 m and a = 2 m + 1 respectively, and three consecutive b locations sepa-
rated by a12.
where x' (t) is the reconstruction which differs from the original signal x( t) by an error
which depends on the values of the frame bounds. The error becomes acceptably
small for practical purposes when the ratio B/ A is near unity. It has been shown,
for the case of the Mexican hat wavelet, that if we use ao == 2 1 / v , where v > 2 and
b o < 0.5, the frame is nearly tight or 'snug' and for practical purposes it may be
considered tight. (This fractional discretization, v, of the power-of-two scale is
known as a voice.) For example, setting ao == 2 1 / 2 and b o == 0.5 for the Mexican hat
leads to A == 13.639 and B == 13.673 and the ratio B/A equals 1.002. The closer this
ratio is to unity, the tighter the frame. Thus discretizing a Mexican hat wavelet
transform using these scale and location parameters results in a highly redundant
representation of the signal but with very little difference between x(t) and x' (t).
The nearly tight Mexican hat wavelet frame with these parameters (ao == 2 1 / 2 and
b o == 0.5) is shown in figure 3.1 for two consecutive scales m and m + 1 and at three
consecutive locations, n == 0, 1 and 2.
3.2.2 Dyadic grid scaling and orthonormal wavelet transforms
Common choices for discrete wavelet parameters ao and b o are 2 and 1 respectively.
This power-of-two logarithmic scaling of both the dilation and translation steps is
known as the dyadic grid arrangement. The dyadic grid is perhaps the simplest and
most efficient discretization for practical purposes and lends itself to the construction

68 The discrete wavelet transform
of an orthonormal wavelet basis. Substituting ao == 2 and b o == 1 into equation (3.2),
we see that the dyadic grid wavelet can be written as
1 ( t - n2 m )
'ljJm,n (t) = V2fi1 'ljJ 2 m
(3.7a)
or, more compactly, as
1)Jm,n (t) == 2- m / 2 1)J(2- m t - n)
( 3 . 7b )
Note that this has the same notation as the general discrete wavelet given by equation
(3.2). From here on in this chapter we will use 1)Jm,n(t) to denote only dyadic grid
scaling with ao == 2 and b o == 1.
Discrete dyadic grid wavelets are commonly chosen to be orthonormal. These
wavelets are both orthogonal to each other and normalized to have unit energy.
This is expressed as
,[ 00 { I if m == m/ and n == n/
1)Jm,n (t)1)Jm',n' (t) dt == .
-00 0 otherwIse
(3.8)
That is to say, the product of each wavelet with all others in the same dyadic system
(i.e. those which are translated and/or dilated versions of each other) are zero. This
means that the information stored in a wavelet coefficient T m n is not repeated
,
elsewhere and allows for the complete regeneration of the original signal without
redundancy. In addition to being orthogonal, orthonormal wavelets are normalized
to have unit energy. This can be seen from equation (3.8) as, when m == m/ and
n == n', the integral gives the energy of the wavelet function equal to unity. Ortho-
normal wavelets have frame bounds A == B == 1 and the corresponding wavelet
family is an orthonormal basis. (A basis is a set of vectors, a combination of which
can completely define the signal, x(t). An orthonormal basis has component vectors
which, in addition to being able to completely define the signal, are perpendicular to
each other.) The discrete dyadic grid wavelet lends itself to a fast computer algorithm,
as we shall see later.
Using the dyadic grid wavelet of equation (3.7a), the discrete wavelet transform
(DWT) can be written as:
Tm,n = .[00 x(t)'ljJm,n(t) dt
(3.9)
By choosing an orthonormal wavelet basis, 1)Jm,n(t), we can reconstruct the original
signal in terms of the wavelet coefficients, T m n, using the inverse discrete wavelet trans-
,
form as follows:
00
x(t) == 
00
 Tm,n1)Jm,n(t)
(3.1 Oa)
m=-oo n=-oo
requiring the summation over all integers m and n. This is actually equation (3.5), with
A == 1 due to the orthonormality of the chosen wavelet. Equation (3.10a) is often seen

Frames and orthogonal wavelet bases 69
written in terms of the inner product
00
x(t) == 
00
 (x, 1)Jm,n)1)Jm,n (t)
(3.10b)
m=-oo n=-oo
where the combined decomposition and reconstruction processes are clearly seen:
going from x( t) to T m,n via the inner product (x, 1)Jm,n) then back to x( t) via the infinite
summations. In addition, as A == B and A == 1, we can see from equation (3.4) that the
energy of the signal may be expressed as
.C00 1x (t)1 2 dt= moo n00ITm,nI2
(3.11)
Before continuing it is important to make clear the distinct difference between the
DWT and the discretized approximations of the CWT covered in chapter 2. The
discretization of the continuous wavelet transform, required for its practical
implementation, involves a discrete approximation of the transform integral (i.e. a
summation) computed on a discrete grid of a scales and b locations. The inverse
continuous wavelet transform is also computed as a discrete approximation. How
close an approximation to the original signal is recovered depends mainly on the
resolution of the discretization used and, with care, usually a very good approxima-
tion can be recovered. On the other hand, for the DWT, as defined in equation (3.9),
the transform integral remains continuous but is determined only on a discretized grid
of a scales and b locations. We can then sum the DWT coefficients (equation (3.10a))
to get the original signal back exactly. We will see later in this chapter how, given an
initial discrete input signal, which we treat as an initial approximation to the
underlying continuous signal, we can compute the wavelet transform and inverse
transform discretely, quickly and without loss of signal information.
3.2.3 The scaling function and the multiresolution representation
Orthonormal dyadic discrete wavelets are associated with scaling functions and their
dilation equations. The scaling function is associated with the smoothing of the signal
and has the same form as the wavelet, given by
cjJm,n (t) == 2- m / 2 cjJ(2- m t - n)
(3.12)
They have the property
.[00 <Po,o( t) dt = 1
where cjJo,o(t) == cjJ(t) is sometimes referred to as the father scaling function or father
wavelet (cf. mother wavelet). (Remember from chapter 2 that the integral of a wavelet
function is zero.) The scaling function is orthogonal to translations of itself, but not to
dilations of itself. The scaling function can be convolved with the signal to produce
approximation coefficients as follows:
(3.13)
Sm,n = .[00 x(t)<Pm,n(t) dt
(3.14)

70 The discrete wavelet transform
From the last three equations, we can see that the approximation coefficients are
simply weighted averages of the continuous signal factored by 2 m / 2 . The approxi-
mation coefficients at a specific scale m are collectively known as the discrete
approximation of the signal at that scale. A continuous approximation of the signal
at scale m can be generated by summing a sequence of scaling functions at this
scale factored by the approximation coefficients as follows:
00
xm(t) == 2:: Sm,nCPm,n(t)
n= -00
(3.15)
where X m (t) is a smooth, scaling-function-dependent, version of the signal x( t) at scale
index m. This continuous approximation approaches x( t) at small scales, i.e. as
m ----7 -00. Figure 3.2(a) shows a simple scaling function, a block pulse, at scale
index 0 and location index 0: CPo,o(t) == cp(t)-the father function-together with
two of its corresponding dilations at that location. It is easy to see that the convolution
of the block pulse with a signal (equation (3.14)) results in a local weighted averaging of
the signal over the nonzero portion of the pulse. Figure 3 .2(b) shows one period of a
sine wave, x(t) contained within a window. Figure 3.2(c) shows various approximations
of the sine wave generated using equations (3.14) and (3.15) with the scaling function set
to a range of widths, 2 0 to 2 7 . These widths are indicated by the vertical lines and arrows
in each plot. Equation (3.14) computes the approximation coefficients Sm n which are,
,
as mentioned above for this simple block scaling function, the weighted average of the
signal over the pulse width. The approximation coefficients are then used in equation
(3.15) to produce an approximation of the signal which is simply a sequence of scaling
functions placed side by side, each factored by their corresponding approximation co-
efficient. This is obvious from the blocky nature of the signal approximations. The
approximation at the scale of the window (==2 7 ) is simply the average over the whole
sine wave which is zero. As the scale decreases, the approximation is seen to approach
the original waveform. This simple block pulse scaling function used in this example
here is associated with the Haar wavelet, which we will come to shortly.
We can represent a signal x(t) using a combined series expansion using both the
approximation coefficients and the wavelet (detail) coefficients as follows:
00 mo
x( t) == 2:: Smo,n CPmo,n (t) + 2::
00
2:: Tm,n1jJm,n(t)
(3.16)
n=-oo
m=-oo n=-oo
We can see from this equation that the original continuous signal is expressed as a
combination of an approximation of itself, at arbitrary scale index mo, added to a
succession of signal details from scales mo down to negative infinity. The signal
detail at scale m is defined as
00
dm(t) == 2:: Tm,n1jJm,n(t)
(3.17)
n= -00
hence we can write equation (3.16) as
mo
x(t) == xmo (t) + 2:: dm(t)
(3.18)
m=-oo

Frames and orthogonal wavelet bases 71
<p(t) ..
<Po o(t)
,
/ <P 10 (t)
,
1 / <P 2 o(t)
0.707 ,
/
0.5
(a)
.....
II"'"
-1
o
1
2
3
4
t
(b)
m=O m=l
I
m=2 m=3
I -1  I I I \ I
m=4 m=5
I \ I I  I
m=6 m=7
(c)
Figure 3.2. Smooth approximation of a sine wave using a block pulse scaling function. (a) Simple block
scaling function shown at scale 1 (scale index m = 0) and location n = 0, i.e. CPo,o(t) (shown bold),
together with its dilations at that location. (b) Sine wave of one period. (c) Selected smooth approxi-
mations, xm(t), of the sine wave at increasing scales. The width of one of the scaling functions CPm,n(t) at
scale index m is depicted in each figure by the arrows. Note that the example has been set up so that one
From this equation it is easy to show that
X m -1 ( t) == X m ( t) + d m ( t)
(3.19)
which tells us that if we add the signal detail at an arbitrary scale (index m) to the
approximation at that scale we get the signal approximation at an increased resolution
(i.e. at a smaller scale, index m - 1). This is called a multiresolution representation.

72 The discrete wavelet transform
3.2.4 The scaling equation, scaling coefficients and associated wavelet equation
The scaling equation (or dilation equation) describes the scaling function cjJ(t) in terms
of contracted and shifted versions of itself as follows:
cjJ( t) == 2:: CkcjJ( 2t - k)
k
where cjJ(2t - k) is a contracted version of cjJ(t) shifted along the time axis by an integer
step k and factored by an associated scaling coefficient, Ck. (Take note of the similar
but different terminology-scaling equation and scaling function.) Equation (3.20)
basically tells us that we can build a scaling function at one scale from a number of
scaling equations at the previous scale. The solution to this two-scale difference
equation gives the scaling function cjJ(t). For the sake of simplicity in the rest of the
chapter we concern ourselves only with wavelets of compact support. These have
sequences of nonzero scaling coefficients which are of finite length. Integrating
both sides of the above equation, we can show that the scaling coefficients must satisfy
the following constraint:
(3.20 )
2:: Ck == 2
k
In addition, in order to create an orthogonal system we require that
{ 2 if k' == 0
2:: ck c k+2k! == .
k 0 otherwIse
This also tells us that the sum of the squares of the scaling coefficients is equal to 2.
The same coefficients are used in reverse with alternate signs to produce the
differencing of the associated wavelet equation, i.e.
(3.21)
(3.22 )
1jJ( t) == 2:: ( _1)k Cl- kcjJ(2t - k)
k
This construction ensures that the wavelets and their corresponding scaling functions
are orthogonal. This wavelet equation is commonly seen in practice. In this chapter,
however, we will consider only wavelets of compact support which have a finite
number of scaling coefficients, N k . For this case we can define the wavelet function as
(3.23)
1jJ( t) == 2:: ( _1)k cN k -1-kcjJ(2t - k)
k
This ordering of scaling coefficients used in the wavelet equation allows for our wave-
lets and their corresponding scaling equations to have support over the same interval
[0, N k - 1]. (The ordering of equation (3.23) leads to wavelet and scaling functions
displaced from each other, except for the Haar wavelet where N k == 2.) Note that,
if the number of scaling coefficients is not finite, we cannot use this reordering and
must revert back to an ordering of the type given by equation (3.23). We will stick
to the ordering specified by equation (3.24) in this text.
Often the reconfigured coefficients used for the wavelet function are written more
compactly as
( 3 .24 )
b k == (_I)k CN k -1- k
(3.25)

Frames and orthogonal wavelet bases 73
where the sum of all the coefficients b k is zero. Using this reordering of the coefficients,
equation (3.24) can be written as
Nk-l
1)J(t) == 2:: b k cjJ(2t - k)
k=O
(3.26)
From equations (3.12) and (3.20) and examining the wavelet at scale index m + 1, we
can see that for arbitrary integer values of m the following is true:
r (m+ 1)/2cp ( 2 mt + I - n) = r m / 2 2 -1/2  Ckcp ( 2 t2m - 2n - k)
(3.27a)
which may be written more compactly as
1
CPm+ I,n(t) = yI2  CkCPm,2n+k(t)
(3.27b)
That is, the scaling function at an arbitrary scale is composed of a sequence of shifted
scaling functions at the next smaller scale each factored by their respective scaling co-
efficients. Similarly, for the wavelet function we obtain
1
'l/Jm+ I,n(t) = yI2  b k CPm,2n+k(t)
(3.28)
3.2.5 The Haar wavelet
The Haar wavelet is the simplest example of an orthonormal wavelet. Its scaling equa-
tion contains only two nonzero scaling coefficients and is given by
cjJ(t) == cjJ(2t) + cjJ(2t - 1)
(3.29)
that is, its scaling coefficients are Co == CI == 1. We get these coefficient values by
solving equations (3.21) and (3.22) simultaneously. (From equation (3.21) we see
that Co + CI == 2 and from equation (3.22) CoCo + CI CI == 2.) The solution of the
Haar scaling equation is the single block pulse shown in figure 3.3(a) and defined as
{ I O<t<1
1)J( t) == 0 el se where
(3.30)
This is, in fact, the scaling function used earlier in figure 3.2 to generate the signal
approximations of the sine wave. Reordering the coefficient sequence according to
equation (3.25) we can see that the corresponding Haar wavelet equation is
1)J(t) == cjJ(2t) - cjJ(2t - 1)
(3.31)
The Haar wavelet is shown in figure 3.3(b) and defined as
1 O < t<!
1)J( t) == -1 ! < t < 1
o elsewhere
(3.32)

74 The discrete wavelet transform
<I>(t) <I> (2t) <I>(2t-1)
SCALING 1
FUNCTION - +
-
0 1 0 1/2 0 1/2 1
(a)
1!J (t)
<I> (2t)
WAVELET
FUNCTION
-
-
+
(b)
-<I> (2t-1)
(c)
(d)
(e)
Figure 3.3. Discrete orthonormal wavelets. (a) The Haar scaling function in terms of shifted and
dilated versions of itself. (b) The Haar wavelet in terms of shifted and dilated versions of its scaling
function. (c) Three consecutive scales shown for the Haar wavelet family specified on a dyadic grid,
e.g. from top to bottom: 'l/Jm,n(t), 'l/Jm+ 1,n(t) and 'l/Jm+2,n(t). (d) Three Haar wavelets at three consecu-
tive scales on a dyadic grid. (e) Three Haar wavelets at different scales. This time the Haar wavelets
are not defined on a dyadic grid and are hence not orthogonal to each other.
The mother wavelet for the Haar wavelet system, 1)J(t) == 1)Jo,o(t), is formed from two
dilated unit block pulses sitting next to each other on the time axis, with one of them
inverted. From the mother wavelet we can construct the Haar system of wavelets on a
dyadic grid, 1)Jm,n (t). This is illustrated in figure 3.3( c) for three consecutive scales. The

Frames and orthogonal wavelet bases 75
1
1
o
o
-1
-1
(f)
-5
o
5
-5
o
5
Figure 3.3 (continued). (f) A Meyer wavelet and associated scaling function (right).
orthogonal nature of the family of Haar wavelets in a dyadic grid system is obvious
from figure 3.3( d), where it can be seen that the positive and negative parts of the
Haar wavelet at any scale coincide with a constant (positive or negative) part of
the Haar wavelet at the next larger scale (and all subsequent larger scales). In
addition, Haar wavelets at the same scale index m on a dyadic grid do not overlap.
Hence, it is obvious that the convolution of the Haar wavelet with any others in
the same dyadic grid gives zero. Figure 3.3(e) shows three Haar wavelets which are
not specified on a dyadic grid. The non-orthogonal nature of the Haar wavelets
across scales, when specified in this way, is obvious from the plot. In addition, if
these wavelets overlap each other along each scale this also destroys orthogonality.
(Although this is not the case for other orthonormal wavelets-see later.) Finally,
note that the Haar wavelet is of finite width on the time axis; that is, it has compact
support. As was stated in the last section, wavelets which have compact support have
a finite number of scaling coefficients and these are the type of wavelet we concentrate
on in this chapter. Not all orthonormal wavelets have compact support. Figure 3.3(f),
for example, shows a Meyer wavelet which is orthonormal with infinite support,
although, as with all wavelets, it is localized, decaying relatively quickly from its
central peak.
3.2.6 Coefficients from coefficients: the fast wavelet transform
From equation (3.14) we can see that the approximation coefficients at scale index
m + 1 are given by
Sm+ I,n = '[00 x(t)CPm+ I,n(t) dt
Using equation (3.27b) this can be written as
Sm+ I,n = '[00 x(t) [   CkCPm,2n+k(t)] dt
(3.33)
(3.34)
We can rewrite this as
Sm+l,n =   Ck[J:oo X(t)CPm,2n+k(t)dt]
(3.35)

76 The discrete wavelet transform
The integral in brackets gives the approximation coefficients Sm,2n+k for each k. We
can therefore write this equation as
1 1
Sm+l,n = V2  Ck S m,2n+k = V2  Ck-2n S m,k
Hence, using this equation, we can generate the approximation coefficients at scale
index m + 1 using the scaling coefficients at the previous scale.
Similarly the wavelet coefficients can be found from the approximation coefficients
at the previous scale using the reordered scaling coefficients b k as follows:
(3.36)
1 1
Tm+l,n = V2  b k S m ,2n+k = V2 bk-2nSm,k
(3.37)
We can see now that, if we know the approximation coefficients Sm n at a specific scale
0,
mo then, through the repeated application of equations (3.36) and (3.37), we can
generate the approximation and detail wavelet coefficients at aU scales larger than
mo. Notice that, to do this, we do not even need to know exactly what the underlying
continuous signal x(t) is, only Smo,n. Equations (3.36) and (3.37) represent the multire-
solution decomposition algorithm. The decomposition algorithm is the first half of the
fast wavelet transform which allows us to compute the wavelet coefficients in this
way, rather than computing them laboriously from the convolution of equation
(3.9). Iterating equations (3.36) and (3.37) performs respectively a highpass and lowpass
filtering of the input (i.e. the coefficients Sm,2n+k) to get the outputs (Sm+ l,n and
Tm+l,n). The vectors containing the sequences (1/V2)ck and (1/V2)b k represent the
filters: (1/ V2) Ck is the lowpass filter, letting through low signal frequencies and hence
a smoothed version of the signal, and (1/ V2)b k is the highpass filter, letting through
the high frequencies corresponding to the signal details. We will come back to the
filtering process in more detail in later sections of this chapter.
We can go in the opposite direction and reconstruct Sm,n from Sm+ l,n and T m + l,n.
We know already from equation (3.17) that X m _ 1 (t) == X m (t) + d m (t); we can expand
this as
Xm-l(t) == 2:: Sm,nCPm,n(t) + 2:: Tm,n1jJm,n(t)
(3.38)
n
n
Using equations (3.27b) and (3.28) we can expand this equation in terms of the scaling
function at the previous scale as follows:
1 1
Xm-I (t) = 2:: Sm,n 12 2:: CkCPm-I,2n+k(t) + 2:: Tm,n 12 2:: b k CPm-I,2n+k(t)
n YL k n YL k
(3.39)
Rearranging the summation indices, we get
1 1
Xm-I (t) = 2:: Sm,n 12 2:: Ck-2nCPm-l,k(t) + 2:: Tm,n 12 2:: b k - 2n CPm-l,k(t)
n YL k n YL k
(3.40 )

Discrete input signals of finite length 77
We also know that we can expand X m _ 1 (t) in terms of the approximation coefficients
at scale m - 1, i.e.
Xm-l (t) == 2:: Sm-l,ncPm-l,n(t)
n
(3.41 )
Equating the coefficients in equation (3.41) with equation (3.40) we note that the
index k at scale index m relates to the location index n at scale index m - 1. In
addition, location index n in equation (3.40) is not equivalent to location index n in
equation (3.41), as the former corresponds to scale index m, with associated discrete
location spacings 2 m , and the latter to scale index m - 1, with discrete location
spacings 2 m - 1. Hence the n indices are twice as dense in the latter expression. The
simplest way to proceed before equating the two expressions is to swap the indices
k and n in equation (3.40) which, after some algebra, produces the reconstruction
algorithm:
1 1
S m - 1 n == V2 '""'" C n - 2k S m k + V2 '""'" b n - 2k T m k
, 2 ' 2 '
k k
(3.42)
where we have reused k as the location index of the transform coefficients at scale
index m to differentiate it from n, the location index at scale m - 1. Hence, at the
smaller scale, m - 1, the approximation coefficients can be found in terms of a com-
bination of approximation and detail coefficients at the next scale, m. Note that if
there are only a finite number of nonzero scaling coefficients (== N K), then C n _ 2k has
nonzero values only when in the range 0 to N k - 1. The reconstruction algorithm is
the second half of the fast wavelet transform (FWT). Note that in the literature the
fast wavelet transform, discrete wavelet transform, decomposition/reconstruction
algorithms, fast orthogonal wavelet transform, multiresolution algorithm, pyramid
algorithm, tree algorithm and so on, are all used to mean the same thing. It becomes
even more confusing when other discretizations of the continuous wavelet transform
are referred to as the discrete wavelet transform. Take care!
3.3 Discrete input signals of finite length
3.3.1 Approximations and details
So far we have considered the discrete orthonormal wavelet transform of a con-
tinuous function x(t), where it was shown how the continuous function could be
represented as a series expansion of wavelet functions at all scales and locations
(equation (3.10a)) or a combined series expansion involving the scaling and wavelet
functions (equation (3.16)). In this section, and from here on, we will consider discrete
input signals specified at integer spacings. To fit into a wavelet multiresolution
framework, the discrete signal input into the multiresolution algorithm should be
the signal approximation coefficients at scale index m == 0, defined by
SO,n = '[00 x(t)cp(t - n) dt
(3.43)

78 The discrete wavelet transform
which, as we now know from equations (3.36) and (3.37), will allow us to generate all
subsequent approximation and detail coefficients, Sm nand T m n, at scale indices
, ,
greater than m == o. In this section we will assume that we have been given So n.
,
Section 3.4 considers further the question of discrete input data which may not
be So n.
,
In practice our discrete input signal So n is of finite length N, which is an integer
,
power of 2: N == 2 M . Thus the range of scales we can investigate is 0 < m < M.
Substituting both m == 0 and m == M into equation (3.16), and noting that we have
a finite range of n which halves at each scale, we can see that the signal approximation
scale m == 0 (the input signal) can be written as the smooth signal at scale M plus a
combination of detailed signals as follows:
2 M - m _ 1

n=O
M
SO,ncPO,n(t) == SM,ncPM,n(t) + 
m=l
2 M - m _ 1

n=O
T m,n 1jJm,n (t)
( 3 .44 )
This is the form we use to describe our finite length discrete signal in terms of its
discrete wavelet expansion. The covering of a finite length time segment with wavelets
is illustrated in figure 3.4 for Daubechies D4 wavelets at two successive scales. The
lower scale covers the time window using eight wavelets, and the larger scale uses
four wavelets. One of the wavelets in each plot is shown bold for clarity. The wavelets
shown which spill over the end of the window have been wrapped around back to the
(a)
(b)
Figure 3.4. Covering the time axis with dyadic grid wavelets. (a) Eight Daubechies D4 wavelets
covering the time axis at scale m. (b) Four Daubechies D4 wavelets covering the time axis at scale
m + 1. These wavelets are twice the width of those in (a).

Discrete input signals of finite length 79
beginning. Known as wraparound, it is the simplest and one of the most common
treatments of the boundary for a finite length signal and we will employ it throughout
the rest of this chapter. However, note that, by employing this method, we assume
that the signal segment under investigation represents one period of a periodic
signal and we are in effect pasting the end of the signal back on to the beginning.
Obviously, if the signal is not periodic, and in practice it usually is not, then we
create artificial singularities at the boundary which results in large detail coefficients
generated near to the boundary.
We can rewrite equation (3.44) as
M
xo(t) == XM(t) + 2:: dm(t)
m=l
(3.45)
where the mean signal approximation at scale M is
XM( t) == S M,nCPM,n (t)
(3.46)
As the approximation coefficients are simply factored, weighted averages of the signal
then, when wraparound is employed to deal with the boundaries, the single approx-
imation component S M n is related to the mean of the input signal through the
,
relationship SO,n == SM,n/vf2M, where the overbar denotes the mean of the sequence
So n. In addition, when wraparound has been used to deal with the boundaries, the
,
mean signal approximation at the largest scale, XM(t), is a constant valued function
equal to the input signal mean. (We will see why this is so later in section 3.5.1.)
The term on the far right of equation (3.45) represents the series expansion of the
fluctuating components of the finite length signal at various scales in terms of its detail
coefficients. The detail signal approximation corresponding to scale index m is defined
for a finite length signal as
2 M - m _ 1
d m (t) == 2:: T m,n 1jJm,n (t)
n=O
(3.47)
As we saw above (equation (3.45)), adding the approximation of the signal at scale
index M to the sum of all detail signal components across scales 0 < m < M gives
the approximation of the original signal at scale index o. Figure 3.5(a) shows the
details of a chirp signal with a short burst of noise added to the middle of it. A
Daubechies D20 wavelet was used in the decomposition (see an example of this
wavelet later in figure 3.15(e)). The original signal is shown at the top of the plot.
Below the signal the details for ten wavelet scales, d l (t) to dIO(t), are shown. The
bottom trace is the remaining signal approximation XIO(t). Adding together all
these details plus the remaining approximation (which is the signal mean) returns
the original signal. Two things are noticeable from the plot. First, there is a shift to
the left of the large amplitude details with increasing scale, as we would expect as
the chirp oscillation increases in frequency from left to right. The second thing to
notice is that the high frequency burst of noise is captured at the smallest scales,
again as we would expect.
We saw in equation (3.19) that the signal approximation at a specific scale was a
combination of the approximation and detail at the next lower scale. Ifwe rewrite this

80 The discrete wavelet transform
(a)
12
xo(t) = original signal
14
200
400
d l (t)
d 2 (t)
d/t)
dit)
ds(t)
d 6 (t)
dlt)
dgCt)
d 9 (t)
dlO(t)
XlO(t) = signal mean
800 1000
10
8
6
4
2
o
o
600
(b) 14
12 xo(t) = original signal
10 xl (t)
xlt)
8 x 3 (t)
xi t )
6 xs(t)
x 6 (t)
4 xlt)
xgCt)
2 x 9 (t)
xlO(t)
0
0 200 400 600 800 1000
Figure 3.5. Multiresolution decomposition of a chirp signal containing a short burst of noise. ( a) Signal
details d m (t). (The signals have been displaced from each other on the vertical axis to aid clarity.) (b)
Signal approximations xm(t).
equation as
X m ( t) == x m - I ( t) - d m ( t)
(3.48)
and begin at scale m - 1 == 0, that of the input signal, we can see that at scale index
m == 1, the signal approximation is given by
Xl (t) == xo(t) - d l (t)
at the next scale (m == 2) the signal approximation is given by
X2 (t) == Xo (t) - d l (t) - d 2 (t)
(3.49a)
(3.49b)
and at the next scale by
X3(t) == xo(t) - dl(t) - d 2 (t) - d 3 (t)
(3.49c)

Discrete input signals of finite length 81
and so on, corresponding to the successive stripping of high frequency information
(contained within the dm(t)) from the original signal. Figure 3.5(b) contains successive
approximations xm(t) of the chirp signal. The top trace is the original signal xo(t).
Subsequent smoothing of the signal takes place throughout the traces from the top
to the bottom of the figure. As we saw in equation (3.48), the difference between
each of the approximation traces X m -l (t) and xm(t) is the detail component dm(t).
We can view these differences in the detail component traces of figure 3.5(a).
We have glossed over much of the mathematical detail of multiresolution analysis
here. Most mathematical accounts of the subject begin with a discussion of orthogo-
nal nested subspaces and the signal approximations and details, X m (t) and d m (t), as
projections on to these spaces. This tack has not been followed here; see for example
Mallat (1998), Blatter (1998), Sarkar et al (1998) or Williams and Armatunga (1994).
In this chapter we concentrate on the mechanics, rather than the mathematics, of
multiresolution analysis.
3.3.2 The multiresolution algorithm-an example
Once we have our discrete input signal So n, we can compute Sm nand T m n using the
, "
decomposition algorithm given by equations (3.36) and (3.37). This can be done for
scale indices m > 0, up to a maximum scale determined by the length of the input
signal. To do this, we use an iterative procedure as follows. First we compute SI n
,
and T 1 n from the input coefficients So n, i.e.
, ,
1
S',n = V2  Ck S O,2n+k (3,50a)
and
1
T1,n = V2  b k S O ,2n+k
In the same way, we can then find S2 nand T 2 n from the approximation coefficients
, ,
SI n, i.e.
,
(3.50b)
1
S2,n = V2  Ck S ,,2n+k
(3.51a)
and
1
T2,n = V2  b k S I ,2n+k
Next, we can find S3 nand T3 n from the approximation coefficients S2 n, and so on, up
, , ,
to those coefficients at scale index M, where only one approximation and one detail
coefficient is computed: SMO and T MO . At scale index M we have performed a full
, ,
decomposition of the finite-length, discrete input signal. We are left with an array
of coefficients: a single approximation coefficient value, S M 0, plus the detail co-
,
efficients, T m n, corresponding to discrete wavelets of scale a == 2 m and location
,
b == 2 m n. The finite time series is of length N == 2 M . This gives the ranges of m and
n for the detail coefficients as respectively 1 < m < M and 0 < n < 2 M -m - 1.
(3.51 b)

82 The discrete wavelet transform
Notice that the range of n successively halves at each iteration as it is a function of
scale index m for a finite length signal. At the smallest wavelet scale, index m == 1,
2 M /2 1 == N /2 coefficients are computed, at the next scale 2 M /2 2 == N /4 are computed
and so on, at larger and larger scales, until the largest scale (m == M) where only one
(==2 M /2M) coefficient is computed. The total number of detail coefficients for a
discrete time series of length N == 2 M is then, 1 + 2 + 4 + 8 + . . . + 2 M - 1, or
2::==-/ 2 m == 2 M - 1 == N - 1. In addition to the detail coefficients, the single approx-
imation coefficient S M 0 remains. This is related to the signal mean as we saw above,
,
and is required in addition to the detail coefficients to fully represent the discrete
signal. Thus a discrete input signal of length N can be broken down into exactly N
components without any loss of information using discrete orthonormal wavelets.
In addition, no signal information is repeated in the coefficient representation. This
is known as zero redundancy.
The decomposition of approximation coefficients into approximation and detail
coefficients at subsequent levels can be illustrated schematically thus
s - - -
O,n
,
,
,
,
"
TIn
,
s ---
1,n
,
,
,
,
"
T 2n
,
s ---
2,n
,
,
,
,
"
T 3n
,
s ---
3,n
,
,
,
,
"
. . .
- - - S
M,O
,
,
,
,
"
T MO
,
Figure 3.6 shows an alternative schematic of the same process, illustrating the
decomposition and insertion of the approximation and detail coefficients at each
Scale index
m=O
S
O,n
....... .... .........
m=l
S
I,n
m=2
m=3
m=4
m=5
/
n=O,...,N/32-1
n=O, 1 ,..,N/8-1
n=O,1,...,N/16-1
n=O, 1 ,..,N/4-1
original signal
/n=O,1,..,N-I
T
I,n
T
I,n
T
I,n
wavelet transform
+- vector at full
decomposition
\
n=O,..,N/2-1
Figure 3.6. Schematic diagram of the Haar filtering algorithm. - - - ---+ Sm,n used to derive Sm + I,n;
 Sm,n used to derive T m + I,n; - Tm,n taken to next level as Tm,n, i.e. no further manipulation.

Discrete input signals of finite length 83
iteration within the wavelet transform vector for an arbitrary input signal vector
containing 32 components. The wavelet transform vector after the full decomposition
has the form W(M) == (SM, T M , TM-I'...' Tm,..., T 2 , T 1 ), where Tm represents the
sub-vector containing the coefficients Tm n at scale index m (where n is in the range
,
o to 2 M -m - 1). We can halt the transformation process before the full decomposi-
tion. If we do this, say at an arbitrary level mo, the transform vector has the form
W(mo) == (Smo' Tmo' Tmo-l'...' T 2 , T 1 ), where mo can take the range 1 < mo < M - 1.
In this case, the transform vector does not contain a single approximation component
but rather the sequence of approximation components Smo,n. However, the transform
vector always contains N == 2 M components. For example, we can see from figure 3.6
that stopping the algorithm at m == 2 results in W(2) == (S2, T 2, T 1). Remember, the
range of n is a function of scale index m, hence this vector contains eight S2 n
,
components, eight T 2 n components and 16 TI n components, matching the 32 compo-
, ,
nents of the original input signal vector. The range ofn is indicated below the full decom-
position vector in figure 3.6. Notice also that we can express the original input signal as
the transform vector at scale index zero, i.e. W(O).
An example of a wavelet decomposition using a discrete wavelet is shown in
figure 3.7. The input signal is composed of a section of a sine wave, some noise and
a flatline. The signal is decomposed using a Daubechies D6 wavelet. A member of
this family is shown in figure 3.7(b). (We will look more closely at the Daubechies
wavelet family later in this chapter.) The discrete transform plot is shown in figure
3. 7 (c), where the dyadic grid arrangement may be seen clearly. This plot is simply a
discretized dyadic map of the detail coefficients, Tm n, where the coefficients at
,
larger scales have correspondingly longer boxes (as the wavelets cover larger segments
of the input signal). In addition to the detail coefficients, T m n, the remaining approx-
,
imation coefficient SMO is added to the bottom of the plot. As we would expect it
,
covers the whole time axis. We can see from the transform plot that the dominant
oscillation is picked up at scale index m == 6 and the high frequency noise is picked
up within the middle segment of the transform plot at the smaller scales. We can
use the reconstruction algorithm (equation (3.42)) to get back the original input
signal So n from the array of detail coefficients shown in figure 3.7(c). Alternatively,
,
as with the continuous transform, we can reconstruct a modified version of the
input signal by using only selected coefficients in the reconstruction. This is shown
in figures 3.7(d) and (e), where only the coefficients corresponding to scales m == 5 to
8 are kept (the others are set to zero) and the signal is reconstructed. This has removed
a significant amount of the noise from the signal although the sinusoidal waveform is
less smooth than the original in figure 3.7(a). We will come across more sophisticated
ways to remove noise and other signal artefacts later, in section 3.4.2 of this chapter.
3.3.3 Wavelet energy
After a full decomposition, the energy contained within the coefficients at each scale is
given by
2 M - m _ 1
Em == 2:: (Tm,n)2
n=O
(3.52)

84
The discrete wavelet transform
2
o
-2
o
(a)
200
1000
(b)
scale index
m= 1
scale index
m=5
m==
m= 10
remaining approximation
coefficient
(c)
(d)
2
-2
o
(e)
500
1000
Figure 3.7. Discrete wavelet transform of a composite signal. (a) Original composite signal. (b) A
member of the Daubechies D6 wavelet family. (c) Discrete transform plot. (Note dyadic struc-
ture-large positive coefficient values are white and large negative values black.) (d) Coefficient
removal. (e) Reconstructed signal using only retained coefficients in (d). The original composite
signal (a) is composed of three segments: a sinusoid, uniformly distributed noise and a fiatline.
The signal is decomposed using Daubechies D6 wavelets (b) to give the dyadic array of transform
coefficients plotted in (c). The coefficients corresponding to scales 5 to 9 are kept (d) and used to
reconstruct the signal in (e). Note that a grey scale is used to depict the coefficient values, where
the maximum value is white and the minimum value is black.
A wavelet-based power spectrum of the signal may be produced using these scale-
dependent energies. To do so, we require a frequency measure which is a reciprocal
of the wavelet dilation, e.g. the passband centre of the power spectrum of the wavelet.
A wavelet power spectrum can then be produced for the signal which is directly
comparable with both its Fourier and continuous wavelet counterparts (see chapter
2, section 2.9). This topic is dealt with in detail in chapter 4, section 4.2.1, in connec-
tion with the statistical measures used to analyse turbulent fluid signals.
The total energy of the discrete input signal
N-I
E == 2:: (SO,n)2
n=O
(3.53)

Discrete input signals of finite length 85
is equal to the sum of the squared detail coefficients over all scales plus the square of
the remaining approximation coefficient, S M 0, as follows:
,
M
E == (SM,0)2 + 2::
m=1
2 M - m _ 1
2:: (Tm,n)2
n=O
(3.54 )
In fact, the energy contained within the transform vector at all stages of the multire-
solution decomposition remains constant. We can, therefore, write the conservation
of energy more generally as
N-l
E == 2:: (W/ m ))2
i=O
where W/ m ) are the individual components of the transform vector W(m) ordered as
described in the previous section. When m == 0, this equation corresponds to the
summation of the component energies of the input signal (equation (3.53)) and
when m == M it corresponds to the summation of the energies within the coefficients
at full decomposition (equation (3.54)).
(3.55)
3.3.4 Alternative indexing of dyadic grid coefficients
There are three main methods used in practice to index the coefficients resulting from
the discrete wavelet transform. It is worth discussing these now. All three methods are
popular in the scientific literature and appear often in many of the examples of the
practical application of wavelet analysis in the subsequent chapters of this book.
We will use the full decomposition of a 32 component input signal as an illustration.
Scale indexing: The scale indexing system (m, n) corresponding to an input signal
of length N == 2 M is shown schematically on a dyadic grid in figure 3.8(b) for the
discrete signal shown in figure 3.8(a). The lowest scale on the grid m == 1 corresponds
to a spacing of 2 1 == 2 on the data set. The discrete input signal is at scale index m == o.
We have already come across this grid structure in the plot of the transform co-
efficients in figure 3.7(c). Such plots give a good visual indication of the covering of
the timescale plane by the wavelets and their relative importance in making up the
signal.
Sequential indexing: Again, we have already come across this form of indexing
(figure 3.6). Figure 3.8(c) contains a schematic of the transform coefficients given in
sequential format. We know that a signal N samples long produces N wavelet
vector components. It makes sense, therefore, to find an ordering of these coefficients
to fit into a vector of length N. The discrete time series, So n, n == 0, 1,2, . . . , N - 1, can
,
be decomposed into N - 1 detail coefficients plus one approximation coefficient
where N is an integer power of 2: N == 2 M . After a full decomposition these transform
coefficients can be resequenced from the two-dimensional dyadic array T m n to the
transform vector W(M), where the components W/ M ) have the same range as the origi-
nal signal (i == 0, 1, 2, . . . , N - 1). The vector component index i is found from the
dyadic grid indices m and n through the relationship i == 2 M -m + n. In addition,
the superscript M in parentheses denotes a full decomposition of the signal over M
scales. The transform vector components, W/M)are plotted in figure 3.8(c). The last

86
The discrete wavelet transform
(a)
X 2
(N=32)
T ,2
scale index 1
scale index 2
scale index 3
scale index 4
scale index 5
T 2 ,0 T 2 ,1
T 3 ,0
T 2 ,2
T 3 ,1
T 3 ,2
T 2 ,7
T 3 ,3
T 4 ,0
T 4 ,1
T S,O (=T M,O)
8S,0 _ signal mean component
(b)
sequential indexing
W o , W b W 2 , W 3 , W 4 , W s ,.......................W N _ 3 ' W N - 2 , W N - b
8 M ,0
TM,o
T M - 1 ,0 T M - 1 ,1
T M-2,0 T M-2, 1 T M-2,2 T M-2,3
(c) corresponding scale indexing
T 1 ,0-T 1 ,lS
T T 4 ,2
T 4,1
4,0
level index 4
level index 3
level index 2
level index 1
level index 0
level index -1
T 3 ,0 T 3 ,1
T 2 ,0
T 3 ,2
T 2 ,1
T 2 ,2
T 3 ,7
T 2 ,3
T 1 ,0
T 1 ,1
TO,O
signal mean component
(d)
Figure 3.8. Schematic diagram of alternative indexing methods for dyadic grid wavelet transform co-
efficients. (a) Original signal containing 32 components (scale index m = 0). (b) Scale indexing,
T m,n. ( c) Sequential indexing, W/ M ), where i = 2 M - n + n (corresponding scale indexing is shown
at the bottom of the plot). (d) Level indexing, TZn.
,
half of the series in the figure represents the coefficients corresponding to the smallest
wavelet scale (index m == 1). The next quarter, working backwards, represents the co-
efficients corresponding to scale index m == 2, the next eighth to the m == 3 coefficients,
and so on, back to wi M ), which is the single coefficient for scale index m == M.
The remaining component W6 M ) is the single approximation coefficient (8 M,O). As

Discrete input signals of finite length 87
mentioned in section 3.3.2, if we halt the decomposition algorithm at scale mo (before
full decomposition) this results in an intermediate transform vector W(mo) which
contains a number of approximation coefficients, Smo,n at its beginning.
Level indexing: Level indices, I, are often used instead of scale indices, m. The level
index I is simply equal to M - m. In this case, the number of wavelets used to cover
the signal at each level at a specific level is 2 1 , e.g. level I == 0 corresponds to a single
wavelet and the scale of the whole time series, level I == 1, corresponds to two wave-
lets, and the scale of one half of the time series, and so on. The number of wavelets
used at a level is simply 2 1 . In addition, it is standard in this nomenclature to
denote the remaining approximation coefficient at level I == -1. The wavelet array
becomes T1n. Figure 3.8(d) shows the l,n indexing system. Level indexing is used
,
when the signal is specified over the unit interval and the analysis is set up in terms
of resolution rather than scale (i.e. the lowest resolution I == 0 corresponds to the
length of the whole signal, whereas the smallest scale m == 0 corresponds to the
distance between each signal point).
3.3.5 A simple worked example: the Haar wavelet transform
Now we will illustrate the methods described above using a Haar wavelet in the
decomposition of a discrete input signal, So n: n == 0, 1,2, . . . , N - 1. To do so, we
,
employ the decomposition algorithm given by equations (3.36) and (3.37). The
Haar wavelet has two scaling coefficients, Co == 1 and CI == 1. Substituting these into
equation (3.36) we can obtain the approximation coefficients at the next scale through
the relationship
1
Sm+ I,n = 0 [Sm,2n + Sm,2n+ I] (3.56)
Similarly, through equation (3.37) we can obtain the detail coefficients at subsequent
scales using
1
T m + I,n = 0 [Sm,2n - Sm,2n+ d (3.57)
U sing these equations, we will perform the Haar wavelet decomposition of the (very)
simple discrete signal, (1, 2,3,4). As the signal contains only four data points, we can
only perform two iterations of the Haar decomposition algorithm given by equations
(3.56) and (3.57). After two iterations we expect to obtain four transform coefficients:
three wavelet coefficients Tmn-two at scale index m == 1, (T I 0, T II ), and one at scale
, , ,
index m == 2, (T 20 ), plus a signal mean coefficient at scale index m == 2, (S20). This is
, ,
illustrated through the iteration of the transform vector in figure 3.9 and also through
a schematic of the coefficients in figure 3.10.
The first iteration of the decomposition algorithm gives
1 1
T I 0 == - [1 - 2] == --
, 0 0
1 3
SI,O = 0 [0 + 3] = 0
1 1
T II == - [3 - 4] == --
, 0 0
1 7
SI,I = 0 [3 +4] = 0

88 The discrete wavelet transform
scale m=O
1
2
3
4
So,o SO,1 SO,2 SO,3
1 8t iteration D
scale m= 1
3 7 1 1
,J2 ,J2 ,J2 ,J2
SI,O SI,1 T 1,0 T 1,1
2 nd iteration D
scale m=2
5
-2
1 1
,J2 ,J2
S2,0 T 2,1 T 1,0 T 1,1
Figure 3.9. Iteration of the wavelet transform vector in the decomposition of a simple signal. - - - ---+ Sm n
,
used to derive Sm+l,n;  Sm,n used to derive T m + 1 ,n; - Tm,n taken to next level as Tm,n, i.e. no
further manipulation.
The transform coefficient vector, after this first iteration, is then
W(1) - ( 8 1 0 8 1 1 T l 0 T l 1 ) - (   -  -  )
- " " " , - yI2'yI2' yI2' yI2
The second iteration (only involving the remaining approximation coefficients 8 1 0
,
and 8 1 1) yields
,
T 2 ,o =  [  -  ] = -2 and S2,O =  [  +  ] = 5
The transform coefficient vector after this second iteration is now
(2) _ _ ( 1 1 )
Wi -(S2,o,T 2 ,o,T I ,o,T I ,I)- 5,-2,- yI2 ,- yI2
The signal mean is found from the remaining approximation coefficient, i.e.
82,0/ (yI2)2 == 2.5. We can also see that the energy of the original discrete signal
(1 2 + 2 2 + 3 2 + 4 2 == 30) is equal to the energy of the transform coefficient vector
after the first iteration
(  y + (  y + ( -  y + ( -  y = 30
and the second iteration, giving the full decomposition
2 2 ( 1 ) 2 ( 1 ) 2
5+(-2)+ - y12 + - y12 =30
which is a result we expect from the conservation of energy condition expressed in
equations (3.52) to (3.55).
To return from the transform coefficients to the original discrete signal we work
in reverse. The inverse Haar transform can be written simply as: at even locations 2n:
1
Sm,2n = yI2 [Sm+ I,n + T m + I,n] (3.58)

Discrete input signals of finite length
89
SO,O SO,1 SO,2 S 0,3
   
5 5
i f r iteration 1 0
c=>
0 -5
0 0
(a) discrete input (b)
signal xi = (1,2,3,4)
S2,0 T 2,0 T 1,0 T 1,1
 .  
5-
,.-
--
...-
--'"
--
".,.-
-,,-
......
k--'"
,
/0
c=> .-/
,
.-
.-
,
.-
,
,
....-
...
......
-......
......
...--
....--
iteration 2
3
II
-5/ 0
.-
,-
,
./
/" (c)
,
,.
,.
./'
II
,
,.
"
"
,-
.-
,

,
....
....
.-
,
I
I
I
,
I
I
I
I
I
I
I
j
I
I
I
4
3
3
0 0 0
-3 I -3 I -3
0 0 0
(d) xz(t) = signal mean (e) dz(t) (f) d 1 (t)

Tz,o
5 z ,0
(g)
+
+
ILJ ILJ

xo(t)
x (2.5)
x (-1.0)
x (-0.5) x (-0.5)
(h)
Figure 3.10. Haar decomposition and reconstruction of a simple ramp signal. The original discrete
input signal is decomposed through an initial iteration to (b), the first decomposition, then further
to ( c), the second and final decomposition. From the coefficients at the full decomposition ( c), the
following can be constructed: the scale index 2 approximation (the signal mean) (d), the scale
index 2 detail ( e) and the scale index 1 detail (f). (g) The transform coefficient plot associated with
the transform values given in (c). (h) Schematic diagram of the addition of the rescaled Haar wavelets
to reconstruct the original signal. (Adding the reconstructions in (d), (e) and (f) returns the approx-
imation of the signal at scale index m = 0.)
and at odd locations 2n + 1:
1
Sm,2n+1 = yI2 [Sm+l,n - Tm+l,n]
For the Haar wavelet transform, we can derive these reconstruction equations directly
from equations (3.56) and (3.57). Alternatively, we could derive them from the recon-
struction algorithm of equation (3.42).
(3.59)

90 The discrete wavelet transform
We will perform the reconstruction on the transform vector we have just
computed:
W (2) - (S T T T ) - (5 _ 2 -  -  )
- 2,0, 2,0, 1,0, 1,1 - , , 0' 0
The first iteration using the reconstruction pair (equations (3.58) and (3.59)) yields
131 7
SI,O = 0 (S2,O + T 2 ,o) = 0 SI,I = 0 (S2,O - T 2 ,o) = 0
which results in
( 3 7 1 1 )
(SI,O, SI,I, T1,o, T1,1) = 0 ' 0 ' - 0 ' - 0
I tera ting again gives
1
Soo == M ( SI0 + T 10 ) == 1
, y2' ,
1
S02== M ( SII+ T ll ) ==3
, y2' ,
1
So 1 == M ( SI 0 - T 1 0 ) == 2
, y2' ,
1
So 3 == M ( s 1 1 - T 1 1 ) == 4
, y2' ,
hence we get back the original signal (So,0,SO,I,SO,2,SO,3) == (1,2,3,4).
Figure 3.10 attempts to show visually the decomposition of the signal into the
Haar wavelet components. Figure 3.10(a) contains the original signal, figure
3.1 O(b) plots the coefficients after the first iteration of the decomposition algorithm
and figure 3.10(c) plots the coefficients after the second iteration of the algorithm.
The coefficients contained in figure 3.10(c) correspond in turn to a single scaling func-
tion at scale index m == 2, a wavelet at scale index m == 2 and two wavelets at scale
index m == 1. These are shown respectively in figures 1 O( d)-(f). The transform co-
efficient plot for this signal is equally simplistic, consisting of four coefficient values
partitioned as shown in figure 3.10(g).
We can find the corresponding approximation and details of the signal by taking
the inverse transform of the coefficients at each scale. First the approximation co-
efficient is used to determine the signal approximation at the largest scale:
X2(t) == S2,OcP2,0(t) (3.60)
where the scaling function cP2,0(t), for the Haar wavelet, is simply a block pulse of
length 4 and amplitude 1/ (0)2 ==!, beginning at t == O. Thus X2 (t) is simply a
block pulse of length 4 and magnitude S2,0 x ! == 2.5, which is, in fact, the signal
mean. A plot of X2(t) is shown in figure 3.10(d). Similarly, we can obtain the signal
detail component at scale index 2 as follows:
d 2 (t) == T 2 ,01jJ2,0(t) (3.61)
This detail component (shown in figure 3.10(e)) is simply a single Haar wavelet span-
ning the data set with coefficient value T 2 ,0 == -2, hence magnitude -2/ (0)2 == -1.
Next the detail signal component at scale index 1 is found from
1
d 1 ( t) == 2:: T 1 ,n 1jJ 1 ,n ( t) == T 1 ,0 1jJ 1 ,0 ( t) + T 1, I1jJ 1 , 1 ( t)
n=O
(3.62)

Everything discrete 91
which is simply two Haar wavelets at scale index 1, side by side, with amplitudes given
by Tl,O/ (0) == (-1/0) / (0) == -0.5 and T1,1 / (0) == (-1/0) / (0) == -0.5.
The detail component d 1 (t) is plotted in figure 3.1 O(f). We already know from equa-
tion (3.45) that the approximation of the signal at scale index 0 can be found by
adding together all the detail components plus the signal approximation at scale
index M, i.e.
M
xo(t) == XM(t) + 2:: dm(t)
m=l
(3.63a)
We have already seen an example of this partitioning of the signal into approximation
and detail components in figure 3.5 for a chirp signal. For the Haar case considered
here M == 2, hence
Xo(t) == X2(t) + d 2 (t) + d 1 (t)
(3.63b)
This is simply the addition of the approximation and detail components shown in
figures 3.10(d)-(f). This is shown schematically in figure 3.10(h).
3.4 Everything discrete
3.4.1 Discrete experimental input signals
We will now consider the case where we have a discrete experimental signal collected,
say, using some piece of experimental apparatus, and we want to perform a wavelet
decomposition of it using the multiresolution algorithm. In addition, we would also
like to represent the approximation and detail signal components discretely at the
resolution of the input signal. This discrete signal, which we will denote Xi, is of
finite length N, i.e. i == 0, 1, . . . , N - 1. It has been acquired at discrete time intervals
t (the sampling interval) to give the discrete time signal x( t i ): i == 0, 1,2, . . . , N - 1.
The sampling of the signal provides a finite resolution to the acquired signal. This
discretization of the continuous signal is then mapped on to a discrete signal Xi,
where the sampling interval has been normalized to 1. In doing so we must remember
t and add it when required, for example in real applications when computing signal
frequencies.
Common practice is to input the discretely sampled experimental signal, Xi,
directly as the approximation coefficients at scale m == 0, and begin the multi-
resolution analysis from there. However, it is not correct to use the sampled time
series Xi directly in this way. We should really use the approximation coefficients
So n obtained from the original continuous signal at scale m == 0, as defined by equa-
,
tion (3.43). In practice we usually do not know exactly what x(t) is. As SO,n is a
weighted average of x( t) in the vicinity of n, then it is usually reasonable to input
Xi as So n if our signal is slowly varying between samples at this scale. That is we
,
simply set
So n == X n
,
( 3 .64 )

92 The discrete wavelet transform
where, at scale index m == 0, both the coefficient location index n and signal discreti-
zation index i have the same range (0 to N - 1) and are equal to each other. In the rest
of this chapter we will assume that the sampled experimental signal has been input
directly as So n. The literature is full of studies of real data where this has been
,
done and, if anything, it is the rule rather than the exception. Obviously, it does
not make it any more correct! In fact, Strang and Nguyen (1996) call the use of the
sampled signal directly within the transform a 'wavelet crime' and suggest various
ways to preprocess the sampled signal prior to performing the analysis.
In practice, continuous approximations, xm(t), and details, dm(t), of the signal are
not constructed from the multiresolution analysis, especially when inputting the
signal Xi as the detail coefficients at scale index m == o. Instead, either the approxi-
mation and detail coefficients, Sm nand T m n, are displayed at their respective scales
, ,
or, alternatively, they are used to construct representations of the signal at the
scale of the input signal (m == 0). The latter is sometimes preferable as the scalings
of the displays are visually comparable. As an example, let us consider the Haar
wavelet decomposition of a simple discrete signal containing eight components:
Xi == (4,5, -4, -1,0,1,2,0). The signal is shown in figure 3.11(a). We can see
that the first half of the signal has a marked step from 5 to -4, whereas the second
half of the signal appears much smoother. The transform vector after a full
decomposition is
(3) _ ( )
W - S3,0, T3,0, T 2 ,0, T 2 ,1, Tl,O' T1,1, T 1 ,2, T 1 ,3
== (2.475,0.354,7.000, -0.500, -0.707, -2.121, -0.707,1.414)
( 7 1 14 1 1 3 1 2 )
== (y!2)3'(y!2)3'2'-2'- y!2 ,- y!2 ,- y!2 ' y!2
(You can find these values for yourself using the procedure described in section 3.3.5.)
This vector is shown schematically in figure 3.11(b). Performing the inverse transform
on this vector leads us back to the original discrete input signal. Referring back to
the schematic of the wavelet decomposition given in section 3.3.2, we can similarly
represent the reconstruction from the transform vector, W(3), as
S --- S --- S --- S
3,n 2,n 1,n O,n
, , ,
/ / /
/ / /
/ / /
/ / /
T 3 ,n
T 2 ,n
T 1 ,n
Remember that the original signal Xi was input into the multiresolution algorithm
as So n.
,
Let us now look at what happens to the reconstructed signal when we remove the
smallest-scale (i.e. highest-frequency) detail coefficients in turn (figures 3.11(c)-(e)).
Removing the components, Tl n, the modified transform vector becomes (2.475,
,
0.354, 7.000, -0.500, 0, 0, 0, 0), where the coefficients set to zero are shown bold.
Performing the inverse transform on this vector with the coefficients at the smallest
wavelet scale set to zero removes the high-frequency details at this scale and returns
a smoother version of the input signal. We will denote this smooth version of the

Everything discrete 93
o
o
o
5
o
5
(a)
(b)
o
o
o
5
o
5
(c)
(d)
o
o
o
5
o
5
(e)
(f)
o
o
o
5
o
5
(g)
(h)
Figure 3.11. Multiresolution decomposition as scale thresholding. ( a) XO,i (= Xl,i + d1,i). (b) Wavelet
coefficients (S3,O, T3,O' T 2 ,o, T 2 ,1, Tl,O' T1,1, T 1 ,2, T 1 ,3). (c) Xl,i (=X2,i+ d 2,i), (=XO,i-dl,i). (d) d1,i.
(e) X2 i ( =X3 i + d 3 i ) , ( =Xl i - d 2 i ) . (f) d 2 i. (g) X3 i (signal mean), ( =X2 i - d 3 i ) . (h) d 3 i.
, , , , , " ",
input signal as Xl i. This operation is shown schematically as
,
s --- S --- S --- x .
3,n 2,n 1,n 1,z
, , ,
/' /' /'
/' /' /'
/' /' /'
/' /' /'
T 3 ,n
T 2 ,n
zeros
The reconstructed discrete signal becomes xI,i == (4.5,4.5, -2.5, -2.5,0.5,0.5, 1, 1).
That is, the signal is smoothed by the averaging of each pair of signal values as shown
in figure 3.11(c). Note that we use the nomenclature Xl i for the approximation signal
,
Xl (t) expressed in terms of discrete coefficients at the scale of the original input signal
(i.e. m == 0). Remember from section 3.2.3 that the approximation coefficients, 8m n,
,

94 The discrete wavelet transform
provide a discrete approximation of the signal at scale index m. By passing the
coefficients through the reconstruction filter we can express the contributions of
these discrete approximations at the scale of the original signal (scale index 0). In
the rest of this section, we will see how to express all the discrete approximations
and details at the scale of the input signal with index m == o.
N ow we remove the detail coefficients associated with the next smallest scales
from the transform vector to get (2.475, 0.354, 0, 0, 0, 0, 0, 0). Reconstructing the
signal using this modified transform vector, we get x2,i == (1, 1, 1, 1, 0.75, 0.75,
0.75,0.75) shown in figure 3.11(e). Again we can show this schematically as
S --- S --- S' --- x .
3,n 2,n 1,n 2,l
, , ,
/ / /
/ / /
/ / /
/ / /
T 3n
,
zeros
zeros
where we have put the prime on S n to indicate that it has different valued com-
,
ponents from SI n above in the full reconstruction as it is computed with the T 2 n
, ,
components set to zero. We can see that the nomenclature is beginning to get a bit
awkward here as X2 i is in effect sb n: a modified version of the input signal. In X2 i,
" ,
the subscript 2 relates to the largest scale of the original approximation coefficients
used at the beginning of the reconstruction, and the index i reminds us that we are
back at the input resolution of the original signal (m == 0) with N components.
Removing the last detail coefficient, the modified transform vector becomes
(2.475, 0, 0, 0, 0, 0, 0, 0) which, when used to reconstruct the signal, produces
x3,i == (0.875, 0.875, 0.875, 0.875, 0.875, 0.875, 0.875, 0.875). This can be shown as
S --- S' --- S' --- x .
3,n 2,n 1,n 3,l
, , ,
/ / /
/ / /
/ / /
/ / /
zeros
zeros
zeros
Actually, we have now removed all signal detail and all the components of vector X3 i
,
are of constant value. The signal has been progressively smoothed until all that is left
is a row of constant values equal to the signal mean (figure 3.11(g)).
We can by see comparing figures 3.11(c), (e) and (g) that the high-frequency infor-
mation has been successively stripped from the discrete input signal (figure 3.11(a)) by
successively removing the discrete details at scale indices mo == 1, 2 and 3 in turn.
These discrete signal details are shown in figures 3.11(d), (f) and (h). If we want to
generate the details of the discrete signal at any scale using the multiresolution
algorithm, we perform the inverse transform using only the detail coefficients at
that scale (we could also subtract two successive discrete approximations). For
example, to compute the detail at the lowest wavelet scale (index m == 1), which we
will denote d 1 i, the following multiresolution operation is required:
,
zeros - - - zeros - - - zeros - - - d .
1,l
, , ,
/ / /
/ / /
/ / /
/ / /
zeros
zeros
TIn
,

Everything discrete 95
dI,i is in fact the contribution of the wavelets at scale index m == 1 (i.e.1)JI,O(t), 1)JI,I (t),
1)JI,2(t) and 1)JI,3(t)) to the original signal Xi expressed at discrete points at the scale of
the input function m == 0 at locations i == 0, 1, . . . , N - 1. Similarly we can calculate
the contribution d 2 i using the following operation:
,
zeros
--- zeros --- Sf
1,n
'" '"
/' /'
/' /'
/' /'
/' /'
--- d .
2,z
'"
/'
/'
/'
/'
zeros T 2,n zeros
where, obviously, the S; n vector has all of its components equal to zero as it comes
,
from S3 nand T3 n with elements set to zero. Then d 2 i is the contribution of the wave-
, , ,
lets at scale index m == 2 (i.e. which for this example come from the two wavelets
1)J2,O(t) and 1)J2,I (t)) to the original signal Xi given at discrete points at the scale of
the input function m == 0, i.e. at locations i == 0 to N - 1. Similarly, beginning with
T3 n we can calculate the contribution d 3 i.
, ,
In the above example we were effectively taking the information stored in the
approximation and detail coefficients Sm nand T m n and expressing it at scale index
, ,
m == O. The discrete approximation and detail contributions X m i and d m i are related
, ,
to their continuous counterparts X m (t) and d m (t) through the scaling equation at scale
m == 0 as follows:
N-I
X m (t) == 2:: Xm,icPO,i( t)
i=O
2 M - m _ I
[=  Sm,nq)m,n(t)]
(3.65)
and
N-I
dm(t) == 2:: dm,icPo,i(t)
i=O
2 M - m _ I
[=  Tm,nm,n(t)]
(3.66)
Notice that the scaling function cPO,i(t) is not only used to compute the continuous
approximation from X m i but also the detail components from d m i. This is because
, ,
the scaling coefficient sequence for the wavelet, b k , has already been used in the initial
stages of the reconstruction sequence to take T m n to S _ In. Thereafter the contribu-
, ,
tions can be expressed in terms of scaling functions, i.e. from S _ I n to S _ 2 n and so
, ,
on. You can verify equations (3.65) and (3.66) using the reconstruction algorithm of
equation (3.42) (noting that one of its terms will be zero) and the scaling function rela-
tionship given by equation (3.27b).
Figure 3.11(c) shows Xl i, i.e. where only the detail signal d l i corresponding to the
, ,
smallest scale wavelets (m == 1) has been removed (figure 3.11 (d)). Figure 3.11 (e)
shows X2 i, where both detail signals d l i and d 2 i, corresponding to the smallest and
, , ,
next smallest scale wavelets, have been removed from the original discrete input
signal. d 2 i is shown in figure 3.11 (f). Figure 3.11 (g) shows X3 i, where the detail signals
, ,
d m i corresponding to all the wavelet scales (m == 1,2 and 3) have been stripped from
,
the original signal leaving only the signal mean component. This can be written as
3
X 3 . == X o . - '"" d .
,l ,l  m,l
m=I
(3.67a)

96 The discrete wavelet transform
or, in general, over M scales as
M
X M '==X O '- '"" d .
,l ,l  m,l
m=l
(3.67b)
where Xo i is simply the input signal Xi. This equation stems directly from its contin-
,
uous counterpart of equation (3.45) using equations (3.65) and (3.66). We have now
seen how we can express everything discretely. We can use a suitable discrete input
signal within the multiresolution algorithm, and we can express the approximation
and details of this signal at discrete points at the input resolution; and, as we will
see later in section 3.5.1, even the wavelet and scaling functions are expressed discre-
tely within the multiresolution algorithm, built up by the repeated iteration of the
scaling coefficient vectors.
3.4.2 Smoothing, thresholding and denoising
Let us look again at the discrete input signal used in the previous section (figure
3.12(a)). Once we have performed the full decomposition, we are free to alter any
of the coefficients in the transform vector before performing the inverse. We can
set groups of components to zero, as we did in the last section (figure 3.11), or set
selected individual components to zero. We can reduce the magnitudes of some
components rather than set them to zero. In fact, we can manipulate the components
in a variety of ways depending on what we want to achieve. Notice, however, that the
transform vector contains a range of values: some large and some small. Let us see
what happens if we throwaway the smallest valued coefficients in turn and perform
the inverse transforms. We start with the smallest valued coefficient 0.354. Setting it
to zero, the modified transform vector becomes (2.475, 0, 7.000, -0.500, -0.707,
-2.121, -0.707, 1.414). Performing the inverse on this vector gives the reconstructed
discrete signal (3.875, 4.875, -4.125, -1.125, 0.125, 1.125, 2.125, 0.125), as shown in
figure 3.12(b). There is no discernible difference between the original discrete signal
and the reconstruction. We now remove the next smallest valued coefficient (smallest
in an absolute sense), the -0.5 coefficient. The transform vector becomes (2.475, 0,
7.000,0, -0.707, -2.121, -0.707,1.414) and the reconstructed signal is (3.875,
4.875, -4.125, -1.125,0.375,1.375,1.875, -0.125). Again we have to look carefully
at figure 3 .12( c) to discern any difference between the original discrete signal and the
reconstruction. Next we remove the two -0.707 components to get (2.475,0,7.000,0,
0, -2.121,0, 1.414). Reconstructing we get (4.375,4.375, -4.125, -1.125,0.875,
0.875, 1.875, -0.125). This reconstruction is shown in figure 3.12(d) where we can
now notice obvious smoothing between both the first and second signal point and
the fifth and sixth. Next setting the last transform vector component to zero, we
reconstruct to get (4.375, 4.375, -4.125, -1.125,0.875,0.875,0.875,0.875). This is
shown in figure 3.12(e). Removing the -2.121 component leaves the coefficient
vector as (2.475, 0, 7.000, 0, 0, 0, 0, 0) and the reconstruction becomes (4.375,
4.375, -2.625, -2.625,0.875,0.875,0.875,0.875) as shown in figure 3.12(f). Remov-
ing the 2.475 component (the signal mean component) leaves the coefficient vector
with only a single component: (0, 0, 7.000, 0, 0, 0, 0, 0). Reconstructing using this

Everything discrete 97
o
o
o 5
(a) (2.475,0.354,7.000,-0.500,-0.070,-2.121,-0.707,1.414)
o 5
(b) (2.475,0,7.000,-0.500,-0.707,-2.121,-0.707,1.414)
o
o
o 5
(c) (2.475,0,7.000,0,-0.707,-2.121,-0.707,1.414)
o 5
(d) (2.475,0,7.000,0,0,-2.121,0,1.414)
o
o
o 5
(e) (2.475,0,7.000,0,0,-2.121,0,0)
o
5
(f) (2.475,0,7.000,0,0,0,0,0)
o
o
(9) (0,0,7.000,0,0,0,0,0)
Figure 3.12. Signal reconstruction using thresholded wavelet coefficients. The coefficient vectors used
in the reconstructions are given below each reconstructed signal. Note that (a) is the original signal as
it uses all of the original eight wavelet coefficients.
5
component leaves a single (wavelet shaped) fluctuation contained within the first half
of the signal (figure 3.12(g)).
Can you see what has happened? The least significant components have been
smoothed out first, leaving the more significant fluctuating parts of the signal
intact. What we have actually done is to threshold the wavelet coefficients at increas-
ing magnitudes. First, all the components whose magnitude was equal to or below
0.354 were removed; i.e. 0.354 was the threshold. Then 0.5 was set as the threshold,
then 0.707 and so on. By doing this we removed the least significant influences on
the signal first. Hence, the shape of the reconstructed signal resembles the original
even with a large number of coefficients set to zero. Compare this magnitude thresh-
olding of the coefficients to the scale-dependent smoothing of figure 3.11, which

98 The discrete wavelet transform
removed the components at each scale in turn, beginning with the smallest scales.
Note that in practice we would normally deal with the signal mean coefficient
separately, either retaining it regardless of magnitude or removing it at the beginning
of the thresholding process.
We can define the scale-dependent smoothing of the wavelet coefficients as
Tscale == { 0
m,n T
m,n
m > m*
(3.68 )
*
m<m
where m* is the index of the threshold scale, or the transform vector. Figure 3.13(a)
shows a schematic diagram of scale-dependent smoothing for sequentially indexed
coefficients Wi. (Assuming a full decomposition, we have dropped the M superscript
from wf1.) In this case the thresholding criterion is defined as
Wcale == { 0
l W.
l
i>2M-m*
i<2M-m*
(3.69)
where the range of the sequential index i is from 0 to N - 1 and N is the length of the
original signal; hence i == 2 M -m* is the first location index within the transform vector
where the coefficients are set to zero. (Remember that the smallest scale coefficients
are placed at the right-hand end of the transform vector and hence have the highest
index values.) Note that in practice the coefficients at the very largest scales are some-
times also set to zero to remove drift effects from the signal. The reconstructed signal
using the scale thresholded coefficient vector with scale threshold m* is simply the
smooth approximation of the signal at scale m* expressed at the scale of the input
signal, m == 0, i.e. X m * i. (See previous section.)
,
Magnitude thresholding is normally carried out to remove noise from a signal, to
partition signals into two or more (and not necessarily noisy) components, or simply
to smooth the data. It involves the reduction or complete removal of selected wavelet
coefficients in order to separate out the behaviour of interest from within the signal.
There are many methods for selecting and modifying the coefficients. The two most
popular are hard and soft thresholding. Unlike scale-dependent smoothing, which
removes all small scale coefficients below the scale index m* regardless of amplitude,
hard and soft thresholding remove, or reduce, the smallest amplitude coefficients
regardless of scale. This is shown schematically in figure 3.13(b). To hard threshold
the coefficients, a threshold, A, is set which is related to some mean value of the
wavelet coefficients at each scale, e.g. standard deviation, mean absolute deviation,
etc. Those coefficients above the threshold are deemed to correspond to the coherent
part of the signal, and those below the threshold are deemed to correspond to the
random or noisy part of the signal. Hard thresholding is of the form:
W?ard == { 0
l W.
l
IWil < A
IWil > A
(3.70 )
Hard thresholding makes a decision simply to keep or remove the coefficients. (Hard
thresholding was performed in the example of figure 3.12.) Soft thresholding

Everything discrete
99
Wi
keep
....
remove

l
(a)
/
Scale cutoff at
scale index m *
i.e. sequential index i = 2M-m *
Wi
retain
A -. -----
remove
remove l
-A -. -----
retain
(b)
o
wi° ft
w?ard
o
(c)
-A 0 A
.
-A 0 A
Wi
Figure 3.13. Scale-dependent smoothing and coefficient thresholding. (a) Scale-dependent smoothing.
(b) Amplitude thresholding. ( c) The relationship between the original coefficients and hard (left) and
soft (right) thresholded coefficients.

100 The discrete wavelet transform
recognizes that the coefficients contain both signal and noise, and attempts to isolate
the signal by removing the noisy part from all coefficients. Soft thresholding is of the
form
Woft == { 0
l sign(WJ(1 Wi I - A)
IWil < A
IWil > A
(3.71)
where all coefficients below the threshold, A, are set to zero and all the coefficients
whose magnitude is greater than A are shrunk towards zero by an amount A.
Figure 3.13(c) shows a schematic diagram of the relationship between the original
and thresholded coefficients for both hard and soft thresholding. The retained
coefficients shown in the figure are kept as they are when hard thresholding is
employed and their magnitude is reduced by A when soft thresholding is employed.
Figure 3.14 shows examples of both hard and soft thresholding of a test signal
composed of two sinusoids plus Gaussian white noise (figures 3.14(a) and (b)). The
wavelet coefficients obtained from a Daubechies DI0 decomposition are shown in
figure 3.14(c). These have been hard thresholded for thresholds set at A == 2, 4, 6
and 8 respectively. The reconstructions corresponding to each of the thresholds are
shown in figures 3.14(d)-(g). We can see that for low thresholds some of the high
frequency noise is retained, whereas for high thresholds the signal is excessively
smoothed. From visual inspection, an optimum threshold would seem to lie
somewhere in the region of A == 4. One commonly used measure of the optimum
reconstruction is the mean square error between the reconstructed signal and the
original signal and, in fact, it is found to be minimum near to this value of the thresh-
old. The corresponding soft thresholded reconstructions are shown in figures 3.14(h)
and (i) for A == 2 and 4 respectively.
Often we do not know the exact form of either the underlying signal or the
corrupting noise. The choice of threshold is therefore non-trivial and we can apply
a number of signal and/or noise based criteria in producing a value pertinent to the
problem under consideration. The threshold can, for example, be a constant value
applied to the coefficients across all scales, some of the scales, or its value can vary
according to scale. One of the most popular and simplest thresholds in use is the
universal threshold defined as
AU == (21n N) 1/2 ()
(3.72 )
where (21n N) 1/2 is the expected maximum value of a white noise sequence of length N
and unit standard deviation, and () is the standard deviation of the noise in the signal
under consideration. Thus if the underlying coherent part of the signal is zero (i.e. the
signal contains only noise), using AU as the threshold gives a high probability of
setting all the coefficients to zero. For large samples the universal threshold will
remove, with high probability, all the noise in the reconstruction, but part of the
underlying function might also be lost, hence the universal threshold tends to over-
smooth in practice. (This method is often known as the VISUSHRINK method as
the resulting smooth signal estimate is visually more appealing.) In addition, in
practice it is usual to leave the coefficients at the largest scales untouched even
if they do not pass the universal threshold. The example of the sinusoids plus

Everything discrete 101
o
(a) 0 100 200 300 400 500 600 700 800 900 1000
o
(b) 0 100 200 300 400 500 600 700 800 900 1000
10
o
o
-1
-10 -to -5 0 5 to
(c) 0 100 200 300 400 500 600 700 800 900 1000
10
o
-10
o 100 200 300 400 500 600 700 800 900 1000
o
(d) 0 100 200 300 400 500 600 700 800 900 1000
Figure 3.14. Hard and soft thresholding. (a) Original time series composed of two sinusoids, both with
unit amplitude, one twice the periodicity of the other. (b) Time series in (a) with added Gaussian
noise (zero mean and unit standard deviation). (c) Wavelet coefficients in sequential format derived
using the Daubechies DIO shown. (d) Hard thresholded coefficients, A = 2 (top), and corresponding
reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.)
noise shown in figure 3.14 contains Gaussian noise with (J" == 1 and N == 1024, the
universal threshold for this data is then Au == 3.723. Within a wide range of practical
data sizes (2 6 - 219), only in about one tenth of the realizations will any pure noise
variable exceed the threshold. In this sense universal thresholding method gives a
'noise-free' reconstruction.

102 The discrete wavelet transform
10
o
-10
o 100 200 300 400 500 600 700 800 900 1000
o
o 100 200 300 400 500 600 700 800 900 1000
(e)
10
o
-10
o 100 200 300 400 500 600 700 800 900 1000
o
(f)
o 100 200 300 400 500 600 700 800 900 1000
10
o
-10
o 100 200 300 400 500 600 700 800 900 1000
. . .. ".
. . . .
0 ..
. " . .
'. : . .
.. ".. . . .
-
(g) 0 100 200 300 400 500 600 700 800 900 1000
Figure 3.14 (continued). (e) Hard thresholded coefficients, A = 4 (top), and corresponding
reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.)
(f) Hard thresholded coefficients, A = 6 (top), and corresponding reconstructed time series
(bottom). (The original time series of figure (a) is shown dashed.) (g) Hard thresholded coefficients,
A = 8 (top), and corresponding reconstructed time series (bottom). (The original time series of figure
(a) is shown dashed.)

Everything discrete 103
10
o
-10
o 100 200 300 400 500 600 700 800 900 1000
o
o 100 200 300 400 500 600 700 800 900 1000
(h)
10
o
-10
o 100 200 300 400 500 600 700 800 900 1000
(i)
o 100 200 300 400 500 600 700 800 900 1000
Figure 3.14 (continued). (h) Soft thresholded coefficients, A = 2 (top), and corresponding
reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.) (i) Soft
thresholded coefficients, A = 4 (top), and corresponding reconstructed time series (bottom). (The
original time series of figure (a) is shown dashed.)
Universal soft thresholding usually produces a signal reconstruction with less
energy than that for hard thresholding using the same threshold value, as the retained
coefficients are all shrunk towards zero. This can be seen by comparing figure 3.14(e)
with 3.14(i) for A == 4. Hence, it is often the case in practice that the universal
threshold used for hard thresholding is divided by about 2 when employed as a
soft threshold. Another problem encountered when implementing the universal
threshold in practice is that we do not know the value of (J" for our signal. In this
case, a robust estimate a- is used, typically set to the median of absolute deviation
(MAD) of the wavelet coefficients at the smallest scale divided by 0.6745 to calibrate
with the standard deviation of a Gaussian distribution. Thus the universal threshold
becomes
A = (2 In N) 1/2 M AD = ( 21n ) 1/2 A
U 0.6745 N (J"
(3.73 )

104 The discrete wavelet transform
Many other thresholding methods have been proposed including the mznzmax
method, the SURE method, the HYBRID method, cross-validation methods, the
Lorentz method and various Bayesian approaches. Some of these produce a global
threshold of constant value across scales and others provide a scale-dependent
threshold. We do not consider these in detail here. More information is provided at
the end of the chapter.
3.5 Daubechies wavelets
As we saw with the Haar wavelet transform earlier in section 3.3.5, the coefficients are
ordered in two distinct sequences: one acts as a smoothing filter for the data, the other
extracts the signal detail at each scale. The Haar wavelet is extremely simple in that it
has only two scaling coefficients and both are equal to unity. In this section we will
look at a family of discrete wavelets of which the Haar wavelet is the simplest
member-the Daubechies wavelets. The scaling functions associated with these
wavelets satisfy the conditions given in section 3.2.4, as all orthogonal wavelets do.
In addition to satisfying these criteria, Daubechies required that her wavelets had
compact support (i.e. a finite number, N k , of scaling coefficients) and were smooth
to some degree. The smoothness of the wavelet is associated with a moment condition
which can be expressed in terms of the scaling coefficients as
Nk-l
2:: (_I)k ckk m == 0
k=O
(3.74)
for integers m == 0, 1, 2, . . . , N k /2 - 1. These wavelets have N k /2 vanishing moments,
which means that they can suppress parts of the signal which are polynomial up to
degree N k /2 - 1. Or, to put it another way, Daubechies wavelets are very good at
representing polynomial behaviour within the signal. Some examples of Daubechies
wavelets, their scaling functions and associated energy spectra are given in figure 3.15.
The support lengths of the Daubechies wavelets are N k - 1, i.e. the D2 (Haar)
wavelet, as we already know, has a support length of 1, the D4 wavelet has support
length of 3, the D6 a support length of 5 and so on. We can see from figure 3.15 that
the scaling function lets through the lower frequencies and hence acts as a lowpass
filter, and the associated wavelet lets through the higher frequencies and hence acts
as a highpass filter. In addition, we can see that the spectra are oscillatory in
nature, with bumps decreasing in amplitude from the lower to higher frequencies.
The magnitudes of the secondary bumps in the spectra reduce as the number of scaling
coefficients, and hence the number of vanishing moments of the wavelet, increase.
In this chapter we have already looked in detail at the Daubechies wavelet
with only two scaling coefficients (the D2 or Haar wavelet). Let us now look at the
Daubechies wavelet which has four scaling coefficients, the D4. The 'D' represents
this particular family of Daubechies wavelet and the '4' represents the number of
nonzero scaling coefficients, N k . (Note that Daubechies wavelets are also often
defined by the number of zero moments they have, which equals Nk/2, in which
case the sequence runs Dl, D2, D3, D4, . . ., etc.)

-2 -2 0 0
o 1 2 3 4 5 0 1 2 3 4 5 012345678910 012345678910
(c) D6
2 2 1 1
WAVELET
2
I
o -
-2
o
(a) D2 (Haar).
2
o
-2
o
(b)D4
2
2
o
Daubechies wavelets 105
SCALING FUNCTION
SCALING FUNCTION
SPECTRUM
WAVELET SPECTRUM
2
1
1
I
o I-
- 0.5
0.5
2
1 0
o 0
012345678910 012345678910
1
2
1 1
o
0.5
0.5
-2
3 0
o 0
012345678910 012345678910
2
3
2
1 1
o
0.5
0.5
o 0 0.5 0.5
-2 -2 0 0
0123456789 0123456789 012345678910 012345678910
(d) D 1 0
2 2 1 1
o 0 0.5 0.5
-2
o
(e) D20
10
-2
20 0
10
o 0
20 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Figure 3.15. A selection of Daubechies wavelets and scaling functions with their energy spectra. All
wavelets and scaling functions are shown only over their respective support -outside their support
they have zero value. Note also that only the positive part of the spectrum is given; an identical
mirror image is present at negative frequencies.

106 The discrete wavelet transform
From equation (3.20), we know that the scaling equation for a four-coefficient
wavelet is
cP(t) == cOcP(2t) + Cl cP(2t - 1) + c2cP(2t - 2) + c3cP(2t - 3)
(3.75)
and, from equation (3.23), that the corresponding wavelet function is
1jJ(t) == c3cP(2t) - c2cP(2t - 1) + Cl cP(2t - 2) - cOcP(2t - 3)
(3.76)
To find the values of the scaling coefficients for the D4 wavelet we use equations
(3.21), (3.22) and (3.74). From equation (3.21) we get
Co + Cl + C2 + C3 == 2
(3.77)
from equation (3.22) we get
2 2 2 2 2
Co + Cl + C2 + C3 ==
(3.78 )
and from equation (3.74) with m == 0 we get
Co - Cl + C2 - C3 == 0
(3.79 )
and again using equation (3.74), this time setting m == 1, we get
-1 Cl + 2C2 - 3C3 == 0
F our scaling coefficients which satisfy the above four equations are
(3.80)
1 +V3
Co ==
4
3+V3
Cl ==
4
3 - V3
C2 ==
4
1- V3
C3 ==
4
and so do
1- V3
Co ==
4
3 - V3
Cl ==
4
3+V3
C2 ==
4
1 +V3
C3 ==
4
One set leads to cP(t) and the other to cP( -t). We will adopt the first set, which are
Co == 0.6830127, Cl == 1.1830127, C2 == 0.3169873 and C3 == -0.1830127 respectively.
The scaling coefficients for the Daubechies wavelet system for larger numbers of
coefficients are found by numerical computation. The coefficients for Daubechies
wavelets up to D20 are given in table 3.1.
We can compute the scaling function from the D4 coefficients using equation
(3.75). To do this we must rewrite it as
cPj(t) == cocPj-l(2t) + ClcPj-l(2t - 1) + C2cPj-l(2t - 2) + C3cPj-l(2t - 3)
(3.81)
where the subscript} is the iteration number. Then choosing an arbitrary initial shape
for the scaling function cPo(t) we find cPl (t), then using cPl (t) we find cP2(t), and so on,
iterating until cPj(t) == cPj-l (t) (or at least until cPj(t) is close enough to cPj-l (t) for our
purposes). Once we have an approximation for cP(t) we could define the wavelet
directly using equation (3.76). However, to perform the D4 wavelet transform of a
discrete signal in practice we are not required to compute the wavelet or scaling
functions directly; rather we simply employ the scaling coefficients within the multi-
resolution algorithm (equations (3.36) and (3.37)) in the same manner as we did for

Daubechies wavelets 107
Table 3.1. Daubechies wavelet coefficients D2 to D20
D2 -0.34265671 0.01774979 0.04345268
1 -0.04560113 6.07514995e - 4 -0.09564726
1 0.10970265 -2.54790472e - 3 3.54892813e - 4
-0.00882680 5.00226853e - 4 0.03162417
D4 -0.01779187 -6.67962023e - 3
0.6830127 4.71742793e - 3 D16 -6.05496058e - 3
1.1830127 0.07695562 2.61296728e - 3
0.3169873 D12 0.44246725 3.25814671e - 4
-0.1830127 0.15774243 0.95548615 -3.5632975ge - 4
D6 0.69950381 0.82781653 -5.5645514e - 5
0.47046721 1.06226376 -0.02238574
1.14111692 0.44583132 -0.40165863 D20
0.650365 -0.31998660 6.68194092e - 4 0.03771716
-0.19093442 -0.18351806 0.18207636 0.26612218
-0.12083221 0.13788809 -0.02456390 0.74557507
0.0498175 0.03892321 -0.06235021 0.97362811
-0.04466375 0.01977216 0.39763774
D8 7.83251152e - 4 0.01236884 -0.35333620
0.32580343 6.75606236e - 3 -6.88771926e - 3 -0.27710988
1.01094572 -1.52353381e - 3 -5.5400454ge - 4 0.18012745
0.89220014 9.55229711e - 4 0.13160299
-0.03957503 D14 -1.66137261e - 4 -0.10096657
-0.26450717 0.11009943 -0.04165925
0.0436163 0.56079128 D18 0.04696981
0.0465036 1.03114849 0.05385035 5.10043697e - 3
-0.01498699 0.66437248 0.34483430 -0.01517900
-0.20351382 0.85534906 1.97332536e - 3
D10 -0.31683501 0.92954571 2.8176865ge - 3
0.22641898 0.10084647 0.18836955 -9.69947840e - 4
0.85394354 0.11400345 -0.41475176 -1.64709006e - 4
1.02432694 -0.05378245 -0.13695355 1.32354367e - 4
0.19576696 -0.02343994 0.21006834 -1.87584156e - 5
the Haar wavelet. In this case, the approximation coefficients are computed using
Nk-l
S -  S
m+ln V2 C k m,2n+k
, - 2
k=O
1
= V2 [COS m ,2n + Cj S m,2n+ j + C2 S m,2n+2 + C3 S m,2n+3]
== 0.483S m ,2n + 0.837 Sm,2n+ 1 + 0.224S m ,2n+2 - 0.129S m ,2n+3 (3.82a)
We therefore take the product of a four-digit sequence of the signal by the scaling
coefficient vector (1/V2)ck (the lowpass filter) to generate the approximation
component at the first scale. The mechanics of this are shown schematically in
figure 3.16. The four-digit sequence, (1/ V2)Ck, slides across the signal at scale
index m in jumps of two generating, at each jump, an approximation component at

108 The discrete wavelet transform
So,o SO,l SO,2 SO,3 SO,4 so,s SO,6 SO,7
1 1 1 1
...fi Co ...fi C 1 ...fi C 2 ...fi C 3
1
SI,O = -fi [eo So,o + e 1 SO,1 + c 2 SO,2 + e 3 SO,3]
SO,O SO,l SO,2 SO,3 SO,4 SO,S SO,6 SO,7
1 1 1 1
...fi Co ...fi C 1 ...fi C 2 ...fi C 3
1
Sl,l = -fi [co SO,2 + e 1 SO,3 + e 2 SO,4 + c 3 So,s]
SO,O SO,l SO,2 SO,3 SO,4 SO,S SO,6 SO,7
1 1 1 1
...fi Co ...fi C 1 ...fi C 2 ...fi C 3
1
SI 2 = M [eo So 4 + c 1 So 5 + c 2 So 6 + c 3 So 7]
, 'V2 , , , ,
Sl,O Sl,l Sl,2 Sl,3 Sl,4 Sl,5 Sl,6 Sl,7
Figure 3.16. Filtering of the signal: decomposition. The original signal at scale index m = 0 (i.e. So n) is
,
filtered to produce the approximation coefficients Sl n. This is done by sliding the lowpass filter along
,
the signal one step at a time. Subsampling removes every second value. The diagram shows only the
retained values, i.e. effectively the filter coefficient vector jumps two steps at each iteration. Next, the
sequence Sl n is filtered in the same way to get S2 n and so on. The detail coefficients are found by
, ,
using the same method but employing highpass wavelet filter coefficients.

Daubechies wavelets 109
the next scale (m + 1). To generate the corresponding coefficients for the wavelet we
use the reconfigured scaling coefficient sequence (1/ yI2)b k (the highpass filter), where
b k == (_I)k cN k -1- k, as follows:
1 N k - 1
T m + I,n = "2 L b k S m ,2n+k
Y L k=O
1
= yI2 [C 3 S m ,2n - C2 S m,2n+1 + C,Sm,2n+2 - C O S m ,2n+3]
== -0.129S m ,2n - 0.224S m ,2n+ 1 + 0.837 Sm,2n+2 - 0.483S m ,2n+3
(3.82b)
As an example, we will now consider the D4 wavelet decomposition of the signal
Xi == (1,0,0,0,0,0,0,0) which we input into the multiresolution algorithm as SO,n.
It has N == 8 components, so we iterate the transform decomposition algorithm
three times. After the first iteration the transform vector becomes (0.483, 0.000,
0.000, 0.224, -0.129, 0.000, 0.000, 0.837) containing four approximation components
followed by four detailed coefficients, i.e. (SI,O, SI,I, SI,2, SI,3, T 1 ,0, T 1 ,1, T 1 ,2, T 1 ,3).
Remember that we employ wraparound to deal with the edges of the signal. When
either of the filter pairs is placed at 2n == 6, the 2n + 2 and 2n + 3 locations are not
contained within the signal vector (i == 0, 1, . . . , N - 1). Thus, the last two filter
coefficients of 0.224 and -0.129 (approximation) or 0.837 and -0.483 (detail) are
placed back at the start of the signal vector, the first of each coinciding with the
unit value at the beginning of the signal, hence producing respectively the values
0.224 or 0.837. These values are placed at location SI 3 and T 1 3 respectively. The
, ,
next iteration is performed on the remaining four approximation components S10,
,
SII, S12, SI 3 producing two approximation components followed by two detailed
, , ,
wavelet components. These are added to the four detailed coefficients obtained at
the previous iteration to give the vector (0.204,0.296, -0.171,0.354, -0.129,0.000,
0.000, 0.837). The third and final iteration produces (0.354, -0.065, -0.171, 0.354,
-0.129, 0.000, 0.000, 0.837). This vector has the form (S30, T 30 , T 20 , T 2 1, T 10 ,
, , , , ,
T 1 1, T 12 , T 1 3)' where the signal mean can be found from the first coefficient, i.e.
S3:0/ 23 / 2 == 0354/23/2 == 0.125.
We know that the Daubechies D4 wavelet has two vanishing moments, thus it can
suppress both constant signals and linear signals. This means that for an infinitely
long constant or linear signal all the coefficients are zero. However, end effects
occur if the signal is of finite length. We will use the linear ramp signal
(0, 1,2, . . . ,31), shown in figure 3.17(a), as an example. The first iteration produces
the transform vector
w(1) == (SI,O, SI,I, SI,2, . . . , SI,15, T 1 ,0, T 1 ,1, T 1 ,2, . . . , T 1 ,15)
== (0.897,3.725,6.553,9.382,12.21,15.039,
17.867, 20.696, 23.524, 26.352, 29.181, 32.009,
34.838,37.666,40.495,40.291,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0, 0, -11.314)

110 The discrete wavelet transform
40
o
2: _nnmm I
o 16 32
(a)
- 50 0 8 16 24 32
(b)
10
10
- 50 0 8 16 24 32
(c)
-100 0 8 16 24 32
(d)
10
100
0
-100 8 16 24 32
-100 0
0 8 16 24 32
(e) (f)
10
10
-100 0 8 16 24 32
(g)
-100 0 8 16 24 32
(h)
Figure 3.17. The multiresolution decomposition of a ramp signal using Daubechies wavelets. ( a) Origi-
nal signal. (b) Decomposition 1 (D4 wavelet). (c) Decomposition 2 (D4 wavelet). (d) Decomposition
3 (D4 wavelet). (e) Decomposition 4 (D4 wavelet). (f) Decomposition 5, full decomposition (D4
wavelet). (g) Full decomposition using D6 wavelets. (h) Full decomposition using D12 wavelets.
(Note change in scale of the vertical axes.)
where all the detail coefficients at the lowest scale, T I n, are zero except for the end
,
coefficient T I 15. This nonzero coefficient is caused by the signal wraparound where
,
the wavelet filter vector (1/ y!2)b k placed at the end of the signal encounters the
sequence 30, 31, 0, 1. We can see that the computation for the corresponding detail
component is
T II5 == -0.129 x 30 - 0.224 x 31 +0.837 x 0 - 0.483 x 1 == -11.314
,
A histogram of the coefficients is shown in figure 3.17(b). Notice also the edge effect in
the approximation coefficients where they all increase linearly in value except the last

Daubechies wavelets 111
one due to the wraparound with the scaling filter. The second iteration produces the
transform vector
W(2) == (3.804, 11.804, 19.804, 27.804, 35.804, 43.80, 52.196, 52.981,
0, 0, 0, 0, 0, 0, 1.464, - 15.321 ,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11.314)
where we see that the next set of detail coefficients (T 2 0 to T 2 7) are zero except those
, ,
at the end (figure 3.17(c)). There are two nonzero coefficients now due to the edge
effect in the smooth coefficients at the previous scale. If we repeat the decomposition
to the largest scale we get the fully decomposed signal
W(S) == (87.681, - 31.460, 6.405, - 29 .530, 0, 0, 3.624, - 21.149,
0,0,0,0,0,0, 1.464, -15.321,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,11.314)
As we get 'edge' coefficients at all scales, we get no zero coefficients at scales m == 4
and m == 5 where there are respectively only two (T 4 0, T4 I) and one (Ts 0) coefficient.
" ,
The full D2 (Haar) decomposition of the ramp signal is
W(S) == (87.7, -45.3, -16.0, -16.0, -5.7, -5.7, -5.7, -5.7, -2.0, -2.0,
- 2.0, - 2.0, - 2.0, - 2.0, - 2.0, - 2.0, -0.7, -0.7, -0.7, -0.7, -0.7,
-0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7)
Notice that there are no zero coefficients, as we would expect, since the Haar wavelet
has only one vanishing moment and therefore does not suppress linear signals.
On the other hand, all Daubechies wavelets with more than four scaling
coefficients have more than two vanishing moments and all can therefore suppress
linear signals. For example, the D6 wavelet can suppress mean, linear and quadratic
signals. The D6 decomposition of the ramp signal, which employs the scaling
coefficients (co, CI, C2, C3, C4, cs) == (0.470,1.141,0.650, -0.191, -0.121,0.0498), is
W(S) == (87.681, -2.259, -29.201,25.344,0.149,9.909,
-22.224,6.2484,0,0,0,0,0,4.22, -17.636,4.766,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -15.1 75, 3.861 )
Notice that this time the two end coefficients are nonzero. This is because the filter is
longer. It now has six components and hence two end locations of the filter on the
discrete signal overlap and wraparound to the beginning of the signal. A D8 decom-
position would generate three end coefficients, a D 1 0 would generate four and so on.
The full decompositions of the ramp signal using both the D6 and D12 wavelet are
shown in figures 3.17(f) and (g) for comparison. The more scaling coefficients the
wavelet has, the higher the number of its vanishing moments and hence the higher
the degree of polynomial it can suppress. However, the more scaling coefficients
that a wavelet has, the larger its support length and hence the less compact it becomes.

112 The discrete wavelet transform
This makes it less localized in the time domain and hence less able to isolate
singularities in the signal (including edge effects). This is the trade-off which must
be considered when selecting the best wavelet for the data analysis task.
3.5.1 Filtering
Let us revisit the filtering process described in the last section and shown in figure
3.16. In signal processing, the approximation coefficients at resolution m, Sm n, are
,
convolved with the lowpass filter. This done by moving the filter along the signal
one step at a time. The approximation coefficients are then subsampled (or down-
sampled) where every second value is chosen to give the approximation coefficient
vector at scale m + 1. The approximation coefficients at resolution m, Sm n, are also
,
convolved with the highpass filter and subsampled in the same way to give the
detail signal coefficients at scale m + 1. The detail components T m + l,n are kept and
the approximation components Sm+ l,n are again passed through the lowpass and
highpass filters to give components Tm+2,n and Sm+2,n. This process is repeated
over all scales to give the full decomposition of the signal. Each step in the decompo-
sition filtering process is shown schematically in figure 3.18(a). The sequences
(1/ V2)Ck and (1/ V2)b k are contained respectively within the lowpass and highpass
filter vectors used within the wavelet decomposition algorithm. The filter coefficients
are sometimes referred to as 'taps'.
F or signal reconstruction (equation (3.42)), the filtering process is simply
reversed, whereby the components at the larger scales are fed back through the filters.
Sm,n
0--@----. sm+ 1,0
I LP I -----@-- Sm+2,0
(a)
T m + 1 ,n
T m+2,n
Sm+2,0 
S @ 
m+l,n .
Sm,n
T m+2,n
(b)
Figure 3.18. Schematic of the signal filtering. (a) Schematic diagram of the filtering of the approxima-
tion coefficients to produce the approximation and detail coefficients at successive scales. The
subsample symbol @ means take every second value of the filtered signal. HP and LP are, respec-
tively, the highpass and lowpass filters. In practice the filter is moved along one location at a time on
the signal, hence filtering plus subsampling corresponds to skipping every second value as shown in
figure 3.16. (b) Schematic diagram of the filtering of the approximation and detail coefficients to
produce the approximation coefficients at successively smaller scales. The upsample symbol @
means add a zero between every second value of the input vector. The coefficients in the HP' and
LP' filters are in reverse order to their counterparts used in the decomposition shown in (a). Figures
(a) and (b) represent a subband coding scheme.

Daubechies wavelets 113
..h [e 3 sm,l + 0 + e l sm,2 + 0]
..h [h 3 sm,O + 0 + hI Sm,l + 0]
ADD
Figure 3.19. Filtering of the signal: reconstruction. The smooth and detail coefficients at scale index
m are passed back through the filters to reconstruct the smooth coefficients at the higher
resolution Sm _ 1 n. This is done by sliding the lowpass and highpass filters along their respective
,
coefficient sequence with zeros inserted between the coefficients. (Refer back to equation (3.42)
where, remember, 'n' is the coefficient location index at the lower scale and 'k' is the index at the
higher scale.)
This is shown in figure 3 .18(b). The approximation and detail components at scale
index m + 1 are first upsampled (zeros are inserted between their values) and then
passed through the lowpass and highpass filters respectively. This time, however,
the filter coefficients are reversed in order and are shifted back along the signal. An
example of the reconstruction filtering is shown in figure 3.19 where the component
Sm,5 is found from the sequences Sm+l,n and Tm+l,n. Note that the leading (right-
hand) filter coefficient defines the location of the Sm n coefficient. Hence the com-
,
putation of the Sm 0, Sm 1 and Sm 2 coefficients involve components of the filter
" ,
vector lying off the left-hand end of the upsampled signal at scale m + 1. These are
simply wrapped back around to the right-hand end of the signal.
We can produce a discrete approximation of the scaling function at successive
scales if we set all the values of the transform vector to zero except the first one
and pass this vector repeatedly back up through the lowpass filter. This is illustrated
in the sequence of plots contained in figure 3.20. Figure 3.20(a) shows the initial trans-
form vector, 64 components long, with only the first component set to unity, the rest
set to zero, i.e. (1,0,0,0,...,0). After the first iteration, four components can be seen.
These are equal to (co/V2, cl/V2, c2/V2, c3/V2, 0, 0, 0,...0) (notice that this is the
decomposition filter). The next iteration (figure 3.20(c)) produces 10 nonzero values
in the transform vector, the next, 22 nonzero values, and so on. The transform
vector components take on the appearance of the scaling function quite quickly.
What we are actually doing is reconstructing So n based on the initial vector
,
(1,0,0,0, . . . ,0) equal to the following transform vectors: (SI,O, SI,I, SI,2, . . . , T 1 ,0,
T 1 ,1, T 1 ,2,...) for figure 3.20(b); (S2,0, S2,1, S2,2,..., SI,O, SI,I, SI,2,..., T 1 ,0, T 1 ,1,
T 1 ,2,. ..) for figure 3.20(c); (S3,0, S3,1, S3,2, . . . , S2,0, S2,1, S2,2,. . . , SI,O, SI,I, SI,2,. . . ,
T 1 ,0, T 1 ,1, T 1 ,2,...) for figure 3.18(d), and so on. Notice that, as we employ
wraparound to deal with the edge effects, the final signal shown in figure 3.20(g) is
a constant value equal to the signal mean. Figure 3.20(h) shows a high-resolution

114 The discrete wavelet transform
- <> I I I I I I I I -
I I I I I I I I I
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 3.20. The reconstruction of a Daubechies D4 scaling function. (a) The transform vector (1, 0,
0, 0, 0, 0, . . . ). (b) One iteration, i.e. S 1 0 = 1. (c) Two iterations, i.e. S20 = 1. (d) Three iterations,
, ,
i.e. S30 = 1. ( e) Four iterations, i.e. S40 = 1. (f) Five iterations, i.e. Ss 0 = 1. (g) Six iterations of
, , ,
the transform vector with 64 components, i.e. S60 = 1. This results in a constant signal equal to
,
the mean due to wraparound. (h) After nine iterations of a transform vector (1, 0, 0, 0, 0, ...)
with 4096 components.

Daubechies wavelets 115
representation of a Daubechies scaling function generated by iterating nine times with
an initial transform vector containing 4096 components.
You can verify that the constant signal of figure 3.20(g) stems from the wrap-
around employed when reconstructing from the full decomposition wavelet vector
with S M 0 set to unity. In this case, for the first iteration, adding a zero to get
,
(SM,O'O) and running the reconstruction filter over both components leads to two
approximation coefficients (S M -1,0, S M -1,1) equal to ( (CO + C2) / vfi, (Cl + C3) / vfi) ==
(0.707,0.707). Adding zeros to this vector and iterating again produces four
approximation coefficients (0.5,0.5,0.5,0.5) and so on until we obtain the full
reconstruction after the sixth iteration which contains constant values of So n equal
,
to (l/vfi)6 == 0.125. We can see this is true for the general case as, from equation
(3.74) with m == 0 together with equation (3.21), it can be shown that
2:: Ck == 2:: Ck == 1
k even k odd
(3.83)
Hence, the combined effect of wraparound plus the zeros inserted in the reconstruction
process causes S M,O to be multiplied by both (1/ vfi) 2::keven Ck and (1/ vfi) 2::k odd ck
in turn to find S M _ 1 0 and S M _ 1 1, which both have the same value simply equal to
, ,
sM,o/vfi. This transform vector now has two constant values (the rest zero). The
next iteration produces four constant nonzero values and so on. It is easy to prove
to yourself that using Daubechies wavelets with different numbers of nonzero scaling
coefficients, N k , will always result in constant valued transform vector components.
If we repeat the reconstruction filtering, this time setting the first detail coefficient
to unity, we can generate an approximation to a wavelet. Notice that, in doing so,
only the first iteration of the reconstruction algorithm requires the reordered scaling
coefficient sequence for the wavelet, b k , and subsequent reconstruction filtering then
uses the scaling coefficient sequence Ck. Figure 3.21 shows the wavelet reconstruction
process. From these examples, where a single component was set to unity, we can see
that the transform vector components, when passed back through the filters during
the reconstruction process, increasingly spread their influence (information) over
the transform vector in terms of either discrete scaling functions (if the original
component was an approximation coefficient) or discrete wavelet function (if the
original component was a detail coefficient). Figures 3.20 and 3.21 illustrate the
discrete approximations of the wavelet and scaling functions at various scales implicit
within the multiresolution algorithm. The discrete scaling and wavelet functions at
scale 1 are, in fact, the filter coefficient vectors (1/ vfi)b k and (1/ vfi) Ck respectively.
3.5.2 Symmlets and Coiflets
Looking back at figure 3.15 we can see that Daubechies wavelets are quite asym-
metric. To improve symmetry while retaining simplicity, Daubechies proposed
Symmlets as a modification to her original wavelets (also spelled symlets). Symmetric
wavelet filters are desirable in some applications, e.g. image coding, as it is argued that
our visual system is more tolerant of symmetric errors. In addition, it makes it easier
to deal with image boundaries. Daubechies came up with Symmlets by 'juggling
with their phase' during their construction. Figure 3.22 contains two examples of

116 The discrete wavelet transform
(a)
(b)
(c)
(d)
Figure 3.21. The reconstruction of a Daubechies D4 wavelet. (a) One iteration with only Tl 0 initially
,
set equal to 1. (b) Two iterations with only T 20 initially set equal to 1. (c) Three iterations with only
,
T3 0 initially set equal to 1. (d) Four iterations with only T 40 initially set equal to 1.
, ,
Symmlets together with their scaling functions. They have N k /2 - 1 vanishing
moments, support length N k - 1 and filter length N k . However, true symmetry (or
anti symmetry) cannot be achieved for orthonormal wavelet bases with compact
support with one exception: the Haar wavelet which is antisymmetric. Coiflets are
(a)
(b)
Figure 3.22. Symmlets and their associated scaling functions. ( a) S6. (b) S 10. (The scaling functions
are shown dotted.)

Translation invariance 117
(a)
(b)
(c)
Figure 3.23. Coiflets and their associated scaling functions. (a) C6 . (b) C 12 . (c) C 18. (Scaling functions
are shown dotted in figures.)
another wavelet family found by Daubechies. They are also nearly symmetrical and
have vanishing moments for both the scaling function and wavelet: the wavelet has
N k /3 moments equal to zero and the scaling function has N k /3 - 1 moments equal
to zero. They have support length N k - 1 and filter length N k . Three examples of
Coiflet wavelets are shown in figure 3.23. The number of coefficients, N k , used to
define Coiflets increase in multiples of six.
3.6 Translation invariance
Translation in variance (or shift in variance ) is an important property associated with
some wavelet transforms but not others. It simply means that if you shift along the
signal all your transform coefficients simply move along by the same amount.
However, for the dyadic grid structure of the discrete wavelet transform this is clearly
not the case: only if you shift along by the grid spacing at that scale do the coefficients
become translation invariant at that scale and below. Even for the discretization of
the continuous wavelet transform, the transform values are translation invariant
only if shifted by any integer multiple of the discrete time steps. This is illustrated
in figure 3.24 where a simple sinusoidal signal is decomposed using the discrete
orthonormal Haar wavelet and a continuous Mexican hat wavelet discretized at
each time step. The original signal shown on the left-hand side of figure 3.24(a) is
composed of 10 cycles of a sinusoid made up of 1024 discrete data points. The
middle plot in figure 3.24(a) contains the Haar transform coefficients of the signal
plotted in their dyadic grid formation. The right-hand plot contains the continuous
wavelet transform using the Mexican hat wavelet. The coarse structure of the
dyadic grid is evident when compared with the smooth, high resolution Mexican
hat transform plot.
We will assume the signal continues indefinitely and shift the signal window
along to see what happens to the transform plots. The variation in the coefficients
of the Haar transform with displacement of the time window along the signal can
be seen by looking down the central plots in the figure. Shifting the window forward
(or back) along the signal by a scale of 2 m produces the same coefficients in the
transform plot at and below that scale shifted in time by 2 m . This can be observed
in figures 3.24(b )-( d) where the size of the shift is indicated in the transform plot

118 The discrete wavelet transform
(a)
(b)
(c)
(d)
(e)
scale index
m=l-
Haar wavelet
coefficient plot
Mexican Hat wavelet
transform plot
m=5-
m=10 -
I II L.. Cd)
signal mean L==; ( b ( J c J
component
L
L
L
Figure 3.24. Translation invariance. (a) Original signal, 1024 data points. (b) Shift by 64 data points.
(c) Shift by 128 data points. (d) Shift by 256 data points. (e) Shift by 1/8 cycle.
of figure 3.24(a). The coefficient level at and below which the coefficients remain the
same as the original is indicated by the arrows at the left-hand side of each transform
plot. Shifting the signal by an arbitray scale (not a power of two) leads to a completely
different set of coefficients, as can be seen in figure 3.24( e), where the signal is shifted
by an eighth of a sinusoidal cycle (12.8 data points). The translation invariance of the

Biorthogonal wavelets 119
continuous transform plots is obvious by looking down the right-hand figures where,
for any arbitrary shift in the signal, the wavelet transform values are simply translated
with the shift.
We can make a translation invariant version of the discrete wavelet transform.
Known by a variety of names including the redundant, stationary, translation
invariant, maximal overlap or non-decimated wavelet transform, we simply skip the
subsampling part of the filtering process described in section 3.5.1. This results in
the same number of wavelet coefficients generated at each step, equal to the
number of signal components N. The decomposition is the same as that shown in
figure 3.16 except that every value is retained as the filter moves one step at a time
along the signal. In addition, the filters have to be stretched through the addition
of zeros between coefficients, hence this algorithm is called the 'i1 trous' algorithm
from the French 'with holes'. An average basis inverse can be performed which
gives the average of all possible discrete wavelet transform reconstructions over all
possible choices of time origin in the signal. This is sometimes useful for statistical
applications including denoising. In addition to being translation invariant, the
redundant discrete wavelet transform is extendable to signals of arbitrary length.
We do not consider the redundant discrete wavelet transform herein, rather the
reader is referred elsewhere at the end of this chapter.
3.7 Biorthogonal wavelets
For certain applications real, symmetric wavelets are required. One way to obtain
symmetric wavelets is to construct two sets of biorthogonal wavelets: 1Pm n and its
 ,
dual, 1Pm n. One set is used to decompose the signal and the other to reconstruct it.
,
For example, we can decompose the signal using 1Pm n as follows:
,
Tm,n = '[00 x(t)'lj;m,n(t) dt
(3.84)
and perform the inverse transform using 1Pm n,
,
00
x(t) == 
00
 Tm,n;jJm,n(t)
(3.85)
m=-oo n=-oo
Alternatively, we can decompose the signal using
Tm,n = '[00 x(t);f;m,n(t) dt
(3.86)
and reconstruct using
00
x(t) == 
00
 Tm,n1P(t)
(3.87)
m=-oo n=-oo

120 The discrete wavelet transform
Biorthogonal wavelets satisfy the biorthogonality condition:
J oo  { I if m == m/ and n == n/
1Pm n (t)1Pm' n' (t) dt == .
-00" 0 otherwIse
(3.88)
Using biorthogonal wavelets allows us to have perfectly symmetric and anti symmetric
wavelets. Further, they allow certain desirable properties to be incorporated separately
within the decomposition wavelet and the reconstruction wavelet. For example, 1Pm n
 ,
and 1Pm n can have different numbers of vanishing moments. If 1Pm n has more vanishing
,  ,
moments than 1Pm n, then decomposition using 1Pm n suppresses higher order poly-
, , 
nomials and aids data compression. Reconstruction with the wavelets 1Pm n with
,
fewer vanishing moments leads to a smoother reconstruction. This can sometimes be
a useful property, for example in image processing. Figure 3.25 shows three examples
of compactly supported biorthogonal spline wavelets and their duals commonly used in
practice, together with their associated scaling equations.
(a)
(b)
(c)
Figure 3.25. Biorthogonal spline wavelets. (a) Biorthogonal (1,5) spline wavelets: cjJ(t), 'ljJ(t) (left) and
(t), ;j;(t) (right). (b) Biorthogonal (2,4) spline wavelets: cjJ(t), 'ljJ(t) (left) and (t), ;j;(t) (right). (c)
Biorthogonal (3,7) spline wavelets: cjJ(t), 'ljJ(t) (left) and (t), ;j;(t) (right). (Scaling functions shown
dotted.)

Two-dimensional wavelet transforms 121
3.8 Two-dimensional wavelet transforms
In many applications the data set is in the form of a two-dimensional array, for
example the heights of a machined surface or natural topography, or perhaps the
intensities of an array of pixels making up an image. We may want to perform a
wavelet decomposition of such arrays either to compress the data or to carry out a
wavelet-based parametric characterization of the data. To perform a discrete wavelet
decomposition of a two-dimensional array we must use two-dimensional discrete
wavelet transforms. We can simply generate these from tensor products of their
one-dimensional orthonormal counterparts. The most common (and simplest)
arrangement is to use the same scaling in the horizontal and vertical directions.
These give square transforms. (Other forms are possible, e.g. rectangular transforms
where the horizontal and vertical scaling vary independently and also two-
dimensional transforms which are not simply tensor products. These are outside
the scope of this text.) The two-dimensional Haar scaling and wavelet functions are:
two-dimensional scaling function
cjJ(t 1 , t 2 ) == cjJ(t 1 )cjJ(t 2 )
(3.89a)
two-dimensional horizontal wavelet
1P h ( t 1 , t 2) == cjJ ( t 1 ) 1P ( t 2 )
(3.89b)
two-dimensional vertical wavelet
1P V (tl, t2) == 1P(tl)cjJ(t2)
(3.89c)
two-dimensional diagonal wavelet
1P d (t 1, t 2 ) == 1P( t 1 )1P( t 2 )
(3.89d)
Remember that in both the last chapter and this we have been using t as our
independent variable, either temporal or (less common) spatial. In this section t 1
and t 2 represent spatial coordinates.
The multiresolution decomposition of the two-dimensional coefficient matrices
can be expressed as
1
Sm + 1,(nj,n2) = "2   ck j Ck 2 S m(2nj +k j , 2n 2 +k 2 )
k 1 k 2
T h -  '"" '"" b S
m+ 1,(nl,n2) - 2   k 1 ck 2 m(2nl +k 1 , 2n 2 +k 2 )
k 1 k 2
T):,+ 1,(nj,n2) =    Ck j b k2 Sm(2nj +kj,2n2 +k 2 )
k 1 k 2
(3.90a)
(3.90b)
(3.90c)
T+ 1,(nj,n2) =    b kj b k2 S m (2nj +k j ,2n2 +k 2 )
k 1 k 2
(3.90d)
where k 1 , k 2 are the scaling coefficient indices and nl, n2 are the location indices at scale
m + 1. (Compare with the one-dimensional case of equations (3.36) and (3.37).) We can
simply use discrete versions of the two-dimensional wavelets at scale index 1 to perform

122 The discrete wavelet transform
L
(a) (b)
 
Figure 3.26. The two-dimensional discrete Haar wavelet at scale index 1. (a) Scaling function. (b)
Horizontal wavelet. (c) Vertical wavelet. (d) Diagonal wavelet.
the multiresolution analysis of the array. The discrete two-dimensional Haar scaling
and wavelet functions in matrix form at scale index m == 1 are
scaling function
1
 [ ]
vertical wavelet
1
 [ 1 -1 ]
2 1 -1
horizontal wavelet
1
[- -]
diagonal wavelet
1
 [ 1 -1 ]
2 -1 1
As these are at scale index 1, they are simply tensor products of the scaling and wave-
let filter coefficients Ck/ V2 and b k / V2. These discrete functions are shown schemati-
cally in figure 3.26. Note the! factor before each matrix is simply the square of the
1/ V2 factor preceding the corresponding one-dimensional functions. For discrete
scaling and wavelet functions at larger scales, this normalization factor becomes
1 /2 m . These four 2 x 2 matrices are required for the Haar wavelet decomposition
of the two-dimensional array. The general idea of the two-dimensional wavelet
decomposition is shown schematically in figure 3.27. The original input array is
represented by Xo defined at scale index m == o. As with the one-dimensional case,
its components are input as the approximation coefficients at scale index 0, i.e. the
matrix So. After the first wavelet decomposition, a decomposition array is formed
at scale index 1 which is split into four distinct submatrices: the vertical detailed

Two-dimensional wavelet transforms 123
2 M .2 M components
ORIGINAL
ARRAY
scale index 1
r-------------------
I
I
I
I
I
I
I
Sl
TV
1
I
I
I
I
I
I
I
I
I
I
I
I
L____________________
T h
1
T d
1
scale index 2
r-------- r--------
: S :: TV :
I I I I
I I I
I 2:: 2:
:_-------- :_--------
TV
1
r-------- r--------
I I I
h :: d
: T 2 :: T 2
I I I
I I I
1______---.. 1_________
T h
1
T d
1
etc.
Figure 3.27. Schematic diagram of the matrix manipulation required to perform wavelet decomposition
on a two-dimensional grid.
components TY, the horizontal detailed components T7, the diagonal detailed com-
ponents T1 and the remaining approximation components Sl. The decomposition
array is the same size as the original array. As with the discrete wavelet decomposition
of one-dimensional signals, the detail coefficients are subsequently left untouched and
the next iteration further decomposes only the approximation components in the

124 The discrete wavelet transform
0.8
0.6
0.4
0.2
00
1
2
3
Figure 3.28. A two-dimensional signal containing a single spike.
submatrix Sl. This results in detail coefficients contained within submatrices T 2 , T
and T at scale index 2 and approximation coefficients within submatrix S2. This
procedure may be repeated M times for a 2 M x 2 M array to get a number of coefficient
submatrices T, T and T of size 2 M -m x 2 M -m, where m == 1,2, . . . , M.
Let us look at a very simple example, a matrix with a single nonzero component
Xo ==
o 0
o 1
o 0
o 0
o 0
o 0
o 0
o 0
(3.91)
This array could represent a single pixel turned on in a display or perhaps a spike
protruding from a flat surface. We can visualize it in figure 3.28. The two-dimensional
Haar multiresolution decomposition of this array is carried out by scanning over it
with each of the four discrete wavelet matrices in turn. This is shown schematically
in figure 3.29. On the first iteration (from scale m == 0 to 1), the scaling function
2 x 2 matrix is scanned over the input array. We require four of these matrices to
cover this 4 x 4 array. The components of the original array are multiplied in turn
by the scaling function array to give the resulting matrix product Sl which is
placed in the top right-hand quadrant of the first iteration matrix. Similarly, the
vertical wavelet 2 x 2 wavelet matrix is scanned over the array and each value is
placed in the top right-hand quadrant of the first iteration array. This is illustrated
in the TY matrix marked on the right-hand of the figure. The procedure is repeated
for the horizontal and diagonal wavelet components, placing the products in the
bottom left-hand and bottom right-hand quadrants respectively. The normalization
factor of 1/2 has been left out of the matrix manipulation until the end to aid clarity.
Hence, after the first iteration, the resulting wavelet transform matrix is
...--..-- ..--..--.
 0.5 O -0.5 0
W{]) O OJ 0 0 (3.92)
- - - . . - - . . - - . . - - . . --
-
0.5 0 o.s 0
0 0 0 0
The next iteration uses only the approximation components in the top left-hand
quadrant (shown within the dotted box). The four component values of this array

Two-dimensional wavelet transforms 125
1
-.
2
1-. [: :l[: :]
2 / [ 1 0 Pl' [ 10 10 ]
/ 10,/10J 10 10
, / '"
S I :' // ",,,,,,,/' --
, I I" , ---_
I / / ,1/ /// /_______:o>.<c:::::--------:-:------------------------,
1t;/fF;or':/; 1    : ] I:   :  ]
," t'" - . l
_Li-----J l-i,<-5l, ::::::::::::::---,----, ,I:   :  II:   : ]
o 0 0 0 ""'"",,_--------------------------
original input array
cornponentsshovvn
shaded
D
wavelet transform array
0.5 0 -0.5 0 after the first iteration
0 0 0 0 WI
-0.5 0 0.5 0
0 0 0 0
Figure 3.29. Schematic diagram of the matrix manipulation required to perform the Haar wavelet
decomposition of the spike signal. Components of the discrete scaling and wavelet matrices are
shown in bold, components of the original discrete signal are shown shaded.
are then interrogated by each of the scaling and wavelet matrices in turn. This
produces the second iteration matrix
0.25 0.25 -0.5 0
W(2) == 0.25 0.25 0 0
(3.93)
-0.5 0 0.5 0
0 0 0 0
This is the second and last iteration we can perform in the decomposition of the 4 x 4
(i.e. M == 2) array. This matrix is made up ofTY, T7, T1, T 2 , T, T and, in the top left-
hand corner, the approximation component S2 which is related to the mean of the
original array. Notice that, at each iteration, the resulting matrix has the same
energy as the original matrix where the energy is the sum of the squared components

126 The discrete wavelet transform
(a)
4
2
o
O 2
4 6
(b)
10
5
o
O 2
4
6
(d)
I '. I
" .
.... -
.....-:--..;.,... l1li
I( .,'" ... ..
"....:........:...:.....
"fI::: '-;."-: ........ ::...
.... ..  '( 'I)). ........
T..:{;,.:;.........
2
1
o
O 2
4 6
Figure 3.30. An 8 x 8 'step' array together with its associated Haar decompositions. (a) Original step.
(b) Iteration 1. (c) Iteration 2. (d) Iteration 3. Note that black dots have been added to the non-zero
coefficients in plots (c) and (d) to aid clarity.
of the matrix, i.e.
2 M - 1 2 M - 1 2 M -1 2 M -1
E==   (X O ,i,j)2 ==   (W i :7))2 ( 3 . 94 )
i=O j=O i=O j=O
where XO,i,j and W i :7) are, respectively, the elements of the input and wavelet
decomposition matrices located on row i and column j. E is equal to 1 for the
array considered above. We can easily verify that E is equal to 1 for the intermediate
decomposition matrix as well as the full decomposition matrix. You can confirm this
conservation of energy property in all the examples which follow.
Let us look at another simple example, the step shown in figure 3.30(a) and given
in matrix form as
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
Xo == 2 2 2 2 2 2 2 2 (3.95)
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
This 8 x 8 array can be decomposed through three iterations using the Haar wavelet
as follows:
first decomposition W(1)
4 4 4 4 0 0 0 0
4 4 4 4 0 0 0 0
3 3 3 3 0 0 0 0
2 2 2 2 0 0 0 0
0 0 0 0 0 0 0 0 (3.96a)
0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0

Two-dimensional wavelet transforms 127
second decomposition W(2)
8 8 0 0 0 0 0 0
5 5 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 (3.96b)
0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0
third (full) decomposition W(3)
13 0 a 0 0 0 0 0
..--..--..--..
  r ll  ...--.
; 3; 0 0 0 0 0 0 0
; JU..U;..- . ,
, , ...--.
..--..--..--.. ...--..--..--
Ilr'lllr'lllr'lll 10 0 0 0 0 0 0 0
: T h :
: 2 ..".".  1 1 0 0 0 0 0 0
11....11....11....1.
...--..--..-- (3.96c)
III IIII IIII II IIIII IIII III
O 0 0 0 0 0 0 0
..11....11....11..... O 0 0 0 0 0 0 0
 T 1l  :
; l . U.. -:-..  1 1 ) l 0 0 0 0
...--..--..--.
O 0 0 0 0 0 0 0
11....11....11....11....11.....11....11
Note that, after the three iterations, only the approximation component and the
components in regions corresponding to the horizontal wavelet decomposition
(labelled T7, T and T) contain nonzero elements. This is because the original
array only contains a single horizontal feature-the horizontal discontinuity or
step. Another thing to notice is that the 64 nonzero elements of the original array
have been transformed into only eight nonzero elements, i.e. a compression of the
data has taken place. The original arrays, together with the three levels of decomposi-
tion, are shown as histograms in figure 3.30.
We can now use the detail coefficients to reconstruct the array at different scales.
This is shown in figure 3.31 and is performed using the detail coefficients at each scale
of interest. In the same way as we did for the one-dimensional signals earlier in section
3.4, discrete detail arrays D, D and D can be reconstructed at the scale of the input
array using each of the detail coefficients, T, T and T. A combined detail can be
found at scale index m, given by
Dm == D + D + D (3.97)
As with the one-dimensional case, the array can be represented as a sum of the
discrete details plus an array mean.
M
Xo == X M + 2:: Dm
m=l
(3.98)
where X M is the smooth version of the input array at the largest scale index, M,
expressed at the scale of the input array m == O. For example, to reconstruct the

128 The discrete wavelet transform
13[U o irtj""'oi O....O....O....O!
r"3"1:=01l0 01 0 0 0 01
ro::":o:iro::::ot 0 0 0 01
i..J.......! i.Q....Q. i.Q....Q....Q....Q.
.............................. ............ ................
ooooooooi
OOOOOOOO!
1 1 1 11 0 0 0 01
o OOOOOOO
, ... ... ... ... ... " \ ... ... ... "'............................
scale 1
vertical detail
coefficients
TV
1
scale 1
diagonal detail
coefficients
T d
1
scale 1
horizontal detail
coefficients
T h
1
(a)
detail
coefficients
sub matrix
T h
1
f'I[![[[I
[::II!:I:[!
000 0 0 0 0 0
000 0 0 0 0 0
000 0 0 0 0 0
000 0 0 0 0 0
...............................................................
. " .
 0.5 0.5!! 0.5 0.5  0.5 0.5 0.5 0.5
1- Q.:.?....QJl Q.:.?.:..Q.:.!- 0.5 - 0.5 - 0.5 -0.5
000 0 0 0 0 0
o 0 0 0 0 0 0
detail matrix
/ D
/1
detail coefficient \
Haar wavelet at
scale index m= 1
1 [ 1 [ 1 1 ]]
2 1 - 1 - 1
(b)
Figure 3.31. Schematic diagram of the matrix manipulation required to derive the detail matrices at
scale index m = 1. (a) The three detail coefficient submatrices at scale index 1 for the step. (b)
Computing the horizontal discrete detail from the detail coefficients at scale index 1.
detail array components at the smallest scale (m == 1), each corresponding component
in each of the three coefficient submatrices, T7, T1 and T 1 , are used in conjunction
with the corresponding Haar wavelets (horizontal, diagonal and vertical) to produce
the array details at this level: D7, D1 and Dl. The m == 1 submatrices are shown in

Two-dimensional wavelet transforms 129
figure 3.31(a). These three 4 x 4 coefficient submatrices are each expanded into an
8 x 8 array corresponding to the horizontal, diagonal and vertical details of the
original array using the corresponding discrete wavelets. This is shown schematically
in figure 3.31 (b) for the horizontal detail at scale index 1, i.e. D7. The expansion of the
transform coefficients through the discrete Haar wavelet into the detailed components
is shown explicitly for two of the T7 components in the figure. We can see that the
detail matrices D1 and Dl will have all elements equal to zero as both T1 and Tl
contain only zero elements. Hence the combined detail component matrix is
DI == D7 + D1 + Dl ==
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
o
o
o
o
0.5
-0.5
o
o
( 3.99 )
Similarly, we can get the detail matrices at scale index 2, D, D and D 2 , using the
scale 2 coefficients, respectively T, T and T 2 and the discrete wavelet at scale
index 2. This is shown for D in figure 3.32. Note that the normalization factor at
scale 2 is 1/2 2 . The scale 3 coefficient submatrices, T, T and T 3 , are simply [3], [0]
and [0] respectively. It is easily seen that this gives the detail at the largest wavelet
1!!jlHIl1:IIII
III!]I!II!1
detail ............/
coefficient
1 1 1 1
1  1 1 1 1
l -1-1-1-1
-1-1-1-1

horizantal Haar wavelet
at scale index 2
000 0 0 0 0 0
000 0 0 0 0 0
000 0 0 0 0 0
000 0 0 0 0 0
rO'25"'''O'j5'''''0':'2"5''''''6':25'1 0.25 0.25 0.25 0.25
1 0.25 0.25 0.25 0.25 i 0.25 0.25 0.25 0.25
1- 0.25 - 0.25 - 0.25 - 0.25! - 0.25 - 0.25 - 0.25 - 0.25
!...9.:.??.....9..:.?..?.....9..:.?....9.:??.J - 0.25 - 0.25 - 0.25 - 0.25
Figure 3.32. Schematic diagram of the matrix manipulation required to compute the horizontal detail at
scale index m = 2.

130 The discrete wavelet transform
scale equal to
0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375
0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375
0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375
D 3 == 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375
-0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375
-0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375
-0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375
-0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375 -0.375
( 3. 1 00 )
where each component is either 3/2 3 or -3/2 3 . In fact, this matrix is simply a single large
horizontal discrete Haar wavelet scaled by 3/2 3 . The approximation coefficient (== 13)
leads to a matrix of values all equal to the mean of the original array (== 13/23), i.e.
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
X 3 == 1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625 (3.101)
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625
Figure 3.33 shows the reconstructions at each scale together with the mean component.
From the figure we can see that if we add them all together, DI + D 2 + D3 + X 3 , we get
back the original input array Xo.
Figure 3.33. The details of the step array. (a) Original step array Xo. (b) Dl. (c) D 2 . (d) D3. (e) X 3 . (f)
Sum of (b), (c), (d) and (e) giving original signal (a).

Two-dimensional wavelet transforms 131
scale (w)
(a)
(b)
(c)
Figure 3.34. An image and its first two decomposition matrices. (a) Original array So. (b) Transform
array V(l) containing submatrices Sl, T 1 , T7 and Tf. (c) Transform array V(2) containing submatrices
S2, T 1 , T7, Tf, T 2 , T and Tf
100 100 100 100
50 50 50 50
0 0 0 0
0 50 100 0 50 100 0 50 100 0 50 100
(a:
100 100 100 100
50 50 50 50
0 0 0 0
0 50 100 0 50 100 0 50 100 0 50 100
(e:
100 100
50 50
0 0
0 50 100 0 50 100
(i) G)
Figure 3.35. Haar decomposition of a surface data set. (a) Original data set 128 x 128 array of surface
heights taken from a river bed. (b) Scale index m = 1 discrete detail D 1 . (c) Scale index m = 2 discrete
detail D 2 . (d) Scale index m = 3 discrete detail D3. (e) Scale index m = 4 discrete detail D4. (f) Scale
index m = 5 discrete detail Ds. (g) Scale index m = 6 discrete detail D6. (h) Scale index m = 7 discrete
detail D7. (i) Sum of first three discrete details D 1 + D 2 + D3 (= (b) + (c) + (d)). (j) Scale index
m = 3 discrete approximation X 3 = Xo - (D1 + D 2 + D 3 ) (= (a) - (b) - (c) - (d)). (Greyscale
used in all images: maximum = white, minimum = black.)

132 The discrete wavelet transform
Figures 3.34 and 3.35 show examples of much larger data sets, both using the
Haar wavelet transform. Figure 3.34 shows an image ('Lena') together with its first
two decomposition matrices, where the approximation and detail coefficient sub-
matrices can be clearly seen. Figure 3.35 shows an example of the Haar decomposition
of a more irregular array. The 128 x 128 array shown in figure 3.35(a) contains the
heights of a measured rough surface. The details of this array, at scale indices
m == 1-7, are shown in figures 3.35(b )-(h). Figure 3.35(i) contains the summation of
the first three details and figure 3.35(j) shows the resulting approximation at scale
index m == 3 when these details are subtracted from the original array. The blocky
nature of the Haar decomposition is noticeable from the plots. The two-dimensional
Haar wavelet is very simple in its form and, as with their one-dimensional counter-
parts, there are more complex forms of Daubechies wavelets and, of course, many
other wavelet families to choose from. Some examples of these wavelets are given
in figure 3.36. Using these wavelets will result in overlap of the wavelet at the array
edge and therefore would require the use of wraparound or other methods to deal
with the data edges.
Figure 3.36. Examples of two-dimensional orthonormal wavelets: Daubechies, Symmlets and Coiflets.
(a) D2 Haar wavelet. (b) D4. (c) D8. (d) D12. (e) D12, another perspective. (f) D20. (g) Symmlet S8.
(h) Coifiet C6. (Wavelets shown viewed at various angles and elevations.)

Adaptive transforms: wavelet packets 133
3.9 Adaptive transforms: wavelet packets
As we saw in chapter 2 (figure 2.29) the resolution of the wavelet transform is not
uniform in the time-frequency plane. The Heisenberg boxes expand in frequency
and contract in time as fluctuations at smaller and smaller scales are explored. The
short term Fourier transform (STFT), on the other hand, covers the time-frequency
plane with tiles of constant aspect ratio (figure 2.30). We also looked briefly at match-
ing pursuits which offer another way of extracting time-frequency information. In
this section we will consider another method which can adapt to the signal and
hence allows for more flexibility in the partitioning of the time-frequency plane:
the wavelet packet transform.
Wavelet packet (WP) transforms are a generalization of the discrete wavelet
transform. Wavelet packets involve particular linear combinations of wavelets and
the wavelet packet decomposition of a signal is performed in a manner similar to
the multiresolution algorithm given earlier for the discrete wavelet transform. The
difference is that, in the WP signal decomposition, both the approximation and
detailed coefficients are further decomposed at each level. This leads to the
decomposition tree structure depicted at the top of figure 3.37. Compare this with
the schematic of the wavelet decomposition given in figure 3.6. At each stage in the
scale index
m=O
original
time
/ series
X n
m=l
m=3
wavelet
packet
vectors
m=5
SSSS TSSSS etc
n, n,
111111
wavelet packet tiling of time- frequency plane at each scale
fteqE 1111
t.; time
Figure 3.37. Schematic diagram of wavelet packet decomposition.

134 The discrete wavelet transform
wavelet algorithm, the detailed coefficients are simply transferred down, unchanged,
to the next level. However, in the wavelet packet algorithm, all the coefficients at each
stage are further decomposed. In this way, we end up with an array of wavelet packet
coefficients with M levels each with N coefficients. A total of N coefficients from this
M x N array can then be selected to represent the signal. The standard wavelet trans-
form decomposition coefficients are contained within the WP array, shown by the
bold boxes in figure 3.37. A new nomenclature is employed in the figure to indicate
the operations that have been performed on each set of coefficients. S produces the
approximation components of the previous set of coefficients by lowpass filtering,
and T the detail components through highpass filtering. We simply add the letter S
or T to the left-hand end of the coefficient name to indicate the most recent filtering
procedure. SSTS n , for example, corresponds to the original signallowpass filtered,
then highpass filtered then passed twice through the lowpass filter. Notice also that
the subscript contains only the location index n. The scaling index m is omitted as
it is obviously equal to the number of letters Sand T in the coefficient name. As
with the original wavelet transform, the number of coefficients at each scale depends
upon the scale, with one coefficient in each coefficient group at the largest scale M and
N /2 coefficients at the smallest scale m == 1. Hence, the coefficient index spans
n == 0, 1, . . . , 2 M - m - 1.
At each stage in the decomposition, the wavelet packet algorithm partitions the
time-frequency plane into rectangles of constant aspect ratio. These become wider (in
time) and narrower (in frequency) as the decomposition proceeds. This is shown
schematically at the bottom of figure 3.37 for each scale. A variety of tilings of the
time-frequency plane is possible using the wavelet packet coefficients. For example,
we could keep all the coefficients at a level and discard all the others. This would tile
the plane in boxes of constant shape, just like one of those shown at the bottom of
figure 3.37. Other tilings are possible, some examples of these being shown in figure
3.38. The standard wavelet transform is just one of all the possible tiling patterns.
Figure 3.39 contains the wavelet packet coefficient selections corresponding to the
tiling patterns of figure 3.38.
wavelet transfonn tiling
high fiequency
small scales. - - - -_
frequency
low frequency
large scales -- - - - - -. --
time
(a)
(b)
(c)
Figure 3.38. Schematic diagrams of allowable wavelet packet tiling of the time-frequency plane. The
right-hand tiling is that used in the wavelet transform algorithm and corresponds to the components
contained in the bold boxes shown in the previous figure.

Adaptive transforms: wavelet packets 135
scale index
m=O
m=l
m=2
m=3
m=4
m=5
(a)
scale index
m=O
m=l
m=2
m=3
m=4
m=5
(b)
scale index
m=O
m=l
m=2
m=3
m=4
m=5
¥' original time series
¥' original time series
¥' original time series
(c)
Figure 3.39. The choice of wavelet packet components leading to the tHings shown in figure 3.38.

136 The discrete wavelet transform
The optimal or 'best' coefficient selection (hence tiling arrangement) is chosen to
represent the signal based on some predefined criterion. This criterion is normally
based on an information cost function which aims to retain as much information
in a few coefficients as possible. The most common measure of information used is
the Shannon entropy measure. This is defined for a discrete distribution Pi,
i == 0, 1, . . . , N - 1, as
S(p) == - 2:: Pi log(pJ
(3.102 )
where, for this case, Pi are the normalized energies (i.e. squared magnitudes) of the
wavelet packet coefficients under consideration. Low entropies occur when the
larger coefficient energies are concentrated at only a few discrete locations. The
minimum possible entropy of zero occurs when Pi == 1 for only one value of i, the
other probabilities being zero. In this case all the information needed to represent
the signal is condensed within a single coefficient. The maximum entropy occurs
when there is an equal distribution of coefficient energies. In this case Pi == 1/ Nand
the signal information is evenly spread throughout all the coefficients. We can see
that Pi acts as a discrete probability distribution of the energies. (More information
on the Shannon entropy measure together with an illustrative figure is given in
chapter 4, section 4.2.4.)
The set of N wavelet packet coefficients which contain the least entropy are
selected to represent the signal. That is, we want the signal information to be
concentrated within as few coefficients as possible. To find these coefficients the
WP array, such as the one we saw in figure 3.37, is inspected from the bottom
upwards. At each scale, each pair of partitioned coefficient sets (the 'children') are
compared with those from which they were derived (their 'parent'). If the combined
children's coefficients have a smaller entropy than those of their parent then they are
The entropy of this block,
together with the entropy
assigned to its pair is less
than its parent above hence
it is selected.
The entropy of this block, plus
the entropy assigned to its pair
have less entropy than their
parent above and hence is
selected. The combined
entropy of these children is
assigned to their parent in order
that entropy comparisons can
be made further up the tree.
These two children
blocks have more
entropy than the one
directly above them,
hence they are not
selected.
These two children blocks have more
entropy than their parent directly above
them, hence they are not selected.
Similarly, the next pair and the end pair
at this level are not selected for the same
reason.
The combined entropy of these
children is less than their parent
above and hence they are
selected. However, the value
of their combined entropy is
assigned to their parent in order
that entropy comparisons can
be made further up the tree.
Figure 3.40. Wavelet packet coefficient selection.

Adaptive transforms: wavelet packets 137
2
o
2 0
10
20
30
40
50
60
x.
I
r-//
-\ 11 r  JL
..... JLi1
________ --..fL lr----t-r-IL D- 
-
 J1J ...............   Lr-U -----I""" ----I"""  -----r--  ...............   
--0..- -- -- --..... -- -- --0..- --..... --  """'I-   .I'""" -- ---0...- r-- --
--0..- - -
r--
- -
- -
-  """"1....- r---- -
- -
-----.r  - --IL
1
2
3
4
5
6
WP
WT
(a)
m=2
m=5
--
m=3
m=6
(b)
Figure 3.41. Wavelet packet decomposition of a simple signal. (a) Signal (top) with wavelet packet
decomposition (below). The coefficient values are plotted as heights. The scale indices, m = 1 to 6, are
given down the left-hand side of the plot. Trace WP contains the best selection of wavelet packets and
trace WT contains the wavelet transform decomposition for comparison. (b) The time-frequency tiling
associated with each wavelet packet decomposition in (a). Larger coefficient values are plotted darker.
Original Signal
m=1
m=4

138 The discrete wavelet transform
retained. If not, the parent's coefficients are retained. When the children are selected
their entropy value is assigned to their parent in order that subsequent entropy
comparisons can be made further up the tree. This is shown schematically in figure
3.40. Once the whole WP array has been inspected in this way, we get an optimal
tiling of the time-frequency plane (with respect to the localization of coefficient
energies). This tiling provides the best basis for the signal decomposition.
(c)
(d)
(e)
Figure 3.41 (continued). (c) The time-frequency tiling associated with the best wavelet packet
decomposition (left) and wavelet decomposition (right). (d) The 16 largest coefficients from (c): wave-
let packet decomposition (left) and wavelet decomposition (right). (e) The reconstruction of the
signal using only the 16 largest coefficients given in (d): wavelet packet (left) and wavelet (right).

Adaptive transforms: wavelet packets 139
Figure 3.41 illustrates the wavelet packet method on a simple discrete signal
composed of a sampled sinusoid plus a spike. The signal is 64 data points in
length. A Haar wavelet is used to decompose the signal. Figure 3.41(a) shows the
wavelet packet coefficients below the original signal for each stage in the WP algo-
rithm. The coefficients are displayed as histograms. The bottom two traces contain
the coefficients corresponding to the best wavelet packet basis and the 'traditional'
discrete wavelet basis respectively. The WP tiling of the coefficient energies in the
time-frequency plane for each scale is given in figure 3.41(b). The larger coefficient
x. 
I
 .

...-.. ..--.--....-....-..--....--.. ".1

J
1 J- /
2VJ
3"J

1

r--"L-r
  L......
4
....................
J"'1-........---- --,
 
"-
5 - ----
r-
-
""L- -
6 --
--
WP --

WT-
....
(a)
-

Figure 3.42. Wavelet packet decomposition of a simple signal using a Daubechies D20 wavelet. ( a) The
wavelet packet decomposition of the signal shown at the top of figure 3.31(a). (b) The time-frequency
tiling associated with the best wavelet packet decomposition (left) and wavelet decomposition (right).

140 The discrete wavelet transform
energies are shaded darker in the plot. In figure 3.41 ( c) the optimal WP tiling is
compared with the wavelet transform tiling of the time-frequency plane. The plots
in figure 3.41 (d) outline the 16 largest coefficients in both time-frequency planes of
figure 3.41(c). These are used to reconstruct the signals shown in figure 3.41(e). The
16 largest wavelet packet coefficients contain 98.6% of the signal energy, whereas
the 16 wavelet transform coefficients contain 96.5% of the signal energy. The wavelet
packet reconstruction using the selected coefficients is visibly smoother than the
reconstruction using the traditional wavelet transform coefficients. Figure 3.42
contains the same signal as figure 3.41. However, this time the WP decomposition
is performed using a Daubechies D20 wavelet (refer back to figure 3.15). Using this
wavelet results in a different tiling of the time-frequency plane for the WP method
(compare the left-hand plots of figures 3.42(b) and 3.41(c)). Again, the 16 largest
coefficients are used in the signal reconstruction. The oscillatory parts of both recon-
structions shown in figure 3.42( d) are visibly smoother than their Haar counterparts
in the previous figure (figure 3.41(e)). We expect this as the D20 is more smoothly
oscillatory than the Haar. Note, however, comparing figures 3.42(c) and 3.41(d),
we see that the signal spike leads to a single high frequency tile for both Haar
decompositions but respectively five and four high frequency tiles for the D20
wavelet. The more compact support of the Haar wavelet has allowed for a better
localization of the signal spike, but it does makes it less able than the D20 to represent
the smooth oscillations in the signal. The energies of the reconstructed signals for the
D20 decompositions using only the largest 16 coefficients are 99.8% (WP) and 99.7%
(WT), indicating the data compression possibilities of the techniques.
I
I
I
I
I
(c)
(d)
Figure 3.42 (continued). The 16 largest coefficients from figure 3.31(c): wavelet packet decomposition
(left) and wavelet decomposition (right). (d) The reconstruction of the signal using only the 16 largest
coefficients given in figure 3.31(d): wavelet packet (left) and wavelet (right).

Endnotes 141
3.10 Endnotes
3.10.1 Chapter keywords and phrases
(Y ou may find it helpful to jot down your understanding of each of them.)
discrete wavelet transform
wavelet / detail coefficients
wavelet frames
tight frame
orthonormal basis
dyadic grid
scaling function
approximation coefficients
inverse discrete wavelet transform
discrete approximation
continuous approximation
signal detail
multiresolution
scaling equation
scaling coefficients
compact support
fast wavelet transform
decomposition algorithm
reconstruction algorithm
wraparound
scale index
sequential indexing
level indexing
hard thresholding
soft thresholding
scale thresholding
Daubechies wavelets
subsampled
upsampled
SymmIe ts
Coiflets
translation invariance
redundant / stationary/translation
in varian t / non -de cima t e d / max imal
overlap/discrete wavelet transform
biorthogonal wavelets
wavelet packet transforms
Shannon entropy measure
3.10.2 Further resources
Papers describing the discrete wavelet transform at an introductory level include
those by Kim and Aggarwal (2000), Depczynski et al (1997), Asamoah (1999) and
Graps (1995). The paper by Williams and Armatunga (1994) contains a good
explanation of the derivation of the Daubechies D4 scaling coefficients and multi-
resolution analysis. They present a clear account of the wavelet filtering of a signal
using matrices. Other useful introductory papers are those by Sarkar et al (1998),
Meneveau (1991a), Jawerth and Sweldens (1994), Newland (1994a-c), Strang
(1989, 1993) and the original paper on multiresolution analysis is by Mallat (1989).
In his book, Newland (1993a) gives a more detailed account of the conditions that
must be satisfied for discrete orthonormal wavelets. Diou et al (1999) and Toubin
et al (1999) provide some useful information on projecting the details and approxi-
mations determined at one scale at another scale. Chui (1997) provides a little
more mathematical detail in a readable account of discrete wavelet transforms and
their role in signal analysis. Strang and Nguyens' (1996) text concentrates on the
connection between wavelets and filter banks used in digital signal processing.
Daubechies' (1992) book provides a good grounding in all aspects of wavelet trans-
forms, containing among other things useful text on wavelet frames and wavelet
symmetry. It also contains more information on the construction of Symmlets,
Coiflets and biorthogonal wavelets. This chapter has concentrated on two compact,

142 The discrete wavelet transform
discrete orthonormal wavelets: the Haar wavelet and the Daubechies D4 wavelet.
There are many other wavelets we have not considered, most notably the Shannon,
Meyer, and Battle-Lemarie wavelets. More information on these can be found in,
for example, Chui (1997), Mallat (1998), Daubechies (1992) and Blatter (1998).
More examples of the multiresolution analysis of simple discrete signals can be
found in the book by Walker (1999). Books providing a more mathematical account
of discrete wavelet transforms are those by Hernandez and Weiss (1996), Benedetto
and Frazier (1994), Chui (1992a,b), Walter (1994) and Percival and Walden (2000).
There is also a lot of useful information on the web. The appendix contains a list
of useful websites from which to begin a search.
The book by Starck et al. (1998) contains a lot of good introductory material on
two-dimensional discrete wavelet transforms, covering many of their practical appli-
cations including remote sensing, image compression, multi scale vision models,
object detection and multi scale clustering. A nice illustrative paper on the application
of two-dimensional wavelet transforms is that by Jiang et al (1999), who investigate
the three-dimensional surface of orthopaedic joint prostheses. We considered only
square two-dimensional transforms in this chapter, where the horizontal and vertical
scalings were kept the same. If they are allowed to vary independently, we get
rectangular transforms. Also possible are two-dimensional transforms which are
not simply tensor products but wavelets constructed 'intrinsically' for higher
dimensions (Jawerth and Sweldens, 1994). There are many applications of wavelet
packets cited in the rest of this book. Quinquis (1998) provides a nice introduction
to wavelet packets, while the paper by Hess-Nielsen and Wickerhauser (1996) and
Wickerhauser's (1994) book provide a more in-depth account.
We mentioned briefly the redundant wavelet transform, a variant of the discrete
wavelet transform which produces N coefficients at each level and is translation
invariant. This has been found useful in statistical applications: see for example
Coifman and Donoho (1995), Lang et al (1996) and Nason and Silverman (1995).
A concise summary of the various wavelet thresholding methods developed over
recent years is to be found in the paper by Abramovich et al (2000) together with a
comprehensive list of references on the subject. In addition, the book by Ogden
(1997) provides a more detailed overview together with numerous examples. The
universal threshold method is detailed in the paper by Donoho and Johnstone
(1994) together with another global method, the minimax thresholding method.
Donoho and Johnstone (1995) also developed a scheme which uses the wavelet co-
efficients at each scale to choose a scale-dependent threshold. This method is
known as the SURE or SureShrink method after 'Stein's Unbiased Risk Estimate'
on which it is based. They suggest a HYBRID method to be used in practice for
decompositions where, at some scales, the wavelet representation is sparse, i.e. a
large number of the coefficients are zero (or very near zero). The HYBRID method
uses the SURE method, but defaults to the universal threshold for scales where the
data representation is sparse due to the noise overwhelming the little signal con-
tributed from the nonzero coefficients. A number of other methods have been
proposed: including those based on cross validation, e.g. Nason (1996), Jansen and
Bultheel (1999); Bayesian approaches, e.g. Chipman et al (1997), Vidakovich
(1998), Abramovich et al (1998); and the Lorentz curve (Katul and Vidakovich,

Endnotes 143
1996, 1998). Translation invariant denoising is considered by Coifman and Donoho
(1995) who compute the estimated signal using all possible discrete shifts of the signal.
See also the papers by Donoho and Johnstone (1995, 1998), Donoho (1995), John-
stone and Silverman (1997), Hall and Patil (1996), Efromovich (1999), Downie and
Silverman (1998), Moulin (1994), Krim and Schick (1999) and Krim et al (1999).
We will come across the use ofthresholding extensively in the subsequent application
chapters of this book. In particular, thresholding is revisited in chapter 4, section
4.2.3, where another thresholding method, the Lorentz threshold, is explained.
Often we want to compress the signal into as few detail coefficients as possible
without losing too much information for data compression applications such as
speech and audio transmission. We can do this by finding a wavelet which decom-
poses the signal into a few large amplitude detail coefficients, which we retain, and
many coefficients close to zero, which we discard. How many coefficients we set to
zero affects the subsequent quality of the reconstruction. In practice for larger data
sets and more suitable wavelets we can get good compression ratios without the
loss of significant detail, where the term 'significant' is determined by the user.
Here we have not considered compression schemes, including quantization and
encoding, but rather the reader is referred in the first instance to the simplified
explanation of Walker (1999) and the more detailed treatments in Chui (1997) and
Mallat (1998).
Note that all examples in this chapter have used wraparound to deal with the
signal edges, i.e. the part of the wavelet spilling over the end of the signal is placed
back at the start. This is the simplest and one of the most common treatments of
edge effects for a finite length signal, and it results in exactly the same number of
decomposition coefficients as original signal components. However, it is not always
the best for the application. Other methods were shown in chapter 2, figure 2.35.
In addition, there are families of boundary wavelets which are intrinsically defined
on finite length signals (see for example the discussions in Jawerth and Sweldens
(1994) and Chui (1997)). Take particular care when using off-the-shelf wavelet
software packages as they may employ other boundary methods as the default setting,
for example zero padding, which results in slightly more than N coefficients resulting
from the full decomposition. In addition, we have employed one version of the
scaling coefficient reordering b k (equation (3.25)). Again this is very much software
dependent and you will find alternative ordering employed by various software
packages.

Chapter 4
Fluids
4.1 Introduction
A fluid is a non-rigid interconnected mass which may in general exhibit either laminar
or turbulent flow. Laminar flows are characteristic of slow-moving or highly viscous
flows where the fluid particles move in an ordered fashion, sliding over themselves in
sheets (or laminae, hence 'laminar'). Turbulent flows, on the other hand, are charac-
teristic of fast-moving or low-viscosity flows, where small disturbances in the flow
quickly blow up causing the fluid particles to move in an unpredictable fashion,
mixing themselves up from one point in the flow to the next. Almost all real flows
of interest to scientists and engineers are turbulent: the flow of water in rivers and
pipelines; the flow within hydraulic machinery, e.g. turbines and pumps; atmospheric
wind flows and ocean currents; and the flow or air around buildings, moving vehicles
and aircraft. Turbulence manifests itself as a multi scale cascading phenomenon,
where fluctuations (eddies) over a large range of scales are superimposed on a
mean flow, e.g. the buffeting experienced on a windy day. Over the past decade, the
wavelet transform has emerged as a particularly powerful tool for the elucidation
of fluid signals (e.g. velocities, pressures and temperatures), both temporal and
spatial, covering a variety of pertinent problems from the Kolmogorov scaling of
high-Reynolds-number homogeneous turbulence to the nature of vortex shedding
downstream of bluff bodies. Many researchers have made use of the wavelet trans-
form's ability to probe simultaneously both the spectral and temporal (or spatial)
structure of turbulent fluid flows. This chapter begins with a basic outline of the
wavelet-based statistical methods used extensively in the analysis of fluid flows. The
chapter is then split into two main parts: the first details the wavelet analysis of
jets, wakes and coherent structures in engineering flows and the second considers
geophysical flows.
The choice of the most appropriate wavelet to use in the analysis of fluid data
depends very much on the nature of the data itself. As we shall see in the forthcoming
sections, in general, discrete dyadic grid orthonormal wavelets are used for statistical
measures of turbulence, where ensemble averaging is usually a necessity due to the
translation invariance of this type of transform. Often the Haar wavelet is employed
due to its relationship with velocity structure functions used in traditional turbulence
Copyright @ 2002 lOP Publishing Ltd.

analysis. In addition, it is compact in the time domain, its short support eliminating
edge effects. Due to their high resolution in the wavelet domain, continuous wavelets
are normally employed for feature detection in flows with recognizable coherent
structures. The Mexican hat wavelet is often used when compactness in the time
domain is important or when modulus maxima-based analysis is undertaken.
Complex varieties of wavelet (very often the Morlet wavelet) are used when phase
information is important, for example in vortex shedding flows. As we saw in chapter
2, the standard Morlet wavelet, with its wavepacket structure, has better compactness
in the frequency domain than the single-humped Mexican hat.
4.2 Statistical measures
Traditionally, turbulent statistical measures are often calculated in Fourier space.
However, important temporal information is lost owing to the non-local nature of
the Fourier modes. As a result wavelets have been utilized to quantify the temporal
and spectral distribution of the energy in new statistical terms, such as wavelet
variance, skewness, flatness, etc. These statistical measures are generally computed
for discrete orthonormal wavelet expansions, which some authors believe are prefer-
able because orthogonality both reduces the number of wavelet coefficients and
suppresses undesired relationships between them. Wavelet-based statistics enable
both scale- and location-dependent behaviour to be quantified. In this section we
will look briefly at some basic wavelet-based statistics developed for the analysis of
turbulent flow signals. We will consider mainly the manipulation of discrete trans-
form coefficients Tm n generated from full decompositions using real-valued, discrete
,
orthonormal wavelet transforms of the type we covered in chapter 3. In addition, we
will assume that the mean has been removed from the signal and that it contains
N (==2M) data points.
4.2.1 Moments, energy and power spectra
The pth order statistical moment of the wavelet coefficients T m n at scale index m is
,
defined as
2 M - m -1
2: (Tm,n)P
/TP ) - n=O
\ m,n m - 2 M - m
(4.1 )
where only the coefficients at scale m are used in the summation. The brackets (- . .)m
denote the average taken over the number of coefficients at scale m, hence the
2 M -m term in the denominator. For example, the commonly used coefficient variance
at scale index m is
(T,n)m ==
2 M - m -1
2: (Tm,n)2
n=O
2 M - m
(4.2)
Copyright @ 2002 lOP Publishing Ltd.

The wavelet coefficient variance is simply the average energy wrapped up per
coefficient at each scale m. Do not confuse equation (4.2) with Pm' the mean energy
per unit time in the signal at scale m (given below in equation (4.7)), also known as
the scale-dependent power.
A general dimensionless moment function can be defined as
FP == (T,n)m
m (( T,n)m)P/2
(4.3)
where the pth order moment is normalized by dividing it by the rescaled variance. For
example, the scale-dependent coefficient skewness factor is defined as the normalized
third moment:
F 3 = (T,n)m
m (( T,n)m)3/2
and, similarly, the scale-dependent coefficient flatness factor is defined as
r = (T,n)m
m (( T,n)m)2
The flatness factor gives a measure of the peakedness (or flatness) of the probability
distribution of the coefficients at each level. It is well known that for Gaussian
distributions the flatness factor is equal to 3.
Values higher than 3 occur for distributions with more pronounced tails. The
flatness factor increases as the flow signal becomes more intermittent (e.g. Meneveau,
1991 a; Mouri et aI, 1999).
We now consider the wavelet-based scale-dependent energy defined as
(4.4)
(4.5)
2 M - m _ 1
Em == 2:: (Tm,n)2 t
n=O
(4.6)
Notice that this equation is slightly different in form from that given in chapter 3,
equation (3.52) (where an integer time step was assumed), as the sampling time,
t, has now been added. The scale-dependent energy per unit time, or scale-
dependent power, is Pm == Em/T where T is the total time period of the signal.
Hence, as T == 2 M t, it can be written as
2 M - m _ 1
2: (Tm,n)2
n=O
2 M
(4.7)
P ==
m
Therefore, as long as a signal has zero mean, both the total energy and total power of
the signal can be found by summing Em and Pm respectively over all scale indices m.
We can construct a wavelet power spectrum for direct comparison with the
Fourier spectrum as follows:
1 2 m t2M-m-l 2 12m t
PW(fm) = T Inl  (T m ,n)!1t = T Inl Em
(4.8)
Copyright @ 2002 lOP Publishing Ltd.

The term (2 m t) / In 2 stems from the dyadic spacing of the grid. The temporal scale
of the wavelet at scale index m is equal to 2 m t. Often the Haar wavelet is used in
turbulence studies and this temporal scale is taken as its representative period,
hence the associated frequency is fm == 1/ (2 m t). (However, note that we can
easily modify this expression to take into account a characteristic frequency of the
mother wavelet, such as the spectral peak, h, or bandpass centre frequency, fc. For
example, by employing fe, the scale-dependent frequency becomes fm == fc/ (2 m t)
and hencefc would appear in the denominator of equation (4.8).) A Taylor expansion
of fm == 1/ (2 m t) gives the discrete incremental change in frequency associated with
the discrete change in scale index m:
D.. r = r ' D.. r" (D..m)2 r/ll (D..m) 3 r"" (D..m)4 . . .
Ym Jm m+Jm 2! +Jm 3! +Jm 4! +
(4.9a)
Truncating at the first term with fm == 1/ (2 m t) and remembering also that the
scale index is an integer (i.e. m == 1) we get fm == -ln2/(2m t) == -In(2) xfm,
where the negative sign is ignored in practice. It is common in the literature for the
Taylor expansion to be truncated at the first term in this way-even though it has
a simple limit, i.e.
 I' == 1 [ -I ( 2 ) In(2)2 _ In(2)3 In(2)4 . . . J
Ym 2m t n + 2 6 + 24 +
1 1
2 2 m t
_ fm (4.9b)
2
Again, ignoring the negative sign we see from the previous expression that the
incremental discrete change in the frequency is equal to half the frequency itself. In
fact, this is obvious without employing a Taylor expansion, i.e.
1
I' - I' _ I' -
Ym-Jm+l Jm-2m+lt
1
2 m t
1 1
2 2 m t
(4.9c)
that is, the incremental discrete change in the frequency is equal to half the frequency
itself. This is for the forward difference fm+ 1 - fm. For the backward difference,
fm - fm-l, it is equal to fm (try for yourself). So now we have three options for
fm (fm/ 2 , In(2)fm and fm) which are all equally valid. In the rest of this text we
will use fm == In(2)fm as it is the most commonly used form in practice (e.g. Katul
et aI, 1994; Katul and Parlange, 1995; Kulkarni et aI, 1999). The addition of the
(2 m t) /In(2) term in equation (4.8) is therefore required for the discrete summation
over the frequency range to equal the total power in the signal. The total length of
signal T is present in the denominator of equation (4.8) as the area under the
power spectrum represents the average energy per unit time. As T == 2 M t we can
rewrite the power spectrum as
2 M - m -1
PWUm) = In(2 -m  (Tm,n)2
(4.10a)
or simply in terms of the wavelet coefficient variance as
t 2
PwUm) = In(2) (Tm,n)m
( 4.1 Ob )
Copyright @ 2002 lOP Publishing Ltd.

or in terms of the scale-dependent signal power as
1
PwUm) = In(2)fm Pm
( 4. 1 Oc )
Compare the wavelet power spectrum for the dyadic grid orthonormal transform
given by equations (4.8) and (4.10a)-( 4.10c) with the definition of the power
spectrum for the continuous wavelet given in chapter 2, section 2.9, as
1 .f T 2
PwU) = :t: c T(f, b) db
T egO
( 4.11 )
where C g is the admissibility constant for the particular wavelet used, fc is a
characteristic frequency of the mother wavelet defined at scale a == 1, the frequency
f is equal tofc/a and the derivative df == -da/a 2 .
Figure 4.1 contains a vortex shedding signal taken downstream of a cylinder in
an open channel flow together with associated wavelet and Fourier-based power
spectra. Both continuous Mexican hat and discrete Daubechies D4 wavelets have
(a)
0.17
-..

 0

-0.17
o
10
20
30
40
t (see)
50
60
70
80
(b)
10
1
-..
00
....
..... 0.1
s:::
::s
 0.01
.b
.....
 0.001

I-;
Q)
 1.10-4
0..
1.10- 5
1.10- 6
0.01
0.1
1
o
- Fourier
-0 continuous
-+- discrete
frequency (Hz)
Figure 4.1. Power spectra of a vortex shedding velocity signal. (a) Velocity signal taken downstream of
a cylinder in open channel flow. (b) Fourier and wavelet (Daubechies D4 and Mexican hat) power
spectra of the signal in (a). After Addison et al 2001 ASCE Journal of Engineering Mechanics 127
58-70. (Q) ASCE 2001, reproduced with permission of the publisher.
Copyright @ 2002 lOP Publishing Ltd.

been used to construct the wavelet spectra. Note the distinct peak in the Fourier
spectrum at the vortex shedding frequency of 0.133 Hz. The continuous Mexican
hat spectrum also peaks at this value, although smearing of the spectrum around
the maximum is evident. The Mexican hat spectrum was produced using a fine
resolution, translation-invariant discretization of the temporal location parameter,
b, and a fractional power-of-two scale for the wavelet scale, a. Close inspection of
the plot corresponding to the discrete orthonormal Daubechies wavelet transform,
however, reveals the coarser resolution due to the dyadic grid structure, i.e. integer
power-of-two translations and dilations. Other examples of wavelet power spectra
occur later in this chapter: see for example figures 4.2, 4.14 and 4.33. Qiu et al
(1995a) provide some words of caution in the application and interpretation of
wavelet-based power spectra resulting from dyadic orthonormal wavelet transforms.
They found that orthonormal wavelet analysis can lead to false spectral slopes
and suggested that wavelets of similar shape to the coherent structures are used
and that non-orthonormal wavelet analysis is used when high resolution is important.
They illustrated the problem by employing the Haar, Daubechies D12 and Lemarie-
Meyer-Battle (LMB) wavelets to construct energy spectra from atmospheric
turbulence measurements.
One commonly used statistical measure of the energy distribution across scales is
the normalized variance of the wavelet energy. This is called the fluctuation intensity
(FI) and is defined as
FI = [( T,n)m - (( T';',n)m)2] 1/2
m (T,n)m
(4.12 )
which measures the standard deviation of the variance in coefficient energies at scale
index m. It is also sometimes referred to as the coefficient of variation (CV) (e.g. Katul
et aI, 1994; Kulkarni et aI, 1999). It follows from equation (4.12) that skewness and
flatness measures may also be found for the scale-dependent energies (e.g. Yee et aI,
1996). The fluctuation intensity provides a measure of the variability of the signal
energy at scale index m. Another way to present this variability is by constructing
the dual spectrum, which combines the power spectrum, Pw(fm), and the fluctuation
intensity converted to suitable units:
2 m tEm
DwUm) = In(2)T [I + FIm] = PwUm)[I + FIm]
(4.13 )
The dual spectrum is a plot of both Pw(fm) and Dw(fm) which provides information
concerning both the contribution to the energy at various scales and its associated
spatial variability (i.e. its variance-see Meneveau, 1991a,b). An example of a dual
spectrum is given in figure 4.2. It is derived from the streamwise velocity signal
taken downstream of a cylinder in a wind tunnel experiment using LMB wavelets.
The slight increase in the relative variance of the local energies at the larger wave-
numbers is due to the increasingly intermittent nature of the kinetic energy
distribution here. This is confirmed by the flatness factor plot for the same signal
given in figure 4.3. Note that the wavenumber is used in the plots, where the
Copyright @ 2002 lOP Publishing Ltd.

10 8
10 6
10 4

"-"

 10 2

o

10°
10- 2
10-4
2 -10- 3
....... .6
.... "'G1
......
.... 6.
....0.....
,
,
...... .6
...
'q
\
\
\
\
\
\
\
\
\ 4
\
\
19
10- 2
10- 1
kr1
wavenumber
10°
Figure 4.2. Dual spectrum of the turbulent flow behind a cylinder (streamwise one-dimensional spec-
trum). Solid line-usual Fourier spectrum. Circles-wavelet spectrum. Triangles-wavelet mean
energy at every scale to which one standard deviation (computed from the spatial fluctuations)
has been added. After Meneveau 1991 b J Fluid Mech 232 469-520. With kind permission from
Cambridge University Press.


I-;
o
1:) 2-10 1

00
00
(])


2 -10°
2 -10- 3
o
o
o
a
o
8
8
o
o
o
a
o
o
13
f3
10- 2
10- 1
kr1
wavenumber
10°
Figure 4.3. Flatness factors of the wavelet coefficient computed from laboratory data. The circles are
for the boundary layer flow and the squares are for the turbulent wake (cylinder). After Meneveau
1991b J Fluid Mech 232 469-520. With kind permission from Cambridge University Press.
Copyright @ 2002 lOP Publishing Ltd.

wavenumber is the spatial frequency, k, of the flow structures which has a similar
form to the temporal angular frequency w. That is, the reciprocal of the length
scale, y, multiplied by 27r:
k == 27r
m
y
(4.14 )
The denominator y represents the physical distance between structures. We can esti-
mate y from velocity signals for flows which have relatively low turbulence intensities
compared with their mean advective velocities using Taylor's frozen flow hypothesis.
This states that, if the turbulent field is changing slowly enough with respect to the
mean velocity, then measuring data at a point as the turbulent field advects past is
equivalent to taking a linear section through the field. Thus characteristic time
periods, p, in the velocity signal can be converted to spatial separations y = up,
where u is the mean advective velocity. If, as before, we set the temporal scale
2 m t == P (== l/fm) then y == u2 m t == 2 m y, where y is the spatial increment set
by the mean velocity and the sampling time. In addition, the scale index frequency
is related to the wavenumber through the mean velocity and the 27r factor as
k m == (27r/up) == (27r/u)fm. The power spectrum in terms of wavenumber k m is then
2 M - m -1
Pw(k m ) = [ 27rln2f;M-m ] [ (Tm,n)2]
( 4.15a)
As y == u t, this can be rewritten as
y 1
Pw(k m ) = 27r In(2)2-m Pm
( 4.15b )
where Pm == Em/Tis the scale-dependent energy per unit time, or scale-dependent
power, given above in equation (4.7). It can also be written (most commonly) in
terms of the coefficient variation as
Pw(k m )
y 2
27r In(2) (Tm,n)m
( 4.16)
Remember that (T,n)m is the sum of the coefficient energies at scale m normalized
by 2 M - m, i.e. the energy per coefficient, hence the disappearance of the 2 - m
factor in the denominator going from equation (4.15b) to (4.16), as Pm is the sum
of the coefficient energies at scale m normalized (in a global sense) by 2 M .
The wavelet co-spectrum or cross-spectrum between two variables u and v is
defined as
[ u t ]
Pw(k m ) = 27r In(2)
2 M - m -1
2:: (Tm,n)u(Tm,n)v
n=O
2 M - m
( 4.17)
where (T m,n)u denotes the wavelet coefficients for variable u. (If u v then equation
(4.17) reduces to the standard power spectral density function defined by equation
(4.15a).) Katul et al (1998b) have used wavelet co-spectra to analyse the relationship
Copyright @ 2002 lOP Publishing Ltd.

between turbulent velocity and skin temperature perturbations above a grass-covered
forest clearing.
4.2.2 Intermittency and correlation
The intermittency at each scale can be viewed directly using the intermittency
index proposed by Farge (1992). This allows the investigator to visualize the
uneven distribution of energy through time at a given wavelet scale. The intermittency
index, 1m n, is defined as
,
I = (Tm,n?
m,n /T 2 )
\ m,n m
( 4.18)
1m n is the ratio of local energy to the mean energy at temporal scale 2 m t. For
,
example, a constant value of 1m n = 1 for all m and n means that there is no flow
,
intermittency at all, whereas a value of 10 at a specific set of indices m and n means
that, at that location in the signal, there is ten times more energy contained within
the coefficient at that location than for the temporal mean at that scale. Figure 4.4
contains a plot of the intermittency indices for the vortex shedding signal shown in
figure 4.1(a). High magnitude values of Imn can be observed intermittently at the
,
lower scales.
The correlation between the scales can be measured using the pth moment scale
correlation Rfn defined as
2 M - (m - 1) _ 1
R p - 2 M - m  B p B p
m - [ !!:. ] m-ln
m, 2 '
( 4.19)
n=O
where Bfn,n is the pth order moment function (defined below) and [] requires that
the integer part only be used. In order to pair all the coefficients at the smaller
scale (index m - 1) with those at the larger scale (index m), the sum is taken over
the number of coefficients at the smaller scale, e.g. from n == 0 to 2 M - (m - 1) - 1.
400
s:::
ef

100
00
10
20
30
40
time (t)
50
60
70
80
Figure 4.4. Intermittency indices according to wavelet scale for the flow downstream of a cylinder in an
open channel. The traces from bottom to top correspond to scale indices m = 1, 2, 3, 4 and 5. The
index plots have been displaced by 100 units to aid viewing. After Addison et al2001 ASCE Journal
of Engineering Mechanics 127 58-70. (Q) ASCE 2001, reproduced with permission of the publisher.
Copyright @ 2002 lOP Publishing Ltd.

(For more information see Yamada and Ohkitani (1991) and Yee et al (1996).) Bfn n is
,
the pth order moment function defined as
BP == (Tm,n)P
m,n 2M -m-l
2: (Tm,n)P
n=O
(4.20 )
Note that this has a similar form to the intermittency index when p = 2, except that
Bfn n has a normalized sum at each scale, i.e.
,
 Bfn,n == 1 (4.21)
n
whereas the sum of the intermittency indices at scale m is equal to the number of
coefficients at that scale, i.e.
'"'" I == 2 M - m
 m,n
n
(4.22)
Figure 4.5(c) shows the scale correlation that exists within the turbulent velocity
signal shown in figure 4.5(a) taken from within the atmospheric boundary layer by
Yamada and Ohkitani (1991). At large scales there is no obvious correlation.
However, at small scales there is a general increase in all the pth moment scale
correlations. This is in contrast to the pth moment correlations shown in figure
4.5( d) for the phase-randomized signal of figure 4.5(b). Phase randomization of a
signal is performed by randomizing the phases of the Fourier components of the
signal then taking the inverse Fourier transform. It destroys the correlations between
levels in the signal and provides a useful benchmark signal for detecting such correla-
tions. Moriyama et al (1998) also examined scale correlation in a study of the density
fluctuations in granular flows through pipes at various flow rates. They found that
low density flows exhibited a Gaussian distribution for the wavelet coefficients at
all scales. However, for higher density flows the distribution became noticeably
non-Gaussian. Figure 4.6 shows examples of the two probability density functions
(PDFs). The authors found no correlation between scales for the low density flows
(figures 4.6(a) and 4.7(a)) and concluded that the time series signal from these
flows are equivalent to a random signal. Extended tails were, however, found for
the probability distributions corresponding to the high density flows (figure 4.6(b))
which contain significant correlation across scales (figure 4. 7(b )).
4.2.3 Wavelet thresholding
Thresholding techniques are used extensively in the analysis of fluid flows to partition
the signal into a coherent and 'more random' turbulent part, sometimes referred to as
the strong and weak signal components respectively. We saw a variety ofthresholding
methods in chapter 3 (refer back to chapter 3, section 3.4.2) and many of these have
been employed in the analysis of turbulent fluid signals. Examples of wavelet thresh-
olding used in the analysis of both spatial and temporal fluid signals can be found in
Hagelberg and Gamage (1994), Turner and Leclerc (1994), Higuchi et al (1994), Farge
et al (1996) (see figure 4.31), Katul and Vidakovich (1996, 1998), Briggs and Levine
Copyright @ 2002 lOP Publishing Ltd.

15 15
10
::s
5 5
1 k 2 3 1 k 2 3
-10 4 -10 4
(a) (b)
8 8
6 6
-.. 
 
-- 4 -- 4
t t
N N
OQ OQ
0 2 0 2
...-4 ...-4
0 0
-2  -2
I
0 5 0 5 10
j j
large
scales
small
scales
(c)
(d)
Figure 4.5. The pth moment correlations between scales m and m - 1. ( a) Original signal. (b) Phase
randomized signal. (c) pth moment correlations of original signal (a). p = 1 (squares); p = 2 (circles);
p = 3 (triangles); p = 4 (diamonds). (d) pth moment correlations of phase randomized signal (b).
Note that authors use levels j instead of scale m where j = M - m and the nomenclature C(p) (j)
instead of Rfn, i.e. large levels j correspond to small scales. After Yamada and Ohkitani (1991).
Reproduced with the kind permission of the Physical Society of Japan and the authors.
(1997), Hagelberg et al (1998), Katul et al (1998a), Szilagyi et al (1999) and Kailas and
Narasimha (1999). Figure 4.8 shows an example of the partitioning of the vortex
shedding signal of figure 4.1. The partitioning is performed using both hard thresh-
olding and scale-dependent thresholding. The original coefficients from a Daubechies
D4 decomposition are shown in sequential format in figure 4.8(a). The signal was
8192 data points in length. Figure 4. 8(b ) (top plot) shows the small scale coefficients
set to zero where the threshold scale index was set to m* == 6. Subsequent wavelet
reconstruction using the remaining coefficients gives the strong signal. The weak
signal is reconstructed from the coefficients below m*. The bottom plot of figure
4.8(b) contains the original signal plotted together with the weak and strong parts
of the signal. Figure 4.8(c) plots both the hard thresholded coefficients and
corresponding signal partitions obtained using Donoho and Johnstone's universal
threshold, with a- derived from the median absolute deviation of the wavelet
Copyright @ 2002 lOP Publishing Ltd.

- Gaussian -- Gaussian
a = 15 a = 15
0 = 14 II = 14
.. = 13 10-0 0 = 13
. = 12 I( = 12
. = 11 . = 11
10- 1 a = 10 " = 10
. =9 . =9
 .. =8  10- 2 .. =8
...... ......
...... ......
...... ......
 
..D ..D
0 10- 3 0
I-< I-;
Pot Pot 10-4
'JO
(I).
10- 5 10- 6
-6 --4 -2 0 2 4 6 -10 -5 0 5 10
normalized aO k normalized aO k
J, J,
(a) (b)
Figure 4.6. PDFs of wavelet coefficients for sparse uniform flows. (a) Sparse uniform flow. (b) Dense
uniform flow. The abscissa stands for the wavelet coefficient which is normalized to have unit
variance, the ordinate for the probability of finding the coefficient (j = level index). After Moriyama
et al (1998). Reproduced with the kind permission of the Physical Society of Japan and the authors.
coefficients at the smallest scale. Comparing figures 4.8(b) and (c), we can see that
scale thresholding smoothes the signal in the strong part and leaves remnants from
the vortex shedding process in the weak part. On the other hand, hard thresholding
retains much of the high frequency components of large amplitude in the strong
10- 2 . . 10- 2 I .
g q = 1 oq=1
.q=2 /lq=2
nq=3 uq=3
10- 1 .q=4 10- 1 "q=4
-q=5 .q=5 .
. . . .
5 5 ,. K "
. . . (] (]
0-' . 0-' K K . It (] (] IJ
It "
0 s (] e (] (] (] C /I /I
it. 8 I it.
10° a B . 10°  6 6 a 6
. <) <) 0 e I D e ft 8 <) <) <) <) 0 <) <) 0 f)
. I . . I . .
10- 1 . . 10- 1 I I
0 I 5 10 \ 15 0 5 10 15
J J
large scales small scales
(a) (b)
Figure 4.7. Correlation between adjacent scales of the sparse and dense uniform flows. ( a) Sparse
uniform flow. (b) Dense uniform flow. Note that level indexing is used: j = level index;
q = moment. cjq) is the qth scale moment correlation. cjq) is equivalent to Rfn, given in the text
for pth moment scale indexing. After Moriyama et al (1998). Reproduced with the kind permission
of the Physical Society of Japan and the authors.
Copyright @ 2002 lOP Publishing Ltd.

1
0
(a) -1
0 1000 2000 3000 4000 5000 6000 7000 8000
1
0
-1
0 1000 2000 3000 4000 5000 6000 7000 8000
0.3
o
0.2
0.1
-0.1
-0.2
-0.3
o
10
20
30
40
50
60
70
80
(b)
1
o
-1
o 1000 2000 3000 4000 5000 6000 7000 8000
0.3
o
0.2
0.1
-0.1
-0.2
-0.3
o
10
20
30
40
50
60
70
80
(c)
Figure 4.8. Scale and hard thresholding of a vortex shedding signal. (a) Original coefficients. (b) Scale
thresholded coefficients (top) and associated partitioning of signal (bottom). Top trace = original
signal; middle trace = strong signal component; bottom trace = weak signal component. (c) Hard
thresholded coefficients (top) and associated partitioning of signal (bottom). After Addison et al
2001 ASCE Journal of Engineering Mechanics 127 58-70. (Q) ASCE 2001, reproduced with permission
of the publisher.
Copyright @ 2002 lOP Publishing Ltd.

1
energy
loss relationship for evenly
distributed energy
L --- -----------
o
(= 0.20)
o
o
proportions of
coefficients rejected
Po
(= 0.75)
1
Figure 4.9. A schematic of the Lorentz curve used in wavelet thresholding. The optimal proportion Po
is determined from the tangent parallel to the diagonal.
part of the signal and removes the vortex shedding fluctuations from the weak part,
leaving a more evenly distributed noisy weak signal.
Lorentz thresholding has been suggested as another method for setting a global
threshold for the analysis of coherent structures within turbulent fluid signals (Katul
and Vidakovich, 1996; Katul et aI, 1998a). In contrast to other methods, the Lorentz
threshold does not assume a probabilistic structure for the wavelet coefficients. It
uses the fact that the energy in the wavelet domain is not evenly distributed over
the coefficients. If we plot the proportion of energy loss against the removal of
each of the smallest energy coefficients in turn, we obtain a Lorentz curve. A
schematic of a Lorentz curve is shown in figure 4.9. As the energy is not evenly distrib-
uted throughout the coefficients for turbulent signals this curve is convex. If the
energy were distributed evenly we would get the diagonal line shown in the figure.
The tangent to the Lorentz curve with the same slope as the diagonal locates the
point on the curve where the energy lost by removing a single coefficient is equal to
the average energy in the coefficients. At this point the gain (in parsimony) by thresh-
olding an additional wavelet coefficient is smaller than the loss in energy. This tangent
corresponds to Po and Lo. The example in figure 4.9 shows that thresholding at this
point removes 75% (Po == 0.75) of the coefficients but removes only 20% (Lo == 0.2) of
the energy.
An example of Lorentz thresholding is given in figure 4.10. Figures 4.10(a) and
4.1 O(b) both contain a section of a turbulent time series shown together with its
strong and weak components found through hard thresholding using the Lorentz
threshold. The data were acquired by a laser Doppler anemometer within a turbulent
channel flow downstream of a bluff block obstacle. (For more information see
Addison et al (2001).) A Haar and Symmlet(12) wavelet were used respectively in
figures 4.10(a) and 4.10(b). For the Haar decomposition only 244 coefficients out
of 4096 making up the original signal were used in the strong reconstruction. This
represents only 6% of the coefficients obtained from the decomposition. However,
95.1 % of the signal energy is contained within this signal. The weak signal is
composed of the remaining coefficients and represents only 4.9% of the signal
Copyright @ 2002 lOP Publishing Ltd.

- n-6--
0.4
0.2
o _...IW-_....._,..._.,_..."" __...,.....___-....___....,.---......"'_.._..... "'""..._-,,-
(a) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
0.6
0.4
0.2
o -_..... _-__-v.....,,...........______......""....__P..._",...'W'....,............... ,,- _.......-.......,-'1r__........__..
(b) 0
500
1000
1500
2000
2500
3000
3500
4000
4500
(c)
1
0.5
o
-0.5
o
(d)
500
1000
1500
2000
2500
3000
3500
4000
4500
(e)
Figure 4.10. Lorentz thresholding of a turbulent velocity signal. ( a) Original signal (top) with strong
signal (middle) and weak signal (bottom). Lorentz thresholding used with a Haar wavelet. The
strong signal contains 95.1 % of the energy of the original signal, using only 6% (244 out of 4096)
of the coefficients. (The mean has been removed from the original signal.) (b) As (a) but using a
Copyright @ 2002 lOP Publishing Ltd.

energy. The smoother Symmlet wavelet uses even fewer coefficients in the reconstruc-
tion and contains slightly more energy in the strong signal reconstruction. The
thresholded Symmlet coefficients are shown in figure 4.1 O( d), where it can be seen
that most of the large amplitude coefficients are to be found at lower scales. Figure
4.1 O( e) contains an enlargement of the first quarter of the signal showing the acquired
data points together with the wavelet estimates of the strong signal. The blocky nature
of the Haar reconstruction is obvious from the plot.
4.2.4 Wavelet selection using entropy measures
Ifwe have no preset requirements for the wavelet used in the analysis of a signal, such
as vanishing moments or smoothness, then an entropy measure can be employed for
the selection of the most suitable wavelet. The Shannon entropy is defined for a
discrete probability distribution Pi' i == 1, 2, . . . , N,
S(p) == - 2:: Pi log(Pi)
(4.23 )
where 2::i Pi == 1. (Refer back to chapter 3, section 3.9, where we looked briefly at the
role of entropy measures in selecting the 'best' set of wavelet packet coefficients.) The
maximum entropy possible from a distribution occurs when the data set has an equal
probability distribution Pi at every i, i.e. when the information is evenly spread across
the signal. Any other distribution results in an S(p) less than the maximum. The more
clustered the distribution the lower the entropy. The minimum entropy occurs when
all the information is contained in a single location, i.e. at only one value of i, where
Pi == 1 (see figure 4.11). This entropy measure is extended to the wavelet coefficient
energies where we usually want to contain as much information from the signal in
as few wavelet coefficients as possible. Hence we look for the wavelet which gives
us the maximum entropy for the squared coefficients. To utilize the Shannon entropy
measure in the selection of the optimal wavelet, the normalized wavelet coefficient
energies t n are used where
,
2
_ 2 T m n
T == '
m , n """"'" """"'" T 2
Lm Ln m,n
( 4.24 )
The denominator is the total energy in the signal defined in terms of the wavelet
coefficients, hence 2::m,n t,n == 1. The normalized wavelet coefficient energy t,n
is the relative proportion of the total energy contained within the coefficient Tm n.
,
Symmlet(12) wavelet. The strong signal contains 96.3% of the energy of the original signal, using
only 4.7% (193 out of 4096) of the coefficients. (c) The Lorentz curve corresponding to the Symmlet
decomposition. The whole curve is shown on the left and the last eighth of the curve on the right. (d)
The strong and weak coefficients for the Symmlet decomposition. (e) Close-up of the strong signal
reconstructions using the Haar (top) and Symmlet (bottom) wavelets. The reconstructed signal is
shown as a bold line and the acquired data points are shown as light grey circles. The velocity
signal was acquired within a uniform channel (water) flow using a laser Doppler anemometer.
Lorentz thresholding was carried out following the method of Katul and Vidakovic (1996).
Copyright @ 2002 lOP Publishing Ltd.

Box 1 Box 2 Box 3 Box 4
N
0.25 0.25 0.25 0.25 - L Pi log(p) = 0.60206 (maximum entropy)
i=l
N
0.80 0.10 0.10 0.00 - L Pi log(p) = 0.27753 (an arbitrary
i=l intermediate entropy)
N
1.00 0.00 0.00 0.00 - L Pi log(Pi) = 0 (minimum entropy)
i=l
where Pi probability associated with box i
N = number of boxes, here equal to 4.
base 10 logarithums used.
also define Pi log(P) = 0 when Pi = 0
Figure 4.11. Schematic illustration of Shannon entropy measure. Three signals each containing four
data points, (0.25, 0.25, 0.25, 0.25), (0.80, 0.10, 0.10, 0.00), (1.00, 0.00, 0.00, 0.00), are shown with
each data point contained within a box. The signals have been normalized so that the sum of their
components equals 1, analogous to a discrete probability distribution.
We then define the Shannon entropy measure in terms of the normalized wavelet
coefficient energies as
S(t 2 ) == -   t,n log(t,n)
m n
( 4.25)
Numerous authors have used the Shannon entropy measure to select the best wavelet
in the analysis of turbulent signals; see for example Briggs and Levine (1997) and
Katul and Vidakovich (1996).
4.3 Engineering flows
4.3.1 Jets, wakes, turbulence and coherent structures
Coherent structures are large-scale organized motions that exist in turbulent fluid
flows and which influence a number of fluid-related processes including mixing,
noise, vibration, heat transfer and drag. There has been much research carried out
on the twin problems of separating them from background turbulence and character-
izing their properties. Recently wavelet analysis has joined the toolbox of methods
used in their investigation. Bonnet et al (1998) have described the use of wavelet trans-
forms, together with a large number of other eddy structure identification methods
(conditional sampling, pattern recognition, proper orthogonal decomposition,
stochastic estimation, topological concept-based methods, full field methods) in an
investigation of a shear layer generated at the interface of two fluid streams of
different velocities. They mentioned the utility of the wavelet transform in providing
information concerning the location of vortical structures in the flow, i.e. how their
Copyright @ 2002 lOP Publishing Ltd.

. .
.. ...
-.. -... -..
 0.050  0.050  0.050
 '-" 
;:J 0.025 ;:J 0.025 ;:J 0.025
00 1 2 3 4 5 6 7 8 910 00 1 2 3 4 5 6 7 8 9 10 00 1 2 3 4 5 6 7 8 910
time (s) time (s) time (s)
(a) 10mm
(b) 75mm
(c) 150mm
Figure 4.12. Velocity time series taken at various centreline locations downstream of an orifice plate in
a pipe. This material has been reproduced from Addison P S 1999 Proceedings of the Institution of
Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 213 217-229, figure 4,
by permission of the council of the Institution of Mechanical Engineers.
spatial energy is distributed as a function of physical location and length scale. The
structure of the near field of a three-dimensional wall jet has been investigated by
Sullivan and Pollard (1996). They compared wavelet analysis to three other methods
(proper orthogonal decomposition, Gram-Charlier estimation, linear stochastic
estimation) for the identification of coherent flow structures from multipoint meas-
urements made in the three-dimensional flowfield. They used the peaks in the wavelet
power spectral density plots from a number of simultaneously measured signals to
form cross-sectional maps of the spectral peaks across the flowfield normal to the
jet axis. By doing this, they found that they could locate secondary flows within the
flowfield corresponding to localized vortex motion.
Figure 4.12 shows three velocity signals taken within a low Reynolds number
pulsed flow downstream of a pipe orifice plate (Addison, 1999). The axisymmetric,
periodic vortices shed from the orifice plate produce the regular oscillatory velocity
time series shown in figure 4.12( a). This highly organized motion breaks down to a
more complex flow regime as it advects downstream (figures 4.12(b) and (c)). The
Mexican hat wavelet transform plots relating to the velocity signals are shown as
both contour plots and surfaces in figure 4.13. The regular oscillatory nature of the
initial vortex shedding is clearly seen in the smoothly undulating transform plot of
figure 4.13(a). In addition, high frequency background turbulent activity can also
be seen at the smaller wavelet scales towards the bottom of the plot. Slightly further
downstream, the vortices begin to interact with each other in a process of merging and
disintegration. The subsequent development of larger scale structures within the flow-
field can be seen in the transform plots of figures 4.13(b) and (c). The Fourier and
wavelet power spectra associated with the transform plots of figure 4.13 are given
in figure 4.14. The appearance of the wavelet spectrum as a smoothed version of
the Fourier spectrum is evident in the plots. This is most obvious in figure 4.14(a),
where there is a marked smearing of the spectral peak in the wavelet spectrum.
According to Bonnet et al (1998) the smooth wavelet-based spectrum is an advantage
when analysing single realizations of the flow.
The structure of the eddies in a free jet flow has been investigated using contin-
uous wavelet transforms by Li and Nozaki (1995). Using Mexican hat wavelets
they analysed velocity time series taken from a number of locations within the jet
at both the jet centre line and the jet edge (mixing layer). In subsequent research,
they used a wavelet cross-correlation function based on the Mexican hat (Li and
Copyright @ 2002 lOP Publishing Ltd.

0.64
 0.16


 0.04
0.01
(a)
0.64
 0.16
u


 0.04
0.01
(b)
2.56
 0.64


 0.16
0.04
(c)
0.5 1.0 1.5 2.0 2.5
b (sees)
0.5 1.0 1.5 2.0 2.5
b (sees)
2.0 4.0 6.0 8.0 10.0
b (sees)

.


t-:; .


0.64
0.16
Q Iill 0.04
ec 0.01
2.5

.


t-:; .
..c
!o..

0.64
0.16
Q Iill 0.04
ec 0.01
2.5

.

"'

t-:; .


10.0
2.56
0.64
Q Iill 0.16
ec 0.04
Figure 4.13. Wavelet transform plots of the velocity time series taken downstream of an orifice plate.
(a) 10 mm downstream (0  b  2.56 s, 0.01  a  0.64 s). (b) 75 mm downstream (0  b  2.56 s,
0.01  a  0.64s). (c) 150mm downstream (0  b  10.24s, 0.02  a  2.56s. Note change in the
scale ranges of the axes from plots (a) and (b)). This material has been reproduced from Addison P S
1999 Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering
Science 213 217-229, figure 5, by permission of the council of the Institution of Mechanical Engineers.
Nozaki, 1997) and the Morlet wavelet (Li, 1998a) to determine the relationship
between two simultaneously measured signals at different locations within the jet
flowfield. The wavelet cross-correlation function between two signals x and y is
defined for the continuous wavelet transform as
Cxy(a,/) =  .f T TAa,b)*Ty(a,b+T')db
T 0
( 4.26)
where Tx(a, b) and Ty(a, b) are respectively the wavelet transforms of signals x and y;
the asterisk denotes the complex conjugate; T is the time period of the signal (taken
long enough for the signal statistics to settle down) and T/ is the delay between the
two signals. Cxy(a, T/) is then the cross-correlation between the wavelet coefficients
of each signal over a time delay T at scale a. If the correlation between points in
the same signal is sought (i.e. the wavelet auto-correlation function) then the same
signal is used as x and y in the above equation. Figure 4.15 shows a (Mexican hat)
wavelet cross-correlation function together with the traditional cross-correlation
Copyright @ 2002 lOP Publishing Ltd.

0.01
0.001
1_10--4
1_10- 5
S 1-1O--{j
 1_10- 7
1-10- 8
1_10- 9
o
0.01
(a)
0.01
0.001
1_10--4
1_10- 5
S 1-1O--{j
 1_10- 7
1-10- 8
1_10- 9
o
0.01
(c)
0.01
0.001
1_10--4
1_10- 5
S 1-1O--{j
 1_10- 7
1-10- 8
1_10- 9
o
0.01
100
°0
00
0.1 1 10
frequency (Hz)
100
0.1 1 10
frequency (Hz)
(b)
0.1 1 10
frequency (Hz)
100
Figure 4.14. Wavelet and Fourier spectra of the velocity time series taken downstream of an orifice
plate. (a) 10 mm downstream. (b) 75 mm downstream. (c) 150 mm downstream. The ordinate is
the turbulent energy in arbitrary units and the abscissa is the frequency in Hz. Circles indicate the
wavelet power spectrum curve. This material has been reproduced from Addison P S 1999 Proceed-
ings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 213
217-229, figure 7, by permission of the council of the Institution of Mechanical Engineers.
68
62
56
50
44
 38
32
26
20
14
8
1
P 0.5
 0
r;,. -0.5
-1
o
WR (a;t )
xy
1
0.8
0.6
0.4
0.2
o
-0.2
-0.4
-0.6
-0.8
-1
10 20 30 40 50 60 70 80
"t
Figure 4.15. Wavelet cross-correlation of two test signals. Test signals:
x ( t) = k t [ sin ( 2 ) + sin ( t ) ] e - kt + 0.1 [ sin ( 1f( t  12) ) + sin ( 1f( \  8) ) + sin ( 1f( t ; 3) ) ]
y(t) = kt [ cas ( 2 ) + cas ( t ) ] e -kt + 0.1 [ cas ( 1f2) ) + cas ( 1f1) ) + cas ( 1f;t) ) ]
After Li and Nozaki (1997). Reproduced with kind permission of the Japan Society of Mechanical
Engineers.
Copyright @ 2002 lOP Publishing Ltd.

y
measurement type I
measurement type II
d
x/d x/d y/d hot-wire X
hot-wire hot-wire y/d
x/d hot-wire
x/d
(a)
255
230
205
180

 155
 130
105
80
55
30
5
1
P 0.5
'-' 0

 -0.5
-1
o
WR (a;t)
xy
1
0.8
0.6
0.4
0.2
o
-0.2
-0.4
-0.6
-0.8
-1
50
100 150
't(ms)
200 250
(b)
Figure 4.16. Wavelet cross-correlation jet flow signals. (a) Schematic of the experimental configura-
tion. (b) Wavelet cross-correlation plot corresponding to locations x / d = 13 and 2x / d = 3.
After Li and Nozaki (1997). Reproduced with kind permission of the Japan Society of Mechanical
Engineers.
function for the test signal defined in the figure. Figure 4.16(a) shows a schematic of the
experimental jet flow investigated by Li and N ozaki, and figure 4.16(b) shows the cross-
correlation functions (wavelet and traditional) for locations x/ d == 13 and 2x/ d == 3.
It can be seen from the wavelet cross-correlation plot that the large-scale periodic
motion at scale a == 140 ms contains three small-scale motions with a == 30 ms. Hudgins
et al (1993) have provided an example of cubic spline-based cross-correlation functions
for u and w velocity fluctuations in atmospheric turbulence, and Benaissa et al (1999)
have employed a wavelet cross-spectral density function in an analysis of the dissipation
of a passive scalar in a heated boundary layer.
The vortex shedding downstream of an elliptical plate has been studied by Abe
et al (1999). They used the Morlet wavelet (wo == 6) to analyse velocity signals
downstream of the plate on both the major and minor axes. Figure 4.17 shows a
typical power spectrum obtained from the velocity signals on each of these planes,
together with a diagrammatic sketch of the plate (minor plane denoted z, major
Copyright @ 2002 lOP Publishing Ltd.

z
(a)
L .(a)
(b)
- .Y.
s
B
u
Q)
P-4
00
I-;
Q)

o
P-4
o
100
200
f [Hz]
300
400
Figure 4.17. Power spectrum of the horizontal velocity fluctuation u. (a) Signal taken at
location (xjD,yjD,zjD) = (4.0,0.0,2.0) in the minor plane. (b) Signal taken at location
(xj D, y j D, zj D) = (4.0,2.0,0.0) in the major plane. D is the minor diameter of the elliptic plate.
Disc aspect ratio AR = 3.0. After Abe et al (1999). Reproduced with kind permission of the Japan
Society of Mechanical Engineers.
plane denoted y). From the spectrum we see that the frequency of vortex shedding
from the major plane is approximately 80 Hz and the minor plane 45 Hz. Figure
4.18 shows the Morlet wavelet transform of the velocity signal on the minor plane.
Both the real part of the transform and the modulus are shown. The dominant
vortex shedding scale of a/ t = 24 is evident in the modulus plot. (t == 0.5 ms is
the digitization time interval of the velocity signal.) This corresponds to the vortex
shedding frequency for the minor plane (as seen in figure 4.17). However, a Fourier
spectrum of the wavelet transform values at the level a/ t == 24 (figure 4.19) reveals
a broad spectrum which peaks in the range 15-20 Hz, i.e. approximately one-fifth
of the vortex shedding frequency. A similar low frequency unsteady fluctuation was
also found for the major plane at approximately one-fifth of the vortex shedding
......

 50
100
10

· . J J
-v ,. ...
......
-<:]
...........
 50
100
" L h L.. . . 1 ..Jr-v.r- J.
N  .. v.......... ,'If......... - ..-.......    ..
512
blAt
Figure 4.18. Morlet wavelet transform of u at (4.0,0.0,2.0) in the minor plane for AR = 3.0. Top: real
part. Middle: modulus, where only values greater than 30% of maximum plotted. Bottom: velocity
fluctuation u. After Abe et al (1999). Reproduced with kind permission of the Japan Society of
Mechanical Engineers.
o
1024
Copyright @ 2002 lOP Publishing Ltd.



-
o
o 5
10
50 100
j[Hz]
Figure 4.19. Power spectrum of the fluctuation components of the modulus. (a) In the minor plane at
a/ b..t = 24. (b) In the major plane at a/ b..t = 41. After Abe et al (1999). Reproduced with kind
permission of the Japan Society of Mechanical Engineers.
frequency of 45 Hz. Further analysis by the authors revealed that, although both low
frequency fluctuations were at approximately one-fifth of the vortex shedding
frequency, the unsteadiness is in phase on both sides of the wake but out of phase
in the different planes. Further details of the work are given by Kiya and Abe (1999).
Boniforti et al (1997) have employed the Morlet wavelet in a study of transitional
shear flows downstream of a cylinder in an unsteady flow. They used filtering of the
wavelet energy map to select structured events (i.e. those 'marked by distinct
individuality'). This was done by setting those energy map values far from local
maxima to zero. The conditional frequency spectra they subsequently constructed
gave useful information which allowed the selection of eigenmodes linked to further
degrees of freedom in a dynamical model of the wake. Higuchi et al (1994) made use
of wavelet maps, energy maps and structure maps based on Mexican hat wavelets to
investigate the two-dimensional wake occurring behind a pair of flat plates in water
flow. As the spacing of the plates became closer, the asymmetry of the wake increased.
This increase in asymmetry in the wake is shown in figure 4.20, which contains a flow
visualization and mean velocity map of the wake flow at spacing ratios of s/h == 2 and
s / h == 1, where s is the spacing between the plates and h is the plate width. Figure 4.21
contains the time series taken behind the plates spaced at s / h == 2, together with its
corresponding scalogram and Fourier spectrum. Figure 4.22 shows an energy map
for plates spaced at s / h == 1, together with its corresponding structure map. The struc-
ture map is constructed from the energy maps using a hard thresholding algorithm
where the local threshold, A, is based on the local maxima in wavelet energies
within the scalograms. (Full details of the algorithm are given in Higuchi et al
(1994).) The authors state that the following features of this complex asymmetric
flow can be observed more clearly on such structure maps: the absence of local
maxima at durations between the primary vortices and small scale activity, the lack
of apparent correlation between them, the occasional interruptions in the sequence
of vortices, fluctuations in frequency and duration and the small asymmetry of the
primary shedding frequencies. The authors also used the structure maps in the
construction of conditional power spectra.
Copyright @ 2002 lOP Publishing Ltd.

-..
o

2- 1.6
.,Q 1.2
0c:) 008
 0.4
:> 0
 -0.4
S -6.00
-3.00
-0.15
/JOn' 0
"It 0
'J0.fJ(j 0.15
J.>-q; 3.00
1.6
1.2
0.8
0.4
o
8.0
6.0
5.0
4.0
3.0 ?'
2.0 \;!
S'
1.5 .
1.0 (,j
0.5
6.00
-..
o
2- 1.6
.,Q 1.2
0c:) 0.8
o 0.4
 0
 -0.4
S -5
-2.5
-1.5
-0.5 0
/JOn 0 0.5
"Iii
O.fJ 1.5
6>-q;
1.6
1.2
0.8
0.4
o
8.0
6.0
4.0
3.0 
2.0 
S7
1.5 o

1.0 
0.5
2.5
5
Figure 4.20. Flow visualization and mean velocity map for the flowfield behind two flat plates. Left:
spacing s / h = 2. Right: spacing s / h = 1. After Higuchi et al (1994). Reproduced with kind permis-
sion of the authors.
Mouri et al (1999) employed orthonormal wavelet transform statistics to investi-
gate an experimental velocity signal taken from within isotropic turbulence generated
downstream of a biplanar grid in a wind tunnel. The signal decomposition was
performed using four separate wavelets (Haar, Meyer, Harmonic, Daubechies D20)
and the authors computed a number of measures including flatness factors and
scale correlations. Figure 4.23 shows the results of the flatness factor calculation
obtained for each of the wavelets used. Good agreement between the plots for
each wavelet exists. However, the Haar does exhibit a slight discrepancy which
Mouri and his co-workers put down to the poor localization of the Haar wavelet
in the wavenumber regime. The flatness factor plots all show an increasing trend at
small scales due to the intermittency of the flow, whereby the energy associated
with small scales is dominated by a small number of wavelet coefficients. This is
shown in figure 4.24, where the proportion of wavelet coefficients found above
both one and two standard deviations of the coefficient values at each level are
plotted. This proportion decreases with decreasing scale as expected, since more
energy is concentrated in fewer coefficients at the smaller scales. They also compared
their findings with both those of a numerically generated isotropic turbulence field
and an analytical solution for the field associated with a vortex tube. Using, among
other techniques, the flatness factor and intermittency index, Camussi and Guj
Copyright @ 2002 lOP Publishing Ltd.

[ >rrVs
0.02
20.02
40.02
60.02
80.02
100.02
time (s)
(a)
0.2.10- 1
10 1
-..
00

s:::
o
......

10-4
 10 0
-0.2.10- 1
0.02
20.02
40.02
60.02
80.02
100.02
time (s)
(b)
-..
o
 0.06
00
 0.05
......
 0.04


.;j 0.03
s:::
o
"'0
 0.02
t\S

u
& 0.01
: \j "---
 0.00
o 0.0 1.0

2.0
3.0
lis
(c)
Figure 4.21. (a) Velocity trace, (b) wavelet scalogram and (c) Fourier spectrum in the wake of two fiat
plates spaced at s/h = 2, x/h = 3, y/h = -3.5. The timescale for the signal matches that of the map.
After Higuchi et al (1994). Reproduced with kind permission of the authors. See also colour section.
(1997) found universal properties in fully developed turbulent flows at low Reynolds
numbers. Using Battle-Lemarie wavelets, they developed a conditional averaging
technique based on the wavelet transform which allowed them to identify the time
signatures of coherent structures. They also investigated a jet flow and found both
coherent structures, i.e. vortex rings, and intermittent structures similar to those
found in homogenous grid turbulence. Later, Guj and Camussi (1999) showed
that, when used to investigate the temporal dynamics of coherent structures and
the evolution of energy at each scale, wavelet statistical measures exhibit almost
universal properties. However, when used to investigate the characteristic size and
Copyright @ 2002 lOP Publishing Ltd.

5s
8 0.2-10- 2
.04
.04
.04
.04
.04 0.1-10- 2
(a)
8
.04
.04
.04
.04
.04
(b) 0.1-10 1
 10 0
00
'-'"
s:::
0
....-1
1d
I-;
..a
10- 1
0.5-10 0
61.46
65.46
68.46
73.46
77.46
81.46
(c)
Figure 4.22. (a) Energy map and (b) and (c) structure maps on the narrow side of a wake behind a
double plate at s / h = 1, x / h = 3, y / h = -2. The detailed map (c) corresponds to the fourth line of
(a) and (b). After Higuchi et al (1994). Reproduced with kind permission of the authors. See also
colour section.
time (s)
shape of turbulent structures no universality is observed and, in fact, a strong depen-
dence on the flow Reynolds number and/or flow conditions is observed. See also
Camussi and Guj (1999).
There are many other papers concerning the application of wavelet transform
analysis to jets, wakes and coherent structures. Sutherland and Linden (1998) have
used Daubechies D20 wavelets in a study of both wave radiation and the coupling
of coherent structures in a wake shed from a tall thin vertical barrier in a stratified
fluid (salt water). Ohmura et al (1995) employed Daubechies DI0 wavelets for both
Copyright @ 2002 lOP Publishing Ltd.

10
8
I-; inertial range
0 H
.... 6
u
 A
00 5
00 +
QJ
s::: L
 4
 ,
3
(a)
10
I-;
o
t) 6

 5
QJ
s:::
 4

8 0
. inertial range
H
i
A
+
L
Q .
g Q
3 ............. ........ JL .o.ug","'ui...Q..'g
(b)
10
8
o
I-; · inertial range
6 H
 A
 5  .
 4 i L
  g +
3 ....... ......... .......... .t)..D."O.."'g
(c)
2 4
t
6 8 10 12 14
log2 ( 1i1T 1) \
small scales
large scales
o
. inertial range
H
A
o t
.
L
g g ,
g
... ................ ... .Q.. D...O.. & "0 --'O"'i
(d)
i
A
.
L
g g +
.......... ...... ... ....g. ..g. .-&.. g.. -e.. Q
.
2
4
14
6 8 10 12
log2 (c:,/'Y])
(e)
Figure 4.23. Flatness factor versus wavelet scale. The bases used are (a) Harr's, (b) Daubechies',
(c) Meyer's, (d) harmonic with c.p = 0 and (e) harmonic with c.p = K12. The wavelet scale is normalized
by the Kolmogorov length scale 7]. Filled circles are for the longitudinal velocity component. Open
circles are for the transverse component. Solid lines denote the flatness factor of the velocity
increments. L = integral length scale. A = Taylor microscale. After Mouri et al 1999 Journal of
Fluid Mechanics 389 229-254. With kind permission from Cambridge University Press.
noise reduction and the extraction of frequency components from an experimental
signal of Taylor-Couette flow en route to a region of chaotic turbulence. Van
Milligan et al (1995) have detailed a number of wavelet-based correlation measures
including a delayed cross-coherence and cross-bi-spectrum, applying them to a
model of drift wave turbulence relevant to plasma physics. Within a paper concerning
the analysis of the near field flow structure in a turbulent jet, Li (1998b) defined a
number of wavelet-based flow parameters including a wavelet Reynolds stress func-
tion, wavelet triple velocity correlation function, and wavelet skewness and flatness
factors. luso et al (1996) have also used the Mexican hat in a wavelet formulation
of Reynolds stresses. They found that this wavelet is particularly suitable for finding
Copyright @ 2002 lOP Publishing Ltd.

ooo.n 00 - - nO 0 - 0 o. n ne. 0 -e-.. g.. .0.. .0.. .. e
s::: 0.3 e e e iP. >2(iP).
o J.k J
.,..;
...
u
c\S
tt::
;...
] 0.2

e
. inertial range
o H
A L
+ +
. ."''''''0'''0''-6'''0''-0-'' o. --0- --e. --a n 8
o "'2 2( "'2 )
e U . k > U .
J. J
0.1
2 4 6 8 10 12 14
log2 (/'Y))
Figure 4.24. Number fractions of wavelet transforms with U],k > (u 2 )j and U],k > 2(u 2 )j' Filled circles
are longitudinal components. Open circles are for transverse components. Dotted lines indicate
values expected from Gaussian distribution. Note that the authors use the nomenclature il;'k to
mean the squared wavelet coefficient at location k and level j. In addition, (il 2 )j is the variance
(average energy) of the coefficients at level j. After Mouri et al 1999 Journal of Fluid Mechanics
389 229-254. With kind permission from Cambridge University Press.
the regular structures in images associated with vortex pairs. Yilmaz and Kodal
(2000) have investigated turbulent coaxial jet flows using Morlet wavelet transforms,
as have Walker et al (1997), who used the Morlet wavelet to investigate multiple
acoustic modes and shear layer instabilities which characterize a supersonic jet.
Jordan and Miksad (1998) have examined intermittent events in a wake and
Jordan et al (2000) have used a method based on wavelet ridges to demodulate
transitional wake instability modes where there is no well defined carrier frequency.
Gordeyev and Thomas (1999) have found interesting phase shifting behaviour of
the subharmonic instability within a forced laminar jet shear layer using a Morlet
wavelet decomposition of velocity signals. They provided a Hamiltonian formulation
of the problem, and found good agreement with the experimentally observed
phenomena and this model. Hangan et al (1999) have developed a wavelet pattern
recognition method and used it to investigate the relationship between small
(incoherent) and large (coherent) turbulent scales. Specifically, they employed the
method in the study of velocity data fields taken in the near region of two wake
generators-a solid circular cylinder and a porous mesh strip. Poggie and Smits
(1997) have analysed the wall pressure fluctuations in a Mach 3 flow over a blunt
fin using the continuous wavelet transform. They partitioned the signal into two
parts: one associated with the characteristic timescale of the shock crossing events
and the other associated with the relatively smaller scale turbulent fluctuations.
4.3.2 Fluid-structure interaction
The interaction of fluids with structures is a common phenomenon which manifests
itself, within an engineering context, as a number of problems including wind loading
on buildings, flow-induced vibration of bridge decks and both the loading and
scouring at bridge piers. Gurley et al (1997) have discussed the use of the wavelet
transform in a more general paper concerning analysis and simulation tools in
wind engineering. They illustrated the potential uses of wavelet-based methods for
Copyright @ 2002 lOP Publishing Ltd.

transient and evolutionary phenomena, using the specific example of a wind velocity
signal measured just after the hurricane eye has passed the measuring instrument. In
another related paper, Gurley and Kareem (1999) have discussed the applications of
both the discrete and continuous wavelet transform to earthquake, wind and ocean
engineering, including fluids engineering problems such as the transient response of
buildings to wind storms, the analysis of bridge responses to vortex shedding and
the correlation between pressure measured at a building rooftop and upstream.
Both papers give details of two wavelet methods for the simulation of non-stationary
wind velocity signals. An example of a measured and corresponding synthetic signal is
shown in figure 4.25.
Hajj and Tieleman (1996) have suggested using the wavelet transform to charac-
terize the intermittent nature of wind events to model pressure variations on low rise
structures. The authors detailed the advantages of using a wavelet-based approach
over a conventional Fourier approach to the problem. They illustrated their ideas
briefly using the Daubechies D4 wavelet to decompose a sample wind velocity time
series. In later work using the Morlet wavelet transform, Hajj et al (1998) found
correlations between energetic events in the atmospheric wind and low pressure
25
20
15
10
50 10 20 30 40 50 60
(a)
25
15
,\
20
10
50
10
20
30
40
50
60
(b)
Figure 4.25. Measured and synthesized wind velocity signals. (a) Measured and (b) synthesized using
wavelet transforms. Reprinted from Gurley et al 1997 Probabilistic Engineering Mechanics 12(1)
9-31. Copyright (1997), with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

Figure 4.26. Dimensions of the experimental building and the pressure tap locations. After Hajj et al
(1998). Reproduced with kind permission of Academic Press Ltd.
peaks that occur at pressure taps placed over a large area of a low rise building (figure
4.26). Figure 4.27(a) shows a velocity time trace taken by a cup-vane anemometer at a
site near the building. Figure 4.27(b) shows the pressure trace taken at location 2
during the experiment. Figures 4.28( a) and (b) contain the energy density plots
corresponding to the velocity and pressure signals of figure 4.27. Figure 4.28( c)
shows the cross-scalograms for the velocity and pressure data. The cross-scalogram
is defined by Hajj et al as Tx(a, b)*Ty(a, b), where Tx(a, b) and Ty(a, b) are the trans-
form of two signals x( t) and y( t) respectively and the asterisk represents the complex
16

8
-.. 12
00
'8

4
(a)
0
-1
\jr:::... -2
-3
--4
(b)
Figure 4.27. Velocity and pressure time traces. (a) u velocity time trace. (b) Pressure trace at tap loca-
tion 2. After Hajj et al (1998). Reproduced with kind permission of Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

5 C> 5
0 
4 . 4
 3 1 ,81s r 00",,' 3
s::: '1 1 1,1 I ,I I. I
......-4 2 2
I I I
1 1
0 0
0 200 400 600 800
t(s)
(a) (b)
5
1
(c)
Figure 4.28. Wavelet energy density and cross-scalogram plots. (a) Energy density plot of velocity
signal. (b) Energy density plot of pressure trace. (c) Cross-scalogram plot. After Hajj et al (1998).
Reproduced with kind permission of Academic Press Ltd.
conjugate. This type of scalogram gives peaks where the fluctuations from the two
time series coincide in scale and time. The cross-scalogram shows that there is a
feature occurring at In(a) == 4 (corresponding to a frequency of 0.05 Hz) at between
450 and 550 seconds. The cross-scalograms show the relationship between the time-
localized fluctuations of the velocity and low pressure peaks. This was found to be
the case for all four pressure tappings and both the u and v velocities considered in
the study. Hajj (1999) also analysed data from the same source using Daubechies
D4 wavelets to show that the energy of each of the scales of turbulence within the
wind signals varied significantly with time. He provides quantitative measures of
this intermittency and shows that some of the scales contribute significantly to the
total energy over short periods of time. Marshall et al (1999) have used both the
first and second derivative of Gaussian (DOG) wavelets to study gust-induced bend-
ing moments in trees. After initially attempting to use the bending moment time series
to trigger the conditional sampling of the vertical and horizontal velocities, they
inverted the procedure and used vertical velocity events to trigger the conditional
sampling of the bending moment and horizontal velocities. The trigger for the
conditional sampling was the detection of a 'Honami gust'-an upward-down-
ward-upward wavelike disturbance in the flow. These events were detected using
the zero crossings in the first DOG wavelet transform occurring at a predetermined
wavelet dilation scale. Their study provided an insight into the mechanism of gust-
excited motion of trees. Hamdan et al (1996) have compared the analytical properties
of four wavelets-Morlet, Mexican hat, eighth derivative of Gaussian and a
Daubechies tight frame wavelet-in the elucidation of vibration signals from a
cylinder in a cross flow. Jubran et al (1998a,b) extended this work to cover the chaotic
nature of vortex shedding from a cylinder in a cross flow.
4.3.3 Two-dimensional flow fields
A number of studies have been conducted concerning the wavelet analysis of two-
dimensional flow fields-either imaged data or numerically simulated flow fields. In
this section we review a few of these studies.
Copyright @ 2002 lOP Publishing Ltd.

,--<;>. :G :  . /
/( ./ /....
-81.91
thresholded at : 20.0 a' =0.02
94.58
-193.57
221.54 thresholded at: 44.0 a' =0.04 --428.36
459.25
thresholded at : 90
a' =0.08
. 
thresholded at: 168.8 a' =0.16 -690.80 1086.88 thresholded at: 437.0 a' =0.32
--486.99
722.58
t .. ..
-897.38 1805.38 thresholded at: 1042.0 a' =0.5
Figure 4.29. Structures educed from the wavelet transform of the laminar mixing layer. After Kailas
and Narishima 1999 Experiments in Fluids 27 167-174, figure 3. (Q) Springer-Verlag 1999. Reprinted
with kind permission of the authors and publisher.
The digital images of a mixing layer have been analysed by Kailas and Narasimha
(1999) using two-dimensional Mexican hat wavelets. Figures 4.29(a) and 4.30(a)
contain the original images of the laminar and turbulent mixing layers respectively.
Thresholded versions of the original images are plotted with the originals to reveal
-127.22
A B
116.04 thresholded at: 10.0 a' =0.02
-389.36
thresholded at : 50.0 a' =0.08
425.55
 '
C,f)E
A
-703.29
716.99
thresholded at : -250 a' =0.32
.r .. ., ., ....".. ....-:.;j;;,. .:,....;'  ... ._ ......_ &;-";___
."11a..::!4'ii'{f'::';:'l': t.::--";£
.:i:i' ,  .. "'Jp}' y.."Ji'd  :..rr  " . .i-fJ..i-;..{t.'f
.......,.. .., .. ..."",, "'"i.... .. ..111..,., .. ,........,_"j.-"r. ""'"""".....
,.'""'<t.....:tj: ... '''.. ..r;;_ :.-:
-,.. :"1J>'.-- ... r:. .. ,
A B -" ..... .---...
-65.62
thres.h<} lded at : 5.0 a' =0.01
59.97
-253.14
 ,..
. L
/ .
CD'... ..
A
-606.67
thresholded at : 100.0 a' =0.06
746.86
-718.80
thresholded at : --400.0a' =0.5
579.43
Figure 4.30. Structures educed from the wavelet transform of the turbulent mixing layer. After Kailas
and Narishima 1999 Experiments in Fluids 27 167-174, figure 4. (Q) Springer-Verlag 1999. Reprinted
with kind permission of the authors and publisher.
Copyright @ 2002 lOP Publishing Ltd.

the scale-dependent structure of the images. The threshold was set so as to capture
details unique to that scale, while omitting unnecessary clutter. The roll-up of each
vortex in the laminar mixing layer is obvious in the a/ == 0.02 plot of figure 4.29.
The homogeneous nature of the small-scale structure present within the turbulent
mixing layer is clearly seen at the smallest wavelet scales (a/ == 0.01, 0.02 and 0.04)
in figure 4.30. The large-scale structural organization of the turbulent mixing layer
becomes more obvious at larger scales. Kailas and Narishima followed up this
work in a study of the coherent structures evident in the vorticity fields of numerically
simulated jets, both with and without heating (Siddhartha et aI, 2000). Using a
-6 --4 -2 0 2 4 6
vorticity _ < (wavelet)
---6 --4 -2 0 2 4 6
vorticity _«Fourier linear)
Figure 4.31. Comparison of nonlinear wavelet filtering (left) with linear Fourier filtering (right) of
vorticity for the same compression rate. Top: total vorticity. Middle: coherent part. Bottom: incoher-
ent part. (Note that the original figure is in colour; see colour section.) After Farge et al (1999).
Reproduced with kind permission of the authors.
Copyright @ 2002 lOP Publishing Ltd.

two-dimensional wavelet which imitated the vorticity field topology, V orobieff and
Rockwell (1996) investigated two-dimensional, PIV (particle image velocimetry)
vorticity fields generated beneath a delta wing. In particular they focused their
attention on the distinct topological structures associated with the onset of vortex
breakdown and found the wavelet transform to be an effective pattern recognition
tool for this kind of study. Ishizawa and Hattori (1998) separated out the turbulent
and coherent subregions of two-dimensional magneto hydrodynamic fields using
Meyer wavelets where each of the separated regions could be defined by its own
distinct spectral slope. Rightley et al (1999) have filtered images of shock-accelerated
gas curtains using two-dimensional Mexican hat wavelets. They found that their
wavelet analysis produced clear evidence for the growth of scales both smaller
and larger than initial disturbances imposed on the flow. Everson et al (1990) have
also used two-dimensional Mexican hats to study two-dimensional spatial dye
10 0
10- 1
P(w)
P(w) (WL)
P(w_ <)(WL)
Gaussian
-..
 10- 2

10- 3
10-4
-20 -15 -10 -5 0 5 10 15 20
w
10 0
P(w)
P(w) (fourlin)
10- 1 P(w_ <) (fourlin)
Gaussian
-..
 10- 2


10- 3
10-4
-20 -15 -10 -5
o
w
5
10
15
20
Figure 4.32. PDF of vorticity. Top: nonlinear wavelet filtering. Bottom: linear Fourier filtering. The
solid lines correspond to total vorticity, the dashed lines to the coherent part and the dotted-dashed
lines to the incoherent part, and the dotted lines to a Gaussian fit. After Farge et al (1999).
Reproduced with kind permission of the authors.
Copyright @ 2002 lOP Publishing Ltd.

concentrations in turbulent jets from laser-induced-fluorescence images to gain an
insight into the spatial-scale-dependent structure of the flow field. Do-Khac et al
(1994) have used both continuous and discrete orthonormal wavelet analysis in a
study of two numerically generated, two-dimensional turbulent fields: a growing
planar mixing layer and homogenous turbulence (see also Basdevant et aI, 1993).
Using a third-order B-spline wavelet, Tsujimoto et al (1999) analysed a direct
numerical simulation of channel flow in an investigation concerning the grouping
of quasi-streamwise vortices. They found evidence for clusters of the quasi-
streamwise vortices appearing intermittently within the flowfield.
Farge et al (1999) used an iterative method based on hard thresholding to extract
coherent vortices from two-dimensional turbulent vorticity fields. The method is
designed in such a way that it results in the discarded wavelet coefficients of the
vorticity field exhibiting a Gaussian distribution. This was tested using moment
functions (including skewness and flatness) of the reconstructed signal using the
weak coefficients. When this reconstructed signal is of a Gaussian form, it is assumed
to contain the noisy part of the signal and the remaining strong signal is assumed to
contain the coherent part. Figure 4.31 shows a realization of the two-dimensional
vorticity field, together with its decomposition into coherent and incoherent parts,
using both quintic spline wavelets and conventional Fourier techniques. We can see
from figure 4.32, which contains the PDFs of the coherent and incoherent parts,
that the incoherent part of the vorticity field defined by the wavelet filtering more
closely matches the Gaussian PDF than does the Fourier filtered field.
4.4 Geophysical flows
The meteorological community has been particularly active in the application of
wavelet-based methods to the analysis of fluid flows. Many of the papers already
cited earlier in section 4.2 come from this source. Meyers et al (1993) have provided
an introduction to the use of wavelets (specifically the Morlet wavelet) in Oceanogra-
phy and Meteorology. They illustrated their discussion with examples of the wavelet
decomposition of simple signals into modulus and phase before applying wavelets to
the analysis of Yanai waves. They commented both on the 'nontrivial task' of inter-
preting the phase of complicated signals and on the edge effects of the data on the
transform plots. The rest of this section covers the application of wavelet transforms
to a variety of geophysical flow phenomena.
4.4.1 Atmospheric processes
A number of workers have investigated the power spectra of time series measure-
ments acquired in the atmospheric boundary layer. Some researchers have
investigated the power spectra of the measurements and the deviation from the
expected -5/3 Kolmogrov scaling in the inertial subrange. Others have concentrated
more on the detection and interrogation of coherent structures in the flow. Figure 4.33
shows a plot of the power spectra (Haar, wavelet and Fourier) of three signals
(u, w velocities and temperature) taken in the atmospheric boundary layer by Katul
Copyright @ 2002 lOP Publishing Ltd.

10 4
10 3
 10 2
00
8 10 1

8 10 0
 10 - 1  \:
 . 1
p..
 10- 2

 10- 3
p..
10-4
10- 5
10- 3 2
(a)
10 2 .
10 1
S 10 0
N
U
 10- 1 -
8
B 10- 2
u

p..
00
I-; 10- 3


8. 10-4 -
10- 5
(b)
,
.--...-- 
 
.....
.....
Fourier
. Haar wavelet
-5/3
E
w
...
...
E
u
10- 2 2
10-1 2 10 0 2
wavenumber (m- 1 )
10 1 2 10 2
.....
.....
Fourier
Haar wavelet
-5/3
.....
....
.....
.....
.....
....
....
.....
....
.....
......
.....
......
......
.....
I "'
"'
'",
",
,
....
...
...
...
. ......
...
.
10- 3 2
10- 2 2
10-1 2 10 0 2
wavenumber (m)
10 1 2
10 2
Figure 4.33. Fourier and wavelet power spectra. ( a) Comparison between Fourier ( solid line) and
Haar wavelet (closed circle) power spectra for the longitudinal (u) and vertical velocity (w). The u
spectrum is shifted by two decades to permit comparison with w spectrum at small wave numbers.
Taylor's hypothesis is used to convert the time domain to wave number domain. The -5/3 power
law (dotted line) predicted by K41 is also shown. (b) Same as for (a) but for temperature. After
Katul et al (1994). Reproduced with kind permission of the authors.
et al (1994). There is good agreement between the Fourier and wavelet spectra. The
- 5/3 signature of the inertial sub range is evident in all spectra, especially for that
of the u velocity where it extends over a large portion of the curve. Figure 4.34
shows a plot of the fluctuation intensity (called the coefficient of variation, CV, by
the authors) for the three signals. An increase in the fluctuation intensity with increas-
ing wavenumber indicates increasing turbulent energy activity at smaller scales. Also
noticeable in the plot is that the fluctuation intensity for the temperature signal is
much larger than those of the velocities, possibly indicating that the temperature is
not simply advected by the flowfield, even at small scales. Szilagyi et al (1996)
Copyright @ 2002 lOP Publishing Ltd.

5.5 SD IE T
· U:1J /&
5.0 . SD /b.., I
.. SD \\fE W I
4.5 1': T '
fourier I
4.0 ,6-
3.5 ,'c' .----- W
3.0 - - - .....' / ___ u
G 2.5 ..., .. 'e--::::
,,' > --.-.-.
2.0 ___. 'r - -i
----.
1.5 7e.----'
. ..- ... 
1.0 ./
0.5 -
0.0 - - - - - - - - - - - - - - fourier
-0.5
2 3
. i
10- 1
2 3
100 2 3
wavenumber (m- 1 )
10 1
2 3
10 2
Figure 4.34. Fourier and wavelet power spectra. The coefficient of variation (CV) as a function of
wave number for longitudinal and vertical velocity as well as temperature. The dotted line is the
CV assumed by Fourier analysis. After Katul et al (1994). Reproduced with kind permission of
the authors.
investigated the effect of turbulent intermittency on the shape of the power spectrum
of an atmospheric boundary layer turbulent time series. Using Daubechies D4 wave-
lets, they showed that the local wavelet spectrum in the inertial sub range was sensitive
to intermittency in the flow. They defined the strength of the intermittency in terms of
the wavelet variance. For regions of weak intermittency the local wavelet spectrum
was found to have a slope flatter than that of Kolmogorov's -5/3 law, whereas
regions of strong intermittency exhibited slopes greater than -5/3. The average
slope tended to the -5/3 law. Katul and Parlange (1995) have analysed the -1
power law that arises in heat flux measurements under unstable atmospheric
conditions. Large values of wavelet flatness factor (much greater than 3) indicated
strong non-Gaussian statistics of the signals in the -1 power law range. This they
linked directly to the widening of the gradient probability density function of the
flux measurements. Using the Haar wavelet transform, Katul and Chu (1998)
computed the power spectra associated with velocity signals acquired both from
the atmospheric surface layer and laboratory open channel flow in a study of the
emergence of a universal slope of -1 in turbulent power spectra at physical scales
larger than the inertial subrange (with its own well documented slope of -5/3). See
also Hudgins et al (1993) who make use of cubic spline-based cross-spectra and
cross-scalograms in their investigation of velocity fluctuations in atmospheric
turbulence.
The large-scale intermittent structures involved in the exchange of heat and mass
within and above plant canopies have been investigated using wavelet decomposition
of a variety of signals by a number of research workers. These structures usually
appear in the signals (e.g. temperature, vapour density, velocity) as distinct large
amplitude excursions, or jumps from the mean value. Many researchers have
used the locations of local maxima of the wavelet variance of the continuous wavelet
Copyright @ 2002 lOP Publishing Ltd.

600
straw mulch, z=9.6 em
14:50-15:00 PST, August 23, 1994
,.-..,

'-"
,.-..,

'-"

W(a m )
400
200
o
0.01
0.1
a (s)
1
Figure 4.35. Determination of optimum a = am at the scale of maximum wavelet variance. From
Chen W, Novak M D, Black T A and Lee X 1997 'Coherent eddies and temperature structure func-
tions for three contrasting surfaces. Part 1: ramp model with finite micro front time', Boundary-Layer
Meteorology 84 99-123. Reproduced with kind permission of Kluwer Academic Publishers and the
authors.
transform to identify the timescales of coherent structures. We saw In chapter
2, section 2.9, that the wavelet variance is defined as
(T2(a) =  .f T 1 T(a, b) 1 2 db
T 0
( 4.27)
where T is the duration of the signal segment under investigation. Obviously the
longer T is, the more accurate a value of (J"2(a) is obtained for a stationary signal.
The wavelet variance is, in fact, a measure of the average energy associated with
scale a (refer back to the end of section 2.9 of chapter 2.) Note that some authors
omit T from the denominator, some add C g , and others add C g andfc to the denomi-
nator in their definition of wavelet variance. If C g andfc are both added we get back to
the wavelet power spectral density of equation (4.11). It is reasonable to use any of
these definitions as T, C g and Ie act only as constant rescaling factors, and only the
locations of the peaks in (J"2 (a) are required in order to identify the dominant scales
in the signal. Chen et al (1997b) decomposed temperature signals using the Mexican
hat wavelet in an attempt to detect ramp structures in turbulent flow signals acquired
above a variety of surfaces. The average recurrence times determined from the wave-
let analysis agreed well with those found from the analysis of third-order structure
functions of the temperature. Figure 4.35 shows the wavelet variance plot derived
from one of these temperature signals. The peak in the variance plot was used by
the authors to select the dominant scale, which they denoted am. The wavelet
coefficients at this scale were then plotted together with the original signal and used
to locate coherent structures in the flow. This is shown in figure 4.36 for three
surfaces: bare soil, straw mulch and forest. The dotted line in each of the three
plots corresponds to the wavelet coefficients at scale am for each signal (solid lines).
Copyright @ 2002 lOP Publishing Ltd.

3
0
-3
5
-...
u
a 0
"-'
I

-5
0
5
0
-5
0
T(t)
onmmm F(am,b)
bare soil
straw mulch
1
2
3
4
5
6
Douglas- fir forest
50
100
200
150
t(s)
Figure 4.36. Typical segments of air temperature time series (solid lines) and the wavelet transform
values at that level of am ( dotted lines). Every zero crossing with a negative slope of the dotted line
signifies a ramp event. From Chen W, Novak M D, Black T A and Lee X 1997 'Coherent eddies
and temperature structure functions for three contrasting surfaces. Part 1: ramp model with finite
microfront time', Boundary-Layer Meteorology 84 99-123. Reproduced with kind permission of
Kluwer Academic Publishers and the authors.
Each zero crossing of the coefficient trace with negative slope signifies a ramp event.
Gao and Li (1993) have also used wavelet variances from a Mexican hat wavelet
decomposition of thermal and velocity fields at an atmosphere-forest interface to
identify coherent structures. Their data consisted of simultaneously sampled tem-
peratures and vertical velocities at six heights: two within and six above a deciduous
forest. They plotted scalograms at each height over the measurement duration and
used the scalogram data to construct wavelet variance plots of both temperature
and vertical velocity signals in order to locate coherent structures within the flow field.
Qiu et al (1995b) employed representative sections of flow signals as the basic
shape for pseudo-wavelets in a study of turbulence patterns above three different
vegetation layers: orchard canopy, forest canopy and maize canopy. These pseudo-
wavelets were simple linear ramp shapes which matched with the global structure
of the ramp structures evident in the signal. Although, these pseudo-wavelets do
not satisfy the mathematical constraints required of them to be considered wavelets,
the authors used them in the same way as wavelets to find coherent structures from
within the signal. Collineau and Bruinet (1993a,b) used four continuous wavelets
(Haar, Mexican hat, a wave shape and a ramp shape) to detect coherent motions
in a forest canopy. Lu and Fitzjarrald (1994) used the peaks in the scale-dependent
wavelet variance of the continuous Haar wavelet to identify coherent structures
Copyright @ 2002 lOP Publishing Ltd.

above a midlatitute deciduous forest. It is also worth noting that wavelets have also
been employed to characterize the physical gaps that occur in the forest canopy itself
(see chapter 7, section 7.4.2). Howell and Mahrt (1994b) have extracted energy-
containing events from a wind velocity signal taken above flat terrain in near neutral
conditions using an adaptive multiresolution data filter. Qi and Neumann (1997) used
both the Haar and Daubechies D20 wavelets in both standard multiresolution and
non-decimated formats in an assessment of timescale-dependent errors of the bulk
aerodynamic formula for turbulent heat fluxes used in general circulation models
(GCMs). They focused their investigation on the errors arising between a GCM
simulation and various eddy correlation flux measurements, including those of
wind, sensible heat, latent heat and CO 2 , above a forest canopy. Hayashi (1994)
used Meyer wavelets in an investigation of the turbulent transport of momentum
in the atmospheric surface layer. The study found that the large scale fluctuations
were mainly responsible for the large downward transport of momentum. Yee et al
(1996) used a wide range of wavelet statistical measures to analyse the dynamical
characteristics of concentration fluctuations in a dispersing plume. They used their
results to construct a conceptual model of the turbulent transport, stirring and
mixing regimes in a dispersing plume. Katul and Vidakovich (1996) have used the
Shannon entropy measure to pick the optimal Daubechies wavelet for Lorentz
thresholding-based partitioning of the attached and detached eddy motion in
atmospheric turbulence signals. See also Katul et al (1998a,b). Szilagyi et al (1999)
have provided a method for finding the principal timescale of coherent structures
using the scale-dependent wavelet coefficient energies. These are normalized, to find
the relative contribution at each scale, and corrected for the presence of signal
noise using their own 'global' method. They illustrated their new method using
Daubechies (D4, D6, D8) and Symmlet (S8, S12, S16) wavelets to transform wind
velocity measurements taken above bare and vegetated surfaces. Their method,
which decomposes the energy contribution at each scale into an organized and
random eddy motion, gives very similar results to denoising using Donoho and
Johnstone's (1994) thresholding method. Turner and Leclerc (1994) have used the
Haar wavelet in the conditional sampling of velocity-time series data to identify
coherent eddy structures within the 'more random' flow field of atmospheric turbu-
lence. This was performed by thresholding the time series at arbitrary multiples of
the standard deviations of the wavelet coefficients at each scale. These were used to
produce a strong signal from those coefficients above the threshold together with a
weaker background signal from those coefficients below the threshold.
Kulkarni et al (1999) have used a variety of (Haar) wavelet-based statistical
measures to investigate intermittent turbulent transport in the atmospheric surface
layer over a monsoon trough region including power spectral density, coefficient of
variation of energy (fluctuation intensity), flatness factor, and an isotropy (ISO)
coefficient defined as
( ) 2(km)
ISO k m = 2(k) 2(k)
u m + v m
(4.28 )
where u, v and ware respectively the longitudinal, transverse and vertical components
of velocity; k m is the wavenumber corresponding to wavelet scale m; and (km)
Copyright @ 2002 lOP Publishing Ltd.

1730
0.75
0.20
0.25
0.05
1430
0.65 0.50 0.35
0.75 0.60 0.55
0.10
0.45
0.75
1135
0.80
0.70
0.05
,.-...,
E-4
r/'J.

'-'
<U
. 0830 0.90
0.95
0.85
0.80
0.20
0.90
0.30
0.40 0.20 0.25
0.90
0.15
0330 0.95
0.15
0.35
6 7
scale (m)
Figure 4.37. The isotropy ISO(K m ) with scales at different observational hours. (The bottom axis is
actually scale index m.) From Kulkarni J R, Sadani L K and Murthy B S 1999 'Wavelet analysis
of intermittent turbulent transport in the atmospheric surface layer over a monsoon trough
region', Boundary Layer Meteorology 90 217-239. Reproduced with kind permission of Kluwer
Academic Publishers and the authors.
0030
1
2
3
4
5
8
9
10
11
12
denotes the wavelet variance of the u velocity signal at scale index m. If all three
components exhibit the same variance, the anisotropy coefficient is equal to unity.
The contour plot in figure 4.37 shows the variation in isotropy with scale index m
over a 17 hour period. Kulkarni and his co-workers used regions of sharp decrease
in the isotropy coefficient to define separation between small-scale eddies and
large-scale eddies in the flow field. The small-scale eddies exhibit a high degree of
three-dimensionality, whereas the larger-scale structures become increasingly two
dimensional. The temporal evolution of the flatness factor for the same data is
shown in figure 4.38. The plot is constructed using equation (4.5) to compute the
flatness factor for each scale at various times throughout the observation period. A
contour plot is then generated using this computed data with the scale index as the
horizontal axis, the time as the vertical axis and the flatness factors used to form
the contours. The increase in the flatness factor with smaller scales, indicating
increasing non-Gaussianity, can be seen in the plot together with the variation in
this statistical measure throughout the day. In the study, the sampling frequency of
the sonic anemometer used was 8.42 Hz and the mean of the longitudinal velocity
was 3.025 m S-l. Thus the scale indices m in the flatness plot relate to physical
distances (8.42/3.025) x 2 m metres.
Using both symmetric and anti symmetric spline wavelets and dyadic wavelet
transforms, Hagelberg and Gamage (1994) examined both scale thresholding and
hard thresholding in a study of velocity, temperature and buoyancy flux signals
taken from within the atmospheric boundary layer. They employed a combination
of both thresholds to separate out the coherent and non-coherent part of the signals.
They went on to use wavelet variance to detect the dominant scale in each of the
Copyright @ 2002 lOP Publishing Ltd.

1730
3.0
3.8 3.8 2.2
1430 4.2 3.8 2.6 3.4 1.8
5.4
4.6 6.2 2.6
1135 5.8 7.8 4.6 3.0 1.4
 6.6 3.0
E--4
r.I.) 6.6 3.4
 5.8 2.2
'-'"
(]) 5.0
S
....-1 0830 3.4 2.6 3.8 2.2
+->
0330 4.2 3.4
3.8 4.3 2.6
5.8 3.0
0030 4.2
1 2 3 4 5 6 7 8 9 10 11 12
scale (m)
Figure 4.38. The flatness factor FF(Km) with scales at different observational hours. (The bottom axis
is actually scale index m.) From Kulkarni J R, Sadani L K and Murthy B S 1999 'Wavelet analysis of
intermittent turbulent transport in the atmospheric surface layer over a monsoon trough region',
Boundary Layer Meteorology 90 217-239. Reproduced with kind permission of Kluwer Academic
Publishers and the authors.
signals. The wavelet transform was then interrogated at this scale to form an
intermittency index based on the extrema in the wavelet transform occurring at
this maximum scale of wavelet variance. Hagelberg et al (1998) have used hard
thresholding of two-dimensional Mexican hat wavelet coefficients to analyse the
scale distribution of coherent water vapour structures in the marine atmospheric
boundary layer as measured by a shipboard Raman lidar. Using the minimum
entropy criterion described previously in section 4.2.4, Briggs and Levine (1997)
selected the Daubechies D8 from the Daubechies family D2 to D20 in a study
which used wavelet thresholding methods to filter two-dimensional meteorological
forecast fields.
Mahrt (1991) used both wavelet variance and skewness of a Haar wavelet
decomposition of aircraft-measured vertical velocity data in an investigation of the
eddy asymmetry in the sheared heated atmospheric boundary layer. Gamage and
Blumen (1993) have employed wavelets to inspect temperature records containing
atmospheric cold fronts. They stated that the wavelet transform is particularly appro-
priate for such frontal signals which are characterized by relatively isolated frontal
gradients and by aperiodic and intermittent disturbances. Weng and Lau (1994)
have used both the orthogonal Haar and continuous Morlet wavelets scalograms
to probe satellite-collected infrared radiance data for the organization of convection
processes over the Western Pacific. Grivet- Talocia and Einaudi (1995) have devel-
oped a wavelet-based automatic algorithm for the detection, identification and
extraction of gravity waves from atmospheric pressure traces. Demoz et al (1998)
used the Morlet wavelet in a study of the dynamical processes in cirrus clouds.
They interrogated the Morlet scalograms to elucidate combined information on the
Copyright @ 2002 lOP Publishing Ltd.

intensity and phase of the undulations in vertical air velocity-time series which
they interpreted as convective cells and, at larger scales, gravity wave trains. Other
studies which have applied wavelet-based techniques to large scale geophysical flow
data include the investigation of the El Nino Southern Oscillation (ENSO) by
Kestin et al (1998) and the investigation, using the Mexican hat wavelet, of
meteorological variables (wind speed, pressure and electrostatic field) by Takeuchi
et al (1994).
4.4.2 Ocean processes
A clear and concise study of the grouping characteristics of wind waves using the
Morlet wavelet transform has been presented by Lui (2000). He used the transform
plot to identify local wavegroups in a time series of surface elevation corresponding
to wind-generated waves measured in nearshore areas of the Great Lakes. He
found a linear relationship between group energy and duration. He also showed
that mean maximum group wave height and significant wave height differ by
around 17% and stressed the implications this has for engineering design. Finally,
he used his results to illustrate the non-stationarity of the data, linking his findings
to an earlier study concerning the characteristics of waves on the Atlantic Ocean
(Liu, 1994). Shen et al (1994) studied wind-generated ocean surface waves using
Morlet wavelets and produced visual evidence from the wavelet scalograms which
showed the wave action conservation law not to hold for long fetches. Using a
decomposition of wave data based on complex Morlet wavelets, Savtchenko et al
(1998) studied the relationship between the phase of the dominant waves and
moments of the first appearance of burst and sweep events in corresponding velocity
fluctuation data. Their results allowed them to identify strong contributions to the
Reynolds stress during various atmospheric conditions in the coastal marine
atmospheric boundary layer. However, they found no evidence for the correlation
of wave phase with the bursts and sweeps. This supports their conclusion that it is
not only the wavefield that plays a role in the generation of these ordered motions
but that large scale eddies in the windfield playa part too. Willemsen (1995) has
used wavelet packets to quantify the lag and scaling of maximum wave heights
with wind speed over the ocean and, in smaller-scale laboratory experiments, Jinshan
et al (1998) have employed Morlet wavelet-based decomposition in a novel method to
detect and quantify breaking waves from wave height signals.
The analysis of subtidal sea fluctuations has been performed by Percival and
Mofjeld (1997) using non-decimated discrete wavelet transforms. In their discussion
they outlined the implications for coastal inundation forecasting using such wavelet-
based analysis techniques. Quinquis et al (1996) have used Daubechies DI0 wavelets
to analyse the fine structure extracted from oceanographic celerity profiles. Machu
et al (1999) analysed both sea surface heights and ocean colour (chlorophyll pigment)
data to detect Rossby waves appearing on the ocean surface. R6denas and Garello
(1997) have used both continuous and discrete wavelet analysis to detect and charac-
terize oceanic internal waves from SAR (synthetic aperture radar) ocean image
profiles and Gourdeau (1998) has used wavelet analysis in a study of internal tidal
waves in the ocean. The analysis of sea surface temperature time series in the
Copyright @ 2002 lOP Publishing Ltd.

Indian Ocean has been considered by Meyers and O'Brien (1994) and the analysis of
two-dimensional sea surface temperature fields off the coast of Japan and their effect
on the turbulent mixing processes is described by Ostrovskii (1995). Applications of
wavelet transforms to other geophysical processes including cloud structure and
surface temperatures are given in chapter 7, section 7.4.
4.5 Other applications in fluids and further resources
Two comprehensive introductory papers on the role of wavelet transforms in the
analysis of fluid flows are those by Farge (1992) and Meneveau (1991a). See also
Farge et al (1996) which details the applications of wavelets and wavelet packets
to the analysis, modelling and computation of turbulent flows. A number of
edited texts contain papers concerning the modelling and analysis of fluid flows.
Wavelets in Geophysics edited by Foufoula-Georgiou and Kumar (1994) contains
a number of papers concerning the wavelet analysis of geophysical flows. The
book concerning wavelets in physics edited by van den Berg (1999) contains a
number of papers concerning fluid turbulence. The reader is also directed to the
papers cited in other chapters of this book which concern: turbulent flows within
combustion engines (chapter 5, section 5.8.3); blood flow (chapter 6, sections 6.4.1
and 6.5); subsurface flows, river runoffs and liquid-bubble column flows (chapter 7,
sections 7.4.2, 7.4.4 and 7.5.2 respectively); and turbulent plasmas (chapter 7,
section 7.5.3).
A number of attempts have been made to probe the multifractal nature of turbu-
lent flows using wavelet transforms. See for example Roux et al (1999) who used the
wavelet modulus maxima method to detect vorticity filaments in turbulent swirling
flows. They discriminated between vortex filaments and background pressure fluctua-
tions by examining the profile of the modulus maxima lines: specifically the a-scales
corresponding to the beginning of the modulus maxima line and its peak value,
and the magnitude of the peak value itself. They effectively filtered the signal using
thresholds based on these three criteria. They also considered the multifractal
nature of the data using the modulus maxima. There are many other papers con-
cerning the fractal and multifractal nature of turbulence and the use of wavelet
transforms in describing them. Some of these are cited in chapter 7, section 7.2,
where the link between wavelet analysis and fractal geometry is explored (see
especially figure 7.12). In addition, the reader is referred to the paper by Roux et al
(1999), which contains over 120 references to work in this field. There are many
other examples in the literature of the application of wavelet-based techniques to
fluid flow problems. Liandrat (1996) has described wavelet algorithms for the analysis
and modelling of turbulence and Trevino and Andreas (1996) have discussed the
application (and limitations) of wavelet analysis applied to nonstationary flows.
Liu and Chen (1995) have detailed wavelet-based numerical methods for the simula-
tion of one-dimensional and two-dimensional advection-diffusion phenomena, and
Elliot and Majda (1994) have described a wavelet-based method for the stochastic
simulation of turbulent diffusion. Gerritsen and Olsson (1996) have developed
easy-to-implement mathematical criteria based on anti symmetric quadratic spline
Copyright @ 2002 lOP Publishing Ltd.

wavelets to detect discontinuities, sharp gradients and spurious oscillations within a
numerical model of fluid flows. Anderson and Diao (1995) give details of a
two-dimensional wavelet transform method for the analysis of images used in holo-
graphic particle velocimetry, and Kishida et al (1999) have used three-dimensional
helical wavelets to investigate the local interactions in the nonlinear energy transfer
process within three-dimensional homogeneous, isotropic turbulence.
Copyright @ 2002 lOP Publishing Ltd.

Chapter 5
Engineering testing, monitoring and
characterization
5.1 Introduction
Wavelet analysis has been applied to a variety of pertinent problems in engineering. In
this chapter, we review a selection of these, including the assessment of machine
processes behaviour; condition monitoring of rotating machinery; the analysis of
nonlinear and transient oscillations; the characterization of structural impacting;
the interrogation of NDT signals; and the characterization of rough surfaces. As
with the fluid problems described in the previous chapter, the choice of the most
appropriate wavelet to use in the analysis of engineering problems depends very
much on the nature of the data itself. Both discrete and continuous (usually complex)
wavelets have been used to monitor rotating machinery such as gears, shafts and bear-
ings. Discrete wavelets are favoured when, for example, a small number of data are
required as input to a classifier such as a neural network. Continuous wavelets are
favoured when high temporal resolution is required at all scales. Complex continuous
wavelets are well suited to the free vibrations of plates and beams. The temporal
records of such vibrations quickly exhibit a high degree of complexity due to the
superposition of multiple wavegroups (from multiple reflections at the specimen
edges) whose group velocity is frequency dependent. Complex continuous wavelets
are able to unfold these signals in time and frequency, allowing for the decoupling
of vibration modes. Most surface characterization work has used discrete wavelets,
whose coefficients are used to determine scale dependent surface characteristics
such as a power law (fractal) scaling of a surface. Finally, as we might expect, discrete
wavelet transform coefficients are particularly useful for signal compression problems
. . .
In engIneerIng.
5.2 Machining processes: control, chatter, wear and breakage
The objective of any machining process is the efficient production of a part of specific
shape with acceptable dimensional accuracy and surface quality. The monitoring of
machining processes is therefore an important problem in manufacturing engineer-
ing. There is a considerable economic incentive to develop a reliable monitoring
Copyright @ 2002 lOP Publishing Ltd.

-1.0
I,
(a)
1.0
1.0
-1.5 -1.0
1.0
1.5
(b)
Figure 5.1. Gaussian modulated sinusoid: the wavelet used by Khraisheh et al in their metal cutting
study. (a) Real part. (b) Imaginary part. After Khraisheh et al (1995). Reproduced with kind permis-
sion of Academic Press Ltd.
technique, hence a considerable research effort has centred around this problem over
recent years. Success has been limited by the inherent problems associated with the
monitoring signals which are typically affected by process defects (e.g. chatter and
wear), working conditions, process noise and sampling noise (Wu and Du, 1996).
Recently, a number of researchers have attempted to tackle the problem using wavelet
transforms. Khraisheh et al (1995) used a modulated Gaussian (i.e. Morlet type)
wavelet in an experimental investigation of chatter vibrations occurring during
metal cutting. The complex wavelet they used in the study is shown in figure 5.1.
Figure 5.2 contains one of the time series from the study together with its wavelet
transform plot. The transform plot is partitioned into two regions: white for high
energy regions and black for low energy regions. The cutting process caused built-
up edges (BUEs) to appear on the cutting tool, which broke off intermittently
during the cutting process. The occurrence and breakage of these edges is evident
in the wavelet transform plot of figure 5.2 as white vertical patches extending to
the high frequencies. In addition to the detection and occurrence of BUEs during
the cutting process, Khraisheh and his colleagues showed that the wavelet transform
was good at detecting the boundary of transient regions in these signals. The
frequency information found in the wavelet transform of the signals suggested that
Copyright @ 2002 lOP Publishing Ltd.

200
150
100
50
0
-50
-100
-150
1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88
(a)
1500
-....
N
::q
'-"
 1000
u
s:::
(!)
&
Q)
ct: 500
1.740
1.765
1.790
1.815
time (s)
1.840
1.865
1.890
(b)
Figure 5.2. Time series based analysis of primary chatter in metal cutting. (a) The time history for tool
acceleration in the x direction. (b) The wavelet transform. White represents rich signal energy and
black is poor signal energy. (Feed rate = 0.254 mmlrev, width of cut = 5.08 mm, spindle
speed = 200 rpm.) After Khraisheh et al (1995). Reproduced with kind permission of Academic
Press Ltd.
the cutting process was quadratically nonlinear. The authors went on to detect
chaotic motion in the cutting signals for high feed rates of material.
Biorthogonal spline wavelets were used by Berger et al (1998) to identify chatter
and non-chatter cutting states associated with the orthogonal cutting of stiff metal
cylinders using a CNC lathe. They investigated the effect of both cut depth and
cutting frequency (independently). Figure 5.3 shows the cutting force signal (top)
for a 2.3 mm cut, together with the detail components for scales 1 to 5 (labelled dl,
d2, etc. in the figure) plus the remaining approximation (labelled a5). The authors
used a variety of wavelet-based signal amplitude parameters including the standard
deviation, mean absolute and median absolute deviations of the signal details.
They found that the ratio of the mean absolute deviations of scales 3 and 4 provided
the best way to differentiate between pre-chatter and chatter states. The mean
absolute (m.a.) deviation of a series, Xi, i == 0, . . . , N - 1, is defined as
1 N-l
m.a. == - 2:: IXi - xl
N i=O
(5.1)
where x is the mean value of Xi. (Do not confuse with the median of absolute
deviation often used within thresholding algorithms-see chapter 3, section 3.4.2.)
This statistic was then calculated for the detail signal components at each scale,
denoted m.a.(m). Figure 5.4 shows a plot of the ratio of m.a.(3)/m.a.(4) for these
Copyright @ 2002 lOP Publishing Ltd.

(a)
00 _
5  : - 'j
-50 ' 
20  ' . \ "
 0 . :
"'0 _ 20" " . " '.
-40 . "j ," .
 -8 }rltV\
JJ.MIJHv1
50
o
-50
 -:g
:0 i 8 " " .""
-20 .
-40 "
100 200 300 400 500 600 700 800 900 1000
('f')
"'0
(b)
Figure 5.3. Biorthogonal 6,8 spline wavelet decomposition of a cutting force signal. (Note scale
indexing.) The original signal, s, is at the top of the figure. Immediately below s is the signal
approximation, a5, at scale 5. The detailed signals over the first five scales, d5-dl, are given below
a5. After Berger et al (1998). Reproduced with kind permission of Academic Press Ltd.
18
16
14
12
0 10
..-;

I-<
8
6
4
2
2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80
depth of cut (mm)
Figure 5.4. The ratio m.a.(3) fm.a.( 4) versus cut depth for the signal in the previous figure. After Berger
et al (1998). Reproduced with kind permission of Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

signals taken for a series of cut depths from 2.3 mm to 2.8 mm. The chatter state
occurs at a 2.8 mm depth of cut and it can be seen from the plot that it has an m.a.
ratio markedly higher than the four pre-chatter states at 2.3, 2.5, 2.6 and 2.7 mm.
Berger and his co-workers also used the variation in the kurtosis of the detail signals
at scale 3 to complement the m.a.(3)/m.a.(4) ratio as a chatter indicator.
Kamarthi and Pittner (1997) have compared Fourier transform and wavelet
transform-based neural network schemes to determine the wear of lathe tools from
the force and vibration (acceleration) signals taken at the tool holder. Subsequently,
Pittner et al (1998) proposed a wavelet network scheme to tackle this problem where
the two major tasks in the method-sensor data representation and flank wear
assessment-are combined within a single computational unit. This is done by
determining the wavelet parameters and the neural network weights concurrently
during the neural network training process. In a study of flank wear estimation in
turning, Bukkapatnam et al (2000) used a Daubechies D4 wavelet-based filtering
technique to smooth vibration signals prior to analysis. The fractal characteristics
of the signals were then computed and related to instantaneous flank wear using a
recurrent neural network. See also the earlier study by the same research group
where wavelet packets were employed to decompose acoustic emission signals used
to monitor the machining process (Bukkapatnam et aI, 1999). A combination of
sensory data from CCD camera (images) and microphone (sounds) allowed
Mannan et al (2000) to monitor the condition of cutting tools. Symlet wavelets
were used to decompose the sound signals, and the energy content of the detail co-
efficients was used to differentiate between sharp and worn tools. Luo et al (2000)
have used combination wavelets in a study of vibration signals from machining
processes. Tansel et al (2000) have provided details of a neural-network-based
method for the off-line evaluation of tool condition. They used a number of prepro-
cessing algorithms to reduce the input data to the neural network and found that a
wavelet-based method performed best. See also the earlier work by Tansel et al
(1993) and the wavelet-neural-network-based condition monitoring algorithm
developed by Zhou et al (1995). Wu and Du (1996) have used wavelet packets to
monitor both chatter in turning processes and tool wear in drilling processes. They
developed an automatic feature extraction procedure which sorts the wavelet packets
according to energy content and selects the first few packets containing the most
energy. The signal is reconstructed from the dominant packets and interrogated
using two feature assessment criteria defined by the authors: cross-correlation (time
domain) and cross-coherence (frequency domain). Figure 5.5 shows Wu and Du's
wavelet packet decomposition of machine tool vibration signals for a turning process
during both stable cutting and chatter. Comparing the two figures we can see that
certain packets contain a significantly larger amount of energy during chatter. In
fact, the energy in wavelet packets number 5 and 13 at level 5 became dominant
during chatter. The energy in these packets was described using a peak-to-valley
index (i.e. the difference between the maximum and minimum packet coefficients).
The value of this index was then used to determine whether the vibration signal
indicates a state of chatter or stable cutting.
The Haar wavelet was used by Lee and Tarng to monitor tool failure in end
milling operations (Lee and Tarng, 1999; Tarng and Lee, 1999). They followed this
Copyright @ 2002 lOP Publishing Ltd.

20
250(ms)
o
-20
(a)
AI.. It...AI...It...AI., 1..AI., Ii ._ 'I" ,......_ 1.t.t_l
I' 1 - rr 1- II' "l' · -,. '-r T-lI' If 'J""'Ir r.,r ""II -r
. ,., .. . .i.IL -' . ...... J. ..J...1 A I L I"'L..J.
.. ,- . ...,-.. , I ".-..- T-... O--or"n' , o
 -  .... - ....  ..&.J  L, .R
T ..-- 1"1'"- " r ....,'O 
:- lJU --
-   I
I I I I III I I ) It I I  I:: 1 l I I I I il I ( III 1 I
I :f III
11 ,11 III II III I I I I ! I ! I III III III II I "I
(b)
60
o
-60
(c)
___ .It . .-... ..-..  .....oIL.II
... I'" ,., 'I'"
- - - -- - ...........  - - 
-......  - ,- .....  - ,. .....
" " " " A. A. A .  -- ---
VVV ,,,V,.,   - -  -"".-
III' lauJ
In ' .. ,
. .. I
II , I
H II H ill III J 1 U I ill ,I I 1:1 II 1,1 I ,m I I I I
II ,t III I I 1 I II ,I II I 1:1 I I III 11 f I II
(d)
Figure 5.5. Signal and wavelet packet decomposition during stable cutting and chatter. (a) Stable
cutting signal and (b) associated wavelet packet decomposition. ( c) Chatter signal and (d) associated
wavelet packet decomposition. After Wu and Du (1996). Reproduced with kind permission of
Academic Press Ltd.
up in a later paper (Lee and Tarng, 2000) employing both a Daubechies D2 (Haar)
and D12 wavelet to monitor spindle motor current. They used the approximation
coefficients at level 4 as the best indicator of tool failure. Fu et al (1999) have used
a matching pursuit method to predict the onset of drill breakage in the drilling
process. They found that the matching pursuit method performed satisfactorily in
detecting small drill bit behaviour with three different wavelet dictionaries: Gaussian,
Gabor and Haar. Xiaoli (1999) has employed both continuous and discrete wavelet
analysis of a.c. servo motor currents in a method for the detection of breakage of
small diameter drills. (See also Li, 1998c.) Related studies for the detection of drill
breakage using wavelet-transformed acoustic emission signals are described by Li
et al (1999) and Xiaoli et al (1997). In another study of the prediction of small drill
bit breakage, Mori et al (1999) developed a procedure to extract prefailure
information from the cutting force signal. They developed three index functions
based on Daubechies D12 wavelet coefficients at various scales: an energy index,
waviness index and irregularity index. These indices could differentiate the thrust
behaviour of three tool states: normal, sawtooth and screeching. A discriminant
function was then applied that reduced the three indices to a binary value that
identified the state of the drill: 'normal' or 'pre failure'. Finally, in their paper
concerned with the machining of millimetre scale optics using ion-beams, Shanbhag
et al (2000) employed a wavelet-based deconvolution algorithm to generate an
appropriate dwell function for ion-beam rastering. They tested their method by
machining a one-dimensional sinusoidal depth profile in a prepolished silicon
substrate.
Copyright @ 2002 lOP Publishing Ltd.

5.3 Rotating machinery
The condition monitoring of rotating machinery attempts to detect and diagnose
machinery faults from vibration signals picked up usually from the machine casing.
In this section we begin with some recent applications of wavelet techniques to
gear diagnostics, where the early detection of gear failure is a prime concern, then
we look at wavelet-based detection and diagnosis of signals from other rotating
machinery components, such as shafts, bearings and blades.
5.3.1 Gears
Daubechies D4 wavelets were employed by Paya et al (1997) in the analysis of vi bra-
tion signals acquired from an accelerometer attached to a bearing housing on an
experimental drive line model (figure 5.6). They investigated these signals for various
configurations of faulty gears and bearings, including gears with material added,
material taken away and faulty bearings. The authors interrogated signal segments
consisting of 1024 data points. Figure 5.7(a) shows the wavelet coefficients obtained
for a good gear and good bearing: the reference case. Notice that the coefficients are
indexed sequentially (chapter 3, section 3.3.4). Figure 5.7(b) shows the coefficients
from one of the faulty configurations: a faulty bearing and shaved gear. The dominant
wavelet coefficients derived from the faulty signal were then fed into a neural network
for classification. The authors found that by preprocessing the data using wavelet
transforms prior to using an artificial neural network they could successfully distin-
guish between all the various fault configurations considered. See also the paper by
Sung et al (2000), who used discrete wavelets (D20s) to detect the location of tooth
defects in a faulty gear system.
A brief account of the use of the Morlet wavelet in vibration analysis for
mechanical fault diagnosis is provided by Wang (1996). Dalpiaz et al (2000) have
assessed a number of techniques for detecting cracks in gears from the vibration
signal picked up from the gearbox casing of an experimental test rig. They employed
cepstrum analysis, time-synchronous average analysis, cyclostationary analysis and
disc brake (load)
shaft couplings
accelerometer
2 1
automotive
gearbox
motor
rolling element
bearing housings
Figure 5.6. Schematic presentation of the model drive system. After Paya et al (1997). Reproduced
with kind permission of Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

Figure 5.7. Sequentially indexed wavelet coefficients for two drive line configurations. ( a) Good bearing
and gear. (b) Faulty bearing and shaved gear. (A total of 1024 coefficients were generated. The
smallest scale=wavelet numbers 513-1024, next smallest scale=wavelet numbers 129-512, etc.
The first coefficient is related to the signal mean.) After Paya et al (1997). Reproduced with kind
permission of Academic Press Ltd.
wavelet analysis in the interrogation of the signal. They found that the Morlet-based
wavelet transform is well suited to the detection of transient dynamic effects caused by
these localized faults. A Morlet wavelet (wo == 1.757r == 5.5) was used by Staszewski
and Tomlinson (1994) to detect a damaged tooth in a spur gear. They introduced a
fault detection algorithm which characterized the differences which arose between
damaged and undamaged gears in the wavelet transform modulus plots of their
respective vibration signals. In this way, they were able to differentiate between
damaged and undamaged gears. In addition, visual inspection of the modulus and
phase plots of the wavelet-transformed signals enabled the fault location to be deter-
mined. Staszewski and Worden (1997) used Morlet-based wavelet preprocessing,
among other techniques, in a neural-network-based classification algorithm for
faulty gearboxes. The neural network pattern classifier they developed was applied
to vibration signals from a pair of meshing spur gears with a tooth fault. Yoshida
et al (2000) have also used the Morlet wavelet in the detection of tooth surface failure,
using it to analyse both vibration data from the gearbox and dynamic strain data
taken at the tooth fillet of the gear. They suggested that it may be possible to
detect the failure position on the tooth surface from the strain gauge data using the
wavelet transform, and found the wavelet transform, superior to Fourier methods
when diagnosing the failed tooth and the state of the tooth surface.
A concise account of the application of three orthogonal wavelets (Daubechies
D4, D20 and the harmonic wavelet) to the detection of abnormal signal transients
generated by early gear damage is given by Wang and McFadden (1995). They
used the wavelets to decompose residual gear vibration signals obtained by removing
all harmonics of the tooth meshing frequency from the time domain average. They
noted that, although orthogonal transforms allow for fast algorithms and zero redun-
dancy, there is a distinct lack of resolution in the discrete transform energy scalo-
grams. They suggested the use of non-orthogonal (i.e. highly redundant) wavelets
to overcome this problem and used the Morlet wavelet as an example. Wang and
McFadden (1996) followed this up with a more detailed study of the use of the
Morlet wavelet in the analysis of gearbox signals. They detailed a method to ensure
that the number of scales just covers the frequency band of interest and that the
redundancy of computation, although necessary for fault identification, is minimized.
Copyright @ 2002 lOP Publishing Ltd.

 2
Qj 0
g -2
t\S
synchronous time average
 2

. 0
ob
(!) -2
""0
1
phase modulation
1
wavelet map - amplitude
wavelet map - phase
0.8
0.8
r--""'1
N
:s 0.6
L.......,j
r--""'1
N
:s 0.6
L.......,j
0-
(!) 0.4

0-
(!) 0.4

0.2
0.2
o
o
o
90 180 270 360 0
angular position of shaft [deg.]
90 180 270 360
angular position of shaft [de g.]
Figure 5.8. Vibration of healthy gear. Synchronous time average with its phase modulation, and
amplitude and phase plots. After Boulahbal et al (1999). Reproduced with kind permission of
Academic Press Ltd.
Lin and McFadden (1997) have used cubic B-spline wavelets to decompose gear
vibration data. They analysed a complex time signal generated by removing various
periodic elements of the original vibration signal in the Fourier domain and taking the
inverse Fourier transform. In this way, although the signal became complex, the
reconstructed signal was deemed to carry most of the information describing
the change in the vibration signal caused by the crack. McFadden et al (1999) have
also decomposed gear vibration signals using a generalized S transform which has
many similarities with the wavelet transform.
Boulahbal et al (1999) have used both amplitude and phase information obtained
from a Morlet-based wavelet decomposition to detect cracks in geared systems.
Figure 5.8 shows a synchronous time-averaged vibration signal for a healthy gear
with 16 teeth. Three bands appear in the wavelet transform amplitude plot: one
corresponding to the gear meshing frequency (GMF) of 320 Hz and the other two
corresponding to its harmonics. These are indicated by dashed lines in the plot.
There are 16 jumps in phase along the GMF band corresponding to the 16 teeth of
the gear. Figure 5.9 shows the same plots for a gear which has one of its teeth with
a transverse crack cut into it to a depth of 20% of the tooth thickness. The cut was
made at an angle of 90° to the timing mark on the gear. The location of the crack
can be seen in both the signal and the corresponding wavelet transform plot beneath.
The change in phase can be seen in the wavelet phase plot as a bifurcation in the
vertical phase bands at approximately 90° and at a frequency just above the GMF.
Boulahbal and his colleagues went on to investigate the use of amplitude and phase
maps on 'overall residual' signals obtained by filtering out the GMF and its
harmonics from the signal. In addition, they employed a polar representation of
the amplitude and phase plots (figure 5.10). These are formed by wrapping the
plots in figure 5.8 around a circle and joining up the two ends. According to the
authors, this type of map is 'continuous and very intuitive' due to the periodic
Copyright @ 2002 lOP Publishing Ltd.

 2
Qj 0
g -2
t\S
synchronous time average
 2
........
. 0
on
a) -2
""C
1
phase modulation
1
wavelet map - amplitude
wavelet map - phase
0.8
0.8
1"'"""""'1
N
:s 0.6
L.......,j
1"'"""""'1
N
:s 0.6
L.......,j
ci-
a) 0.4

ci-
a) 0.4

0.2
0.2
o
o
o
90 180 270 360 0
angular position of shaft [deg.]
90 180 270 360
angular position of shaft [deg.]
Figure 5.9. Vibration of gear with cracked tooth. Synchronous time average with its phase modula-
tion, and amplitude and phase plots. After Boulahbal et al (1999). Reproduced with kind permission
of Academic Press Ltd.
nature of the signal which repeats itself every revolution of the gear. In addition, the
polar map squeezes the low frequency components of the map and stretches out the
high frequency components. This is useful as normal wavelet plots become cluttered
at high frequencies as the correspondingly smaller wavelets pick up more and more
detail. The arrows on the two maps show the location of the features in the maps
corresponding to the cracked tooth. Note that the phase map shows up the feature
1/16th of a rotation before the amplitude map. This is because the cracked tooth
causes a local increase in speed which is immediately reflected in the phase. The
local speed increase causes the next tooth entering the meshing region to suffer a
stronger impact. Hence, the stronger impact appears as a feature in the amplitude
map 1/16th of a revolution later than the phase change for a 16-tooth gear.
Figure 5.10. Vibration of gear with cracked tooth. Amplitude and phase wavelet maps of the synchro-
nous time average signal shown in polar representation. After Boulahbal et al (1999). Reproduced
with kind permission of Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

5.3.2 Shafts, bearings and blades
The problem of faulty bearings in rotating machinery is addressed by Li and Ma
(1997) who have investigated two characteristic bearing defects using wavelet
analysis. They employed a decaying exponential (real-only) sinusoidal wavelet to
differentiate between bearings with damaged outer races and damaged rollers. The
variation of the wavelet coefficients at different scales was used to detect localized
bearing defects. For example, the vibration signal from a bearing with a roller
defect is shown in figure 5.11(a). The wavelet decomposition of the signal at five
scales, aI, a2, . . . , as is shown in figure 5 .11 (b). There is a noticeable periodic structure
in the a3 coefficients. The periodicity of this structure is given at 4.76 ms from the
autocorrelation plot of figure 5.11 (c). The frequency of this periodic structure is
210 Hz corresponding well with the frequency of 211 Hz expected from a damaged
1000
0
-1000
0 5 10 15 20 25 30 35 40
(a) time (ms)
0.5
0.4
0.3
0.2
0.1
0
0 5 10 15 20 25 30 35 40 45 50
(b) time (ms)
0.7
0.6
0.5
1 2 3 4 5 6 7 8 9 10
(c) lag (ms)
Figure 5.11. Wavelet-based vibration analysis of defective bearing. (a) Vibration signal of a bearing
with a defective roller. (b) Magnitude of the wavelet transform coefficients for five a scales. (Dilations
in the plot from top to bottom are al > a2 > a3 > a4 > as.) (c) The autocorrelation of the wavelet
magnitudes at dilation a3 in figure (b). Reprinted from Li and Ma, NDT & E International 30(3)
143-149, copyright (1997), with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

roller. The authors were able to develop an algorithm for detecting the onset of loca-
lized defects on roller bearing elements by examining the frequency of structures in
the wavelet coefficient outputs within a narrow frequency band associated with the
defects under consideration. Pifieyro et al (2000) found that a simplified technique
based on the Haar wavelet transform proved to be applicable in the early detection
of the burst generated in modelled signals during a fault development in roller bear-
ings. However, they failed to obtain useful information when using the technique on
laboratory-measured acoustic emission signals.
Mori et al (1996) applied a discrete Haar wavelet transform to vibration signals to
predict the occurrence of spalling in ball bearings. They proposed a method of
spalling prediction based on the trend of the scale dependent wavelet coefficient
maxima which, they noted, increases at small scales just before the occurrence of
spalling. Tandon and Choudhury (1999) have provided an overview of vibration
and acoustic measurement methods for the detection of defects in roller bearings
0.32
0.26
0.20
0.14
0.08
0.02
.. . . - . -"'._.Io-.
7
6
5
4
-
 3
Q)
- 2
1
0
(a) -1
7
6
5
4
-
 3
Q)
- 2
1
0
(b) -1
7
6
5
4
-
 3-
Q) !
-2..J
1 
01
-1
I
1
(c)
L
I'
t 11 I I
28 55 82 109 136 163 190 217 244
r
Figure 5.12 Application of wavelet analysis to the diagnosis of gas turbine faults. Unsteady pressure
wavelet maps. (Note that level indexing is used.) (a) Healthy, (b) one rotor blade twisted and (c)
the difference between maps (b) and (a). After Aretakis and Mathioudakis (1997). Reproduced
with kind permission of the ASME.
Copyright @ 2002 lOP Publishing Ltd.

which sets wavelet transform analysis in context with other methods used in this area.
Yacamini et al (1998) have developed a method to detect the torsional vibrations
associated with a.c. motors and generators from their stator currents. In particular,
they suggested the use of the wavelet transform to analyse the transient torsional
vibration signals which occur during the direct on-line start up of induction
motors. This is a crucial period of operation as large shaft stresses may exist over
short periods of time. Shibata et al (2000) have used discrete Daubechies D8 wavelet
transforms to aid in the visualization of sound signals as a fault diagnosis method for
rotating machinery. In order to detect wrap-up incidents in napping machines used in
the textile industry, Dad et al (1999) performed a discrete wavelet transform
decomposition of vibration signals taken from the bearing housing and input the
coefficients into a multilayer neural network for classification. Tsai et al (2000)
have developed a set of criteria to detect abnormal loads on servomechanisms.
They employed a Haar-based multiresolution analysis to enhance the detection of
abnormal torque events using their method. Lin and Qu (2000) have used a modified
soft thresholding technique based on the continuous Morlet wavelet transform to
analyse vibration signals from both defective rolling bearings and gearboxes. Their
'generalized' soft-thresholding algorithm removes only a fraction of the threshold
from those coefficients above the threshold value, whereas those below the threshold
are set to zero. The technique effectively lies somewhere between soft and hard
thresholding. According to the authors this method is more applicable to impulse
component extraction from mechanical dynamical signals.
0.2
---- Point A (Smaller twist)
Level 4
- PointA
· · · .. Point B
...... Point C
"''',14 .1 Point D
0.1
..s:=
cd....
Ie.,....
cd.... 0.0
-0.1
(a)
-0.2
0.10
Signatures from Pressure KuHte.
(1 Blade Twisted)
Level 4
- PointA
.. . .. Point B
. .. J. Point C
..." ,,£ Point D
0.05
..s:=
cd....
I 0.00
c'\S
-0.05
-0.10
17
Signatures from Pressure KuHte.
(2 Blades Fouled)
(b)
19 20 22 23 24 26 28 29 31
amplitude number (i)
32
Figure 5.13 Unsteady pressure signatures. (a) One rotor blade twisted and (b) two rotor blades
fouled. aif and aih are, respectively, the wavelet amplitudes for the faulty and healthy conditions.
(The signatures are derived from the differences in the level 4 wavelet coefficients.) After Aretakis
and Mathioudakis (1997). Reproduced with kind permission of the ASME.
Copyright @ 2002 lOP Publishing Ltd.

Aretakis and Mathioudakis (1997) used D20 wavelets to diagnose faults in gas
turbines. They calculated discrete wavelet scalograms of signals taken from two gas
turbines: a 'healthy' one and one with a twisted rotor blade. Three different signals
were analysed: wall pressure, compressor case vibration and radiated sound. The
discrete wavelet maps corresponding to the pressure signal from the healthy and
twisted blade turbines are shown in figures 5 .12( a) and (b) respectively. The difference
between the maps was then computed in order to show up the locations of significant
signal departures. Figure 5.12(c) shows this 'difference' map and the locations of the
significant differences are shown on figure 5 .12(b). The authors focused on the differ-
ences of level 4 coefficients as a signature for each type of fault. Two examples of these
signatures are shown in figure 5.13. Figure 5.13(a) contains the signature for four
different operating points, A to D, for a single twisted blade. The signature for
point A corresponds to a rotor blade with smaller twist, i.e. the same fault but less
severe. Figure 5.13(b) contains the signatures at four operating points for two
fouled rotor blades. The results indicated that the signature patterns are independent
of operating conditions and, in addition, each failure type generates a unique failure
signature. The authors suggested that these two properties of the wavelet signature
makes it suitable as a diagnostic tool for such faults.
5.4 Dynamics
A number of papers concerning the application of wavelet-based analytical tech-
niques to the investigation and modelling of dynamical signals have appeared in
recent years. Applications include the evaluation of dynamic properties and system
characteristics; the modelling and control of dynamical behaviour; and the partition-
ing or decoupling of multiple responses within dynamical systems. A few of these are
reviewed briefly in this section.
The detection of system nonlinearities through the identification of damping and
stiffness parameters for multi-degree-of-freedom dynamic systems during transient test-
ing has been carried out by Staszewski (1997, 1998a) using a Morlet wavelet. This wave-
let is very effective for this application as it has good support in both frequency and
time, which allows the decoupling of the system's various modes of vibration with
respect to time. Figure 5.14 shows one of the signals analysed in the study resulting
4.0
-..
C'I
00
'8

s:::
.9 0.0

I-;
(])
-
(])
u
u
c\S
0.2 0.3
time (s)
Figure 5.14. Impulse response function for well separated modes. After Staszewski (1997). Reproduced
with kind permission of Academic Press Ltd.
-4.0
0.0
0.1
0.4
0.5
Copyright @ 2002 lOP Publishing Ltd.

le+OO
,.-...,
N
00

'-'"
Q) le+03
"'d
.s
....-4
s:::
00
cd
S
le+06
frequency (Hz)
Figure 5.15. Frequency response functions for well separated modes. After Staszewski (1997). Repro-
duced with kind permission of Academic Press Ltd.
from the impulse response of a two-degree-of-freedom model system. The frequency
response function and wavelet transform plots-amplitude and phase----corresponding
to this signal are shown in figures 5.15 and 5.16 respectively. Figure 5.17 shows the
ridges of the modulus plot and figure 5.18 shows the real parts of the wavelet transform
skeletons obtained from these ridges. The decoupling of the modes is evident in the plot.
The reconstruction of the modes and subsequent damping parameter estimation is
found to be better for the skeleton reconstructions than a simple reconstruction
based on the wavelet coefficients within a certain frequency range. The ability of the
method to separate modes that are closer in frequency is also tackled in the paper.
Kyprianou and Staszewski (1999) have employed cross-wavelet analysis using the
Morlet wavelet as an alternative approach to classical input-output analysis based
on frequency response functions for nonlinear oscillator system identification. Ruzzene
et al (1997) also used Morlet-based wavelet transforms to identify natural frequencies
and damping ratios of multi-degree-of-freedom (MDOF) systems. They tested the
technique on a four-degree-of-freedom model, then used it to analyse the acceleration
response of a bridge excited by road traffic, wind and low intensity ground motion. In a
subsequent paper, Piombo et al (2000) have reported on dynamic tests performed on a
simply supported bridge in Northern Italy under traffic excitation. They determined
modal parameters using their wavelet estimation technique and verified the accuracy
2
,.-...,
cd
00
o
I, 1
Q)
-
cd
U
00
o
o
0.1 0.2 0.3 0.4 0.5 0
time (s)
(b)
0.1 0.2 0.3 0.4 0.5
(a)
Figure 5.16. Wavelet transform for the impulse function for the well separated modes. (a) Amplitude.
(b) Phase. After Staszewski (1997). Reproduced with kind permission of Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

2
-..

OJ}
o
'I 1

Q)
,......;
cd
U
00
o
o
0.2
0.4
0.6
0.8
1.0
time (s)
Figure 5.17. Ridges of the wavelet transform scalogram. After Staszewski (1997). Reproduced with
kind permission of Academic Press Ltd.
of their modal estimates through a modal quality index based on a wavelet cross
correlation statistic. Recently, Lamarque et al (2000) have introduced a wavelet-
based logarithmic decrement formula to estimate damping in multi-degree-of-freedom
systems from time domain responses. They then employed their method in a study of
the in situ dynamic response of a civil engineering building excited both with harmonic
and shock testing (Hans et aI, 2000). Their results demonstrated that the method
permits the uncoupling of the eigenfrequency modes of the MDOF system and provides
reasonable estimates of the associated damping.
Newland (1999a) applied his own wavelet, the harmonic wavelet, to the analysis
of bending wave propagation within a steel beam. This complex wavelet is defined as a
series of non-overlapping boxes in the Fourier domain (figure 5.19). The inverse
Fourier transform of these box functions gives the wavelets in the time domain
which, with an appropriate choice of spacing, can be forced to be orthogonal to
each other at each scale. Newland applied a smoothed version of this wavelet
(using a Hanning window) to the analysis of experimentally measured impulse
response signals from a suspended mild-steel beam of rectangular cross section.
The windowing of the wavelets improves their temporal localization. However, it
forces them to become nonorthogonal and hence the analysis becomes highly redun-
dant. The response signal from the beam, taken close to the point of impact, is shown
in figure 5.20(a). The corresponding time-frequency map is given in figure 5.20(b).
Copyright @ 2002 lOP Publishing Ltd.

1.0
(a)
-..
00
+""
.
S -1.0
 0
'"0
B
 5.0


"
0.1
0.2
0.3
0.4
0.5
o
(b)
0.10
time (s)
Figure 5.18. Comparison of the real parts of the wavelet transform skeletons (dashed lines) obtained
from the ridges given in the preceding figure and the theoretical impulse response function (solid line).
(a) First mode (20 Hz). (b) Second mode (78 Hz). After Staszewski (1997). Reproduced with kind
permission of Academic Press Ltd.
-5.0
o
0.05
0.15
0.20
Groups of high frequency bending waves travel faster than low frequency ones. This
can be seen in the plots where reflections of the high frequency wave groups occur
more often. To reduce the smearing of the information in the time-frequency
maps, their ridges are determined. These are shown plotted in figure 5.20(c). Deter-
mination of the ridge locations is a non-trivial task in practice and Newland gives
some details on how to tackle this problem. Newland also covers the task of phase
extraction and provides a second example of the application of the harmonic wavelet
to the reflection of acoustic waves in a closed duct. Two more engineering appli-
cations of the harmonic wavelet-ground vibrations from underground trains and
the dynamic behaviour of soil under earthquake excitations-are described by
Newland in a subsequent paper (Newland, 1999b). For more information on the
early development of the harmonic wavelet and the generalized harmonic wavelet,
see Newland (1993a,b) and (1994d) respectively.
Onsay and Haddow (1994) applied the Morlet wavelet to experimental vibration
signals from both semi-infinite and free beams. They showed how the modulus and
phase of the wavelet transform enables good resolution of the signal over a wide
spectral range. This resulted in efficient localization of the complex interference
patterns of transient wave groups within the dispersive medium of the beams.
Kishimoto et al (1995) also applied the Morlet wavelet transform to the time-
frequency analysis of dispersive flexure waves in a simply supported beam. They
found that the dispersion relationship of the group velocities could be evaluated accu-
rately using the information extracted from the wavelet transform. Kishimoto (1995)
Copyright @ 2002 lOP Publishing Ltd.

Level -I
1/2Jt
Level 0
Level I
...-
8
'-'
?
1/4Jt
Level 2
Level 3
Level 4
1/8Jt
o 2Jt4Jt 8Jt
16Jt
32Jt
64Jt
(a)
1
0.5
-0.5

<:'
'-'
? 0
'-v-'
Q)

-3
-2
-1
o
1
2
3
4
t
(b)
0.5

<:'
'-'
? 0
'-v-'
S

-0.5
-3
-2
-1
o
1
2
3
4
t
(c)
Figure 5.19. The harmonic wavelet. (a) Magnitudes of the Fourier transform of the harmonic wavelet
at different levels. (b) Real part of a harmonic wavelet in the time domain. (c) Imaginary part of a
harmonic wavelet in the time domain. Note that the frequency bands of the original harmonic wave-
let shown above are in octaves. The subsequent generalization of the harmonic wavelet by Newland
(1994) for practical applications does not require the adjacent Fourier boxes to be arranged in this
way. In addition, the box spectrum of each wavelet is smoothed and overlapped for practical
applications which improves their localization in time and improves the time-frequency resolution
of the wavelet map (Newland 1999b).
Copyright @ 2002 lOP Publishing Ltd.

s::: 4000
0 2000
.-

I-; 0
Q)
-
Q) -2000
u
u
t\S --4000 0
0.05 0.1 0.15 0.2 0.25
(a) time / seconds
1000 1600 1000
800 1400
N N 800
::r: 1200 ::r:
-- 600 --
 1000  600
u u
s::: 800 s:::
Q) Q)
& 400 600 & 400
Q) Q)
 200 400 
200 200
0 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25
time / seconds time / seconds
(b) (c)
Figure 5.20. Harmonic wavelet analysis of the response at one end of a freely supported elastic beam
subjected to an impulse input. (a) Sample acceleration time history. (b) Original harmonic wavelet
time-frequency map for the sample time-history. (c) Corresponding ridge diagram. After Newland
(1999a). Reproduced with kind permission of the ASME.
suggests the use of the method to determine the velocity and attenuation of ultrasonic
pulse echo signals used in non-destructive testing. In a subsequent related paper,
again using Morlet wavelets, Inoue et al (1996) reported on a time-frequency
analysis of the flexural waves set up by central impacts on a simply supported
beam. A method was developed by the authors to determine both the group velocity
of the structural waves and the impact sites on the beams by utilizing the arrival times
extracted from the wavelet analysis. In a similar study, this time for steel plate
elements, Gaul and Hurlebaus (1997) have determined impact site locations using
the wavelet decomposition of strain sensor data. A wavelet-based study of transient
waves propagating in composite laminate plates has been described by Jeong and
Jang (2000).
An analysis of the non-stationary response of a rigid block resting on a moving
plane has been performed by Basu and Gupta (1999) using a wavelet-based stochastic
linearization technique. This simple model was used to show the potential application
of wavelet-based analytical tools to the response of slipping structures to earthquake
excitation. See also the earlier papers by Basu and Gupta (1997, 1998). Gurley and
Kareem (1999) mention briefly the application of wavelet transforms to ground
motion analysis during earthquakes and building responses to wind events (including
vortex shedding) in their paper concerning analysis and simulation tools for wind
engineering. Robertson et al (1998a,b) have detailed a discrete wavelet transform-
based method for extracting the temporal impulse response functions of structures,
and Chen and Wu (1995) have developed a spline wavelet expansion-based finite
element method for frame structure vibration analysis.
Copyright @ 2002 lOP Publishing Ltd.

A study by Dalpiaz and Rivola (1997) compared the effectiveness and reliability
of different vibration analysis techniques for fault detection and diagnostics in cam
mechanisms used in high-performance packing machines. They compared traditional
analysis methods-amplitude probability density (APD), power spectral density
(PSD) and time synchronous averaging (TSA)-with a wavelet transform method
based on the Morlet wavelet. They found the time-frequency analysis of the wave-
let-based method well suited to detecting and precisely locating transient dynamic
phenomena from the signal. Mastroddi and Bettoli (1999) have performed a wavelet
analysis on the output signal of a nonlinear system in the neighbourhood of a Hopf
bifurcation. They used the Morlet wavelet-based analysis to point out the linear and
nonlinear signatures of the system and suggested its use for aeroelastic applications.
Karshenas et al (1999) compared the wavelet power spectrum smoothing method with
the Welch method in the random vibration control algorithm of an electrodynamic
shaker. They found that the wavelet method achieved twice the power spectrum
resolution of the Welch method. Sjoberg et al (1995) included wavelet transform-
based methods in a comprehensive paper concerning 'black-box' models of nonlinear
dynamical systems. A new wavelet-based method for denoising transient dynamical
signals by first projecting them into a multidimensional state space has been described
by Effern et al (2000) and applied to both model data and event-related potentials
(medical EEG signals).
5.5 Chaos
Nonlinear oscillator systems are capable of the most fascinating behaviour known as
chaotic motion, or simply chaos, whereby even simple nonlinear systems can, under
certain operating conditions, behave in a seemingly unpredictable manner (Addison,
1997). The realization that real systems can exhibit this type of non-periodic response
has prompted much research work in the area over the past two decades. The ability
of wavelet-based methods to characterize chaotic oscillations has received attention
from a variety of workers in the field.
Both Daubechies and Morlet wavelets have been employed by Staszewski and
Worden (1999) to analyse time-series data sets containing a variety of features
including coherent structures (fluid turbulence), fractal structures (devil's staircase
and Mandelbrot-Weierstrass function), chaos (Duffing, Henon, Lorenz and Rossler
systems) and noise (Gaussian white). Their paper provides a wide ranging overview at
an introductory level of the application of wavelets to signals from these types of
systems. Figure 5.21(a) shows the time series of a Duffing oscillator in chaotic
mode. This oscillator is a sinusoidally driven, damped nonlinear oscillator with the
nonlinearity contained in the cubic spring term. The Duffing oscillator investigated
by Staszewski and Worden has the form
x + 0.05x + x 3 == 7.5 cos t
(5.2)
where x and x are, respectively, the first and second derivatives of x (the displacement
if we think of it as a physical mass-spring-damping system). For the set of parameter
Copyright @ 2002 lOP Publishing Ltd.

1"'"""""'1
00
...... 6.0
.-
S
L.......,j
Q) 0.0
""d
.a
.-
......
p..
 -6.0 0 512 1024 1536 2048
(a) data samples
Poincare Map
2.0
1.0
 0.0
.-
u
0
......
(]J
>
-1.0
-2.0
-3.2.5 -1.5 -0.5 0.5 1.5
displacement
(b)
Figure 5.21. Chaotic time series of the Duffing oscillator and corresponding Poincare map. After
Staszewski and Worden (1999). Reproduced with kind permission of World Scientific Publishing
Co Pte Ltd and the authors.
values given in equation (5.2) this forced nonlinear oscillator produces a chaotic
response. The Poincare map of figure 5.21(b) was generated by plotting the velocity
against displacement once every period of forcing of the oscillator. This map is
useful in highlighting the fractal structure of the strange attractor associated with
the chaotic system. Figure 5.22 contains the phase portrait of a Gaussian noise
signal together with that for the Duffing oscillator. The similarity between the two
2.2 2.2
-.. 2.1 -.. 2.1
 
00 00
0 0
...... 2 ...... 2
 
(]J (]J
...... ......
 
u 1.9 u 1.9
00 00
1.8 1.8
0 20 40 60 80 100 120 140
(a) time [s] (b)
o 20 40 60 80 100 120 140
time [s]
Figure 5.22. Wavelet phase for (a) Gaussian noise and (b) Duffing oscillator. After Staszewski and
Worden (1999). Reproduced with kind permission of World Scientific Publishing Co Pte Ltd and
the authors.
Copyright @ 2002 lOP Publishing Ltd.

at small scales is evident, highlighting the self-similarity contained within the two
systems. Wong and Chen (2001) provide a clear, well illustrated introduction to the
Morlet wavelet transform of the nonlinear and chaotic behaviour of multi-degree-
of-freedom systems. They introduce the transform explaining its use, using a variety
of simple test signals before considering single, then multiple, oscillator systems based
on coupled Duffing oscillators. These systems are considered when subjected to both
single impulses and continuous forcing. They conclude by examining the chaotic
response of the single-degree-of-freedom Duffing oscillator, contrasting its modulus
and phase plots with nonchaotic cases. Their paper is well worth consulting for the
many clearly presented diagrams used to illustrate the discussion.
In their comprehensive paper on adaptive strategies for recognition, noise filter-
ing, control, synchronization and targeting of chaos, Arrecchi and Boccaletti (1997)
have shown how to employ the Daubechies D20 wavelet in a noise reduction strategy
to separate noisy contributions from deterministic parts of chaotic data sets. As
an example (Boccaletti et aI, 1997), they used the Mackey-Glass delay differential
equation configured to produce 7.5-dimensional dynamics with both white and
co loured noise added separately. They detailed the effect of wavelet threshold level
on the ability of their method to determine the underlying dynamics of the system
and recommended it for easy implementation in experimental situations as it does
not require information on the correlation properties of the additive noise. Grzesiak
(2000) has also employed a wavelet-based denoising technique to filter chaotic data.
Grzesiak determined the efficiency of the technique by comparing the correlation
dimension of noisy and clean data generated for a variety of chaotic dynamical
systems and found that the wavelet method was comparable with other methods
commonly used to filter chaotic data. Permann and Hamilton (1992) have performed
a wavelet analysis of the time series from a Duffing oscillator in both periodic and
chaotic mode. They were able to detect small-amplitude harmonic forcing terms,
even when the data were highly nonstationary and of short duration. See also the
paper by Permann and Hamilton (1994) who investigated the chaotic behaviour of
a weakly damped and weakly forced Morse oscillator using Daubechies D8 wavelets.
Lamarque and Malasoma (1996) have constructed wavelet-based exponents, similar
to Lyapunov exponents, for the identification of chaotic behaviour. Cao et al (1995)
have used wavelet networks to make both short- and long-term predictions of the
time series from chaotic systems. Systems they investigated include the Mackey-
Glass equation, Lorenz system, and the U shiki and Ikeda maps. In addition, they
modified the Ikeda map by using one of its parameters as a variable. In this way
they were able to investigate parameter-varying systems. Allingham et al (1998)
have used a hybrid system to model time series from chaotic dynamical systems.
Their system combines continuous optimization with a wavelet matching pursuit
method. Gamero et al (1997) have analysed the Lorenz equations displaying chaotic
motion using their multiresolution-based information measures for dynamical
signals. They also considered the Henon map and an EEG signal displaying an
epileptic seizure. Heidari et al (1996) have studied the wavelet transform of
deterministic self-similar signals, suggesting its use as a method to interrogate the
noisy strange attractor of the Henon map. Masoller et al (1998) have interrogated
experimentally measured low-frequency intensity fluctuations from a semiconductor
Copyright @ 2002 lOP Publishing Ltd.

laser operated near threshold. Using the discrete wavelet transform, they compared
their results with the wavelet analysis of a theoretical model, showing that the
differences between the results were confined to the 'fast and short' components of
the signal. Another experimental study by Russo et al (2000) has concentrated on
the dynamical regimes optically induced in a nematic liquid-crystal as the intensity
of an incident laser beam increases. Their paper contains a series of Morlet wavelet
transform plots generated at increasing values of the control parameter, which
illustrates the jump to stochastic behaviour above a known threshold. Their
wavelet-based analysis suggests the presence of a transition towards a chaotic state.
5.6 Non-destructive testing
Non-destructive testing (NDT) is concerned with the interrogation of underlying
structural integrity using procedures which do not impair in any way the intended
performance of the structure during and after examination. Sonic echo testing is a
common method employed in the NDT of structural elements. It involves striking
the test specimen (e.g. structural element or material specimen) with an instrumented
hammer which records both the input pulse (strike) and subsequent response of the
specimen. This response is interpreted as an indirect measurement of the integrity
of the specimen. A typical velocity trace from such a test on a foundation pile is
shown in figure 5.23 (Watson et aI, 1999). A schematic of the sonic pulse transmission
through the pile is shown in the figure. For such a heavily damped system there are
rarely multiple longitudinal reflections and the frequency dependences of the group
velocities are negligible. Therefore, the temporal isolation of the signal features
is more important than their frequency decoupling and hence a Mexican hat was
used which is more temporally compact than the standard Morlet (5 < Wo < 6) or
harmonic wavelets used in the study of free beam and plate vibrations. The scalogram
corresponding to this velocity trace is shown below. The pile is 11 m long and the
velocity of the stress wave through the pile is 3800 m S-l. Thus we would expect to
see the reflection of the end of the pile occur 2 x 11/3800 == 0.0058 s after the initial
impulse. The pile toe reflection shows up particularly well in the top right-hand
quadrant of the scalogram, as it has a distinctively different shape and appears
lower down the scalogram from the initial oscillations which occur at a dilation
around a == 10- 4 . These oscillations, seen to occur in the top left-hand quadrant of
the scalogram just after the input initial pulse, are known as ringdown and are in
fact the surface oscillations of the pile head due to the hammer impact. Figure 5.24
contains the reconstructed traces of both wavelet and Fourier filtered traces.
Simple scale dependent wavelet filtering was employed where the transform compo-
nents at a scales less than 0.0001 were set to zero and an inverse wavelet transform
performed. The wavelet filtered trace is shown in the top left quadrant of figure
5.24. The Fourier low pass filter cut-off frequency was set to 2.25 kHz, as is the
case in practice. The Fourier filtered trace is shown in the top right quadrant of
figure 5.24. The two lower plots in the figure show a zoomed-in section of the
upper traces in region of the pile toe reflection. Comparing the filtered traces of
figure 5.24, it can be seen that the wavelet filtering separates the ringdown oscillations
Copyright @ 2002 lOP Publishing Ltd.

0.01
! .... input pulse
 echo from
OCI
a 0.005 pile toe
'-"
 .".
......
u
0
........
QJ 0
;>
-0.005
o
0.002
0.004
time (s)
0.006
0.008
0.0001
-..
00
"'C
s:::
0
U
QJ
00

=
s:::
0 0.001
......

........
......
"'C
o 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
location 'b' (seconds)
Figure 5.23. Wavelet transform decomposition of a sonic echo signal. Schematic of sonic echo testing
of a foundation pile (top left). A Fourier filtered velocity trace (top right) and corresponding wavelet
transform plot (bottom) of an 11 m pile in stifflvery stiff clay.
3 0 0.002 0.004 0.006 0.008 3 0 0.002 0.004 0.006 0.00
6-10- 6-10-
]' 4-10- 3 rc= J e (!time (s ) ]' 4-10- 3  time (s)
 2_10- 3  2_10- 3
» 0-10 0 » 0-10 0
-2-10- 3 -2-10- 3
0.004 0.005 0.006 0.007 0.008 0.004 0.005 0.006 0.007 0.00
5_10-4 1 /' _' _ ' ..... '/_tirn ( 5 _10-4 1 1 I I I I
]' 3-10-4 time (s) ]' 3-10-4 / \!\/.._tl::\ime (s)
::" 1-10-4 ::" 1-10-4
Q) -4 Q) -4 
::;- -1-10 ::;- -1-10  "-/ 
» -3-10-4 » -3-10-4
-5_10-4 -5_10-4
Figure 5.24. Wavelet and Fourier filtering of the sonic echo pile signal. Wavelet (left) and Fourier
(right) filtered traces for finite element generated pile test data shown in the previous figure. After
Watson et al1999 Journal of Shock and Vibration 6 267-272. With kind permission of IOS Press.
Copyright @ 2002 lOP Publishing Ltd.

o
600
500
400
300
,-, 200
! 100
 0
>, -100
-200
-300
-400
-500
0.002
0.004
0.006
0.008 0.01
time (8)
o 0.002 0.004
200 '
';;' 150
13 100
-::::: 50
 0
>, -50
-100
0.006
0.008 0.01
time (8)
o 0.002 0.004 0.006 0.008 0.01
,-, 300  I
'" 200 time (8)
:g 100
 0
>, -100
-200
Figure 5.25. Filtering of a field test sonic echo signal. Field test result (left) and when filtered: wavelet
(top right) and Fourier (bottom right). After Watson et al1999 Journal of Shock and Vibration 6267-
272. With kind permission of IOS Press.
from the pile toe feature much more effectively than Fourier filtering. Figure 5.25
shows a field signal, together with two filtered versions: one wavelet based, the
other Fourier based. This time a discrete orthonormal Daubechies D8 wavelet was
used. Again the wavelet-based filtering better separates the ringdown oscillations
from the toe feature. However, note that the dyadic nature of this discrete wavelet
transform can present problems in this respect due to its translation invariance.
Figure 5.26 illustrates a more sophisticated filtering method for removing both
erroneous ringdown artefacts from the pile signal based on the modulus maxima of
the scalogram. Figure 5.26(b) contains the finite element (FE) generated velocity
trace of an 11 m pile in stiff clay. A schematic of the pile is given in figure 5.26(a).
A simulated defect in the form of a reduction in section (necking) is present approxi-
mately one third of the way down the pile. The location of both the input pulse and
echo from the pile discontinuity are highlighted in the figure. The transform plot
associated with the wavelet decomposition of the trace is shown in figure 5.26(c).
Modulus maxima are found from the original wavelet transform using a simple
algorithm which scans across the transform plot scale by scale and identifies local
maxima and minima. The modulus maxima obtained for the scalogram of
figure 5.26(c) is shown in figure 5.27(a). We can see from the modulus maxima plot
that the large scale input pulse feature in the signal contains a ridge extending from
high to low frequencies. The other maxima lines do not extend as far down into
the low frequency range. A close approximation to the signal can be reconstructed
using only the maxima lines where the energy contained in the whole scalogram is
reapportioned to the maxima lines in the reconstruction. The reconstruction using
all the maxima lines in figure 5.27(a) is shown in figure 5.27(b). Figure 5.28 illustrates
the filtering of the initial signal using the modulus maxima. This is done in an anti-
clockwise manner from figure 5.28(a) to 5.28(d). The maxima lines are thresholded
at a frequency of 310Hz, shown in figure 5.28(b). All maxima lines which do not
extend down from the higher bandpass frequencies to this threshold level are removed
(figure 5.28(b)). Those which do extend down to and beyond the threshold are
retained. Only the retained lines (figure 5 .28( c)) are used to reconstruct the signal
(figure 5.28(d)). The threshold is chosen to be lower than the ring down artefact in
the signal, hence the maxima lines from the ringdown artefact will fall below the
Copyright @ 2002 lOP Publishing Ltd.

input pulse
/
0.04
3.3m
0.03
0.6m  reflection
rJ::I
! 0.02 /

......
u
0 0.01
..-;
Q)
7.8m ;>
0
-0.01 ;V
H 0 1 2 3 4
0.8m time (s) x 10- 3

(a) 0.4m (b)
10 3


'-"


>.
u

Q)
6-
Q)

10 2
o
1
2 3
time (s)
4
-10- 3
(c)
Figure 5.26. Sonic echo velocity trace from pile head and associated scalogram. (a) Schematic of a
foundation pile. A necking fault has been modelled approximately one third of the way down the
pile. (b) Sonic echo velocity trace taken from the pile head. (c) Wavelet transform plot of the
signal in (b). (Large positive components in black, large negative components in white.) Reprinted
from Watson and Addison (2002), Mechanics Research Communications (in press at time of
publication), with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

0.04

 10 3  0.03
rI.:I
........, !
]. 0.02

i .......
0
0 0.01
-
& (()
>
 102 0
-0.01
0 1 2 3 4 0 1 2 3 4
(a) time (s) x 10 3 (b) time (s) x 10 3
Figure 5.27. The reconstruction from only the scalogram modulus maxima lines. (a) Modulus maxima
plot derived lines from the scalogram in previous figure. (b) Reconstructed trace using only the
modulus maxima lines in (a). Reprinted from Watson and Addison (2002), Mechanics Research
Communications (in press at time of publication), with permission from Elsevier Science.
0.04 0.04
-.. 0.03 -.. 0.03
00
00 -
- S
S 0.02
 0.02 
 
..-
..- u
u 0 0.01
0 0.01 ......-I
......-I Q)
Q) >
:>
0 0
-001 -0.01
. 0 1 2 3 4 0 1 2 3 4
(a) time (s) x10- 3 (d) time (s) x10- 3
10 3 10 3
-.. -..
 
 
u u
]< ]<
>. >.
u u
s::: s:::
Q) Q)
& &
Q) Q)
 10 2  10 2
o
1
2 3
time (s)
4
x10- 3 (c)
(b)
o
1
2 3
time (s)
4
x10- 3
Figure 5.28. The partitioning of the modulus maxima lines for signal filtering. (a) Original data. (b)
Discarded modulus maxima lines. (Threshold shown as horizontal dotted line.) (c) Retained modulus
maxima lines. (d) Filtered data. Reprinted from Watson and Addison (2002), Mechanics Research
Communications (in press at time of publication), with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

threshold and be removed. In addition, noise, which also manifests itself as modulus
maxima restricted to high frequency regions, is also removed from the signal. The
resultant reconstructed trace shown in figure 5.28( d) illustrates how all ringdown
has been eliminated whilst retaining the pertinent signal features. In addition, the
retained features still contain their high frequency components and are not excessively
smoothed, as would be the case if bandpass filtered using Fourier techniques.
Figure 5.29 illustrates the use of the Morlet wavelet with low central frequencies
in the analysis of a highly oscillatory sonic echo signal where the pile toe is not
obvious in the time domain (Addison et aI, 2002a). When using central frequencies,
wo, less than 5 (fo < 0.8) the complete Morlet wavelet given by equation (2.36) in
chapter 2 must be used. Morlet wavelets with low central frequencies result in analyses
that are more 'temporal' than 'spectral' in that they are better at locating short dura-
tion temporal features than those with higher values of Wo (refer back to section 2.12,
chapter 2). This can be seen in figure 5.29, where the pile toe can be located in the
wavelet transform scalogram plots only at lower values of woo The location of the
pile toe is indicated both in the time signal and the lowest scalogram plot which
corresponds to the complete Morlet wavelet with Wo == 1.5 (i.e. fo == 0.238). It is
interesting to note that, although they do seem to be particularly useful for certain
tasks, the literature contains surprisingly very little on the use of complex wavelets
with few oscillations such as the complete Morlet wavelet of low central frequency
or the complex Mexican hat.
A variety of continuous wavelets were used by Abbate et al (1997) in a study of the
signal detection and noise reduction properties of the wavelet transform when used to
elucidate ultrasonic pulse-echo traces in steel specimens. They used a combination of
'pruning' and soft thresholding to reduce the noise from the signals. Pruning simply
sets wavelet coefficients outside a certain a-scale range to zero, i.e.
T(a, b) ==
o
T ( a, b)
o
for a < al
for al < a < a2
for a > a2
(5.3)
where the coefficients outside the range set by al and a2 are thought most likely to
come from noise. Pruning is effectively scale dependent thresholding within a band
of limits al and a2. The soft thresholding employed by Abbate and co-workers had
the usual form:
{ o
T(a,b) ==
s gn [ T ( a, b )] x (I T ( a, b ) I - A)
for IT(a,b)1 < A
for IT(a,b)1 > A
(5.4 )
where A is the threshold. Figure 5.30(a) shows an input signal used in the ultrasonic
testing of steel specimens. Figure 5.30(b) shows the same signal with added noise and
its wavelet transform plot is given in figure 5.30(c). The reconstructed signal after
pruning and thresholding is shown in figure 5.31(a) with the filtered wavelet transform
plot used for the reconstruction shown below in figure 5.31(b). Figure 5.32 contains
the signal from a cast iron sample together with its wavelet-filtered version. A Morlet
wavelet was employed and the filtering used both pruning and thresholding. Both
echoes are clearly detected without any other contribution from the acoustic noise.
Copyright @ 2002 lOP Publishing Ltd.

-..
00

Q) ....-1
"'C d
::s ::s
:E 
.b
 ....-1
C\S .£J


-..
N
::I:


g 10 4
Q)
&
Q)

-..
N
::I:


g 10 4
Q)
::s
C"
Q)

-..
N
::I:


g 10 4
Q)
::s
C"
Q)

-..
N
::I:


g 10 4
Q)
&
Q)

-..
N
::I:


g 10 4
Q)
::s
C"
Q)

o
2
4
6
8 10
time (ms)
12
14
16
Figure 5.29. A complete Morlet wavelet analysis of a sonic echo signal. The figure contains a sonic
echo signal taken from a pile, together with a sequence of scalograms generated using a complete
Morlet wavelet decomposition of the signal with the central frequency set to (from top to bottom)
Wo = 5.5, 4.5, 3.5, 2.5 and 1.5. After Addison et al (2002a). Reproduced with kind permission of
Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

flaw signal y (f)
(a)
1.0
0.5
o
-0.5
-1.0
o
500 1000 1500 2000 2500 3000
(b)
1.5.
1.0
0.5
'::' 0
';; -0.5
-1.0
-1.5-
o
500 1000 1500 2000 2500 3000
.
.

s::: .....
s:::
0 Q)
..... .....
 u
,......; iB
.....
'"0 Q)
(c) 0
u
2 ._...... ..- '_,__. ..,-. _ _. ...
1 ---_.'. ':1(;'.: ..''-o ":. g? "':., ._. ,'- --;0.--' ..s:; .
o ......-.- ''...; ,.-.' '. ',,_.' ...--...... ...,.' - I-
-1
-2
-3
-4
o
.
.
.
500
1000
1500
2000 2500
3000
time
Figure 5.30. Wavelet analysis of a pulse echo reflection signal. ( a) Original signal. (b) Signal in ( a) with
added acoustic noise generated by spherical voids in steel. (c) The wavelet transform contour plot of
the signal in (b). After Abbate et al1997 IEEE Transactions on Ultrasonics and Frequency Control
44(1) 14-26. (Q) IEEE 1997.
Chen et al (1996) have proposed a wavelet-based technique to improve the signal to
noise ratio of ultrasonic inspection signals obtained from coarse-grained stainless
steel specimens. Chen et al (1999c, 2000) have detailed both wavelet and wavelet
packet methods for denoising ultrasonic signals for the non-destructive evaluation
1
0.5
-.. 0,
......


CI.:I -0.5
(a) -1.0
0
 2
s::: ..... 1
s::: 0
0 Q)
..... .....
 u -1
.....
,......; 
..... -2
'"0 Q)
(b) 0 -3
u
0
1\ 
-

.
500 1000 1500 2000 2500 3000
, .
0-
,
500 1000 1500 2000 2500 3000
time
Figure 5.31. Filtered pulse echo reflection signal. (a) The reconstructed signal obtained from the
filtered wavelet transform plot shown in (b). After Abbate et al1997 IEEE Transactions on Ultraso-
nics and Frequency Control 44(1) 14-26. (Q) IEEE 1997.
Copyright @ 2002 lOP Publishing Ltd.

(a)
40
-.. 20
......

00 0
-..
C\$

-20
-40
0
40 '
-.. 20
......

 0
00
-..
,D -20

-40
0
1000
2000
3000
4000
5000
(b)
1000
2000
3000
4000
5000
time
Figure 5.32. Wavelet of a pulse echo reflection signal taken from a cast iron sample. (a) Ultrasonic
signal. (b) Output after wavelet filtering clearly showing the two echoes. After Abbate et al1997
IEEE Transactions on Ultrasonics and Frequency Control 44(1) 14-26. (Q) IEEE 1997.
of steel samples with known defects. Cho et al (1996) employed Morlet wavelets to
detect subsurface defects in steel test specimens from non-contact laser ultrasonic
signals. Staszewski et al (1997) have described a wavelet-based signal processing
method to enhance defect detection in a carbon fibre composite plate interrogated
using ultrasonic Lamb waves and incorporating an optical fibre receiver. Wu and
Chen (1999) have also used the Morlet wavelet to analyse non-contact laser ultrasonic
signals from epoxy-bonded copper-aluminium layered specimens. Their study
focused on the detection of un bonded regions. A Mexican hat wavelet was employed
by Guilbaud and Audoin (1999) in the interrogation of laser-induced ultrasonic
signals used to measure stiffness coefficients in a viscoelastic composite material.
They found that the reliability of their method may justify its use in the field of
material behaviour characterization.
A number of other authors have developed wavelet-based tools to aid the
interpretation of NDT signals. Shyu and Pai (1997) have performed impact-echo
tests on free standing concrete cylinders using Daubechies wavelets. Tang and Shi
(1997) used wavelet techniques to detect and classify a variety of welding defects
from NDT signals. Doyle (1997) has employed a wavelet technique to identify the
impact force on structures using a knowledge of the structure and its response to
the force. Hamelin et al (1996) used wavelets to analyse eddy current signals. They
analysed the complex and real parts of the signal separately in order to develop a
classification scheme to inspect the transformed signals for the location of distinct
maxima and associated phase. Pierri et al (1998) investigated the use of two-
dimensional Haar wavelet transforms in eddy current NDT, and Lingvall and
Stepinski (2000) have described an automatic method for the detection and classi-
fication of cracks located in aircraft riveted lap-joints during eddy current inspection
which employs Coiflet wavelets. Qi (2000) has described a wavelet-based method to
analyse acoustic emission (AE) signals in a study of material fracture behaviour. Qi
employed Daubechies wavelets to analyse experimental AE signals and found that
Copyright @ 2002 lOP Publishing Ltd.

wavelet-based techniques better approximate the relationship between the stress and
stress intensity factor than do classical techniques. The AE behaviour of reinforced
concrete beams tested under flexural loading was investigated by Y oon et al (2000)
using both Fourier- and Morlet-based wavelet methods. They found that the Fourier
spectra and wavelet transforms of the AE signals gave useful information about the
relationship between the damage mechanisms of the concrete (e.g. micro cracking,
localized cracking, flexural cracking and shear bond cracking) and the AE response.
An examination of the integrity of thin coated foil used in the food packaging indus-
try was conducted by Futatsugi et al (1996) using both AE monitoring and micro-
scopic observation. AE signals were omitted by the cracking of the surface coating
(SiO x film) and also from its delamination from the foil. The threshold tensile
strain necessary to cause the first fracture, estimated from the AE signals, agreed
well with the strain determined using a laser microscope.
An experimental study of crack detection in metallic structural elements using
fourth-order Daubechies wavelets has been carried out by Biemans et al (1999).
They instrumented a pre-cracked rectangular metal plate with piezoceramic sensors
and subjected it to both static and dynamic tensile loading. A statistical measurement
based on the logarithmic wavelet variance of the strain data was used as a damage
index. This parameter provided an insight into the scale dependent changes in
energy of the strain data from the piezoceramic sensors. Wang and Deng (1999)
have used wavelets to probe the spatial profiles of damaged cantilevered beams
under static and dynamic loading. Their results clearly show that the location of a
structural crack in the beam can be pinpointed through the Haar wavelet transform
coefficients. They found similar results using Morlet-based transforms. See also the
related papers by Deng and Wang (1998), Liew and Wang (1998) and Quan et al
(1999).
Morlet wavelet transforms have been employed by Li and Berthelot (2000) to
analyse pulse-echo signals from thick annular waveguides. They developed a local
spectral-temporal wavelet energy measure by integrating the energy density
scalogram over a box of limited extent in scale and location. They found this energy
measure to be particularly good at localizing cracks in faulty annular components
and tested it successfully on data from both an annular waveguide with a machined
crack and on a partially annular component of the pitch shaft of an H-46 helicopter.
Marwala (2000) has developed a 'committee of neural networks' technique which
employs frequency response functions, modal properties and wavelet transform
data simultaneously to identify damage in structures. The method was tested on
synthetic data from three coupled oscillators and then used to identify the damage
in seam-welded cylindrical shells. Gros et al (2000) have presented results concerning
the fusion of images from multiple NDT sources gathered during the inspection of a
composite material damaged by impact. The images were fused using a variety of
techniques, including one based on Daubechies D8 wavelet transforms, in order to
improve the defect detection and provide a more accurate measurement of defect
dimensionality. Two-dimensional Morlet wavelets have been used by Li (2000) to
detect partial fringe patterns generated in NDT interferometry induced by defects.
The author found that the wavelet method was suitable for both holographic
interferometry and electron speckle pattern interferometry. Chan et al (2000b) have
Copyright @ 2002 lOP Publishing Ltd.

used wavelet packet denoising within a digital speckle correlation method to detect
defects in multilayer ceramic capacitors in surface-mounted printed circuit boards.
5.7 Surface characterization
The characterization of engineering surfaces is pertinent to a number of engineering
fields providing quantitative information on the formation process of the surface,
for example the manufacturing process used to form a machine component or the
fracture process causing a rugged crack surface. Surface topography is one of the
most important factors affecting the performance of manufactured components. It
can be related to a number of pertinent engineering aspects such as wear, lubrication,
friction, corrosion, fatigue, coating, paintability, etc.
Two-dimensional biorthogonal wavelet transforms have been used by Jiang et al
(1999) to probe the surface topography of orthopaedic joint prostheses. They used
three wavelet-based parameters to characterize the surface: roughness, waviness and
form. The roughness of the surface was defined as the detailed surface found through
the inverse transform of the thresholded wavelet coefficients within a band of the
smallest scale indices. Similarly, the waviness of the surface was defined as the detailed
surface found through the inverse transform of the thresholded wavelet coefficients
within a band of the next smallest scale indices. The thresholding method employed
in both cases simply limited the maximum absolute value of the coefficients to three
times their standard deviation at each scale. This was carried out to disassociate the
overall surface topographic features from localized peaks, pits and scratches, as it was
assumed that each detail coefficient belonging to the roughness or waviness follows a
Gaussian distribution. (Hence, a coefficient is very unlikely to appear outside three
standard deviations of the distribution.) The form of the surface was defined to be
the original surface minus the detail signals over the roughness and waviness scales.
Figure 5.33 contains the original centre profile of a ceramic femoral head together
with the multiresolution approximations of the profile after wavelet decomposition
with the biorthogonal wavelet pair. The detail signals at scales 1 to 4 are shown on
the right-hand column. These scales cover the roughness wavelengths and the
addition of these four detailed signals gives the roughness profile shown at the top
right-hand of the figure. Figure 5.34 shows three three-dimensional plots of
the same surface separated into its roughness, waviness and form components.
M ultiscalar topographical features such as peaks, pits and scratches on the surface
were then defined by Jiang and his colleagues by hard thresholding the coefficients
using two standard deviations of the coefficient amplitude at each scale as the
threshold. An example of the multi scalar topographical features defined in this way
is given in figure 5.35. In a later paper, Jiang et al (2000) have provided details of a
wavelet-based analysis of both rolled steel sheets and ceramic femoral heads. See
also Chen et al (1999d), who have developed a similar wavelet-based method to
decompose surface data sets into a surface roughness component and a wavelet
reference surface.
Chen et al (1995) have analysed surfaces produced by typical manufacturing
processes using Daubechies wavelets, citing the advantage of the space-scale
Copyright @ 2002 lOP Publishing Ltd.

m
1.30
1.15
1.00
0.85
0.70
o
0.0605 0.1210 0.1815 0.2420
the original centre profile mm
A4 = reference profile
o
0.0605 0.1210 0.1815 0.2420
the approximations
j=l
j=2
j=3
j=4

0.006 .
0.003
0.000
-0.003
-0.006
o
0.0605 0.1210 0.1815 0.2420
the roughness centre profile mm
(D! +D2+D3+D4)
o
D4
0.1210
the details
0.1815 0.2420
mm
0.0605
Figure 5.33. Multiscalar decomposition of a ceramic femoral head using a two-dimensional biorthogo-
nal wavelet pair: centre profile. This material has been reproduced from the Proceedings of the Institu-
tion of Mechanical Engineers, Part H, Journal of Engineering in Medicine 1999 213 49-68, figure 6A,
by Jiang et aI, by permission of the council of the Institution of Mechanical Engineers.
localization properties of the wavelet transform over other techniques. A number of
continuous wavelets are considered by Lee et al (1998) in a study of the morphological
characterization of engineering surfaces. They employed the Morlet and Mexican hat
(second derivative of Gaussian) wavelets, as well as the less common Barrat wavelet
and an eighth derivative of a Gaussian function, to investigate the potential applica-
tions of wavelet decomposition in assessing the multi scale features of the engineered
0.05
o
0.10 ]
0.05.
o
Figure 5.34. Reconstructed surface topographies of a ceramic femoral head in different transmission
bands: rough (left), wavy (centre) and form (right). This material has been reproduced from the
Proceedings of the Institution of Mechanical Engineers, Part H, Journal of Engineering in Medicine
1999 213 49-68, figure 7 A, by Jiang et aI, by permission of the council of the Institution of
Mechanical Engineers.
Copyright @ 2002 lOP Publishing Ltd.

slope intensity image
Figure 5.35. Multiscalar topographical features (peaks, pits and scratches) in the equatorial region of
the ceramic femoral head. This material has been reproduced from the Proceedings of the Institution
of Mechanical Engineers, Part H, Journal of Engineering in Medicine 1999 213 49-68, figure 8A, by
Jiang et aI, by permission of the council of the Institution of Mechanical Engineers.
surface, considering both its manufacturing and functional aspects. Song et al (2000)
have developed a technique for the inspection of surface mount devices in the electro-
nic industry using modified Haar wavelets. Dogariu et al (1994) have described the
light scattered by a slightly rough surface in terms of two-dimensional wavelet trans-
forms, and Zhuang et al (1998) have developed a laser-based noncontact system for
the inspection of pipe inner walls where wavelet-based decomposition of the reflected
signal from the pipe wall is used to characterize the surface texture as form, error,
waviness, roughness and sporadic scratches. Jasper et al (1996) have employed
adaptive wavelet bases to capture texture information and locate defects in woven
fabrics from their images, and Lin and Xu (2000) have introduced a wavelet-based
method to quantify the fuzziness of woven and knitted fabrics from their images.
The fractal structure of fractured granite surface profiles is examined using wavelets
by Simonsen et al (1998), and Addison and Watson (1997) have detailed the use of
wavelet transforms to analyse rough surfaces modelled using fractional Brownian
motions. In an analysis of the fractal properties of cracked concrete surfaces,
Dougan et al (2000) have compared both wavelet and Fourier-based power spectral
methods with traditional methods for determining fractal parameters. The multi scale
characterization of pitting corrosion damage has been carried out by Frantziskonis
et al (2000) using Daubechies wavelets with four vanishing moments (i.e. D8s) to
determine the Hurst exponent associated with the geometry of the corrosion pits.
The submicron surface roughness of anisotropically etched silicon has been analysed
using Meyer wavelets by Moktadir and Sato (2000) who found a scaling exponent for
the surface close to 0.5. Srinivasan and Wood (1997) used Daubechies D12 wavelet
transforms as a tool to compute the relevant fractal parameters in a fractal-based
approach to geometrical tolerancing (see also Tumer et aI, 1995). The authors focused
on mechanical tolerances applied to physical-product features, such as machined
parts and assemblies. They used the roundness of ballbearing elements as a design
example. Chapter 7 contains more information concerning the use of wavelet
Copyright @ 2002 lOP Publishing Ltd.

transform-based methods for the analysis of both geophysical topographic features
and surfaces exhibiting fractal structure.
5.8 Other applications in engineering and further resources
5.8.1 Impacting
The impacting of detector tubes in boiling water nuclear reactors has been detected by
Racz and Pazsit (1998) using Haar wavelets. They thresholded the Haar wavelet co-
efficients to remove noise and reconstructed the signals to produce a time series of the
impacts. The threshold was set to four times the standard deviation of the coefficients
whereby all coefficients below this level were set to zero. Reconstruction using only
the remaining large coefficients gave an indication of the impact events in the
signal. Racz and Pazsit investigated both modelled data and in situ measurements.
The top trace in figure 5.36 shows a detector signal taken from the Swedish
Barseback-l reactor where no noticeable vibrations were observed to occur. The
fuel box vibration signal, after filtering out the low frequency detector string vibra-
tions, is shown in the middle of the figure. This signal is then transformed and filtered
using the Haar transform as described above. The reconstructed signal is shown in
the bottom plot. A few intermittent spikes can be seen to occur over the signal
length. The paucity of spikes in this wavelet-filtered signal can be contrasted to the
5
detector si nal LPRM183
o
_--i..-..___
-5
o
500
1000 1500 2000 2500 3000 3500 4000
time
fuel box vibration (after stop filtering)
0.1
o
-0.1
o
500
1000 1500 2000 2500 3000 3500 4000
time
fuel box vibration; severity =0.04069
0.1
o
-Hi. ---.-+t-
-0.1
o
...._. ...1--____.L.___._
500
1000 1500 2000 2500 3000 3500 4000
time
Figure 5.36. Signal from detector tube experiencing no vibrations. The filtered detector signal (middle
trace) is shown below the original signal (top). The reconstructed signal from the thresholded Haar
coefficients is shown in the bottom trace. Reprinted from Racz and Pazsit 1998 Annals of Nuclear
Energy 25(6) 387-400. Copyright (1998), with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

detector signal LPRM033
o
0.5
-0.5
o
500
· .1---__
1000 1500 2000 2500 3000 3500 4000
time
fuel box vibration (after stop filtering)
,
o
0.5
-0.5
o
500
1000 1500 2000 2500 3000 3500 4000
Time
fuel box vibration; severity = 0.765
0.5
o
-0.5
o
500
1000. 1500 2000 2500 3000 3500 4000
time
Figure 5.37. Signal from detector tube which is known to have experienced impacting. Reprinted from
Racz and Pazsit 1998 Annals of Nuclear Energy 25(6) 387-400. Copyright (1998), with permission
from Elsevier Science.
high occurrence of spikes in the bottom plot of figure 5.37. This signal was taken from
a detector tube which was found from subsequent inspection to have experienced
impacting. The authors determined a parameter, S, for the severity of impacting
based on a normalized value of the estimated impact rate. High values of the severity
S indicate the occurrence of strong vibrations and the possibility of impacting.
5.8.2 Data compression
An investigation of the use of wavelet-based methods for the compression of vibration
signals has been carried out by Staszewski (1998b). He compared Fourier-based
compression with wavelet-based compression (using Daubechies D4 and D20 wavelets)
for a variety of signals. Figure 5.38 shows a periodic signal with its associated Fourier
frequency spectrum and wavelet coefficients. We can see by looking at the Fourier and
wavelet domain representations of the signal that the Fourier case is more compact.
In fact, we would expect Fourier representation to favour the concise representation
of a periodic signal. Figure 5.39 shows the normalized mean square error (MSE) of
the reconstructed signal as a function of the number of coefficients (both Fourier and
wavelet) used in its reconstruction. This plot confirms what we can see by eye. That
is, for a periodic signal, a much lower error results from using the same number of
Fourier coefficients as wavelet coefficients. The normalized MSE is defined as
1 00  / 2
MSE(x) ==   (Xi - xJ
NrJ"x i=l
(5.5)
Copyright @ 2002 lOP Publishing Ltd.

1.0
Q)
] 0.5
.....
,.....;
 0.0
Cd
s:::
.-0.5
CI:)
-1.0
o
(a)
128
256
data samples
384
512
Q)
"'C
::s
...
:.& 0.40

S
e 0.20
...
u
Q)
0..
CI:)
0
(b)
2.0
Q)
"'C
B
.....
,.....;
 1.0
...
Q)
,.....;
Q) 0.5


0
(c)
64
128 192
frequency orders
258
128
256 384
wave coefficients
512
Figure 5.38. Periodic data used for compression. (a) Time domain. (b) Frequency domain. (c) Wavelet
domain; dashed line indicates wavelet coefficients decreasing according to the amplitude level. (Note
that Daubechies D20 wavelets are used.) After Staszewski (1998b). Reproduced with kind permission
of Academic Press Ltd.
where Xi are the components of the original signal of length N, () is the variance of
the signal, x is the reconstructed signal using selected coefficients, and the factor of
100 gives MSE(x) as a percentage. The situation depicted in figures 5.38 and 5.39
changes when a transient signal is compressed. An example of such a signal is
given in figure 5.40(a). We can see from figures 5.40(b) and (c) that the Fourier spec-
trum for this signal is relatively broad band and the dominant wavelet coefficients are
relatively localized. For this transient signal, the mean square error plot indicates that
far fewer wavelet coefficients are required to form a good approximation to the
original signal (see figure 5.41). Staszewski went on to use wavelet compression as
a method for feature detection in two data sets: ultrasonic flaw detection signals
and gear vibration data. He used a variety of methods to select the most appropriate
coefficients for this task including: simple thresholding; a priori knowledge of the
Copyright @ 2002 lOP Publishing Ltd.

90.0
30.0
\
\
\
\
\
\
\
\
\
\
"-
....
....
....
..........-.-.
---
---
-.. 60.0





"
o
100
200
300
number of coefficients
Figure 5.39. Compression performance for periodic data. MSE plotted as a function of number of
wavelet coefficients used for compression. Solid line = D4; dashed line = D20; dash-dot line = FFT.
After Staszewski (1998b). Reproduced with kind permission of Academic Press Ltd.
1.0
Q)
'"C 0.5
.a
.....
........
i 0.0
Cd
s:::
OJ)
.(;j -0.5
-1.0
o
(a)
0.03
128
256
data samples
384
512
Q)
'"C
.a
.....
1 0 . 02
S
B 0.01
u
Q)

00
o
(b)
1.5
64
128
frequency orders
192
256
Q)
'"C
.a
;.::: 1.0

cd
........
-+->
Q)
........
Q)
rd

o
(c)
128
256 384
wavelet coefficients
512
Figure 5.40. Transient data used for compression. (a) Time domain. (b) Frequency domain. (c) Wave-
let domain; dashed line indicates wavelet coefficients decreasing according to the amplitude level.
After Staszewski (1998b). Reproduced with kind permission of Academic Press Ltd.
Copyright @ 2002 lOP Publishing Ltd.

\
,
,
"
"-
"
........
.........
.........
........
........
................
"-..-
Figure 5.41. Compression performance for transient data. MSE plotted as a function of number of
wavelet coefficients used for compression. Solid line = D4; dashed line = D20; dash-dot line = FFT.
After Staszewski (1998b). Reproduced with kind permission of Academic Press Ltd.
system; genetic algorithms; and the temporal location of coefficients. Tanaka et al
(1997) have also approached the problem of data compression of mechanical vibra-
tion data. They compressed a variety of signals taken from rotating machinery
using Daubechies wavelets (D4 to D20) and detailed the signal distortion at various
compression ratios. Desforges et al (1998) have described their use of wavelet trans-
forms to compress data for the practical application of probability density function
(PDF) estimation. The original data sets were transformed and only a limited
number of wavelet coefficients were retained and subsequent PDF estimates made
using these values rather than the original data sets themselves. The retained co-
efficients were chosen using a genetic algorithm. They applied their combined data
compression and PDF estimation approach to faulty gearbox signals and military
target radar data.
5.8.3 Engines
Thomas et al (1997) have employed a wavelet network to detect the occurrence of
engine knock from vibration signals taken from spark-ignition engines. They used
40 Morlet wavelets of predetermined scales and locations to decompose the signal
in an attempt to differentiate between three classifications: (1) absence of knock,
(2) increasing knock, and (3) heavy knock. Rasping noises in automotive engine
exhaust ducts due to abrupt accelerations have been characterized by Ayadi et al
(2001) using STFT, CWT and wavelet packet decompositions of the internal tailpipe
pressures. Liu and Ling (1999) have developed a modified matching pursuit method
for machinery fault diagnosis which uses a mutual information measure to select the
best wavelets from the dictionary, i.e. those which carry important information with
little redundancy. They applied their method to the detection of two diesel engine
malfunctions-injection timing advance and 'blow-by'-and found that the tech-
nique performed better than principal component analysis which is widely used in
practice. Sullivan et al (1999) have reported on their examination of the turbulent
velocities within the cylinder head of a spark-ignition engine. They employed
Copyright @ 2002 lOP Publishing Ltd.

continuous Mexican hat wavelet transforms to produce energy maps and investigated
correlations within the flow. In addition, they used Daubechies D4 wavelets in a fast
discrete wavelet transform method to subtract the mean from the instantaneous velo-
city field to give the turbulence velocities, comparing the resulting velocity-time series
with ensemble-averaged and cyclic-averaged methods. See also the related paper by
Ancimer et al (2000). H6ss et al (2000) have detailed a study of the stall inception
behaviour of a jet engine which employed Daubechies wavelets and other analysis
tools to interrogate pressure measurements taken within the engine's compressor
system. They identified three different types of stall inception processes for
undistorted inlet flow and suggested that an active avoidance control strategy may
be devised using wavelet analysis and extended statistical evaluation.
5.8.4 Miscellaneous
Williams and Amaratunga (1994) have provided a comprehensive introduction to the
use of discrete wavelet transforms in engineering which includes their role in data
analysis and the solution of partial differential equations. Piezoelectric accelerometers
have been used by Hale and Adhami (1998) to measure the highly nonstationary
vibration data collected from helicopter missiles during manoeuvres. They analysed
the signals using both wavelets and wavelet packets and found these methods to be
excellent candidates for establishing vibration specifications for ensembles of non-
stationary events. Le- Tien et al (1997) have detailed the wavelet analysis of a radar
echo signal from a spinning rotor. Nygaard and Grue (2000) have described a
wavelet-based method for the computation of wave properties and hydrodynamic
forces on arrays of floating bodies. Sureshbabu and Farrell (1999) have developed
a wavelet-based system identification method for nonlinear control and Patton and
Marks (1996) have detailed a one-dimensional finite element based on Daubechies
D18 wavelets. The application of wavelet methods to data analysis problems in
electrical and chemical engineering is dealt with briefly towards the end of chapter 7.
Copyright @ 2002 lOP Publishing Ltd.

Chapter 6
Medicine
6.1 Introduction
In this chapter we review the many areas of medical science where the wavelet trans-
form has made an impact. We begin with the ECG signal, where attempts have been
made using wavelet methods to determine its characteristic points, compress it, detect
abnormalities, characterize heart rate variability, and probe a variety of arrhythmias.
The next section considers the wavelet as a potential diagnostic tool for neuroelectric
waveforms: the EEG, evoked potentials and event-related potentials. Medical sounds
are then examined, including the sound from the turbulence due to arterial blockage
as well as the sound of the heart itself, respiratory sounds and the acoustic response of
the ear. In addition, ultrasonic time-series signals are covered in this section (although
not ultrasonic images which are left to the next section on medical images). The
section on medical images considers the application of wavelet transform to ultra-
sonic, radiographic and optical images. There then follows a short section on the
analysis of blood flow and blood pressure data before the final section containing a
selection of other medical applications of the wavelet transform, including the wavelet
analysis of DNA, the electromyograph, sleep apnoea syndrome, chronobiological
rhythms, the fractal structure of foetal breathing rates, and more. This final section
ends by providing details of other resources concerned with the application of the
wavelet transform and related methods to medical signals.
6.2 The electrocardiogram
Muscular contraction is associated with electrical changes known as depolarization.
The electrocardiogram (ECG) is a measure of this electrical activity associated with
the heart. The ECG is measured at the body surface and results from electrical
changes associated with activation first of the two small heart chambers, the
atria, and then of the two larger heart chambers, the ventricles. The contraction
of the atria manifests itself as the 'P' wave in the ECG and contraction of the
ventricles produces the feature known as the 'QRS' complex. The subsequent
return of the ventricular mass to a rest state-repolarization-produces the 'T'
Copyright @ 2002 lOP Publishing Ltd.

R
Q S
Figure 6.1. A schematic of the ECG exhibiting normal sinus rhythm. Note that the shape of each
feature can vary depending on the configuration of the ECG leads.
wave. Repolarization of the atria is, however, hidden within the dominant QRS
complex. Figure 6.1 shows a schematic of the ECG waveform for normal sinus
rhythm. Analysis of the local morphology of the ECG signal and its time-varying
properties has produced a variety of clinical diagnostic tools. In this section we
review the application of the wavelet transform to the interrogation of the ECG
signal.
6.2.1 ECG timing, distortions and noise
Producing an algorithm for the detection of the P wave, QRS complex and T wave in
an ECG is a difficult problem due to the time-varying morphology of the signal
subject to physiological conditions and the presence of noise. Recently, a number
of wavelet-based techniques have been proposed to detect these features. Senhadji
et al (1995) compared the ability of wavelet transforms based on three different wave-
lets (Daubechies, spline and Morlet) to recognize and describe isolated cardiac beats.
Sahambi et al (1997a,b) used a first-order derivative of the Gaussian function
(figure 6.2(a)) as a wavelet for the characterization of ECG waveforms. They used
100
80
60
40
20
o
-20
-40
-60
-80
(a)
..I I'" I
x(t)
Wf(2 1 ,t)
Wf(2 2 ,t) --ir.
Wf(2 3 ,t)
Wf(2 4 ,t)
.r- a1r-
I
---. T--
A--
time
(b)
x(t)
zero crossing ..  _ max1Illa
Wf(2 4 ,t) "- '" .or '. .
. . .
-+- mInIma
Figure 6.2. Characterizing the ECG using the wavelet transform. (a) A Gaussian and its first derivative
as a wavelet. (b) ECG signal, its wavelet transforms at scales 2 1 , 2 2 , 2 3 , 2 4 , and maxima, minima and
zero crossing of the wavelet transform at scale 2 4 . The vertical line above the ECG signal indicates the
position of the QRS complex, as detected by the algorithm ofSahambi et al. After Sahambi et al1997a
IEEE Engineering in Medicine and Biology 16(1) 77-83. (Q) IEEE 1997.
Copyright @ 2002 lOP Publishing Ltd.

x(t)
I
1-
I
Wf(2 1 ,t)
,
..ti...
----1,....
r

Wf(2 2 ,t) ---/I,-
+

..
-  ---
Wf(2 3 ,t)
Wf(2 4 ,t)
Figure 6.3. ECG signal with baseline drift and its wavelet transforms at scales 2 1 , 2 2 , 2 3 , 2 4 . The
vertical line above the ECG signal indicates the position of the QRS complex, as detected by the
algorithm of Sahambi et al. After Sahambi et al1997a IEEE Engineering in Medicine and Biology
16(1) 77-83. (Q) IEEE 1997.
modulus maxima-based wavelet analysis to detect and measure various parts of the
signal, specifically the location of the onset and offset of the QRS complex and P
and T waves. Note that for an anti symmetric wavelet, such as the first derivative of
the Gaussian, dominant peaks in the signal correspond to zero crossings in the wave-
let transform. (For symmetric wavelets, dominant peaks in the signal correspond to
extrema in the transform plot.) One of the ECG signals they analysed, together
with its wavelet transform at four consecutive scales, is shown in figure 6.2(b). The
maxima and minima of the wavelet-transformed signal are used to determine the loca-
tion and width of the QRS complex. This is shown in the lower two plots of
figure 6.2(b) containing, respectively, the signal and its corresponding wavelet trans-
form at the largest scale. The vertical lines above the ECG signal at the top of the plot
show the location of the QRS complex determined from the zero crossings of the
modulus maxima of the transformed signal. Figures 6.3 and 6.4 show the same
analysis as figure 6.2, but this time baseline drift and high frequency noise have
been added to the signal respectively. A schematic of the timing intervals is shown
in figure 6.5(a), and figure 6.5(b) shows the values of the timing intervals of one
beat of the ECG signal computed using the wavelet modulus maxima-based
method of Sahambi and his co-workers. The measurements of these intervals give
the relative position of the components in the ECG which are important in delineating
the electrical activity of the heart. Improvements to the technique are described in
Sahambi et al (1998).
An algorithm based on quadratic spline dyadic wavelet transforms and the
modulus maxima method has been developed by Li et al (1995) to detect the
Copyright @ 2002 lOP Publishing Ltd.

I
I
J
I
I
x(t)
Wf(2 1 ,t)
Wf(2 2 ,t)
Wf(2 3 ,t)
Wf(2 4 ,t)
Figure 6.4. ECG signal with 50 Hz interference and its wavelet transforms at scales 2 1 , 2 2 , 2 3 , 2 4 . The
vertical line above the ECG signal indicates the position of the QRS complex, as detected by the
algorithm of Sahambi et al. After Sahambi et al1997a IEEE Engineering in Medicine and Biology
16(1) 77-83. (Q) IEEE 1997.
R
P width
T width
)II'
,41
...
P-R
interval
. r
Ja' Q ,.
I S r S- T
I QRS I interval
, width I .
I. ....
... f
Q- T interval
c
+--. d
III .
III a ..
b
..
e
4
..
...
.
f
II
.,
QRS width
T width
P width
PR interval
ST interval
QT interval
(a): 132 ms
(b): 300 ms
(c): 128 ms
(d): 236 ms
(e): 300 ms
(f): 432 ms
(a)
(b)
Figure 6.5. Timing intervals. (a) Definition of the timing intervals. (b) Measured values of the
timing intervals. After Sahambi et al 1997a IEEE Engineering in Medicine and Biology 16(1)
77-83. (Q) IEEE 1997.
Copyright @ 2002 lOP Publishing Ltd.

characteristic points ofECG signals. The algorithm can distinguish the QRS complex
from a host of other features in the signal, including high P and T waves, noise,
baseline drift and signal artefacts. Sivannarayana and Reddy (1999) have proposed
the use of both launch points and wavelet extrema to obtain amplitude and duration
parameters from the ECG. Kadambe et al (1999) have given details of a wavelet-
based QRS complex detector which also incorporates a modulus maxima algorithm.
Their interest was in the determination of heart rate and hence their study concentrated
on detecting the temporal location of the R wave and measuring the R-R interval.
Other work has been undertaken by Park et al (1998) using a wavelet adaptive filter
to minimize the distortion of the ST segment due to baseline wanderings, and Tikkanen
(1999) has evaluated the performance of different wavelet-based and wavelet packet-
based thresholding methods for removing noise from the ECG.
6.2.2 Detection of abnormalities
Tuteur (1989) was one of the first proponents of the wavelet transform as an analysis
tool for medical signal processing, using a complex Morlet wavelet to detect abnorm-
alities in ECG signals. In particular, Tuteur was interested in an abnormality known
as a ventricular late potential (VLP). This represents low-amplitude electrical activity
due to delayed electrical conduction by the ventricle muscles. VLPs occur in the ECG
after the QRS complex and are often masked by noise. They have been used as a
marker to identify patients at risk from certain types of life threatening arrhythmias.
Tuteur added a synthetic VLP-like segment of signal to ECG data from a child with a
cardiac defect. Figure 6.6(a) shows the original cardiac signal and figure 6.6(b) shows
the original signal together with the synthesized defect. Figure 6.7 shows the wavelet
transforms of the signal at three a scales. The synthesized abnormality can be seen as a
bulge on the middle peak at an a scale of 1/16. Wavelet energy scalograms were used
by Meste et al (1994) as a method of highlighting VLPs and observing temporal and
frequency variability in the ECG from beat to beat. Batista and English (1998)
. t t . . .. . .. . III
I . .  t .. , . .. .
t . ill . III . . . .. /I
1.80 ..,... r' ,.,. I'...'. r....'..!.....' .1'...... r...... r...... r...'......... 1.80
. . ill ... 'II 10 I' . .
t t....!  ...: ....... ..:..... II I II.. II ;.1....1 ..:. ....,.,.. ':111.....: .....1..:... II".:
1.20 ...... . ................. .................. ....,,.......;.. ..... 1.20
.. ........
.. .. t" II .. . .
. 10. .. III '" l1li .
...t.a.... .............tt......... .t.......................oiIt _iIII........".I..t............
. ... III II- . .. . It
110 .. . II 110 . , ..
.. ill. ill II ,. . . III
.....;. ..,.,...;.......... .. ..,.,.. a 111111 1 ..!. ,.,l1li1'... t i.". t. ;:.  1..iIII". .1. .;..........: 0.60
ill ..,.  III I l1li . .
oil ... .. I " . .
.. ... ........ '"
.. It. .....,. t ill
.. .11..1 t ....:-.."".. "" -:....... .... ............:... ..... r. .1....: "11'10111. -: I. .. t._:
.....' . ...:. ....:.;.. . .... . .:. ... l............... 0.00
. III. . l1li .
..... ... ... ....... .-:--..... ..1.: .I.I.I:.. r_. ..... '......I  .."". l.iIII...... .....
III.. ,. '" .
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. II. ........
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. ... t t III ill III
Il1o I.. t........ ...11l1li III..... .... ...1... t..... .. . t............1111 11-... ...... l1li'1....... I........ .........
III ... . II .. . .
. .. 11".....
. .. ... t . ..
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.t'III....... _ ..t.....III.III.......I......... .1.4.. ..... tl. Iot"I.. ....... t........ ..,......1..
II .. f. II III .
... II" .. III . III
l1li .... . I .  III
III III  .. III " ..
.........t1 .IIII."... ... ....... . III. ..:t. ... ..L."''''.. ."..".. .::a.... .....:.......
II III . I . .
..... 1 I .. .
III  ... .
0.60
0.00
0.40 1.20 2.00 2.80 3.60
seconds
seconds
(a)
(b)
Figure 6.6. ECG trace before and after addition of modelled VLP. ( a) Original ECG signal. (b) ECG
with synthetic VLP added just after second peak (synthesized VLP consisted of a 25 Hz sine wave
modulated by a Gaussian envelope). From Tuteur F B 1989 'Wavelet transforms in signal detection',
in Wavelets 132-138. Combs J M, Grossmann A and Tchamitchian P (Eds), Springer-Verlag.
Copyright Springer-Verlag 1989.
Copyright @ 2002 lOP Publishing Ltd.

0.400 1.20 2.00 2.80 3.60 0.400 1.20 2.00 2.80 3.60 0.400 1.20 2.00 2.80 3.60
seconds
seconds
seconds
Figure 6.7. Reconstruction of the ECG signal of figure 6.6(b) at specific wavelet scales. From Tuteur
F B 1989 'Wavelet transforms in signal detection', in Wavelets 132-138. Combs J M, Grossmann A
and Tchamitchian P (Eds), Springer-Verlag. Copyright Springer-Verlag 1989.
employed both the harmonic and closely related musical wavelets in the detection of
VLPs. They performed a wavelet decomposition of the ST and TP segments of the
ECG and compared the relative energies contained at each level in order to detect
VLPs. They reported superior results using their technique over the Simson
method, widely used in clinical practice for the detection of VLPs. In addition, they
found that the reduction in spectral leakage of these wavelets provides better results
than using Daubechies wavelets although there is a reduction in the associated time
resolution.
Couderc et al (1996) employed the Morlet wavelet transform to analyse high-
resolution ECGs in post-myocardial infarction patients both with and without
documented ventricular tachycardia. A discretization of ten wavelet scales covering
the relevant range of the time-frequency plane allowed them to stratify the resulting
time-frequency information concerning ECG abnormalities. In the group of myo-
cardial infarction patients with documented ventricular tachycardia they found
significantly increased high-frequency components corresponding to prolonged QRS
durations and late potentials in the area 80-50 ms after QRS onset. They also applied
their method to the intra-QRS abnormalities in patients with congenital long QT
syndrome. Rakotomamonjy et al (1998) have detailed a method for detecting VLPs
using Morlet wavelet preprocessed data as input to a feedforward neural network.
They tested the technique on simulated ECGs containing VLPs and a range of
additive noise, and found a high degree of accuracy in classification, even for high
levels of noise. Rakotomamonjy et al (1999) have also described a wavelet-based
filtering method for signal-averaged ECGs used for the detection of late potentials.
The detection of myocardial ischaemia in pigs using a wavelet-based entropy
measure is described by Lemire et al (2000). They considered the morphology of
the combined ST segment and T wave, performing a fast wavelet transform using
spline wavelets. The Shannon entropy of the coefficients at each scale was determined
for the combined ST segment- T wave at each beat. An increase in entropy was
detected at certain scales due to coronary occlusion, which led the authors to suggest
a threshold entropy value as an indicator of the occlusion state. The best scale for use
as a marker corresponded to an approximate frequency band of 30-60 Hz. In a pilot
Copyright @ 2002 lOP Publishing Ltd.

study, Gramatikov et al (2000) used Morlet wavelet transforms to analyse the ECG
recordings from patients with left and right coronary stenosis taken before and
after angioplasty. They focused on the morphology of the QRS complex in wavelet
space plotting both two-dimensional contour plots and three-dimensional representa-
tions of the transform magnitude and demonstrated the wavelet's ability to detect
short-lasting events of low amplitude superimposed on large-scale deflections. The
study found changes in the mid-frequency range which reflected the ECG's response
to percutaneous transluminal coronary angioplasty.
6.2.3 Heart rate variability
Rather than consider the morphology of the whole ECG signal, many researchers
have focused on the temporal variability of the heartbeat. To do this, they monitor
the timing interval between beats, taken between each R point on the QRS complex,
and plot this R-R interval against time to give the heart rate. The minute fluctuations
present in the R-R intervals have been used for assessing the influence of the auto-
nomic nervous system on the heart rate. In addition, long-range correlations and
power law scaling have been found through the analysis of heartbeat dynamics.
Much of the current work concerning heart rate variability focuses on its use as a
1.5

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1d upright
I-;
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Q)
...c:
(a) 0.5
0 2 4 6 8 10 12 14
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-..
N
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Q)
1d
 -0.2
Q)
...c: _ 0.4
(b)
-0.6
o
2
4
6 8
time (min)
10
12
14
Figure 6.8. Short term analysis of heart rate variability by adapted wavelet packets. (a) Recorded data
and estimated trend for a normal subject. (b) Detrended HR V signal. The vertical lines indicate the
onset of different procedures with controlled respiration at 6 and 12 breathslmin, and when the
subject is in the upright position after passive tilt. After Wiklund et al 1997 IEEE Engineering in
Medicine and Biology 16(5) 113-118. (Q) IEEE 1997.
Copyright @ 2002 lOP Publishing Ltd.

0.6
lying
6/min 12/min
upright
0.5
0.4
--
N
:::r::
'-" 0.3

u
s:::
Q)
&
Q) 0.2
tt::
0.1
0.0
6 8
time (min)
Figure 6.9. Time-frequency energy distribution of heart rate. The figure shows the decomposition of
the complete recording using wavelet packets. The signal was detrended since the very low frequency
(VLF) component otherwise masks all other components. The sampling frequency was 2.4 Hz. After
Wiklund et al1997 IEEE Engineering in Medicine and Biology 16(5) 113-118. (Q) IEEE 1997.
o
2
4
10
12
14
marker for the prediction and diagnosis of heart disease and assessment of heart
function.
Wiklund et al (1997) used adaptive wavelet transforms (wavelet packets and
cosine packets) to analyse the regulation of heart rate variability (HR V) by the auto-
nomic nervous system. Figure 6.8 contains an original HRV signal used by Wiklund
and co-workers in their study together with a detrended version. The resulting time-
frequency plane decomposition of the detrended signal using wavelet packets is
shown in Figure 6.9. Their results suggested that adapted wavelet transforms can
be used to detect transient changes in the signal and characterize both tonic and
reflex autonomic activity. Thurner et al (1998a) have employed both Daubechies
DI0 and Haar wavelets in the analysis of human heartbeat intervals. They found
that at distinct wavelet scales, corresponding to the interval 16-32 heartbeats, the
scale-dependent standard deviations of the wavelet coefficients could differentiate
between normal patients and those with heart failure. Significantly, they could do
this with 100% accuracy for a standard 27-patient data set. Figure 6.10 shows a
typical series of inter beat intervals against interval number together with its wavelet
coefficients at three scales obtained using the Daubechies D 1 0 wavelet. Figure 6.11
shows plots of coefficient-dependent standard deviations against scale for the 27-
patient data set. Notice how the plots for normal and heart-failure patients decouple
at scales 4 and 5 (i.e. 2 4 _2 5 == 16-32 heartbeats). Further development of the
technique is detailed in a subsequent paper by Thurner et al (1998b).
Ivanov et al (1996) have investigated the ECG signals from subjects with sleep
apnoea. By sampling at an a scale equivalent to eight heartbeats, they performed a
local smoothing of the high-frequency variations in the signal in order to probe
patterns of duration in the interval 30-60 s. The authors used the data to characterize
Copyright @ 2002 lOP Publishing Ltd.

'ti=(R-R)i 'ti+1=(R-R)i+l
t i - 1
t.
l
t i + 1
(a)
1.5
-..
u
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Q)
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Q)
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....-1
o 20000 40000 60000
interval number i
(b)
 1.0
 1.5
Q) 0.0
.0-0.5.

Q) 0.5
8 0.0
Q) -0.5
 0.5
; 0.0
800 -0.5'
400 600 .
200 1J\"e t t 0
.-Tal -n u
i-ntel- ..

.... ... '
Q) s::: .
,......; Q)
Q)....-I
.

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o
u
(c)
20000 40000 60000
interval number i
(d)
Figure 6.10. Multiresolution analysis of heartbeat intervals. (a) Schematic diagram of an electro-
cardiogram segment, showing the beat occurrence times t i and the interbeat (R-R) intervals Ti.
(b) Series of interbeat intervals Ti versus interval number i for a typical normal patient (data set
16265). (Adjacent values of the interbeat interval are connected by straight lines to facilitate viewing.)
Substantial trends are evident. ( c) Three-dimensional representation of the wavelet coefficient W as a
function of scale (1 :::; m :::; 10) and interval number i (n has been rescaled to i), over a portion of the
data set, using the Daubechies 10-tap analysing wavelet. (d) Wavelet coefficient at three scales
(m = 2, 4 and 8) for the data set illustrated in (b). The trends in the original interbeat-interval
time series are removed by the wavelet transformation. After Thurner et al (1998a). Reproduced
with the kind permission of the American Physical Society and the authors.
the nonstationary heartbeat behaviour and elucidate phase interactions. Bates et al
(1998) have compared two Fourier methods (the discrete Fourier transform and
the nonequispaced Fourier transform) of computing the Fourier coefficients used
in the discrete harmonic wavelet transform analysis of heart rate variability. The
same group (Hilton et aI, 1999) have used the discrete harmonic wavelet transform
as well as the discrete Fourier transform to perform spectral analysis of the HRV
signals associated with sleep apnoea/hyponoea syndrome (SAHS). They compared
their spectral analysis of the HRV signals with the current screening method of
pulse oximetry. Their results indicated that spectral analysis ofHRV seems to provide
a better indicator of SAHS than oximetry in non-REM sleep and a comparable indi-
cator in REM sleep. Akay and Fischer (1997) compared a wavelet-based method to
others in a study to determine the fractal nature of HR V signals. Specifically, they
used the method to determine the Hurst exponent of the signal. (See also Fischer
and Akay, 1996, 1998.) Ivanov et al (1999) have reported on the multifractality
Copyright @ 2002 lOP Publishing Ltd.

1.00
1.00

'O 0.10

'O 0.10
0.01
Normal
Heart-failure
SCD
0.01
Normal
Heart-failure
SCD
1 2 3 4 5 6 7 8 9 10
scale m
1 2 3 4 5 6 7 8 9 10
scale m
Figure 6.11. Wavelet coefficient standard deviation against scale. Wavelet coefficient standard
deviation away versus scale m for the standard 27-data-set collection using the Haar wavelet (left)
and the Daubechies wavelet (right). Complete separation of the two groups is achieved at scales 4
and 5, corresponding to 2 4 _2 5 heart-beat intervals. Results for an SCD patient (white squares),
using the same number of interbeat intervals, also exhibit low variability. The outcome is similar
for both analysing wavelets. After Thurner et a11998a. Reproduced with the kind permissions of
the American Physical Society and the authors.
found in the healthy human heart rate signal using a wavelet-based analysis method.
Further, they reported the loss of multifractality for a life-threatening pathological
condition, congestive heart failure. See also Havlin et al (2000) in this regard. (For
more information on fractals and wavelets see chapter 7, section 7.2.) Zhang et al
(1997) employed techniques from nonlinear dynamics (phase space reconstruction,
correlation exponent and Lyapunov exponents) to investigate heart rate variability.
They applied these methods to both the R-R interval time series and to a time
series of the variability of the QRS complex. Both time series were determined
using the wavelet coefficients obtained from a spline wavelet decomposition of the
original ECG signal. J oho et al (1999) analysed heart rate and left ventricular pressure
variability during coronary angioplasty in humans. They presented three-
dimensional, Morlet-based wavelet transform plots which showed clearly a low
frequency response of both signals to coronary occlusion. They concluded that the
regional myocardial ischemia elicited a profound sympathoexcitory response
followed by a gradual suppression over time. This they attributed to the vagal
inhibitory reflex.
6.2.4 Cardiac arrhythmias
A number of wavelet-based techniques have been proposed for the detection, classi-
fication and analysis of arrhythmic ECG signals. Zhang et al (1999) have proposed a
novel arrhythmia detection method, based on a wavelet network. Their system detects
the bifurcation point in the ECG where normal sinus rhythm degenerates into a
pathological arrhythmia. Govindan et al (1997) have detailed an algorithm for
classifying bipolar electro grams from the right atrium of sheep into four groups-
normal sinus rhythm, atrial flutter, paroxysmal atrial fibrillation (AF) and chronic
AF. In their method, they use a Daubechies D6 wavelet to preprocess the ECG
data prior to classification using an artificial neural network. Using a raised cosine
Copyright @ 2002 lOP Publishing Ltd.

wavelet transform, Khadra et al (1997) have undertaken a preliminary investigation
of three arrhythmias-ventricular fibrillation (VF), ventricular tachycardia (VT) and
atrial fibrillation (AF). They developed an algorithm based on the scale-dependent
energy content of the wavelet decomposition to classify the arrhythmias, distinguish-
ing them from each other and normal sinus rhythm. Englund et al (1998) studied the
predictive value of wavelet decomposition of the signal-averaged ECG in identifying
patients with hypertrophic cardiomyopathy at increased risk of malignant ventricular
arrhythmias or sudden death. They concluded, however, that wavelet decomposition
was of limited value in this type of analysis.
Figure 6.12 contains three beats of a normal sinus rhythm of a pig heart together
with its (Morlet) wavelet energy scalogram shown as both a contour plot
250
Q)
00 150
s:::
0
P-
00 50
Q)
I-;
d -50
u

-150
0.080 0.085 0.090 0.095 0.100 0.105 0.110
(a) time (minutes)
10 2
,.....;
I
s:::
0 10 1
.....

.....-4
.....
"'0 1.7Hz band
10°
0.080 0.085 0.090 0.095 0.100 0.105 0.110
(b) time (minutes)
3
-..

ep 2
Q)
s:::
Q)
 1
OJ)
0
.....-4
0
10°

.

'/ 10 2 0.100 0.105 0.110
0.080 0.085 0.090 0.095
(c) time (minutes)
Figure 6.12. Wavelet transform of ECG exhibiting sinus rhythm. ( a) Single channel porcine ECG
showing sinus rhythm. (b) The corresponding energy scalogram of the temporal location against band-
pass frequency of the wavelet. (c) The three-dimensional landscape plot of (b). After Addison et al2000
IEEE Engineering in Medicine and Biology 19(5) 104-109. (Q) IEEE 2000. See also colour section.
Copyright @ 2002 lOP Publishing Ltd.

-200
(]J
00
S  -250
P-4(]J
00"'0
 0 -300
d
u (]J -350
Q)

- 2
gl
'EO 1
..90
-400
125
126
127 128
time (s)
129
1
2
..-..
 5
c>
10
 30
60
125
127 128 129
126 time (s)
Figure 6.13. Wavelet transform of ECG exhibiting ventricular fibrillation. (a) Single channel ECG
showing VF. (b) The corresponding three-dimensional energy scalogram of the temporal location
against bandpass frequency of the wavelet. The arrow points to a region of coherent modulation
of the 10Hz band. After Addison et al 2000 IEEE Engineering in Medicine and Biology 19(5)
104-109. (Q) IEEE 2000.
(figure 6.12(b)) and a three-dimensional surface relief (figure 6.12(c)). Note that the
logarithm of the energy is plotted in the figures as it allows for features with large
differences in their energy to be made visible in the same plot. The QRS complex
of the waveform manifests itself as the conical structures in figure 6.12(b). These
converge to the high frequency components of the RS spike. The P and T waves
are also labelled in the plot. In addition, the continuous band evident in the plot at
a frequency of around 1.7 Hz corresponds to the beat frequency of the sinus
rhythm. The three-dimensional morphology of the signal in wavelet space is shown
in figure 6.12(c). Figure 6.13 shows a portion of a pig heart ECG exhibiting ventricu-
lar fibrillation. The three-dimensional morphology of the energy scalogram reveals
the presence of organized structure contained within the VF signal as seen by the
three undulations within the 10Hz band of the scalogram. This transient modulation
of the 10Hz band is located by the arrow in the plot. Figure 6.14 shows another
portion of VF together with the two-dimensional contour plot of the energy scalo-
gram. Distinct high frequency spiking of a periodic nature can be seen within the
scalogram. This regular structure is not at all evident from the ECG trace nor is it
evident using short time Fourier transform (STFT) analysis due to its fixed window
width (see below). The spiking becomes more prominent with increasing downtime
(duration ofVF). (It is worth noting that this periodic spiking has also been observed
in segments of human VF-see figure 6.15.) A global view of the porcine VF signal in
wavelet space is given in figure 6.16, which contains an energy scalogram for a 5 min
period of VF followed by a 2.5 min period of cardiopulmanory resuscitation (CPR).
The onset of CPR is distinguished by the large amplitude horizontal band appearing
at low frequency at 5 min. Distinct banding can be seen in the scalogram over the first
5 min: a high frequency band at around 10Hz and two lower energy bands at lesser
frequencies, labelled A, Band C. After the onset of CPR, an increase in the passband
frequency of all three bands can be observed in the scalogram. The results depicted in
figures 6.12 to 6.16 show that VF, previously thought to represent disorganized and
unstructured electrical activity of the heart, does in fact contain a rich underlying
Copyright @ 2002 lOP Publishing Ltd.

1---1
1---1
CD
'"d
o

u
CD
Q) -50
CD
00
S -100
Po.
 -150
I-;
8 -200

-250
250
100
50
o
250.5 251 251.5 252 252.5 253 253.5 254 254.5
time (s)
60
high frequency spiking
//\
3.5
30
20
 10
N
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'--"
0 5
p..

3
2
1
3
2.5
2
1.5
1
0.5
o
250 250.5 251 251.5 252 252.5 253 253.5 254 254.5
time (s)
Figure 6.14. Wavelet transform of ECG exhibiting ventricular fibrillation. ( a) Single channel ECG
showing a region ofVF. (b) The corresponding scalogram of the temporal location against bandpass
frequency of the wavelet. Notice the high-frequency periodic spiking observable in the scalogram.
After Addison et al2000 IEEE Engineering in Medicine and Biology 19(5) 104-109. (Q) IEEE 2000.
See also colour section.
structure hidden from traditional Fourier techniques (Addison et aI, 2000; Watson
et aI, 2000). Figure 6.17 illustrates the shortcomings of traditional STFT analysis in
detecting signal features of short duration. The figure contains a scalogram and a
spectogram corresponding to the rhythmic ECG signal shown in figure 6.17(a).
The spectogram is generated from an STFT which has a 3.4 s Hanning window-typi-
cal for this type of analysis. The smearing and hence loss of local information across
the spectogram over these timescales is evident in the plot.
In a pilot study concerning the analysis of pressure traces and ECG correspond-
ing to pig hearts exhibiting VF (Addison et aI, 2002b) some evidence has been found
to suggest that wavelet phase information may be used to interrogate the ECG for
underlying low level mechanical activity in atria. Figure 6.18 shows the pressure in
Copyright @ 2002 lOP Publishing Ltd.

250
-...
00
...
..- 200
s:::
::s
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,b
..-
.e 100

'-"
e,:,
u 50

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'-"
u
0.

10 0
0
1
2
3
4
5
6
time (s)
1
2
3
4
5
6
time (s)
Figure 6.15. Attempted defibrillation of human ventricular fibrillation. Top: 7 s of human ECG
exhibiting VF containing a defibrillation shock event. Bottom: scalogram corresponding to the
ECG signal. Notice the high frequency spiking prior to the shock evident in the scalogram. After
Addison et al 2002b IEEE Engineering in Medicine and Biology 21 58-61. (Q) IEEE 2002. See also
colour section.
the aorta and ECG corresponding to an episode of ventricular fibrillation in a pig heart.
The ECG signal has a typical random or unstructured appearance. The aorta pressure
trace, however, reveals regular low amplitude spikes. On opening the chest of this
animal and observing the heart directly, it became apparent that the ventricles
were fibrillating, but the atria were contracting independently in a coordinated
-..
N
::r:

u
0.

A ....1 0 1
B....
C....
10 0
+- CPR band
5t 6 7
I time (minutes)
onset of CPR
Figure 6.16. The energy scalogram for the first 7 min of porcine ventricular fibrillation. CPR is initiated
at 5 min as indicated. Reprinted from Watson et al (2000) Resuscitation 43(2) 121-127. Copyright
2000, with permission from Elsevier Science. See also colour section.
o
1
2
3
4
Copyright @ 2002 lOP Publishing Ltd.

200
100
0
-100
0 1 2 3 4 5 6
(a)
100
64
50
32
16
8
6
4
3
2
1
(b) 0 1 2 3 4 5 6
18
16
14
12
10
8
6
4
2
0
(c) 2.0 2.5 3.0 3.5 4.5 5.0
Figure 6.17. Wavelet scalogram versus STFT spectogram for a rhythmic signal. (a) Original rhythmic
ECG signal. (b) Morlet-based scalogram corresponding to (a). (c) Spectogram corresponding to (a)
generated using a short time Fourier transform with a 3.4 s Hanning window.
manner. The irregular activity of the much larger ventricular muscle mass completely
obscured this atrial activity in the standard ECG recording shown in the middle of
figure 6.18. The wavelet energy scalogram for this signal is plotted below the ECG
signal. (A Morlet wavelet was used in the study.) The high amplitude band at
around 8-10 Hz is much more compact in extent in frequency than that found for
other traces where no atrial pulsing was apparent. Furthermore, there is some
evidence of 'pulsing' in this band between 1 and 2 Hz in the scalogram. This is
confirmed in figure 6.19 where the location of zero wavelet phase is plotted over a
short range of the bandpass frequencies, between 1.1 and 1.5 Hz. Below this zero-
phase plot is the pressure tracing. The phase plot exhibits a strikingly regular pattern
Copyright @ 2002 lOP Publishing Ltd.

14.5
13.5
12.5
11.5
10.5
0.2
o
-0.2
-0.4
-0.6
-0.8
40
20
10
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2
1
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
729
time (sees)
Figure 6.18. Simultaneous ECG and pressure recordings. The aorta pressure trace (top), with ECG
(middle) and corresponding wavelet energy plot (bottom) obtained using the Morlet wavelet. The
plots correspond to the time period 726.23-731.31 s after the initiation of VF. After Addison et al
2002b IEEE Engineering in Medicine and Biology 21 58-61. (Q) IEEE 2002. See also colour section.
727
728
730
731
with the zero-phase lines aligning themselves remarkably well with the atrial pulsing
of the pressure trace.
Atrial fibrillation (AF) is an arrhythmia associated with the asynchronous
contraction of the atrial muscle fibres. It is the most prevalent cardiac arrhythmia
1.5
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cd 00
t: 00
o 
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time (s)
Figure 6.19. The zero phase lines of the Morlet wavelet transform. (Same times used for horizontal
axis as figure 6.18.) After Addison et al 2002b IEEE Engineering in Medicine and Biology 21 58-
61. (Q) IEEE 2002.
Copyright @ 2002 lOP Publishing Ltd.

-.. 1

 0.5
01)
 0
o
> -0.5
1.0
1.2
1.4
1.6
1.8
2.0
2.2
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1.4
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2.2
2.4
2.6
2.8
3.0
time (seconds)
Figure 6.20. ECG trace exhibiting AF (top), together with its associated scalogram (middle) and
modulus maxima plot (bottom). After Watson et al200!.
in the western world, and is associated with significant morbidity. Figures 6.20 and 6.21
illustrate a technique for the elucidation of AF from within an ECG signal using a
modulus maxima denoising technique (Watson et aI, 2001). Figure 6.20 shows the
wavelet transform decomposition of a 2 s segment of ECG from a patient with atrial
fibrillation. Below the trace is a scalogram plot, obtained using a Mexican hat-based
wavelet transform. This yields high temporal resolution in the wavelet domain, but
generates a very large data set. The modulus maxima of the scalogram are plotted
below the scalogram. As can be seen from the figure, dominant modulus maxima
lines at the scale of 10Hz and below are almost solely associated with the coherent
QRS and T structures. Therefore the modulus maxima lines at this scale with a high
proportion of the total energy within this scale are selected. The selected modulus
maxima lines are then followed across scales and subtracted to leave a residual signal
associated with both system noise and, more importantly, atrial activity. An inverse
transform, performed separately on both sets of retained maxima lines, recovers the
Copyright @ 2002 lOP Publishing Ltd.

1
0.5
0
-0.5
-.. 1
>
---
Q) 0.5
0/,)
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....
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3! 4
...................................
5
6
1 2
_..._-_......_..-._..._................ ....
time (see)
...............
.............................
_ 0 0 . .0 2 2 L · · · · . · · . ,J I
. original signal
-..
>
---
0.04
0.02 QRS and T filtered
o
-0.02
Q)
0/,)

....
.....-I
o
>
o.og 
-0.05
-0.1
2.65 2.75
.: ::: J QRSIDdTsial
2.85 2.95
time (see)
3.05
3.15
Figure 6.21. Wavelet filtering of the ECG exhibiting AF. After Watson et al (2001).
partitioned signals. This time-frequency partitioning of the signal results in two
components: one (1) containing combined low and high frequency components
that correspond to large scale features in the signal, and a second (2) containing
the remaining high frequency components that correspond to small scale AF features
and noise. Most applications are concerned with signal denoising and hence the
retention of component (1). This application, however, is concerned with the removal
of large amplitude features to allow examination of the lower amplitude AF compo-
nents of the signal, and hence component (2) is retained for analysis. Figure 6.21
contains a 7 s segment of ECG taken during a pilot study of patients with AF. The
signal has been partitioned using the modulus maxima technique described above
where the modulus maxima have been separated into large and small scale features.
An enlarged part of the signal is given in the lower three plots in the figure. The
middle plot contains the partitioned signal with the QRS complex and T wave filtered
Copyright @ 2002 lOP Publishing Ltd.

out, revealing regular, coherent features that appear at a frequency of approximately
400 beats per minute, often seen during invasive studies of atrial activity in patients
with AF. The lower plot contains the partition with the filtered-out QRS and T
waves. Although a relatively simple modulus maxima technique was used, whereby
the modulus maximum lines were simply partitioned into two subsets, the ability of
the technique to separate the signal into QRS and T waves and underlying AF is evident
from the preliminary results.
6.2.5 ECG data compression
ECG signals are collected both over long periods of time and at high resolution. This
creates substantial volumes of data. Data compression seeks to reduce the number of
bits of information required to transmit or store digitized ECG signals without
significant loss of signal information. An early paper by Crowe et al (1992) suggests
the wavelet transform as a method for compressing both ECG and heart rate
variability data sets. Using discrete orthonormal wavelet transforms and Daubechies
DI0 wavelets, Chen et al (1993) compressed ECG data sets resulting in compression
ratios up to 22.9: 1 while retaining clinically acceptable signal quality. Thakor et al
(1993a) compared two methods of data reduction on a dyadic scale for normal
and abnormal cardiac rhythms, detailing the errors associated with increasing data
reduction ratios. Karczewicz and Gabbouj (1997) have proposed a novel compression
scheme for ECG data based on B-spline basis functions. Popescu et al (1999) have
developed a multiresolution distributed filtering data reduction method for high
resolution ECG signals (HRECGs) used in the assessment of ventricular tachycardia
risk in post-myocardial infarction patients. The method detects small amplitude late
potentials, which are established arrhythmogenic markers in this group of patients.
The authors found the method superior to hard and soft wavelet thresholding
techniques as well as other established non-wavelet methods. A comparison of the
performance of the many ECG compression methods-wavelets and other-can be
found in the paper by Cardenas-Barrera and Lorenzo-Ginori (1999). More recent
data compression schemes for the ECG include the method using non-orthogonal
wavelet transforms by Ahmed et al (2000) and the set partitioning in hierarchical
trees (SPIHT) algorithm employed by Lu et al (2000).
6.3 Neuroelectric waveforms
The electroencephalogram (EEG) signal is obtained from a set of electrodes which are
usually placed on the scalp. In some cases, however, specially designed subdural elec-
trodes are surgically implanted below the skull to monitor electrical activity obscured
by the skull. The electrical potentials picked up by the electrodes-the EEG signal-
reflects brain electrical activity owing to both intrinsic dynamics and responses to
external stimuli. Experimental external stimuli can take the form of evoked potentials
(EPs), which are in general well defined sensory inputs such as sounds, flashes, smells
or touches, or event-related potentials (ERPs) where experiments are set up to
probe higher cognitive functions such as those associated with memory function or
Copyright @ 2002 lOP Publishing Ltd.

mechanical response. A good place to begin this section is with the comprehensive
introduction to the wavelet analysis of neuroelectric waveforms given by Samar
et al (1999). They provide a concept-driven (minimal mathematics) account of the
use of wavelet techniques in the analysis of EEG and event-related potential wave-
forms. Although the rest of this section deals solely with EEG data and associated
EPs and ERPs, it should be noted that some work has been carried out on the use
of wavelet-based methods in the analysis of magnetoencephalographs (MEGs)
which measure small magnetic fields induced by the electrical activity of the brain.
See for example the papers by Lukka et al (2000) who used wavelet packets within
a stimulus classification algorithm, Nikouline (2000) who studied somatosensory
evoked responses using the Morlet wavelet, or Kneif et al (2000) who investigated
the perception of coherent and non-coherent auditory objects using Morlet wavelets.
6.3.1 Evoked potentials and event-related potentials
Ademoglu et al (1997) investigated the transient response of EEG signals to a set of
brief visual stimuli. The authors studied a specific class of EP due to visual stimuli-
pattern-reversal visual evoked potentials (PRVEPs)-in an attempt to aid the clinical
diagnosis of dementia. Figure 6.22(a) contains plots of 24 normal PRVEPs and 16
from patients with dementia. Using a quadratic B-spline wavelet (figure 6.22(b )),
the researchers decomposed each PRVEP signal containing 512 data points into six
coefficient scales (m == 1-6) plus a residual component. That is, the multiresolution
analysis was halted at scale index m == 6. Thus the transform vector contains eight
approximation coefficients which contain the information from scale indices 7-9.
The authors found that the residual scale coefficients had consistent sign changes
for the normal cases (top of figure 6.23). Reconstructing the waveforms using only
these residual scale coefficients produces identifiable, overlapping waveforms for
the normal cases (middle of figure 6.17). The lower plot of figure 6.23 shows the
synthesized signals, reconstructed from the residual scale, for the PRVEPs obtained
from patients with dementia where no regular pattern is evident.
Huang et al (1999a) have detailed an investigation of auditory evoked potentials
(AEPs) as a measure of the depth of anaesthesia in dogs. These AEPs consist of a
series of waves that represent processes of transduction, transmission and processing
of auditory information from the cochlea to the brain stem. They developed an auto-
mated monitoring system to control the delivery of intravenous anaesthetic based on a
neural network classification of significant wavelet coefficients determined from the
decomposed signal using Daubechies D20 wavelets. A number of other research work-
ers have examined EPs in the EEG. For example, Bartnik et al (1992) proposed a
method for extracting single EPs from background EEG using a multiresolution frame-
work based on cubic spline wavelets. Thakor et al (1993b) used wavelet transforms to
investigate EPs from anaesthetized cats, Betrand et al (1994) examined middle latency
auditory EPs using discrete wavelets and Lim et al (1995) have used wavelets to
examine respiratory-related EPs in human subjects. Tzelepi et al (2000) have described
the findings of a study concerning scalp-recorded visual evoked potentials (VEPs) in
humans where activity in the alpha, beta and gamma range was quantified over the
first 200 ms of the VEP using a discrete Coiflet wavelet-based decomposition of the
Copyright @ 2002 lOP Publishing Ltd.

normal
pathological
. II I
 . ..
o 250 500 0 250 500  250 500 0 250 5000 250 500
--. -:'.
o  O  O  O  O  
 " 
o  O  O  O  O "

o  O  O  O  O  

o  O  O  O  O  
 --: . .t.':-   
o  O  O  O  O .
 ':\r-:---:-:   
o  O  O  O  O  

o  O  O  O  O  
(a)
0.50
0.00
"quadratic B-spine wavelet
1
3
(b) -0.50
Figure 6.22. Analysis of pattern reversal visual evoked potentials by spline wavelets. ( a) The 24 normal
and 16 pathological PRVEPs. (b) The quadratic spline wavelet used in the study. After Ademoglu
1997 IEEE Transactions on Biomedical Engineering 44(9) 881-890. (Q) IEEE 1997.
signal. Slobounov et al (2000) have also probed oscillatory brain activity in response to
visual stimuli. They employed a continuous Morlet wavelet transform decomposition
of the signal and concentrated on the Gamma band activity (30-50 Hz) of the EEG
during visual recognition of non-stable postures of a computer-generated 'virtual
person'. Raz et al (1999) have developed a wavelet packet model of EPs and Dear
and Hart (1999a, b) have reported evidence that stimulus-related synchronized neuronal
discharges in bats are structured to closely resemble some members of the Symmlet
wavelet packet family. Basar et al (1999) used wavelet techniques to investigate the
functional significance of evoked resonance phenomena in the brain, confirming (and
enhancing) previous findings using Fourier methods. Basar et al (2001) have presented
a comprehensive report on new strategies for the wavelet analysis of event-related
Copyright @ 2002 lOP Publishing Ltd.

,
0.0
250.0
500.0
0.0
250.0
500.0
0.0
250.0
I
500.0
Figure 6.23. Coefficient sign pattern and waveform reconstructions for PRVEPs. The coefficient sign
pattern in the residual scale for a consistent (N70-PI00-NI30) complex. The superimposed wave-
forms are the synthesized 8-() activity of the 24 normal (showing this pattern) and 16 pathological
(lacking this pattern) PR VEPs. After Ademoglu 1997 IEEE Transactions on Biomedical Engineering
44(9) 881-890. (Q) IEEE 1997.
oscillations. Using B-spline wavelets, Demiralp et al (1999) studied the wavelet trans-
form decomposition of event-related potentials elicited by human subjects in response
to auditory stimuli to assess differences in cognitive information processing. The
techniques developed in this work were later applied to the analysis of oddball P300
data in a later study (Demiralp, 2001). Devrim et al (1999) have analysed near-
threshold and suprathreshold visual ERPs in order to determine the generation
mechanism of the P300 wave. Quiroga (2000) has presented a wavelet-based denoising
method, using quadratic biorthogonal B-spline wavelets, for the elucidation of single
stimulus EPs. Both auditory and visual evoked potentials were investigated in the
study. Zygierewicz et al (1998) used matching pursuits to extract and quantify EEG
frequency following responses produced in the human primary somatosensory cortex
by 33 Hz vibrotactile stimulation of the right index fingertip in a single subject. In a
later study, the same group (Zygierewicz et aI, 1999) employed matching pursuit
time-frequency decomposition in a parametric study of EEG structures during sleep.
See also the paper by Durka and Blinowska (2001) which gives a general overview of
the matching pursuit method applied to the EEG. Effern et al (2000) have applied
their new wavelet-based method for denoising transient dynamical signals to both
modelled test data and experimentally collected event-related potentials. Their
method was constructed for short and transient time sequences using circular
state space embedding. Sutoh et al (2000) have used Morlet wavelets in a study of
event-related desynchronization during an auditory oddball task.
Copyright @ 2002 lOP Publishing Ltd.

6.3.2 Epileptic seizures and epileptogenic foci
Various wavelet transforms have been compared by Schiff et al (1994a) to character-
ize epileptic seizures exhibited in the EEG. Figure 6.24 shows an EEG signal
(figure 6.24(a)) with six associated transform plots generated using a variety of wave-
let transform methods, from a continuous Mexican hat transform calculated at each
time step (figure 6.24(b)) to a discrete B-spline wavelet critically sampled using cubic
B-spline wavelets and multiresolution framework (figure 6.24(g)). Schiff and his
colleagues found that the use of spline techniques to speed up computation did not
impair feature extraction from the signal. In addition, they pointed out that, since
they do not require a fixed length data window, wavelet transforms provide an
8
0.5 6
0 4
-0.5 2
(a) (e)
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
50 8
40  6
30 '
20 ' . 4 }
10 '. 2
(b) (f)
0 1000 2600 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
50 8
40 6
30
20 4. 
10  2 t
(c) (g)
0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
8
6
4 I
(d) 2 
0 1000 2000 3000 4000 5000 6000
Figure 6.24. Wavelet transforms for electroencephalographic spike and seizure localization. ( a) EEG
recorded at 200 Hz from subdural electrode overlying frontal lobe seizure focus. (b) Continuous
Mexican hat wavelet transform redundantly calculated in standard fashion. Contour lines are
shown at values determined from :::!:1.96 S.D. of the surrogate data wavelet coefficients at each scale.
(c) Continuous Mexican hat wavelet transform redundantly calculated using spline sampling of
EEG and spline interpolant of Mexican hat wavelet. Contour lines are shown at values determined
from :::!:1.96 S.D. of the experimental data wavelet coefficients at each scale for this and subsequent
plots in this figure. (d) Discrete dyadic wavelet transform redundantly sampled using spline interpolant
of Mexican hat and multiresolution framework. (e) Discrete dyadic wavelet transform critically
sampled using spline interpolant of Mexican hat and multiresolution framework. (f) Discrete dyadic
wavelet transform redundantly sampled using cubic B-spline wavelet and multiresolution framework.
(g) Discrete dyadic wavelet transform critically sampled using cubic B-spline wavelet and multi-
resolution framework. For all the plots the abscissas are sample values in units of 1/200 s (30 s
traces), while ordinates are arbitrary units. Reprinted from Schiff S J et al1994 Electroencephalography
and Clinical Neurophysiology 91 442--455. Copyright 1994, with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

improved method for spike detection in the data compared with windowed Fourier
analysis. Blanco et al (1996) analysed the EEG traces from two patients exhibiting
epileptic seizures. They compared the wavelet analysis of the signals with the more
traditional Gabor transformation (i.e. a short term Fourier transform with Gaussian
window). They found that although the Gabor transform provided a good global
average description of the signal, the (cubic spline) wavelet transform provided a
more accurate temporal localization as well as a good detection of short events.
They proposed the wavelet transform as a potentially useful tool for quantifying
and visualizing the time evolution of the frequency content of the EEG. In later
work, Blanco et al (1998) further analysed epileptic seizures using EEG data. The
EEG data taken during tonic-clonic epileptic seizures was inspected using wavelet
packets to filter the signal noise present due to skeletal muscle activity. They
showed that it is possible to obtain useful dynamical parameters and hidden
frequency information from these signals, which are usually neglected by physicians
due to the excessive noise present within them. Sirne et al (1999) have proposed a
data-reduction algorithm for long term EEGs which selects segments of the EEG
in which a transient is detected. Only these segments are presented to the clinician
for review. Akay and Daubenspeck (1999) have attempted to separate contaminating
facial muscle electromyographic (EMG) activity from EEG signals using matching
pursuit analysis. Sun et al (2000) performed soft thresholding ofbiorthogonal wavelet
transform coefficients in a method to preprocess EEG data containing epileptic
seizure events. Using their technique they removed sharp spikes and low amplitude
slow waves prior to displaying the time-frequency properties of the signal as a
pseudo-Wigner distribution. Petrosian et al (2000) have applied recurrent neural
networks combined with signal wavelet decomposition in an attempt to predict the
onset of epileptic seizures in intra- and extracranial EEGs and Gamero et al (1997)
have used the EEG displaying an epileptic seizure to illustrate their discussion on
multiresolution-based information measures for dynamical signals.
A study of temporal lobe epilepsy was carried out by Casdagli et al (1996) to
determine whether wavelet analysis had anything to offer the detection of the origines)
of the epileptic seizure within the brain (the epileptogenic foci). They used wavelet
transforms to analyse nonlinearitites within EEG data taken from electrodes
surgically implanted at various locations in the brain. In addition to wavelet analysis
incorporating the Daubechies D12 wavelet, they employed a variety of other
nonlinear time series analysis techniques. They emphasized the compatibility of the
wavelet transform as a method of characterization for the spiking activity in the
electrode signals due to its localization in both frequency and time. Figure 6.25
shows four of the electrode signals (RST4, L TD3, LSTl, ROF2) together with
their corresponding wavelet coefficients at the second finest resolutions. Figure 6.26
shows the temporal evolution of the kurtosis of the wavelet coefficients at the
second scale kw(2). This is done over an 85 min period computed for two of the
electrode signals, LTD3 and ROF2, and shown in figures 6.26(a) and (b) and figures
6.26(c) and (d) respectively. Significant spiking activity is detected at the seizure
location (61 min) by the ROF2 electrode. The lower plots ((e) and (f)) show a map
of spiking activity where values of kw(2) are plotted for all 28 electrodes on a grey
scale from 2 to 10. It can be seen from the plots that the signals exhibiting
Copyright @ 2002 lOP Publishing Ltd.

RST4
RST4.w
LTD3
LTD3.w
LST1
LST1.w
ROF2
ROF2.w
o
2
4
6
8
10
time (seconds)
Figure 6.25. EEG recordings and wavelet coefficients. RST4, L TD3, LSTI and ROF2: EEG segments
of duration 10.24s recorded 60min before the seizure. RST4.w, LTD3.w, LST1.w and ROF2.w:
wavelet coefficients at the second finest resolution for these segments. Reprinted from Casdagli
et al1996 Physica D 99 381-399. Copyright 1996, with permission from Elsevier Science.
continuous spiking activity are LST3, LTDl,3,5 and RSTl,2,3,4. The authors state
that these are a characteristic of the nonlinearities within the signal. Another study
seeking to locate the epileptogenic foci by Schiff et al (1994b) employed Mexican
hat wavelet transforms to analyse EEG signals exhibiting epileptic seizures. They
performed both a one-dimensional analysis of single EEG signals and a two-
dimensional wavelet analysis on a coarse two-dimensional array of EEG signals
obtained from a 5 x 5 subdural array of electrodes. They emphasized that the wavelet
transform overcomes the requirement of using the fixed window lengths required for
windowed Fourier transforms (the length of which is subject to observer bias). In
order to clarify which electrodes were associated with the seizure onset, the wavelet
coefficients were filtered using a hard thresholding technique. Mizuno-Matsumoto
et al (1999) have used Haar wavelet filtering of electrocorticographic signals prior
to a cross-correlation analysis to determine both the foci and nature of propagation
of epileptiform discharges. They found evidence for two types of focus in their work.
Franaszczuk et al (1998) performed time-frequency analysis of intercranial EEG
recordings during seizure using the matching pursuit method. They found the
matching pursuit method to be a valuable tool in analysing dynamic seizure activity.
Copyright @ 2002 lOP Publishing Ltd.

20
15
--
! 10
5
LTD3
(a)
o
o
10
20
30
40
50
60
70
80 (b) 60
62
20
15
--
(""I 10
!
ROF2
5
(c)
10
o
o
10
20
30
40
50
60
70
80 (d) 60
62
LOF
LST
LTD
RTD
RST
ROF
40 50
(e) time (minutes) (f)
Figure 6.26. Dependence of the kurtosis of wavelet coefficients kw(2) on time. ( a) Solid line: electrode
LTD3. Dotted line: surrogate data. (b) Blowup of (a) in the vicinity of the seizure. (c) Same as (a) for
electrode ROF2. (d) Same as (b) for electrode ROF2. (e) All 28 electrodes. Magnitudes of kw(2)
corresponding to the grey scale are illustrated by the bar to the left of the figure. (f) Blowup of ( e)
in the vicinity of the seizure. Reprinted from Casdagli et al1996 Physica D 99 381-399. Copyright
1996, with permission from Elsevier Science.
2
o
10
20
30
60
70
80
60
62
Figure 6.27 shows selected segments of the seizure obtained from one of the
intercranial contacts used in the study. Figure 6.28(a) shows the time-frequency
energy distribution of an entire seizure acquired by the deepest depth electrode
contact (HI) located near the region of seizure onset. Figures 6.28(b) and 6.28(c)
show the time-frequency distribution in three-dimensional format with respectively
296 and 6000 of the matching waveforms shown.
6.3.3 Classification of the EEG using artificial neural networks
A study by Kalayci and Ozdamar (1995) used the wavelet transform as a pre-
processing tool for neural network analysis of EEG signals, i.e. the neural network
was trained on the wavelet coefficients rather than the original signal. They found
that the size of the input data to the artificial neural network could be drastically
reduced without significantly compromising its performance. They employed both
the Daubechies D4 and D20 wavelets and found that the proper selection of scale
(or combination of scales) to use as the input was critically important. A schematic
diagram of the methodology is shown in figure 6.29. The authors found over 90%
accuracy in the detection of EEG spikes using only eight inputs from scale 3 alone.
Copyright @ 2002 lOP Publishing Ltd.

INI
A1
H1
H2
T17
T23
T32
(a)
A1
H1
H2
T17
T23
T32
(b) Is
A1
H1
H2
T17
T23
T32
(c)
TRA
ORA
IBA
Figure 6.27. Matching pursuit algorithm applied to seizures originating from the mesial temporal lobe.
Selected segments of a complex partial seizure originating from the mesial temporal structures and
recorded with combined depth electrodes and a 32 contact subdural grid array. Two six-depth
contact deep electrode arrays pass through the grid and with the deepest contacts Al and HI, H2
residing in the amygdala and hippocampus, and the most superficial contacts (e.g. A6, H6) at the
lateral temporal neocortex. The seizure begins nearest the deepest hippocampus contacts HI, H2
and subsequently spreads to regionally involve the temporal lobe. To facilitate illustration, only
three depth electrode contacts (those nearest the seizure onset) and three representative subdural
grid contacts (two (TI7, T23) from the anterior region, one (T32) from the posterior region of the
32 contact grid) are shown. Panel (a) shows 20 s of the intercranial EEG that includes the later
portions of the period of seizure initiation (INI; the low-voltage fast activity was preceded by
some periodic spiking as shown) and the period of transitional rhythmic activity (TRA, 14 s). (b),
continuous with (a), shows 20 s of intercranial EEG showing the initial 20 s of organized rhythmic
Copyright @ 2002 lOP Publishing Ltd.

50
45
40
;--. 35
N
e; 30
>-
(,) 25

(1) 20
g.
(1) 15
tt::
10
5
0
(a) 0
INI TRA
ORA
IDA
 
Ii i
 0
50
40
20 40 60 80 100 120 140 160
4J 30
f!
9l.t4. 20
 10

150
(b)
100
50 ,"",e \?")
\\"\
o 0
1000V
 
Ii i
 0
50
40
4J 30
f!
9l.t4. 20
1O
150
100
50 ,"",e \?")
\\"\
o 0
(c)
Figure 6.28. Matching pursuit algorithm applied to seizures originating from the mesial temporal lobe.
(a) Time-frequency energy distribution of the entire seizure (shown in the previous figure) recorded by
contact HI, the deepest electrode contact, located near the region of seizure onset. The intercranial
EEG recording is shown below the time-frequency plot. Only the first 296 matching waveforms,
representing 90% of the total energy, are shown. The periods of seizure initiation (INI), transitional
rhythmic activity (TRA), organized rhythmic activity (ORA) and intermittent bursting (IBA) are
marked. (b) Time-frequency energy distribution of the entire seizure recorder by contact HI. This
contains the same information as (a) now plotted in three dimensions. (c) Another three-dimensional
plot of the seizure. Here 6000 matching waveforms are shown representing 99.996% of the total energy
of the signal. The vertical axis represents the square root of energy rising out of the time-frequency
plane. Reprinted from Franaszczuk et al1998 Electroencephalography and Clinical Neurophysiology
106 513-521. Copyright 1998, with permission from Elsevier Science.
This was comparable to using 20 inputs from the original EEG signal, representing over
50% reduction in the input size. Neural networks trained on wavelet coefficients
were also employed with some degree of success by Hazarika et al (1997) to classify
EEG signals from normal patients and those with a diagnosis of schizophrenia.
Again, the wavelet transform was used as a preprocessing tool for the data sets
before the neural networks learned the data. Although good at differentiating between
the EEGs from normal patients and those with schizophrenia, the authors found that
the technique was quite poor at identifying signals from patients with obsessive compul-
sive disorders. Heinrich et al (1999) have developed a wavelet network to analyse single
ERP responses in a study concerning attention deficit hyperactivity disorder (ADHD)
in children. They studied auditory evoked potentials in a group of 25 ADHD boys,
comparing them with a control group of the same number. They found group-specific
differences using their wavelet network which could not be detected using traditional
activity (ORA). Panel (c) shows 20s of inter cranial EEG late in the seizure beginning 40s after (b),
and illustrates the pattern of intermittent bursting activity (IBA) frequently seen at the conclusion of
a seizure. Reprinted from Franaszczuk et al1998 Electroencephalography and Clinical Neurophysi-
ology 106 513-521. Copyright 1998, with permission from Elsevier Science.
Copyright @ 2002 lOP Publishing Ltd.

50 V
5.12
EEG time see
A
Daub-4 WT Daub-20
scale
.01 .01
1
.05 .01
2
.06 .07
3
.05 .08
4
.1 .1
B 5
NN training
and testing
NN training
and testing
c
Figure 6.29. Wavelet preprocessing for automated neural network detection of EEG spikes. Method-
ology: data selection and generation of ANN training and testing files (stage A), wavelet transforms
of data files computed using Daub-4 and Daub-20 wavelets (stage B), three-layer feedforward neural
networks trained and tested using wavelet transform coefficients (stage C). After Kalayci and
Ozdamar 1995 IEEE Medicine in Engineering and Biology 14(2) 160-166. (Q) IEEE 1995.
extracted latency and amplitude parameters. They concluded that the wavelet network
single sweep analysis is a sensitive tool for clinical ERP studies which should be applied
in addition to the investigation of averaged responses.
6.4 Pathological sounds, ultrasounds and vibrations
This section deals with the wavelet analysis of sound signals of clinical relevance, both
emitted sounds (e.g. turbulent blood sounds, lung sounds and otoacoustic emissions)
and reflected sounds (Doppler ultrasound time series).
Copyright @ 2002 lOP Publishing Ltd.

0%
! l:: i
1fI ; r.< ... :'" '- : I _:C .:>  ;--.:- ;
(a)::: 2 . -
d J :
 : 1 :
J  I  . , . -'  ,- : _  .  . __ _ . : . ..J
 2 - -: - - - -
d 0 I .1-..1.: £ a. "',   .   ...J
2 4
time (see)
.:
- :-1
: I
(b)
Figure 6.30. Recordings of flow sounds for the no occlusion case before the injection of the drug. ( a) Top
panel: flow. Second panel from top: sounds. (b) Wavelet power. Note that in (b), the number 2
was dropped from the detail function D 2j f. After Akay et al1994 IEEE Transactions on Biomedical
Engineering 41(10) 921-928. (Q) IEEE 1994.
6.4.1 Blood flow sounds
The turbulent sounds generated by femoral artery stenosis in dogs have been analysed
by Akay et al (1994) using both wavelet transforms and short term Fourier trans-
forms. They compared signals from the unblocked case (0% occlusion) with 72%
and 85% occlusion. Figure 6.30(a) shows the recording of blood flow rate (top
plot) and the flow sounds (second top plot) taken before the arterial blockage.
Figure 6.30(b) shows the wavelet power plotted at the four smallest scales, where
the wavelet power plot is simply the square of the detail (wavelet) coefficients plotted
against time. The smallest scales correspond to wavelet scale indices of m == 1, 2, 3 and
4, which in turn correspond to frequency bands of 1000-500, 500-250,250-125 and
125-62.5 Hz, all well above the dominant signal frequency of the flow rate of around
2 Hz. It is at these high frequencies that the fluid turbulence in the blood generated by
the blockages can be detected. Figure 6.31 shows corresponding plots for the 85%
occlusion case. By comparing this figure with the previous one (0% occlusion) the
increase in power associated with the higher frequencies can be seen in the wavelet
power plots. The authors also went on to investigate the effect of the introduction
of a vasodilator drug in the performance of this acoustic approach to the diagnosis
of arterial blockage. They found that the power corresponding to the first two wavelet
frequency bandwidths (100-250 Hz) associated with arterial stenoses increased
significantly after the injection of the vasodilator drug papaverine. They concluded
that the diagnostic performance of diastolic heart sounds associated with occluded
arteries can be improved using vasodilator drugs.
In another study, Aydin et al (1999) analysed a large number of embolic signals,
obtained using a trans cranial Doppler ultrasound system, from patients with sympto-
matic carotid stenosis. They employed the Morlet wavelet to detect short duration
Copyright @ 2002 lOP Publishing Ltd.

85%
:s 100
-E
S 50
CY
00  """' 2500
"'Ot3 oo
S :B."!:: 2000
S
(a) '-"
N 2

'I
fi.- 0
N 2


fi.- 0
N
 2
M
I
fi.- 0
N 2

.,.
fi.- 0
(b) 2 4
time (see)
Figure 6.31. Recordings of flow sounds for the 850/0 occlusion case before the injection of the drug. (a)
Top panel: flow. Second panel from top: sounds. (b) Wavelet power. Note that in (b), the number 2
was dropped from the detail function D 2j f. After Akay et al1994 IEEE Transactions on Biomedical
Engineering 41(10) 921-928. (Q) IEEE 1994.
transients in the signal associated with emboli passing through the sample volume.
They found that the method improved both the temporal resolution and temporal
localization of the detected events when compared with short term Fourier tech-
niques. Devuyst et al (2000) have used matching pursuits to detect circulating
microemboli within cerebral arteries using transcranial Doppler ultrasound. Their
method allowed them to discriminate between solid and gaseous brain microemboli
and artefacts. See also the paper by Krongold et al (1999) concerning wavelet-
based detection of microemboli in flowing blood, and the ultrasonic Doppler signal
work by Matani et al (1996) described briefly later in section 6.5 concerning blood
flow and blood pressure.
6.4.2 Heart sounds and heart rates
Chan et al (1997) have used the wavelet transform to detect venous air embolism
(V AE) during surgery. Due to its life-threatening nature, the fast detection of V AE
is essential to ensure prompt clinical treatment. They used a quadratic spline wavelet
(figure 6.32) to analyse Doppler heart sounds (DHS) from dogs. A trace of the heart
sound signal is plotted at the top of figure 6.33. The arrow indicates the point at which
0.02 ml of air was injected. Wavelet transforms of the time series are plotted below
the original signal for scale indices m == 1, 2 and 3. (Note that the authors use j for
scale index.) A distinct increase in the wavelet coefficients at the first scale is obvious
from the plot. A quantitative assessment was made by plotting the normalized
power of single heartbeats at each scale. Figure 6.34 shows the normalized power
plotted against heartbeat for 22 beats. The largest peak corresponds to the first
scale and shows its usefulness as an indicator for V AE. The authors also sought a
relationship between the sum of the normalized power of the heartbeats above a
Copyright @ 2002 lOP Publishing Ltd.

0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1 -0.5 0 0.5 1
Figure 6.32. A quadratic spline of compact support which is continuously differentiable. After Chan et al
1997 IEEE Transactions on Biomedical Engineering 44(4) 237-246. (Q) IEEE 1997.
set threshold, which they called the cumulative embolic power (CEP), and the volume
of air injected. They stated that such a relationship could prove important to the
anaesthetist, allowing him or her to act only when a clinically significant volume of
air embolism is present. The relationship between CEP and air injected is shown in
figure 6.35. The left-hand plots correspond to scale index 1 and the right-hand
plots to scale index 2. The authors concluded that the wavelet transform of the
heart sound signal can provide both fast detection ofVEA and an accurate estimation
of the embolic air present.
The short paper by Bentley et al (1998) describes the use of both continuous and
discrete wavelet transforms, among a range of techniques, to classify the sounds from
2
-..
...
.S 0
::s
. -2
 2
.s 0
Q)
 -2
 2
o
 0
Q)
"8 -2.
:E 2

C\j 0
-2
original signal
I
o
I
1
I
2
time (see)
Figure 6.33. A typical DHS signal containing seven heartbeats and its WT at different scales (j = 1-3).
With 0.02 ml of air injected, the embolic heartbeat (marked by the arrow) was confirmed by an
experienced anaesthetist by listening to the DHS signal. After Chan et al1997 IEEE Transactions
on Biomedical Engineering 44(4) 237-246. (Q) IEEE 1997.
Copyright @ 2002 lOP Publishing Ltd.

I-;
Q)

o -..
0.. .
'"0 s:::
Q) ::s
N .
. .-4 ,.0
Cd 
S 
I-;
o
s:::
5.0
4.0
3.0 threshold level
2.0. - - - - -. -  t - - - _.-
1.0
o
1 3 5 7 9 11 13 15 17 19 21
heart beat
-.- original
_ j=l
...... j= 2
--- j= 3
Figure 6.34. Normalized power of individual heartbeats obtained after WT of a DHS containing the
time segment in figure 6.33. The embolic heartbeat is identified by an increase in power (for j = 1
and 2) above a threshold level (twice the mean power level of the control signal before air
injection). After Chan et al 1997 IEEE Transactions on Biomedical Engineering 44(4) 237-246.
(Q) IEEE 1997.
both native and replacement prosthetic heart valves. They employed the Daubechies
D20 wavelet to extract salient features from the recorded heart sounds for the deter-
mination of valve condition. Zhang et al (1998a) have employed the matching pursuit
method of Mallat and Zhang to both analyse and synthesize phonocardiograms
(PCGs). They showed that the matching pursuit method acts as a powerful filter
for the removal of Gaussian noise. They use the method to develop a time-frequency
scaling technique to enhance these audio signals (Zhang et aI, 1998b) in order to
enhance the diagnosis of heart and heart valve disease. Sava et al (1998) have also
used matching pursuits to determine the important coherent transient components
within phonocardiographic signals. They applied the technique to PCG recordings
from patients with aortic bioprosthetic valves in order to detect specific events
found within the cardiac cycle. They found their method outperforms other methods
and is particularly good for the analysis of PCG signals from patients with malfunc-
tioning bioprostheses.
Masson and Rieu (1998) have analysed signals from artificial heart valves using
Auscher's wavelet in a study which aimed at identifying valve noise associated with
patient discomfort. Simultaneous signals were collected using both a hydrophone
and accelerometer for four different heart valve protheses-three mechanical and
one biological valve-in an in vitro study using a cardiovascular simulator. They
found that, unlike the bioprosthesis, the mechanical valves produced high sound
levels within octave bandwidths centred on 64 Hz up to 512 Hz. Sandham et al
(1998) have analysed seismocardiograms (SCGs) from a male patient, pertaining to
three different physiological conditions: rest, isometric exercise and hyperventilation.
They found statistically significant power changes at the lower wavelet scales which
could be used to determine the physiological condition. In contrast, no discernible
differences were observed in the signal in the time domain.
In order to probe the nonlinear dynamics of foetal heart rate signals,
Papadimitriou and Bezerianos (1999) used a modulus maxima technique to denoise
Doppler ultrasound signals. They found a significant improvement in the 'chaoticity'
detectable in the signal after denoising with their method. Kimura et al (1998) have
also employed wavelet transforms to study the nonlinear dynamics of the foetal
heart rate derived from Doppler ultrasound signals reflected from the moving
Copyright @ 2002 lOP Publishing Ltd.

14 12
,........ 12 ,........ 10
rI) rI)
]. 10 ]. 8
]1 8 ]1 6
0,........ 6 0,........ 4
- N
 II 4  II
'=' '=' 2
CUp.. 2 CUp..
SJ:ij SJ:ij 0
u 0 u
-2 -2
0 0.02 0.04 0.06 0.08 0.1 0
(a) volume of air (ml) (b)
3.0 2.5
"C ,......, "C ,......,
cu1n' cu1n'
.-<;:: 2.5 .-<;:: 2.0
S S
S..c 2.0 S..c 1.5
  1.5 o 
::::: 1. 0
iI N
S '=' 1.0 sb 0.5
..........p.. 0.5 ..........p..
OOJ:ij OOJ:ij 0
..9u ..9u
0 -0.5
-5 -4 -3 -2 -1 0
log [volume of air (ml)] -5
(c) (d)
3.0 2.0
p.. p..
J:ij,......, 25. B  1.5
u .
"C ..... "C .....
.  2.0 cu 
.  1.0
"d-f 1.5 "d-f
S S 
0,........ 1.0 0::::: 0.5
..sil ..s
!'=' 0.5 !'=' 0.0
0 -0.5
-5 -4 -3 -2 -1 0
log [volume of air (ml)] -5
(e) (f)
0.02 0.04 0.06 0.08 0.1
volume of air (ml)
.-+
i
I
-4 -3 -2 -1 0
log [volume of air (ml)]
-4 -3 -2 -1 0
log [volume of air (ml)]
Figure 6.35. Relationship between the CEP and the volume of injected air at two different scale index
(j) values. (a) Linear plot of the mean normalized CEP (j = 1) versus the volume of injected air
(r = 0.83). (b) Linear plot of the mean normalized CEP (j = 2) versus the volume of air
(r = 0.81). Each point shows the mean and standard deviation (error bar) of 12 incidences of air
injections obtained in four dogs. Significant increase: A P < 0.05, #P < 0.01 and * P < 0.0005
compared with control. (c) Linear regression in log-log scale of (a) (y = 0.66r + 4.06, r 2 = 0.98).
(d) Linear regression in log-log scale of (b) (y = 0.94r + 4.35, r 2 = 0.96). (e) Log-log plot of the
normalized CEP (j = 1) versus the volume of injected air for dog 3 (y = 0.51r + 3.78, r 2 = 0.98).
(f) Log-log plot of the normalized CEP (j = 2) versus the volume of injected air for dog 3
(y = 0.81r + 3.84, r 2 = 0.95). Natural logarithms are used for all the log-log plots. Data of (e)
and (f) represent averages of three air injections in dog 3. After Chan et al1997 IEEE Transactions
on Biomedical Engineering 44(4) 237-246. (Q) IEEE 1997.
foetal heart valves. They propose their method as a novel quantitative index of foetal
monitoring to diagnose foetal acidemia.
6.4.3 Lung sounds
Hadjileontiadis and Panas (1997) used wavelet analysis to separate discontinuous
adventitious sounds (DASs) from vesicular sounds (VSs) in pulmonary acoustic
signals. Adventitious sounds are divided into two categories: continuous (wheezes
and rhonchi) and discontinuous (crackles and squawks), and indicate an underlying
physiological malfunction. The algorithm of Hadjileontiadis and Panas combines
multiresolution analysis with hard thresholding to provide a technique to partition
Copyright @ 2002 lOP Publishing Ltd.

volts
xEO
1.00
0.500
o
-0.500
-1.00
1.00
Q) 0.500
"'0
.8
;.:::: 0

 -0.500
-1.00
1.00
0.500
o
-0.500
(a)
(b)
(c)
seconds
0.1 0.3 0.5 0.7 0.9 xEO
time
Figure 6.36. Wavelet filtering of lung sound time series exhibiting fine crackles. (a) A time section of
0.8192 s of fine crackles recorded from a patient with pulmonary fibrosis (case Cl) considered as an
input. (b) The nonstationary output of the wavelet filter, DAS. (c) The stationary output of the wave-
let filter, VS. After Hadjileontiadis and Panas 1997 IEEE Transactions on Biomedical Engineering
44(12) 1269-1281. (Q) IEEE 1997.
DASs from VSs. Figure 6.36 shows a segment of pulmonary acoustic signal from a
patient with pulmonary fibrosis containing fine crackles. The signal was decomposed
using Daubechies D8 wavelets, and then separately reconstructed DAS and VS
signals are shown below the original signal in figures 6.36(b) and (c) respectively.
Figure 6.37 shows a similar decomposition of a signal with squawks recorded from
a patient with interstitial fibrosis. Squawks are short inspiratory wheezes, of longer
duration than crackles, which are heard in association with crackles. Both figures
illustrate the separation achieved between DASs and VSs in the signal. See also the
paper by Sankur et al (1996) who employed a wavelet-based detector to discriminate
crackles in pathological respiratory sounds. They employed a Daubechies D6 wave-
let, which has a similar shape to the crackle waveform, and found their method to be
superior to two existing crackle detection methods. In another study, Charleston et al
(1997) employed discrete wavelet transforms in a method to remove heart sounds
from acquired respiratory signals. Heart sounds often represent severe disturbing
interference and their removal from respiratory signals before clinical analysis is
desirable. The interference cancellation scheme presented by the authors provides
estimates of the location within the signal of the interfering heart signals and then
separates these heart sounds from the original signal.
6.4.4 Acoustic response
Otoacoustic emissions (OAEs) are acoustic signals emitted by the cochlea, either
occurring spontaneously or in response to an acoustic stimulus, and reflect the
Copyright @ 2002 lOP Publishing Ltd.

volts
1.25 ..
0.75
0.25
-0.25
-0.75
1.25
Q) 0.75
'"C
a 0.25
.-
........
p.
 -0.25
-0.75
1.25
0.75
0.25
-0.25
-0.75'
0.05
..
(a)
(b)
(c)
0.15 0.25 0.35 0.45 xEO
time (seconds)
Figure 6.37. Wavelet filtering of lung sound time series exhibiting squawks. (a) A time section of
0.4096 s of squawks recorded from a patient with interstitial fibrosis (case C12) considered as an
input. (b) The nonstationary output of the wavelet filter, DAS. (c) The stationary output of the wave-
let filter, VS. After Hadjileontiadis and Panas 1997 IEEE Transactions on Biomedical Engineering
44(12) 1269-1281. (Q) IEEE 1997.
active processes that are involved in the transduction of mechanical energy into
electrical energy. Their form is related to the status of the cochlea and can be used
to monitor cochlear functionality in patients exposed to prolonged noise and/or
ototoxic agents. Tognola et al (1998) have studied the acoustic response of the cochlea
A030R1 - 80 dB SPL
N360LO - 80 dB SPL
cd

e 99
tr)
0
I repro %
0 5 10 15
95
20 0
(a)
Post-stimulus time (ms)
(b)
5
15
10
20
Post-stimulus time (ms)
Figure 6.38. Click evoked otoacoustic emissions. CEOAEs from (a) a normal hearing adult and
(b) a full-term baby. To reduce the influence of the stimulus artefact, responses have been windowed
2.5/20 ms post-stimulus time. In each row, two replicate recordings from the same ear (A and B
replicate recordings in ILO equipment) are superimposed. Numbers on the left of each panel are
the reproducibility values (in percentage points) between the two replicates. After Tognola et al
1998 IEEE Transactions on Biomedical Engineering 45(6) 686-697. (Q) IEEE 1998.
Copyright @ 2002 lOP Publishing Ltd.

to acoustic stimuli of brief duration-specifically clicks of about 100 JlS duration. The
time-frequency properties of these click-evoked otoacoustic emissions (CEOAEs)
have a close relationship with cochlear mechanisms. The authors compared various
time-frequency analysis methods-the short time Fourier transform, wavelet trans-
forms, the Wigner distribution-and two smoothed Wigner-Ville distributions: the
pseudo-smoothed Wigner distribution and the Choi-Williams distribution. They
found that, although there was no optimal method in an absolute sense, the wavelet
transform method offered the best compromise between time-frequency resolution
and the attenuation of interference terms. Two examples of CEOAEs are shown in
figure 6.38: one for an adult and one for a full term neonate. The OAE response of
the neonate exhibits a typical sustained burstlike behaviour up to 20 ms, whereas
the adult OAE shows clear frequency dispersion (i.e. reduction in high frequency
components in time). The wavelet scalograms corresponding to the CEOAEs in
figure 6.38 are shown in figure 6.39. The frequency dispersion of the adult signal
is evident in the scalogram plot where low frequency components have a longer
subject A030Rl - 80 dB SPL
1 l
00
.

01
0
./}(}o 3
7Q(}1l
c Yh
<j
20
5 2.5
(a)
subject N360LO - 80 dB SPL
1
00
...
.

01
0
20
2
./}e
9lJ(}1l 4
c Yh
<j
6 2.5
(a)
Figure 6.39. Time-frequency energy densities of CEOAEs. (a) Time-frequency distribution (energy
density, normalized arbitrary units) of a CEOAE at 80-dB SPL of subject A030Rl (normal hearing
adult). (b) Time-frequency distribution (energy density, normalized arbitrary units) of a CEOAE at
80-dB SPL of subject N360LO (full-term neonate). After Tognola et al1998 IEEE Transactions on
Biomedical Engineering 45(6) 686-697. (Q) IEEE 1998.
Copyright @ 2002 lOP Publishing Ltd.

subject P300P4 - 83 dB SPL
1
1
/j.e 2
9lJ e l}, 3
c Yu
<) 4
2.5
00

. ...-I
s:::
::s

o
o
Figure 6.40. Time-frequency distribution (energy density, normalized arbitrary units) of a CEOAE at
83 dB SPL of subject P300P4 (suffering from noise-induced hearing loss). After Tognola et al1998
IEEE Transactions on Biomedical Engineering 45(6) 686-697. (Q) IEEE 1998.
duration in the scalogram plot and reach maximal amplitude at longer latencies than
the high frequency components. Figure 6.40 shows a scalogram for an adult suffering
from noise-induced hearing loss. This hearing-impaired patient had hearing loss
greater than 30 dB above a frequency of 2.5 kHz. The lack of OAE response at
frequencies above this 2.5 kHz threshold is evident in the scalogram plot associated
with this patient.
Zheng et al (1999a) employed Morlet wavelets in a comparative study of
modelled and clinically collected OAEs. Wavelet decomposition was used as it was
considered suitable for detecting the mixed frequency components present in both
data sets. Their results indicated that the modelled OAEs were similar to the clinically
detected ones, not only in the time domain waveform but also in the frequency-
latency relationship. Molenaar et al (2000) have also employed wavelet-based
methods in a novel study of OAEs in which both click stimuli and wide band noise
bursts were presented simultaneously. Heneghan et al (1994) have used the Morlet
wavelet to analyse the motion of the hair cells in the inner ear in response to acoustic
signals applied at the ear canal. Yang et al (1992) and Wang and Shamma (1995) have
produced a model for the processing of acoustic signals within the auditory system in
which the spatio-temporal pattern of displacements along the basilar membrane of
the cochlea due to the acoustic signal entering the ear may be considered an affine
wavelet transform. Complex wavelets have been employed by Wang and Shamma
(1994) to analyse a one-dimensional acoustic signal within a model of the auditory
functions in the primary cortex.
6.5 Blood flow and blood pressure
An attempt to shed light on the cardiovasular control mechanisms using wavelet-
based spectral methods has been made by Bracic and Stefanovska (1998). They
Copyright @ 2002 lOP Publishing Ltd.

40
35
-..
::s
 30
S 25
c.S
 20
I-;
::: 15
Q)
-
 10
 5
o
2
1.5
1 0.5
log 1/frequency
900
800
700
600
500 time (s)
400
o
-..
00

Q) 600
S
.....
+-'
400
200
2.0 1.5 1.0 0.5 0
log scale = logl/f
14.
-.. I I
 12 I I
"-' I I
 10 I I
I-; J I
OJ.) 8
0 0.13 Hz,
Cd I
u 6 I I
00
Q) I I
OJ.) 4 I
c\S
I-; 0.27 Hz I
Q)
 2 I
0
2.0 1.5 1.0 0.5 0
log scale = logl/f
Figure 6.41. Wavelet-based analysis of human blood flow dynamics. Morlet-based wavelet transform
(top plot), local maxima (middle plot) and time-averaged values (lower plot). After Bracic and
Stefanovska (1998). Reproduced with the kind permission of the Society of Mathematical Biology.
measured the peripheral blood flow in human skin over 20 min periods using laser
Doppler flowmetry (LDF). Subsequent decomposition of these signals using the
Morlet-based wavelet transform to form energy density scalograms (figure 6.41)
revealed five characteristic frequency peaks. These local maxima in the wavelet-
based energy spectrum, they hypothesized, can be attributed to the heart rate
(1 Hz), respiratory activity (0.3 Hz), blood-pressure regulation (0.1 Hz), neurogenic
(0.04 Hz) and metabolic activity (0.01 Hz). They proposed a variety of statistical
Copyright @ 2002 lOP Publishing Ltd.

measures to characterize their wavelet-based power spectra and used them to reveal
differences in the dynamics of the blood flow between two distinct groups: a control
group of healthy young subjects and a group of athletes. They showed that the
increased blood flow in the trained subjects resulted from both the greater stroke
volume and increased compliance of the peripheral vessels. In a related article
Kvernmo et al (1998) compared the wavelet-based spectral analysis of these signals
before and after exercise. In another related study, Kvernmo et al (1999) used their
methods to determine the effect of vasodilators (endothelium dependent and endothe-
lium independent) on the oscillatory components present in these human cutaneous
blood perfusion signals.
Humeau et al (2000) have presented a model of LDF signals produced when an
arterial occlusion is removed. In such cases the LD F signal increases then returns to
its initial value; this phenomenon is known as reactive hyperaemia and its study is
important in the evaluation of the functional aspects of arterial blood flow.
Humeau and his co-workers obtained their model parameters from experimental
LDF signals obtained from a healthy subject after a 2 min vascular occlusion.
These signals were denoised, prior to parameter estimation, using fourth-order
Symmlet wavelets in a multiresolution decomposition. Reconstruction of the
denoised signal was performed only with those wavelet coefficients from scales
which were noise free. Matani et al (1996) compared short time Fourier transforms
with wavelet transforms in the analysis of ultrasonic Doppler signals from both
simulated and experimental cardiac blood flows. They found that although both
techniques could reproduce the slow changes in velocity, only the wavelet transform
could reproduce the fast changes in velocity with sufficient resolution. Hence they
concluded that the wavelet transform would be more useful in the analysis of
blood flow disorders such as regurgitation.
A method based on wavelet transform analysis was presented by Sato et al (1996)
for the non-invasive determination of the left ventricular (LV) end diastolic pressure
in an isolated canine preparation. The determination of this pressure is clinically
significant for the assessment of ventricular function. Honda et al (1998) extended
this work to humans and developed a non-invasive tool which produced approximate
estimates of LV pressure. Their method used wavelet decomposition of small
amplitude LV vibrations to determine the oscillation frequency of the LV wall.
U sing this information, together with the internal radius and wall thickness, they
were able to estimate LV pressures around end-diastole. Karrakchou et al (1995)
have detailed a wavelet packet method for the analysis of post-occlusion pressure
transients used in the determination of pulmonary microvascular pressure, important
in treating lung oedema. Marrone et al (1999a) have performed Haar wavelet analysis
on blood pressure waves in a study of vasovagal syncope, a rapid and reversing loss of
consciousness due to a reduction of cerebral blood flow caused by dysfunction of the
cardiovascular control. They interrogated the standard deviations and multifractal
characteristics of the wavelet coefficients obtained from a time-series of pressure
maxima and concluded their results indicate a tentative, diagnostic methodology
for this syndrome. Ookawara and Ogawa (2000) have studied the behaviour of
Newtonian and non-Newtonian (bloodlike) fluid within a straight pipe with a
sudden contraction as a model for arterial stenosis. They found that the introduction
Copyright @ 2002 lOP Publishing Ltd.

of pulsations to the flow enhances the recovery of the velocity profile downstream of
the stenosis and their wavelet analysis revealed that the structure of the momentum
transfer within the two fluids was substantially different.
6.6 Medical imaging
There has been much research concerning the application of wavelet methods to the
denoising, visual enhancement and compression of medical images. A wide variety of
medical images, including MRI, PET, SPECT, CT, mammograms, ultrasonic and
optical images, have been considered by numerous research groups. In this section,
only a very brief selection of examples is given of the application of wavelet methods
to the elucidation of medical images.
6.6.1 Ultrasonic images
A method has been developed by Setarehdan and Soraghan (1998) for the measure-
ment and assessment of left ventricular (LV) performance from echocardiographical
images. Their spline-based wavelet method detects the edges of the LV endocardium.
It defines the edge in terms of the global maxima of the wavelet transform of intensity
profiles taken along radial search lines centred within the LV endocardium. Wavelet
image analysis has also been used by Mojsilovic et al (1997) to decompose ultrasonic
images of the heart in order to determine the success of thrombolitic therapy after
acute myocardial infarction. Lee (1996) used a combination of wavelet transforms
and genetic algorithms to detect the edge of tumours in ultrasonic images. Lee's
algorithm incorporates a radial searching routine, whereby the image grey scales
are determined along radial lines from within the tumour. Figure 6.42 shows the
spatial variation in the grey scale along one such radial search line, together with
its wavelet decomposition at three scales. The wavelet decompositions are used to
detect the tumour edge shown in figure 6.43. Fu et al (2000) have detailed a discrete
(Daubechies D6) wavelet-based method for the enhancement of gastric sonogram
images and to compensate for information loss due to histogram equalization.
6.6.2 Magnetic resonance imaging, computed tomography and other
radiographic images
In an early paper, Healy and Weaver (1992) reported on the use of wavelets to
facilitate magnetic resonance imaging and stated that wavelet transforms offer
tangible benefits over traditional Fourier-based imaging (see also Lu et aI, 1992).
Xu et al (1994) introduced a spatially selective noise reduction method for magnetic
resonance (MR) images. In their method, edges and other significant image features
are identified from their strong correlation across scales in the wavelet domain and
noise is identified as having weak correlation across scales. The correlation is
determined from the simple multiplication of the dyadic transform coefficients
across a limited number of scales. Due to their localization in the wavelet domain,
image features remain sharp after filtering. They illustrated their method on both a
Copyright @ 2002 lOP Publishing Ltd.

70
60
Q) 50
::s
ca
;> 40
I
;>..
 30
OJ)
20
10
0
1 11 21 31 41 51
index
(a)
150
100
-.. 50

 0

-50
-100
-150
0 10 20 30 40 50 60
index
(b)
Figure 6.42. Example profile and its three-scale WT. ( a) Example profile. (b) Three-scale WT. After
Lee (1996). Reproduced with kind permission of the lEE.
(a)
(b)
Figure 6.43. Contour extraction performance. (a) Image where contour of tumour has been detected
by conventional radial search with edge detector. (b) Result using proposed GA-based algorithm.
After Lee (1996). Reproduced with kind permission of the lEE.
Copyright @ 2002 lOP Publishing Ltd.

(a)
(b)
(c)
Figure 6.44. MR image from an axial head scan of a volunteer (SNR = 18 dB). (a) Before two-
dimensional filtering; (b) after two-dimensional wavelet filtering; ( c) after two-dimensional Wiener
filtering. After Xu et al1994 IEEE Transactions on Image Processing 3(6) 747-758. (Q) IEEE 1994.
synthetic image and an MR image from an axial head scan. Figure 6.44 shows a noisy
MR image from an axial head scan before and after filtering. The figure also compares
wavelet filtering with Wiener filtering. It was found that wavelet filtering preserved
better high frequency data around the image edges when compared with Wiener
filtering. The authors stated that, although their technique is slightly less accurate
than others, it is more straightforward, easier to implement and significantly more
ro bust.
Wood and Johnson (1999) have outlined a wavelet packet-based method for
denoising MR images containing non-Gaussian (Rician) noise. An image com-
pression method for both computed tomography and magnetic resonance has
been proposed by Wang and Huang (1996) that uses a separable, non-uniform
three-dimensional wavelet transform. The transform uses sets of image slices as
a three-dimensional data set, rather than the working on each slice individually
(two-dimensional method). This removes interslice redundancy. They found
that their technique produces a 70% increase in data compression over that for
Copyright @ 2002 lOP Publishing Ltd.

two-dimensional CT image sets and 35% increase for MR image sets. Knoll et al
(1999) have detailed a method for localized multi scale contour parametrization of
the prostate in both digital CT scans and digitized trans rectal ultrasound images.
Their automated method enables the prostate to be differentiated from adjacent
tissues in low contrast, noisy images. Zheng et al (2000) tested the effect of the digital
compression of CT images on the detection of coronary artery calcifications (CACs).
They found that images compressed up to 20: 1 using both JPEG and wavelet algo-
rithms were acceptable for primary diagnosis of CACs by experienced radiologists.
Other work concerning the use of wavelet transforms in the analysis of MR and
CT images include the wavelet-based approaches to tomographic image reconstruc-
tion described by Kolaczyk (1996) and Bhatia et al (1996), the computer aided
support for the localization of pathological tissues such as brain tumours in MRI
and CT images by Busch (1997), the new MR image guidance dynamic tracking
method proposed by Wendt et al (1998), the denoising and contrast enhancement
method for MR images by Alexander et al (2000) and the wavelet-based feature
extraction method within the neural network algorithm for the automatic windowing
of MR images by Lai and Fang (2000).
In order to improve the visualization of breast pathology, Laine et al (1994)
developed a wavelet-based feature enhancement method to make more obvious
unseen or barely visible features of significance in mammograms. Dhawan et al
(1996) employed wavelet packet transform decomposition of grey level mammo-
graphic images to represent the local texture of the digitized microcalcification
areas associated with the tumour cells. Li et al (1997) have compared a fractal
model of microcalcifications in digital mammograms with both a wavelet-based
method and a morphological operations approach. Heine et al (1999) developed
multiresolution wavelet methods to perform statistical analysis of highly correlated
non-Gaussian random fields and applied them to the analysis of digital mammo-
graphic images. A modulus maxima-based method has been developed by Bruce
and Adhami (1999) to classify mammographic mass shapes. A new wavelet-based
compression scheme for use in coronary angiographic image compression has been
developed by Munteanu et al (1999). Millet et al (2000) have described a wavelet-
based filtering method for dynamic PET (positron emission tomography) data
which improves signal-to-noise ratios without loss of spatial resolution. See also
the paper by Bruckmann and Uhl (2000) which considers medical image compression
techniques for telemedical and archival applications.
6.6.3 Optical imaging
Carmona et al (1995) have developed a procedure for the analysis of brain images
using dyadic wavelets and modulus maxima thresholding. Figure 6.45 shows a
typical brain image (top left) together with the locations of the extrema (modulus
maxima) of the wavelet coefficients at various a-scales. Their method allows a multi-
resolution representation of the image by discarding those wavelet extrema greater
than the 80th percentile of all extrema values. By doing so they filter out the high
gradients produced by the walls of the blood vessels. They go on to tackle the problem
of the large vibratory movement over time of the blood vessels in such images.
Copyright @ 2002 lOP Publishing Ltd.

Figure 6.45. Example of a partial reconstruction from a selective set of extrema of the wavelet
transform. (Left) Top: a typical image. Bottom: image reconstructed from the values of the wavelet
transform at the locations of the local modulus maxima which are smaller than their 80th percentiles
and from the values of the coarse scale image (shown at the bottom of the right column). Notice that
the blood vessels have been removed and replaced by low-frequency surfaces (compare with original
image). (Right) The scale increases from top to bottom. Positions of the local modulus maxima at the
scales s = 2 1 , 2 2 , 2 3 , 2 4 after removing the local maxima with the modulus larger than the 80th per-
centile of the maxima sizes at each scale. Note that the modulus maxima near the boundaries of the
blood vessels are removed. The larger the value of a pixel, the brighter it shows. Bottom: coarse scale
approximation at scale s = 2 1 . After Carmona et al1995 IEEE Transactions on Medical Imaging
14(3) 556-564. (Q) IEEE 1995.
Witkowski et al (1998) used wavelet denoising of image pixels to reduce the noise
present within CCD camera images of both in vitro frog and dog hearts in a study
concerning the nature of ventricular fibrillation. Using voltage sensitive dyes to convert
transmembrane potentials into optical signals, the researchers were able to visualize the
spatial propagation of wavefronts during ventricular fibrillation. Van de W ouwer et al
(2000) have described an automatic classification scheme for the diagnosis and grading
of invasive breast cancer from digitized microscopic images of cell nuclei. The scheme
incorporates a number of image parameters including wavelet energy-based texture
parameters from both the low resolution and detail images.
Copyright @ 2002 lOP Publishing Ltd.

6.7 Other applications in medicine
6.7.1 Electromyographic signals
Electromyographic (EM G) signals represent the electrical activity of muscle during
contraction. A small number of authors have used wavelet methods to gain an insight
into these signals. EMG signals were probed using both the Mexican hat and Morlet
wavelets by Laterza and Olmo (1997). They showed that because of its similarity to
the constituent electrical impulses within the signal, the Mexican hat wavelet
produced a better decomposition of the signal when compared with the Morlet
wavelet, especially in the presence of noise. The maximum value of the Mexican
hat transform was four times larger than that for the Morlet-based transform due
to its better matching with the shape of the signal feature under investigation-the
motor unit action potential (MUAP). Fang et al (1999) have developed a wavelet-
based tool for the identification of single motor unit (SMU) potentials within
EMG signals. They use wavelet-based methods both for denoising the signal via
thresholding and the subsequent identification of SMU spikes in the signal. The
reduction of motion artefacts from surface EMG signals has been investigated by
Conforto et al (1999) using a number of methods including a novel wavelet-based
technique. The four methods they compared were: (1) high pass filtering, (2) a
moving average procedure, (3) a median average procedure and (4) a wavelet-based
procedure they developed themselves. They designed an experimental protocol
which allowed them to evaluate the performance of each method with respect to a
number of criteria and found the wavelet method performed better both with
regard to timing detection and reduced distortion of clinically useful signal content.
A number of other papers have appeared recently reporting on studies involving
the wavelet analysis of the EMG, including the scheme by Karlsson et al (1999)
which uses wavelet packets combined with a wavelet thresholding method for the
spectral analysis of surface EMG signals, the study by the same group (Karlsson,
2000) of a variety of nonstationary signal analysis methods (including the continuous
wavelet transform, for the interrogation of the myoelectric signal during dynamic
contraction), the study of motor unit action potentials using a variety of discrete
wavelets by Pattichis and Pattichis (1999), and the analysis of uterine EMG by
Khalil and Duchene (2000).
6.7.2 Sleep apnoea
Sleep apnoea is a phenomenon characterized by prolonged interruptions of normal
respiration during sleep, caused by the collapse of the upper airway. We have already
come across a couple of studies concerning sleep apnoea in section 6.2.3 in relation to
the study of heart rate variability. Using a spline function wavelet, Figliola and
Serrano (1997) used the distribution of the energy of discrete wavelet coefficients
across scales to characterize three physiological time series associated with sleep
apnoea: heart rate, lung volume variation and blood oxygen saturation. They differ-
entiated between three physiological states-'pre-apnoea', 'periodic breathing' and
'regular'-using wavelet scale-dependent energy distributions which they quantified
Copyright @ 2002 lOP Publishing Ltd.

using an information cost function (ICF) (essentially the Shannon entropy of the
normalized energies-see chapter 4, section 4.2.4). They were able to show that the
ICF decreases from the pre-apnoea to the apnoea state for all three time series.
Another study of sleep apnoea by Kermit et al (2000) used a Haar wavelet decom-
position of airflow signals, where the resulting wavelet coefficients were fed into a
neural network-like classification algorithm which determined whether the signal
represented apnoea events, hyper-apnoea events or normal airflows. (Hyper-apnoea
is the condition where an irregular breathing pattern occurs rather than a complete
cessation of breathing.) It was found by Kermit and his co-workers that apnoea
events were easier to detect than hyper-apnoea events.
6.7.3 DNA
A number of investigators have detailed the use of the wavelet transform in the
analysis of the correlations that exist within DNA sequences. Arneodo et al (1996,
1998a,b) have probed the multifractal nature of DNA sequences using the wavelet
transform modulus maxima method to analyse mapped representations of the
sequence. Altaiski et al (1996) have also provided evidence of the multifractal
nature of DNA sequences. Tsonis et al (1996) investigated the localized structure
of DNA sequences using wavelet transforms and managed to decompose seemingly
homogeneous regions in non-coding sequences into distinct subregions with their
own repetition and construction rules. (More information on fractals and wavelets
is given in chapter 7, section 7.2.)
6.7.4 Miscellaneous
Dickhaus and Heinrich (1996) have presented the components of a system for the
classification of biosignals using 'wavelet networks'. These combine the feature
extraction and selection properties of the wavelet transform within the decision
capabilities of the artificial neural network. Maksimovic and Popovic (1999) have
used both classical neural networks and wavelet networks to classify functional
movements in humans with spinal cord injuries. They found that a combination of
both techniques provides a suitable method for this kind of geometric feature
analysis. Wavelet analysis is detailed, together with a variety of other techniques, in
a comprehensive review of pattern recognition approaches in magnetic resonance
spectroscopy by EI-Deredy (1997). Other applications of wavelet transform methods
in medicine include: the investigations of the effect of morphine on foetal breathing
rates using matching pursuits by Akay and Szeto (1995) and the fractal properties
of foetal breathing rates by Akay and Mulder (1998); the analysis of sleep micro-
strucure through the wavelet decomposition of the polysomnogram by Kubicki
and Herrmann (1996); the interrogation of neuronal activities of the recticular forma-
tion of the lower brainstem and the nucleaus tractus solitarii from an anaesthetized
dog by Lambertz et al (2000); a wavelet-based method for the detection of ultrasound
scattering centres in the liver by Tang and Abeyratne (2000); a method to detect
respiratory and cardiac rhythm disorders from pressure measurements from an
inflated wrist cuff by Dupuis and Eugene (2000); and the analysis by Chan et al
Copyright @ 2002 lOP Publishing Ltd.

(2000a) of chrono biological rhythms through the response of the locomotion of mice
to a change in the light-dark cycle.
6.7.5 Further resources
The special issue of the journal Engineering in Medicine and Biology, edited by Akay
(1995), contains a collection of papers covering a wide range of biomedical signals
(EMG, EEG, ECG, breathing rates and pulmonary capillary pressures) analysed
using a variety of time-frequency methods including wavelet transforms. In a later
paper, Akay (1997) provides a brief summary of selected areas of biomedical research
which have benefited from wavelet analysis, including the detection of coronary heart
disease, turbulent blood flows, irregular heartbeats, the effect of alcohol on foetal
breathing, hearing aids, mammography and medical image compression. In their
paper, Unser and Alrdoubi (1996) present a much more comprehensive overview of
wavelet applications in biomedicine. They include details of wavelet properties that
they consider most important for biomedical applications and provide over one
hundred references. The book edited by Aldroubi and Unser (1996) contains many
detailed papers concerning the application of the wavelet transform to medical
imaging, medical signal processing and biological models. Another collection of
papers, edited by Akay (1998), presents a comprehensive review of the application
of various time-frequency methods to a wide range of biomedical signals. Wavelet
methods feature heavily in this text.
Copyright @ 2002 lOP Publishing Ltd.

Chapter 7
Fractals, finance, geophysics
and other areas
7.1 Introduction
In this last chapter we cover a variety of subject areas in briefer detail than the preced-
ing chapters. Most of the chapter is devoted to three main topics-fractals, finance
and geophysics. First we will look at how we can use wavelet transforms to character-
ize the scaling properties of self-similar fractal and multifractal objects. After this, we
will consider the emerging role of wavelet analysis in financial analysis. The largest
section of the chapter is then devoted to geophysics-where wavelet transform
analysis began with the analysis of seismic signals. Finally, in the last section we
take a brief tour of a selection of other areas where wavelet analysis has made an
impact but has not been covered within the rest of the book.
7.2 Fractals
Fractals are objects which display self-similarity over scales. These objects can be
exactly self-similar, as in figure 7.1(a), where the exact form of the object is repeated
at smaller and smaller scales, or they can be statistically self-similar (figure 7 .1 (b)),
where the statistical properties of the object are consistent across scales (Addison,
1997). Many natural fractals are statistically self-similar, e.g. coastlines, cracking,
tree branching, stock market indices, permeabilities in the subsurface, the distribution
of galaxies, and so on. Previous chapters have already touched upon some natural
phenomena whose fractal properties have been interrogated using wavelet methods:
e.g. fluid turbulence (chapter 4), engineering surface characterization and chaotic
attractors (chapter 5), heart rate variability, breathing rate variability, DNA
sequences and mammographic images (chapter 6). All natural fractals exhibit self-
similarity only over a finite range of scales and hence, unlike their regular mathema-
tical counterparts, their fractal description eventually breaks down. They do,
however, exhibit these fractal properties over a sufficiently large range of scales to
allow fractal geometric methods to be usefully employed in their description. The
property of self-similarity across scales makes wavelet transform analysis a natural
candidate for the interrogation of such objects.
Copyright @ 2002 lOP Publishing Ltd.

150
125
100
75
Y 50
25
o
-25
-50
-50-25 0 25 50 75 100 125 150
(a)
(b)
self similarity
everywhere
25
s
Y 0
-25
-25
o
25
x
x
Figure 7.1. Exactly self-similar and statistically self-similar fractals. (a) The Sierpinski gasket: a
fractal object which is exactly self-similar over all scales. (Each of the circles contains self-similar
parts of the whole gasket at different scales.) (b) The two-dimensional trajectory of ordinary
Brownian motion. (The right-hand plot contains the first 1/16th of the trajectory of the left-hand
plot enlarged to maintain the same degree of resolution between each plot.) After Addison 1997
Fractals and Chaos: An Illustrated Course, P S Addison 1997, Institute of Physics Publishing, Bristol
and Philadelphia.
7.2.1 Exactly self-similar fractals
Figure 7 .2( a) shows another common exactly self-similar fractal, the triadic Cantor
set. The construction method is shown in the plot where, at each step in the generation
of the fractal set, the middle third is removed from the remaining line segments. The
iteration process begins on the unit line and proceeds ad infinitum to construct the set.
Copyright @ 2002 lOP Publishing Ltd.

o
1
step
r-
unit interval
----1
k=O
k=l
k=2
k=3
k=4
k=5
'initiator'
, generator'

Cantor
set
k=oo
3 times magnification
of the left hand third
of the original Cantor set
Cantor
set
9 times magnification
Cantor
set
(a)
(b)
1 
Figure 7.2. Wavelet analysis of the triadic Cantor set. (a) Construction of the triadic Cantor set.
(b) Transform plots for the triadic Cantor set. Note that (sgn(T)IT(a,b)1 1 / 2 ) is plotted against
In(a) and b. (a) After Addison 1997 Fractals and Chaos: An Illustrated Course, P S Addison 1997,
Institute of Physics Publishing, Bristol and Philadelphia. (b) From Arneodo A et aI, Wavelet
transform analysis of invariant measures of some dynamical systems, in Wavelets, Combes J M,
Grossmann A and Tchamitchian P (eds.) Springer-Verlag 1989 pp 182-196. Copyright Springer-
Verlag 1989. Reprinted with kind permission of the authors and publisher.
Copyright @ 2002 lOP Publishing Ltd.

Tg(a,b)
InS /ln3
(b)
»,.;
.<?..:>;...,
V "--"-:'_.:' ...>.:.:.: <-:":
" II\/;,:;' /: .;, 'I - ,/ 1;\ ".../ ,:'1;\ "
/, . II \ ;'/ oJ: I I  '1/  "II i! J/ '1\ :.:.'; ,II! \"1,,
... ,,1111 l/.' II '/;{/." ,.,' :l1'nl' 1/'1' '1\', .
, " ',iA/,i, II" t1 '.ldI,';.'i;;', ; ,'I,' II ,III',' ,Ii!,
.;; 0 ;' ,;,//,I {; '\ ':';': 0 "4  !;' ./';\ ::c :'/ /1\ :I:I\' \111111:!:\ ,"
',. ,\1 ' IN II/, 'III j/ I/f" 11,111,1"11 II' ,I, "/'I, \
I, " " j, ' 111\ \ "1,"", /II /j,'.: ,,/Ii, '/"/1 . ",' ,,'II 'ff
',', /' 1l.'NIII I , I, 'II Ii!" \I!I/ I'if.ll',,;:/; 1 11 1,\",::::.
. . ;' '., " ,.,";/1'// 'II ;'I,i, i,'i ': III, Ii/' ! )/ , 'I 1/ / 1./1 ,/I'/! '1 , / 11 ":' ,
'/>:,; ;;;;;: ,/I/ j /I' ,HI',! h j l'I\,1 1,\//1 I/N ,!IN' .J//\"
I . y: ,:;';;, ';;; "d" J 1,1 W(, ,1/11', /. .,1 ill 11\ II,,' fIN,I;\,
" '1'1 II II' I' II I' ,1,111 II ',' I,'.'
"/H ,I 1/ .,//NI' }',I\ ", I.'.
I},.,.<,.':. " . ).'
Figure 7.3. The wavelet transform of a regular snowflake. The snowflake is shown in ( a). The
scale parameter a is successively divided by the same factor 1=3: (b) a = a*, (c) a = a* /3,
(d) a = a* /32. T(a,b) is expressed in a 1og (5)/log(3) units in order to reveal the self-similarity of
the geometry of the snowflakes. (See Argoul et al (1989) for more details.) Reprinted from
Argoul et al 1989 Physics Letters A 135(6/7) 327-336. Copyright 1989, with permission from
Elsevier Science.
Figure 7.2(b) shows a three-dimensional plot of a Mexican hat wavelet transform for
the triadic Cantor set. The branching structure of the set is easily seen in the transform
plot (Arneodo et aI, 1989). Figure 7.3 contains a regular snowflake fractal generated
in the plane, together with transform plots at three a scales using the radial (two-
dimensional) Mexican hat wavelet (Argoul et aI, 1989). Contour lines, set at an
arbitrary value, show the construction rule of the fractal snowflake in figure 7.4. As
the scale of the wavelet tends to zero, the transform plot approximates more and
more the snowflake itself. See also Antoine et al (1997) who analysed a fractal
Koch curve in their paper concerning the characterization of shapes using the infor-
mation gained from the maxima lines (both modulus maxima and ridges) of the
continuous wavelet transform. The Sierpinski gasket, Cantor set, snowflake fractal
and the Koch curve are exactly self-similar. However, very few examples of exact
self-similarity are to be found in nature (e.g. some fern shapes exhibit nearly exact
self-similarity over a few scales). Most natural fractals exhibit stochastic self-
similarity and consequently most research concerning fractal objects and processes
in nature focuses on stochastic fractals.
Copyright @ 2002 lOP Publishing Ltd.

*
a=a
a*/3
by
(a)
bx
a*/3 2
a*/3 3
(c)
(d)
Figure 7.4. Isocontour lines of the wavelet transform of the snowflake fractal. The isocontour line
T(a, b)/a 1og (5)/log(3) = k (k arbitrarily chosen) for different values of the scale parameter. (a)
a = a* , (b) a = a* /3, (c) a = a* /3 2 , (d) a = a* /3 3 . Reprinted from Argoul et al1989 Physics Letters
A 135(6/7) 327-336. Copyright 1989, with permission from Elsevier Science.
7.2.2 Stochastic fractals
Figure 7.5 shows a sequence of shots for a diffusion limited aggregation (DLA) cluster
and associated wavelet transforms. DLAs are essentially stochastic snowflakes 'grown'
using a numerical technique which allows particles to wander randomly about the plane
(in a Brownian motion) until they encounter the DLA aggregate, sticking to it on
impact (Addison, 1997). DLA has been used to model a variety of physical phenomena
including bacterial colonies, viscous fingering and electrochemical deposition. See also
an extension to this type of work contained in the paper by Pei et al (1995).
One area where wavelet transform analysis has been used extensively is in the
determination of the scaling properties of fractional Brownian motion (fBm) (and
its derivative, fractional Gaussian noise, fGn). Over recent years, these correlated
random functions, first proposed by Mandelbrot and Van Ness (1968) as a general-
ization of Brownian motion, have been suggested as models for a whole range of
natural phenomena, including DNA sequences, geometrical tolerancing in mechani-
cal design, risk analysis, polymer models, landscape surfaces, image textures, the
dynamics of nerve growth, crack profiles, permeability fields in porous media and
non-Fickian diffusive processes (Addison and Ndumu, 1999). The 'smoothness' of
the fBm function increases with the Hurst exponent H, which can vary in the range
o < H < 1. Three examples of fBm, denoted BH(t), are shown in figure 7.6 for
different H. Fractional Brownian motions are nonstationary random processes
Copyright @ 2002 lOP Publishing Ltd.

. ""."1j;!t;,'.311i!;W  .@"',; . ... .
..:MII"fII/!c,.;.,<...'..,",'/,'::: ":.:-i1I1.1l 'j {/':' ..'. .' .....
">rr" 'J 1'1" "," ',.',' d, ','J 'ltJlJliJ:"" ,1..' '.
.:"(;1::?t{'::1tf;;;' .
: .t;...l'l" 1> ,,"', \ I), . "I ,Ii.,:.' t (
;;::.':"." r".,.;.;."J,f i. ....... ", ':J'R{ /.
'>... . .f!jJY;;:: «::.. . ,.rf.1;?':::';.:' .:::.::';;;/<,":'" '
/./ ".
- :. .
. ." .
.':' ,.......'.. 1
.)t;Ai'{'g&!;:1. .L.'
,.:,:,?jjJi  . k11illr," "til';c"
': .......:..)\J;..  . r'j/\{!l\:,,"'t:.,, ..-
..  ... Il;;.' I ; ", < \,."I..  '! ' 1 'I J,.. n.'
. . .....>. .' ...... I,' f J ':(.' "'{ ,'.., r' r-fffi:,',; ''':'..
. .. . ..../ . /,' li6k' ' I I (II   I) . .. , '. .' !j,lt. ,,  r. . .
. . .:y no't:" f If,' · 1:' ,'I_. . .tI' r
'/s, "h/ "1" I, ",. I" H; I.>
. "'t . .., ,'I "":' i ' h . t . ,; tJtrt' #  ' I ' . , . 'i '!I .  J}lll
'., ..,r., i ( 1 J(,'" . \1,., \' I, ,'l' ';\'" )..
, ". ".,' ''/i,t//: I" I, I'r ' .
". . ./I./
-+
T gCa,b)
a 1.60
(b)
Figure 7.5. Wavelet transforms of a DLA cluster. The DLA cluster is shown in (a). The scale para-
meter a is successively divided by the same factor 8 = 1.55: (b) a = a*, (c) a = a* /8, (d) a = a* /8 2 ,
(e) a = a* /83, (f) a = a* /84. T(a, b) is expressed in a1.60 units in order to reveal the self-similarity
of the geometry of the DLA clusters. (See Argoul et al (1989) for more details.) Reprinted from
Argoul et al1989 Physics Letters A 135(6/7) 327-336. Copyright 1989, with permission from Elsevier
Science.
where the standard deviation, (J", of the fBm trace deviations BH (figure 7.6(c)) taken
over a sliding window of length s scales as
H
(J"rxs
(7.1)
H > 0.5 corresponds to persistent fBm where the trace has a tendency to persist in its
progression in the direction in which it was moving, H < 0.5 corresponds to antiper-
sistent fBm where the trace has a tendency to turn back upon itself, and H == 0.5
corresponds to regular Brownian motion where the trace is free to move in either
direction from step to step (for more information see Mandelbrot, 1982; Addison,
Copyright @ 2002 lOP Publishing Ltd.

4
2
.-
+-'
- 0
I
CD
-2
-4
0
(a)
40
0
-40
.-
+-'
- -80
I
CD
-120
-160
-200
0
(c)
10
o
Z' -10
-
I
CD -20
-30
256
512
time 't '
768
1024
-40
o
256
512
time 't'
768
1024
(b)
L\B H
256
512 768
time 't'
1024
Figure 7.6. Fractional Brownian motion traces at various Hurst exponents. (a) fBm trace, H = 0.2,
antipersistent. (b) fBm trace, H = 0.5, neutrally persistent (regular Brownian motion). (c) fBm
trace, H = 0.8, persistent. After Addison 1997 Fractals and Chaos: An Illustrated Course, P S Addison
1997, Institute of Physics Publishing, Bristol and Philadelphia.
1997). The fractal dimension of an fBm trace function can be found from the Hurst
exponent through the simple relationship
D==2-H
(7.2)
There are many methods used in practice to determine either H or D for experimental
data suspected of fBm scaling. One common method uses the Fourier power spec-
trum PF(f), which for an fBm, should scale as
PF(f) rxf- (2H + 1) (7.3)
Hence, a logarithmic plot of power against frequency allows H to be determined from
the slope of the spectrum. Wavelet power spectra, Pw(f), both continuous and
discrete, also exhibit this scaling and can therefore also be used to determine H
(and hence D if required). Figure 7.7 shows a single realization of an fBm trace
with H == 0.6, together with both its Fourier and wavelet power spectra (refer back
to chapter 2, section 2.9). A continuous Mexican hat wavelet was used in the decom-
position. The slope corresponding to -2.2 (i.e. H == 0.6) is plotted on the graph for
comparison. We can see from the plot that the wavelet spectrum is much smoother
due to the finite bandwidth of the spectral components associated with the wavelets.
This can have an advantage when trying to abstract the fractal information from such
Copyright @ 2002 lOP Publishing Ltd.

(a)
1.10 7
1.10 6
-.. 1.10 5
00

.
s:::
::s 1.10 4

1:
. 1.10 3


I-; 100
Q)

0
0.-
10
1
1.10- 4
(b)
slope = -(1 +2H)
1.10- 3 0.01 0.1
frequency (arbitrary units)
1
Figure 7.7. Fourier and wavelet spectra of fractional Brownian motion. (a) A single realization of an
fBm trace function H = 0.6. (b) Fourier (dashed line) and wavelet (smooth line) power spectra of the
trace in (a).
data sets when only a limited number of data sets are available-too few to smooth
the Fourier spectrum through ensemble averaging.
We can compute the fractal scaling characteristics of a data set suspected of
exhibiting fBm scaling from its wavelet decomposition: continuous or discrete.
Simonsen et al (1998) used a discrete wavelet coefficient-based method to analyse
two real data sets: fracture surface profiles and stock market indices. They showed
that the mean absolute discrete wavelet coefficient value scales as
(ITm,nl ex a}:+!)
(7.4)
where
(I Tm,nl)m ==
2 M - m -1
2: ITm,nl
n=O
2 M - m
(7.5)
Copyright @ 2002 lOP Publishing Ltd.

10 3
. Daub 4
10 2 . 'Daub 8
. Daub 12
10 1 . Daub 16
10 0 .. Daub 20
-..

 .. Daub 24
r---1
...c::
L......oI 10- 1

10- 2
10- 3
10- 4
10- 5 - 
10- 4 10- 3 10- 2 10- 1 10 0
a
Figure 7.8. The Awe function versus scale for various choices of wavelet order (Daubechies family) for
self-affine profiles with Hurst exponent H = 0.7. Wavelet order indicated in the figure. The number of
samples used was N = 100. The length of the profiles was L = 4096. Note that the data are rescaled
so that the largest scale is equal to unity (100). The extracted Hurst exponents were H = 0.68 ::f:: 0.01
for Daubechies D4 and H = 0.70::f:: 0.01 in all other cases. The curves are shifted relative to D12 for
clarity. After Simonsen et al (1998). Reproduced with the kind permission of the American Physical
Society and the authors.
is the mean absolute value of the wavelet coefficients at scale am. The authors used
dyadic grid Daubechies wavelets, hence am ex 2 m , and also set the maximum a scale
to unity. The authors called this measure the average wavelet coefficient (A WC)
function. Figure 7.8 shows the logarithmic plots of the A WC against wavelet
scale for an fBm profile computed using a variety of Daubechies wavelets. The
plots are obtained using the average values of 100 realizations of fBm, each 4096
data points in length and with a Hurst exponent H == 0.7. There is very good
agreement between the Hurst exponents of the original traces and those found by
the method. This can be seen from figure 7.8 where the slope of the best-fit lines
is approximately 1.2. The authors proceeded to test the stability of the method
against noise, drift and distortion before using it to interrogate two real data
sets-fracture profiles and stock prices. Figure 7.9(b) shows the A WC analysis of
211 granite profiles, one example of which is plotted in figure 7.9(a). The Fourier
power spectral analysis (figure 7.9(c)) and the A WC analysis are in good agreement.
Figure 7.10 contains the stock market index for shares taken from the Milan stock
exchange over a two and a half year period. Both the A WC and Fourier analysis of
this single time series are plotted below the index. The large spread in the Fourier
spectrum is evident in the plot. The authors concluded that for large numbers of
samples the wavelet method and Fourier (power spectrum) method perform equally
well. However, when only a small number of examples are available the wavelet
method outperforms the Fourier method. This was also concluded by Dougan
et al (2000).
Copyright @ 2002 lOP Publishing Ltd.

24
(a)
22
r---1
 20
L-...I
-..
 18

16
14
0 20 40 60 80
 (mm)
10 8
(c)
10 6
8 104

10 2
10°
10- 2
10- 4 10- 3 10- 2 10- 1
f
10 4
(b)
10 2
-..


r---1 10°

L-...I

10- 2
10- 4
10- 3 10- 2 "10- 1 10°
a
Figure 7.9. Wavelet and Fourier analysis of a granite fracture profile. (a) One single representative
profile from the granite fracture. The number of points in the profile is 2050. (b ) Average wavelet
coefficient (A WC) analysis of the entire set of (2050 x 211) data points. The solid line is the regression
fit to the scaling region. The corresponding Hurst exponent is H = 0.81 ::f:: 0.02. (c) Fourier power
spectral (FPS) analysis of the data. Here the solid line corresponds to a Hurst exponent of
H = 0.79 ::f:: 0.03. After Simonsen et al (1998). Reproduced with the kind permission of the American
Physical Society, the authors and J Schmittbuhl.
Another related method of computing H directly from the wavelet coefficients
using the scaling of the variance of discrete wavelet coefficients at scale index m
which we saw in chapter 4, section 4.2.1, is given by
2 M - m -1
2: (T m,n)2
/ T2 ) _ n = 0
\ m,n m - 2 M - m
(7.6)
From chapter 4, equation (4.1 Ob), we know that coefficient variance is simply related
to the power spectrum as
Pw(fm) ex (T,n)m
(7.7)
Combining this expression with the power law relationship for fBm of index H given
above in expression (7.3) and the fact that frequency fis inversely proportional to the
wavelet scale a (== 2 m ), we obtain the scaling relationship
( T 2 ) (2H + 1)
m n m ex am
,
Copyright @ 2002 lOP Publishing Ltd.
(7.8a)

200 400
..,.
10M
1012
Stoll
... to'
1(1
Jr
Figure 7.10. Wavelet and Fourier analysis of share prices. (a) Fiat share prices taken from the Milan
Stock Exchange for the period from September 1988 (day 1) to May 1991, with three observations per
day. (b) The result of the wavelet analysis for the data in (a). The estimated Hurst exponent,
corresponding to the solid line, is H = 0.65 ::f:: 0.03. (c) The result of the Fourier analysis for the
data in (a). The Hurst exponent in this case is H = 0.62 ::f:: 0.06. Note the more well behaved scaling
region for the wavelet method compared with the Fourier method. After Simonsen et al (1998). Repro-
duced with the kind permission of the American Physical Society, the authors and B Vidakovic.
In the literature, (J" is often used as a compact notation for the variance of the discrete
wavelet coefficients at index m, hence the relationship is written
2 a (2H + 1)
(J"m ex m
(7. 8b )
If we take the square root of both sides of expression (7 .8b) to get
H+l
(J"m ex am 2
(7.9)
we can see that the scaling of the coefficient standard deviations is consistent with that
given by expression (7.4). We expect this, as both the mean absolute value of the
coefficients and the standard deviation of the coefficients are first-order measures
of spread. Furthermore, for an orthonornmal multiresolution expansion using a
dyadic grid, the scale a is proportional to 2 m . We can, therefore, take base 2 loga-
rithms of both sides of expression (7.8b) to get the often-seen expression
log2 ((J") == (2H + l)m + constant
(7.10)
where the constant depends both on the wavelet used and the Hurst exponent. See, for
example, Flandrin (1992) who defines the constant and gives it explicitly for the Haar
Copyright @ 2002 lOP Publishing Ltd.

wavelet. Figure 7.11(a) shows plots oflog2((}) against m for two fBms, one of index
H == 0.6 and the other H == 0.8. These were computed using Haar wavelets. Accord-
ing to expression (7.12), the plotted points should fall on lines of slopes 2.2 and 2.6
respectively. These gradients are superimposed on the plots. Good agreement is
found between the experimental plots and the theoretical line. Remember that this
is for a single realization of the fBm trace, and ensemble averaging over many
traces will produce a much more accurate result. Figure 7 .11 (b) contains a plot of
the coefficients in sequential format for the fBm of index H == 0.6 in figure 7.11(a).
The increase in the variance with scale is evident from the plot. Figure 7 .11 (c) contains
example signal reconstructions using only the coefficients at the levels specified. Li et al
(1996) have used this approach in a geophysical context to examine the fractal struc-
ture of velocity logs.
Note that, from the above arguments, we can also see that the wavelet scale-
dependent energy
2 M - m -1
Em == 2:: (Tm,n)2
n=O
(7.11 )
has the fractal scaling law
log2(E m ) == 2Hm + constant
(7.12)
that is
2H
Em ex am
(7.13 )
This is the same scaling as that given by equation (7.1). This makes sense, as Em is a
measure of the scale dependent variance of the signal. Figure 7.11 (d) shows the
log2(E m ) against m plot for the traces shown in figure 7.11(a). These plots have
slopes of around 1.2 and 1.6 respectively as we would expect. We have concentrated
on discrete transform coefficients T m n as they are prevalent in the fractal-wavelet
,
literature. However, continuous transforms, T (a, b), obviously exhibit the same
scaling law as (7.13), i.e.
E(a) ex a 2H
(7.14 )
(refer back to chapter 2, equation (2.22)) but now the a scale parameter is continuous
(or at least for practical purposes a discretized approximation based on a non-dyadic
grid) and the slope of the plot of log(E(a)) against log(a) is used to find H.
There are now large numbers of scientific papers concerning the identification
and modelling of fractional Brownian objects and processes using wavelets. Meharabi
et al (1997) compared seven different methods for analysing fBm, including one based
on equation (7.10). They found that the wavelet decomposition method offers a highly
accurate and efficient tool for characterizing long-range correlations in complex
distributions and profiles. Kaplan and Kuo (1993, 1995) have presented a method
based on the Haar wavelet to estimate the fractal dimension of fBm from fractal
signals using the scaling relationship between the wavelet coefficients of the discrete
fGn increments of the fBm signal. The fractal feature extraction technique they
developed was used for texture segmentation of synthetic test images and aerial
Copyright @ 2002 lOP Publishing Ltd.

30
25

N E: 20
$ 15
N
 10
........
5
o
o 1 2 3 4 5 6 7 8 9101112
m
30
25

N E: 20
$ 15
N
00 10
.Q
5
o
o 1 2 345 6 7 89101112
m
(a)
m=6,7,8,9,10
m=5
50
m=2
m=l
100
T.
l 0
-50
-100
o
200
400
600
800
1000
1200
(b)
i (=2 M - m +n)
Figure 7.11. The relationship between the wavelet coefficient variance and Hurst exponent for
fractional Brownian motion. (a) fBm traces (H = 0.6 leftlH = 0.8 right) with their corresponding
log2((}) against m plots given below. The slopes of 2H + 1 = 2.2 (left) and 2.6 (right) are shown
on the plots above the data points. (Arbitrary axis units.) (b) The sequentially indexed wavelet
coefficients for the H = 0.6 fBm trace in (a). Note that the coefficient values have been cut off at
the largest scales in the plot as the coefficients at these scales dominate.
images of natural scenes. The paper by Malamud and Turcotte (1999) contains results
from the variance analysis of both fractional Brownian motion and fractional Gaus-
sian noises. They compared the wavelet technique with three others for determining
the scaling relationship and noted that wavelet variance analyses lack many of the
inherent problems associated with Fourier power analysis. The identification of
fBm within white noise is considered by Hwang (1999) and a method for the detection
of transients within l/f Gaussian noises using undecimated discrete wavelet trans-
forms is described by Liu and Fraser-Smith (2000). Chen et al (1997a) used wavelets
to define the fractal dimensions of satellite images. They then employed neural
networks to classify the images, e.g. urban, bare soil, ocean, forest, etc. The energy
content and self-similarity of the detail signals of fractional Brownian images is
Copyright @ 2002 lOP Publishing Ltd.

100 50 50
0 0 0
-100 -50 -50
0 500 1000 0 500 1000 0 500 1000
m=10 m=9 m=7
20 5
0 0
-20 -5
0 500 1000 0 500 1000
m=5 m=2
(c)
30 30
25 25
S 20 -.. 20
 
 15  15
OJ) OJ)
0 10 0 10
....... .......
5 5
0 0 I I I I I I I I I I I I
0 1234567 8 9101112 0 1 2 3 4 5 6 7 8 9101112
m m
(d)
Figure 7.11 (continued). (c) Reconstruction using only those coefficients at the indicated scale. Note
the reduction in the vertical axis scale with decreasing coefficient scales. (d) log2(E m ) against m plots
corresponding to the fBm traces given in (a) (H = 0.6 left/ H = 0.8 right). The slopes of
2H + 1 = 1.2 (left) and 1.6 (right) are shown on the plots above the data points.
discussed briefly by Mallat (1989). Tumer et al (1995) have employed wavelet-based
techniques to determine fractal characteristic measures of precision-machined
surfaces and Ikeda et al (1999) used two-dimensional wavelets to estimate the
dimension of fractal surfaces. Frantziskonis et al (2000) have determined the Hurst
exponent of corrosion pits using Daubechies wavelets. Arneodo et al (1998c) found
fBm scaling in DNA sequences using Mexican hat wavelets. Li and Ulrych (1999)
have used fBms synthesized using wavelets to model geological processes. Jones
et al (1996) have suggested a method for the computation of Hurst exponents
using wavelet packet analysis. They tested their method on synthetic fBm traces
before using it to characterize the spatial distribution of local enzyme concentration
in fungal colonies. Zeldin and Spanos (1996) have outlined a wavelet method for the
synthesis of random fields including fBm, Elliot et al (1997a) have detailed a Fourier-
wavelet (Meyer) method for synthesizing fractal random fields and Pesquet-Popescu
(1999) has detailed a wavelet packet-based method for analysing two-dimensional
fBm fields. Masry (1998) has described the spectral properties of two-dimensional
self-affine fBm fields (and other types) in terms of the discrete wavelet transform.
Copyright @ 2002 lOP Publishing Ltd.

Vergassola and Frisch (1991) and Vergassola et al (1993) have discussed the applica-
tion of wavelet transforms to self-similar processes, considering both turbulent
flow and fBm signals. Elliot et al (1997b) considered a wavelet-based technique to
study the relative spreading of pairs of particles for a family of anisotropic velocity
fields. Thresholding methods for use with fractional Gaussian noise (the derivative
of fBm) are discussed by Wang (1996). A number of methods, including wavelet-
based methods, for the determination of H from fGn signals and its relationship to
fBm is discussed in the context of heart rate variability by Fischer and Akay (Fischer
and Akay, 1996, 1998; Akay and Fischer, 1997). Liu et al (2000b) have developed a
two-dimensional fractal parameter estimation method for natural scenes and
textures. They demonstrated their technique by performing coastline detection and
texture segmentation of both synthetic and natural images. There are many
other scientific papers which detail in greater depth the analysis and synthesis of
fractional Brownian motion using wavelet transforms. See for example Masry
(1993), Hirchoren and Attellis (1997, 1998), Tewfik and Kim (1992), Dijkerman
and Mazumdar (1994), Ramanathan and Zeitouni (1991), Kato and Masry (1999)
and Veitch and Abry (1999).
7.2.3 Multifractals
Multifractal theory concerns itself with fractal objects which cannot be completely
described using a single fractal dimension (monofractals). They have in effect an
infinite number of dimension measures associated with them. This section presents
a short summary of wavelet-based multifractal characterization.
The multifractal scaling of an object is characterized by
N c ex c-!(a) (7.15)
where N c is the number of boxes of length c required to cover the object andf(a) is
the dimension spectrum, which can be interpreted as the fractal dimension of the set
of points with scaling index a (Hillborn, 1994). We can find the multifractal spectrum
of a signal by partitioning it into Nboxes of length c. A probability density, P(c, i), of
the signal in each box, labelled i, is calculated where P( c, i) is the fraction of the total
mass of the object in each box. The qth-order moments M(c, q) are then calculated as
follows:
N(c)
M(c, q) == 2:: P(c, i)q
i= 1
(7.16)
For a multifractal object this moment function scales as
M(c, q) ex cT(q)
(7.17)
From this scaling both a and the f (a) spectrum can be calculated from
a(q) = dT(q)
dq
(7.18)
Copyright @ 2002 lOP Publishing Ltd.

and
f(a) == qa(q) - T(q)
(7.19)
In the wavelet-based method for calculating the f( a) spectrum, the function of
equation (7.16) is replaced by the wavelet-based moment function
M(a,q) == 2:: IT(a,bi)l q
(7.20 )
where IT (a, b i ) I is the ith wavelet transform modulus maxima found at scale a. By
summing only over the modulus maxima, this incorporates the multiplicative struc-
ture of the singularity distribution into the calculation of the partition function
(Muzy et aI, 1991). For a multifractal object this wavelet-based moment function
scales as
M(a, q) ex aT(q)
(7.21)
This relationship is then used to calculate a and the f(a) spectrum and hence
characterize the multifractal object or process under investigation. Note that for
negative q values equation (7.20) becomes unstable in the neighbourhood of points
on maxima lines where the wavelet transform is close to zero. However, this can be
remedied by replacing the value of the wavelet transform modulus at each maximum
by the supremum value along the corresponding maxima line at all scales smaller than
a (Muzy et aI, 1993).
Haase and Lehle (1998) have described a modulus maxima method based on
Gaussian derivative wavelets for determining multifractal spectra. They have
illustrated their method on time series from a circle map and a turbulent velocity
signal. Figure 7.12 contains the turbulent velocity signal, together with its multifractal
spectrum. The signal is shown in figure 7 .12( a), acquired from within an axisymmetric
jet. Figure 7 .12(b) shows the energy spectrum of the signal showing the classic - 5/3
power law of the inertial range of turbulence. Figure 7 .12( c) contains the wavelet
transform plot with its modulus maxima shown directly below in figure 7.12(d). A
Mexican hat wavelet was used in the decomposition. Finally, the f(a) spectrum,
computed from the modulus maxima lines, is shown in figure 7.12(e). Note that if
the velocity signal were a monofractal, such as fBm, the multifractal spectrum
would collapse to a single point.
Full coverage of multifractal theory is outside the scope of this text. For more
information, including the use of T( q) to determine the generalized fractal dimension
Dq, see for example Degaudenzi and Arizmedi (1999) who have studied the multi-
fractal character of an airborne pollen time series using a wavelet transform
modulus maxima method incorporating the third derivative of a Gaussian as the
analysing wavelet. Arrault et al (1997) have used a wavelet-based multifractal
analysis technique to probe high resolution satellite data of marine stratocumulus,
comparing their results with the analysis of monofractal Brownian surfaces. Riedi
et al (1999) have developed a wavelet-based multifractal model for use in computer
traffic network modelling (see also Gilbert and Willinger, 1999). They suggested
the use of their model in other disciplines such as finance and geophysics. Muzy
et al (1991) considered wavelet transforms and multifractal signals, comparing a
Copyright @ 2002 lOP Publishing Ltd.

(a)
10 15
Z' 10 10

 10 5
10°
10 2
k
10 4
(b)
(c)
1.0
2 -4 0.8
.e
-..
l  0.6

0.4
4 -2 0.2
.e
f 1.0 1.2 1.4 1.6
a
(d) (e)
Figure 7.12. Multifractal spectrum of a turbulent velocity signal. (a) Turbulent velocity signal. (b) The
energy spectrum of the signal in (a) showing the -5/3 law. (c) Wavelet transform of (a) using a
Mexican hat wavelet. (d) Skeleton of maxima lines from (c). (e) The multifractal spectrum f(a)
resulting from the wavelet modulus maxima method. Figures kindly provided by Dr Maria
Haase, lnstitut fur Computeranwendungen, UniversiUit Stuttgart.
multifractal turbulent signal with a monofractal fBm signal both with the same
spectral power law. A number of practical examples of the connection between wave-
lets and (multi)fractals can be found in the book on wavelets, fractals and Fourier
transforms edited by Farge et al (1993).
7.3 Finance
This is a very interesting application of wavelet analysis which concerns a pertinent
problem (which there is often a large financial incentive to solve!). The data sets
(financial indices, census data, spatially distributed econometric measures, etc.) are
typically highly nonstationary, exhibit high complexity and involve both (pseudo-)
random processes and intermittent deterministic processes. A good example would
be financial indices generated from, among other effects, a large number of small
scale (pseudo-random) share dealings combined with large scale ( deterministic)
Copyright @ 2002 lOP Publishing Ltd.

Coffee
sample period: 1960.1 until 1995.12
:::--:-:-- - .:-:::::itt.::
m:  ;":I:$::$:
:"«: .: ;:.:-;=-:
300
250
200
150
100
50
o
o
I
0.1
I
0.2
I
0.3
I
0.4
I
0.5
trend
I
0.6
I
0.7
I
0.8
I
0.9
I
1.0
Figure 7.13. Wavelet analysis of coffee price index. Top, dyadic wavelet scalogram. Bottom, original
time series with trendline obtained using the wavelet-transformed data at the base level of resolution.
After Davidson et a11998. Reproduced with kind permission of Academic Press Ltd and the authors.
interest rate adjustments. The application of wavelets to finance is still in its infancy
when compared with other subject areas. However, wavelet theory is beginning to
make a number of inroads into the area. A good overview of the application of wave-
lets in economics and finance is given by Ramsey (1999). The paper provides selective
details of recent advances in the analysis of economic data using wavelet transforms.
More specific examples of applications are detailed below.
We have already come across the work of Simonsen et al (1998) in the previous
section, who showed fractal scaling of stock market indices using a method based on
wavelet coefficient scaling (figure 7.10). Davidson et al (1998) have used the orthogo-
nal dyadic Haar transform to perform semi-nonparametric regression analysis of
commodity price behaviour. In their study, they present dyadic scalograms and
take account of edge effects in the data. An example of one of their commodity
price scalograms is given in figure 7.13. See also Kim (1999) who presented brief
results for the Korean stock price index. Ramsey et al (1995) searched for evidence
of self-similarity in the US stock market price index. They investigated the power
law scaling relationship between the wavelet coefficients and scale and found some
evidence of quasi-periodicity in the occurrence of some large amplitude shocks to
the system. In addition, they concluded that there may be a modest amount of
predictability in the data, and that it may be more than just a simple Brownian
motion. In Ramsey and Lampart (1998), the authors highlighted the importance of
Copyright @ 2002 lOP Publishing Ltd.

timescale decomposition in analysing economic relationships. Using an S12 Symmlet
they analysed the different relationships that can occur between two economic
variables at different levels of decomposition. Wavelet-based methods to remove
hidden cycles from within financial time series have been developed by Arifio et al
(1995). Their methods first decompose the signal into its wavelet coefficients then
compute the energy associated with each scale. The dominant scales are defined as
those with the highest energies. New coefficient sets are then produced related to
each of the dominant scales by either one of two methods developed by the authors.
These reapportion the coefficient values according to the scales of the dominant
energies. Hence, for the signal containing two dominant scales, two new complete
sets of wavelet coefficients are computed. These are then used to reconstruct two
separate signals, corresponding to each dominant scale. Arifio and Vidakovich
illustrated their method using a time series of Spanish cement production. This is
shown in figure 7.14, where the original time series (shown dashed in the figure)
is partitioned into a business cycle component and a seasonal component. A
Daubechies D4 wavelet was used in the decomposition of the original time series.
Evidence for a cascade mechanism in market dynamics has been found by
Arneodo et al (1998d) using wavelet transforms. They suggested that this may be
attributed to a variety of mechanisms, including the heterogeneity of traders and
their different time horizons causing an 'information' cascade from long to short
timescales, the lag between stock market fluctuations and long-run movements in
dividends, and the effect of the release (monthly, quarterly) of major economic
indicators which cascades to fine timescales. In a related paper, Muzy et al (2000)
have proposed a multifractal 'stochastic volatility' model that captures the features
of financial fluctuations where wavelet transforms are employed to determine the
multifractal nature of the time series. A multifractal random walk is then employed
as the 'stochastic volatility' model, whose parameters were estimated for real financial
data. Aussem et al (1998) have used wavelet-transformed financial data as the input to
a neural network which was trained to provide five-days-ahead forecasts for the
S&P500 closing prices. They performed the analysis on the wavelet coefficients over
the four smallest scales using a B-spline wavelet. In addition, they examined each
wavelet series individually to provide separate forecasts for each timescale and recom-
bined these forecasts to form an overall forecast. Morehart et al (1999) have employed
wavelets as a spatial analysis tool for economic and financial measures used in
agriculture. They illustrated their method using decompositions of wheat dependency
data and debt utilization ratio data displayed on two-dimensional maps of the United
States. Figure 7.15 shows one of the plots from the study containing the spatial
distribution of the decomposition of the debt capacity utilization ratio. Morehart
and co-workers recommended the method for the graphical presentation of informa-
tion, density estimation and wavelet-based non-parametric regression. Matching
pursuits have been used by Ramsey and Zhang (1996) to decompose the S&P500
index. The data are characterized by periods of quiet interspersed with intense activity
over short periods of time. The authors found that fewer coefficients are required to
specify the data than for a purely random signal signifying some form of deterministic
structure to the signal. Ramsey and Zhang have also applied matching pursuits to
foreign exchange rate data sets, specifically the Deutschmark-US dollar, yen-US
Copyright @ 2002 lOP Publishing Ltd.

Figure 7.14. Spanish cement production. Top, total series (dashed) and business cycle series. Bottom,
seasonal component. After Arino et al (1995). Reproduced with kind permission of the authors.
dollar and yen-Deutschmark (Ramsey and Zhang, 1997). Their analysis revealed
underlying traits of the signal. However, they went on to state that although most
of the energy of the system occurs in localized bursts of activity, there seems to be
no way of predicting the occurrence of these random events and hence little oppor-
tunity to improve forecasting. Shin and Han (2000) have also investigated exchange
rate forecasting using a method which combines wavelet transforms, genetic
algorithms and artificial neural networks. They used their method to forecast the
daily Korean won/US dollar returns one day ahead of time and found that the genetic
algorithm-based wavelet thresholder in their method performed better than three
other wavelet thresholding algorithms: cross-validation, best level and best basis.
Finally, a method for the filtering out of intraday periodicities in exchange rate
time series has been proposed by Gencay et al (2001) using the maximal overlap
discrete wavelet transform.
Copyright @ 2002 lOP Publishing Ltd.

Figure 7.15. Wavelet representation of debt capacity utilization ratio (DCUR) data. From top to
bottom: DUCR data; detail or wavelet coefficients (non-negative values shown) at successive
resolution levels scales 1, 2, 3 and, bottom, the smoothed version of the data. After Morehart
et al 1999 Int. J. Geographical Information Science, published by Taylor and Francis
http://www.tandf.co.uk.journals/. Reproduced with the kind permission of both the publisher and
authors.
7.4 Geophysics
We have already covered the application of wavelet analysis to geophysical flows in
chapter 4, including canopy flows, cloud formation processes, wind-generated
ocean surface waves, and large scale oceanic and atmospheric flow phenomena. In
Copyright @ 2002 lOP Publishing Ltd.

this section we cover some of the other areas of geophysics that have employed wave-
let transform methods including seismology, well logging, topographic feature
analysis and the analysis of climatic data. A good place to begin a literature search
is with the comprehensive summary paper by Kumar and Foufoula-Georgiou
(1997). A more detailed overview can be found in the earlier collection of papers
edited by the same authors (Foufoula-Georgiou and Kumar, 1994).
7.4.1 Properties of subsurface media
It is generally accepted that modern wavelet transform analysis began in the early
1980s with the Morlet wavelet, developed to aid in the interrogation of seismic signals
(e.g. Goupillaud et aI, 1984). Seismology can, therefore, be thought of as the birth-
place of the modern wavelet transform. Since then, many other wavelets have been
developed and used to analyse these and many other signals in geophysics. There
are, in fact, large numbers of techniques available to measure the properties of sub-
surface media including geophysical techniques (seismology, gravimetry, magnetic
methods, electrical methods, radioactive methods), ground penetrating radar
(GPR) and well logging. A few examples of the wavelet analysis of these signals
are covered in this section.
The role of wavelets is discussed briefly in the review of recent developments in
the processing of seismic data by Talwani and ZeIt (1998). The use of orthonormal
discrete wavelet transforms as a tool for characterizing seismic time series has been
examined by Grubb and Walden (1997). They decomposed seismic signals using
both Daubechies wavelets (D4 to D20) and Symmlets (S8 to S20) providing simple
illustrative examples of the wavelet decompositions in their explanatory paper.
Deighan and Watts (1997) have employed Battle-Lemarie wavelets to filter seismic
signals in order to suppress unwanted surface waves (ground roll) present within
the signals. Fedorenko and Husebye (1999) have developed a method, which includes
the Daubechies D20 wavelet transform, for the automatic detection ofP and S arrival
times in seismic records. Li et al (1996) have used a wavelet-based synthesis technique
to model fractal-like velocity logs of a zero offset vertical seismic profile. They
compared their results with a classical damped least squares method for modelling
this type of data and found that the wavelet-based technique reduces significantly
the artefacts associated with the classical technique. (Note that in this paper the
authors used a 'scale index' which is actually a 'level index' as we have defined it in
chapter 3, section 3.3.4.) In a later paper, Li and Ulrych (1999) used wavelet trans-
form-based models of fractal fBm processes (refer back to section 7.2.2) for the
modelling and analysis of processes with features of geological and physical interest.
Kalcic et al (1999) have used Coiflets to perform wavelet analysis of acoustic imagery
from shallow seismic data taken at the sea floor. By examining the wavelet transform
coefficients of the signals at different scales they could highlight different features
below the surface such as steel cables and methane gas pockets. Hachiya and
Amao (1996) have presented a wavelet-processing method to isolate pertinent
features from acoustic images of subsurface media containing buried artefacts.
They proposed the use of their technique for separating the desired signal from
archaeological acoustic survey data at shallow depth. Bergeron et al (1999) have
Copyright @ 2002 lOP Publishing Ltd.

frequency (Hz)
scale index
frequency (Hz)
4.8 0 20 40 60 80 100
o 20 40 60 80 10001.02.03.0 4.0
0.00
A
0.25
0.50
B
0.75
-..
r.f.)
'-'
Q)
8 1.00
.-
-+-- C
1.25
1.50
D
1.75
2.00
(a)
(b)
(c)
(d)
Figure 7.16. Decomposition of a seismogram using STFT, CWT and MP. (a) A synthetic seismogram
used in the comparison study. (b) A time-frequency decomposition produced by an STFT. (c) A
scale-index translation plot produced by the CWT. (d) A time-frequency energy distribution
produced by the MP. After Chakraborty and Okaya (1995). Reproduced with the kind permission
of the Society of Exploration Geophysicists and the authors.
recently applied three-dimensional Mexican hat wavelet transforms to a three-
dimensional seismic tomographical model. They used the maximum wavelet energies
and associated wavenumbers as proxy quantities for viewing seismic velocity
anomalies (SV As). They found that the distribution of these wavelet quantities
reveals information which is not obvious from direct visual examination of SV As,
e.g. depth extent of tectonic boundaries and inference of plumelike objects.
Nagano and Niitsuma (2000) have reported on a Morlet wavelet-based study to
measure the dispersion of crack waves in subsurface media. These waves are trapped
seismic waves which propagate along fluid-filled crack interfaces and their waveforms
(e.g. velocity-frequency dispersion and amplitude-space distribution) are strongly
dependent on both the geometry and physical properties of the crack.
Chakraborty and Okaya (1995) have compared short term Fourier transforms,
continuous (Morlet) wavelet transforms and matching pursuit (MP) decomposition
in a study of seismic data. Figure 7.16 shows a synthetic seismogram together with
three decompositions using short time Fourier transforms, wavelet transforms and
matching pursuits. Comparing the three signals we can see that the MP decomposi-
tion localizes the signal events best in the time-frequency plane. Figure 7.17 shows
Copyright @ 2002 lOP Publishing Ltd.

18
56
1.0
0.0
0.0
0.5
0.5
1.0
-..
00

Q)
8
.-
...
1.5
1.5
2.0
2.0
2.5
2.5
3.0
3.0
Figure 7.17. Shot Gather 19, Siljan, Sweden. Reflections A, B, C caused by dolerite sills of 60m,
layered 30 m and 20 m, respectively. After Chakraborty and Okaya (1995). Reproduced with the
kind permission of the Society of Exploration Geophysicists and the authors.
a reflection seismogram from Slijan, Sweden. Several major reflections can be identi-
fied in the figure. These reflections, labelled A, Band C in the figure, were identified as
being caused by dolerite sills: one of 60 m thickness, a layered 30 m sill and a 20 m sill.
Figure 7.18 contains the matching pursuit decomposition for two of the traces (18 and
56) of the seismogram. Their locations are identified in figure 7.17. Four different
feature shapes were identified in the matching pursuit time-frequency plane by
the authors. The first type, elliptical shapes elongated in the frequency direction,
represent events that are localized in time but contain many frequencies. These
are associated with reflections. A second type, elongated in time but narrow in
frequency, is identified with low-frequency surface waves. The third type is circular,
representing events which have only one or two frequencies present and exist for
only a short period of time. The fourth type is a long streak in the time direction.
This represents a single frequency such as 60 Hz noise which occurs over a long
time duration. Figure 7.18(a) contains the MP decomposition of signal 18. Reflected
Copyright @ 2002 lOP Publishing Ltd.

18
frequency (Hz)
o 25 50 75 100 125
56
frequency (Hz)
o 25 50 75 100 125 150 175
0.0
0.0
S
A
0.5
A
0.5
1.0
1.0
B
B
,.-..,
00 1.5
'-'" 1.5
(],)
S
....-1 C
+->
C
2.0
2.0
2.5
2.5
Figure 7.18. Time-frequency energy distribution of the Siljan data using MPD. (a) Trace 18. (b) Trace
56. (The boxes on top of the time-frequency plots represent the one-dimensional Fourier spectra of
the respective traces.) After Chakraborty and Okaya (1995). Reproduced with the kind permission of
the Society of Exploration Geophysicists and the authors.
events A, Band C can all be identified in the time-frequency plane by elliptical atoms
with shorter time axes and longer frequency axes. The circular atoms are caused by
random noise present in the data and the vertical peaks are due to shot-generated
noise. Events Band C can also be located in figure 7 .18(b) for signal 56. Verhelst
(1998) has also interrogated seismic data using the matching pursuits method with
Gabor atoms and found phase attributes that could be related to facies types in a
delta system.
The properties of subsurface media, especially the permeabilities of soils and
rock, have in many cases been found to be multi scale in nature and hence particularly
amenable to both fractal and wavelet analysis. Prokoph and Barthelmes (1996) have
used Morlet wavelet transforms both to detect and localize abrupt changes and to
differentiate between periodic and chaotic cycling sequences in marine sedimentary
Copyright @ 2002 lOP Publishing Ltd.

wavelet scalogram
Taaken -
Ottersberg 1
00
......
....-1
3 stages
trans-/regressions
(tr./re.)
,
ill
i
,
I
upper
1 V · Ca. .r-"""- polyplocum reo
lower
 V.Ca.
Q) S 
....... Q)
 t)
..... :>.
00 00
3 L.Ca.
4
Sa.
.5
mucronata tr.
pilula tr.
M.Santonian tr.
koeneni reo
(llsede tectoev.)
Hyphanto-
ceras tr.
80
27.5
- .
6.9 4.8 period (m)
plenus tr.
Early Ceno-
manian tr.
Figure 7.19. Wavelet scalogram (left) of the SP logging data (centre) of borehole Taaken-Ottersberg 1
near Bremen. After Niebuhr and Prokoph (1997). Reproduced with kind permission of Academic
Press Ltd.
successions. They tested their analysis on sample data sets containing periodic
and chaotic sequences, before examining gamma-ray well logs from marls of the
Cretaceous North German basin. Again using Morlet-based transforms, Niebuhr
and Prokoph (1997) investigated self-potential logging data taken from sites in
North Germany. Figure 7.19 shows an example of the borehole logging data together
with its corresponding scalogram. Features at 80 m, 6.9 m and 4.8 m dominate the
scalogram plot. In addition, an abrupt cut-off in the 27.5 m feature in the scalogram
can be seen to occur at the Turonian/Conician (Tu/Co) boundary. Using this type of
analysis, the authors identified regions of cyclicity and chaos in the borehole data.
Prokoph and Agterberg (1999) have further examined the use of wavelet transforms
in the detection of cycles, trends and discontinuities in sedimentary successions.
Moreau et al (1996) proposed a filtering method for non-stationary geophysical
data using orthogonal D20 wavelets and a thresholding technique based on chi-
squared statistics. The authors compared their method with two other thresholding
criteria and applied it to both synthetic data and field data composed of thermistance
measurements made in a limestone underground quarry. Li (1996) has presented
preliminary results concerning the use of continuous complex wavelets in the analysis
of synthetic sonic log data. In a later paper, Li (1998d) presented the method in more
detail, applying the wavelet transform to both synthetic and real data. He performed
Copyright @ 2002 lOP Publishing Ltd.

. -
6x10 4 /
C /
I
I
Q)
u '/
s:::
C\S
.....
 / '.
>

Q) /
.......
Q)
 3x10 4 /

2x10 4
..
-
1x10 4 A
./
0.5
1
1.5
2
2.5
3
3.5
4
scales ( meter)
Figure 7.20. Characteristic multiscales of spatial homogeneity of permeability determined by wavelet
analysis. The data are from the north-east wall of area 5, pit 3 in the alluvial fan deposit, and
were gathered from four sets of vertical transect data: group A at I-12m, group B at 12-24m,
group C at 24-36 m and group D at 36-52 m. Curves indicate the magnitude of the variability as
a function of scale, arrows indicate scale of local maximum variance. Taken from Li B- Land
Loehle C 1995 'Wavelet analysis of multi scale permeabilities in the subsurface' Geophysical Research
Letters 22(23) 3123-3126. Copyright 1995 American Geophysical Union. Reproduced by permission
of the American Geophysical Union.
wavelet power spectral analysis of the data sets to reveal the structural properties of
the underground heterogeneities. Saito and Coifman (1997) have analysed acoustic
well-logging signals using a wavelet dictionary employing two different methods to
select the best basis. Their methods allow them to both differentiate between sand
and shale and estimate the volume fractions of minerals at each depth. Li and
Loehle (1995) have investigated the spatial series of permeabilities taken during a
geological cross-section of alluvial fan deposits. They used the wavelet variance of
(Mexican hat) transformed data to characterize the multi scale heterogeneity of the
subsurface permeabilities. A plot of the wavelet variance for four subsets of the
vertical transect permeability data is shown in figure 7.20. The authors argued that
the plot shows the heterogeneity of the spatial structure in permeability across
scales and between transects.
Panda et al (1996) have applied wavelet transforms and wavelet packets to both
one- and two-dimensional permeability data to determine both the location of layer
boundaries and other discontinuities and reduce the amount of data required to
represent the signal. Figure 7.21 shows their original permeability signal together
with a denoised version. The denoising was performed using a wavelet threshold
Copyright @ 2002 lOP Publishing Ltd.

Figure 7.21. Denoising Page sandstone permeability using wavelet threshold method. After Panda et al
(1996). Copyright 1996 The Society of Petroleum Engineers. Reproduced with kind permission of the
Society of Petroleum Engineers.
method and allowed the authors to identify the true discontinuities in the data. Figure
7.22 shows the original data set (bottom) containing 4096 data points, together with two
reduced permeability data sets. The top plot was reduced to 32 points using a wavelet
packet scheme and the middle plot was reduced, again to 32 data points, using a
traditional pressure-solver scaleup tool. Comparing the two figures we can see that
the wavelet method preserves better the most important features of the permeability
field, especially the low and high permeability streaks. Chu (1996) has outlined a
technique based on multiresolution analysis for upscaling one- and two-dimensional
petroleum rock reservoir parameters for single and multiphase flows. Both Mehrabi
and Sahimi (1997) and Gunal and Visscher (1997) have detailed wavelet-based numer-
ical methods for modelling transport in heterogeneous disordered media, and Sahimi
(2000) has developed a fractal-wavelet neural-network method that can characterize
and model fractured subsurface reservoirs. Fenton (1999) has shown how wavelet
coefficient variances may be used to estimate the characteristic parameters in spatially
stochastic soil variation. In another study of the spatial variability of soil characteristics,
Lark and Webster (1999) used wavelet-based statistics, based on the Daubechies D4
wavelet, to characterize two soil transects taken from contrasting landscapes. Vermeer
and Alkemade (1992) have suggested the use of Mexican hat wavelets for the analysis of
well logs. They used the zero crossings in the scalogram to detect edges in the data which
they interpreted as boundaries between different geological units. In this way they were
able to perform a multi scale segmentation of gamma-ray well log data.
7.4.2 Surface feature analysis
We have already considered the use of wavelets in the analysis of engineering surfaces
in chapter 5 and fractal surfaces earlier in this chapter. The analysis of geophysical
Copyright @ 2002 lOP Publishing Ltd.

 2D.
.@I 10
I:
wavelet solved results, number of points = 32
pressure solver result, number of points = 32
f : ,
I :
....
'-\.J
I
,
I
raw data, Dvmber of points 4098
f:
I:
o
10
20
depth (m)
3Cr
40
Figure 7.22. Comparing wavelet-based local variability preserving scaleup results with pressure
solver scaleup data. Fine grid = 4096, coarse grid = 32. After Panda et al (1996). Copyright 1996
The Society of Petroleum Engineers. Reproduced with kind permission of the Society of Petroleum
Engineers.
surface feature distributions is carried out for a number of reasons including the
monitoring of pollution, vegetation growth, the form and spread of towns and
cities or the analysis of natural topographic features. The topography of the surface
also affects the fluid dynamic properties of the fluid-surface boundary layer (oceanic
and atmospheric).
The interannual variability of normalized difference vegetation index (NDVI)
data has been interrogated by Li and Kafatos (2000) using an S8 Symmlet wavelet
decomposition. They found a possible relationship between the El Nino/Southern
Oscillation Index and an NDVI variation signal from an II-year data set. They
proposed the variability of the NDVI as a good proxy for climate variations. Brad-
shaw and Spies (1992) have considered both Haar and Mexican hat wavelets in a
study to characterize the nature of canopy gaps within forests. They used wavelet
variance to identify the dominant scales in forest canopy structure. In another
study of the response of vegetation to landscape structure, Brosofske et al (1999)
used Mexican hat wavelets to interrogate two diversity indices. Plotting the wavelet
variance resulting from the transformed data, they found that quite different scales
were dominant in the two indices. A fractal-wavelet approach has been employed
by Solka et al (1998) for the identification of man-made regions in unmanned
aerial vehicle imagery. Adolphs (1999) has employed a number of techniques includ-
ing fractal and wavelet methods to analyse the roughness variability of sea ice and
snow cover thickness profiles. The study found that wavelets proved to be superior
Copyright @ 2002 lOP Publishing Ltd.

to visual inspection in objectively finding hidden segments and substructures within
the profiles. The scale analysis of the surface properties (surface temperature and
albedo) of sea ice has been carried out by Lindsay et al (1996) using the discrete
wavelet transform. The techniques developed in this earlier investigation were
later used in a study of the temporal variability of the energy balance of thick
Arctic pack ice by Lindsay (1998). Using the Morlet wavelet, Little et al (1993)
have detected a 200 km long anomalous topographic zone in a 1600 km bathymetric
profile, which they suggest is the site of a short-lived, abandoned spreading centre
associated with a thermal topographic swell. Also using a Morlet wavelet, Cazenave
et al (1995) have analysed medium-wavelength (rv 1 000 km) geoid anomalies over the
central Pacific.
In an early paper on the application of wavelet transforms, Teti and Kritikos
(1992) analysed azimuth cuts taken from two different SAR ocean images using
Morlet wavelets in both a running discretized form and a tight frame form. More
recently, Fukuda and Hirosawa (1999) investigated the smoothing effect of wave-
let-based speckly filtering of SAR images, and Ferretti et al (1999) have exploited
multibaseline SAR interferometry using a wavelet base weighted average method
for the reconstruction of high-quality digital evaluation models. The method leaves
important peaks in the signal much more intact than traditional highpass filtering.
Simhadri et al (1998) have developed a wavelet-based feature extraction computa-
tional scheme to extract fine edges from oceanographic images. Wang et al (1999)
have developed a wavelet-based technique (using a redundant tight wavelet frame)
for the removal of clouds and their shadows from Landsat TM images. Ranchin
and Wald (1993) have reviewed the application of wavelet transforms to remotely
sensed images (specifically SPOT HRV panchromatic images). Later, Wald and
Baleynaud (1999) employed wavelet-based techniques in a study of air quality in
cities using Landsat thermal infrared data. Zhou et al (1998) used a multiresolution
wavelet-based method to merge two types of remote sensing image (SPOT pan-
chromatic images and Landsat TM images). They did this to produce a hybrid
image containing both good terrain detail and useful spectral information. There
are many other papers in the literature concerning the fusion of mulitsensor
geophysical data images using wavelet methods. See for example Nunez et al (1999),
Zhukov et al (1999), Ranchin and Wald (1997) and the references contained therein.
7.4.3 Climate, clouds, rainfall and river levels
An introductory tutorial on the use of wavelet transforms in climatic time series
analysis is given by Lau and Weng (1995). In their paper, an analogy between the
wavelet transform representation and music is used to illustrate the difference
between the local and global information in climatic time series before applying the
methods to Northern hemisphere surface temperature measurements (over a period
of 140 years) and a deep sea sediment record (over 2.5 million years). Baliunus et al
(1997) have employed both Mexican hat and Morlet wavelets in a study of tem-
perature data taken over three centuries in central England. The better temporal
resolution of the Mexican hat is put to use in filtering trends in the time domain,
whereas the superior frequency resolution of the Morlet wavelet is used for spectral
Copyright @ 2002 lOP Publishing Ltd.

studies of the data. In a paper covering the monitoring and analysis of multiple
microclimate variables for characterizing the ecology of the physical environment,
Chen et al (1999b) used Mexican hat wavelets to look at the scaling in temperature
and overstorey cover at ground surface along a transect within a pine forest. Hu
and Nitta (1996) used Mexican hat wavelets to detect dominant timescales in the
rainfall records over North China and India over the period 1891-1992. They
found that the dominant timescales in the rainfall variations are located in two time-
scale bands: shorter than 10 years and 14-28 years. Fraedrich et al (1997) have
compared a number of analysis techniques to identify abrupt climate changes from
historic time series of the flood levels of the river Nile. They found that a multi scale
moving t-test has advantages over the wavelet transform (using the first derivative of
Gaussian wavelet) in detecting and specifying the degree of significance of the abrupt
climate change. Labat et al (2000) have provided a well illustrated, comprehensive
account of the use of a variety of wavelet methods to analyse rainfall-runoff relation-
ships from karstic springs. They employed both discrete and continuous wavelet tools
in their investigation including wavelet spectra and cross-spectral analysis, wavelet
coherence, hard thresholding and multiresolution covariances. They studied periodic
pumping and intermittent runoff processes in two separate karstic spring outflow data
sets. The wavelet methods allowed them to separate different subprocesses connected
with the data. They then analysed rainfall and springflow rates in three karstic basins
and found that the wavelet cross-analyses gave meaningful information on the
temporal variability of the rainfall-runoff relationship. Jay and Flinchem (1997)
used continuous wavelet analysis to interrogate the modulation of the external tide
in a river by variations in streamflow. They found the wavelet transform method
superior to Fourier and harmonic techniques for studies of nonstationary river
tides because of its time-frequency resolution properties. Nakken (1999) used
Morlet wavelet transforms to investigate the rainfall and runoff records of a river
catchment in order to differentiate components due to climate change from those
due to man-made effects (e.g. land use changes). The method is proposed for the
detection of streamflow response to climate change, especially large-scale circulation
phenomena.
Serio and Tramutoli (1995) found evidence for the existence of scaling laws in a
cloud system generated and advected by strong baroclinic instability. Using two-
dimensional Mexican hats to decompose infrared cloud images collected by satellite,
they were able to discern two scaling behaviours in this type of frontal instability.
Using purpose-built wavelets whose shape was inspired by the general shape of the
signatures of the aircraft echoes within the time series, Boisse et al (1999) have
developed an automated wavelet-based method for removing aeroplane and other tran-
sitory echoes in strato-tropospheric (ST) radar measurements. Grecu and Krajewski
(1999) have used wavelet and fractal preprocessing techniques within a neural
network-based methodology for the detection of anomalous propagation echoes in
weather radar data. Low total ozone events over northern Sweden have been analysed
using Mexican hat wavelet transforms by Weinberg et al (1996). See also the interesting
discussion on the merits of the wavelet-based analysis of the data ensuing
from this paper involving Flynn and Celarier (1997) and Weinberg et al (1997).
Kumar and Foufoula-Georgiou (1993) have decomposed two-dimensional rainfall
Copyright @ 2002 lOP Publishing Ltd.

fields using two-dimensional orthogonal Haar wavelets in a multiresolution represen-
tation. They proposed the use of wavelets as a consistent method for the decomposi-
tion of inhomogeneous and anisotropic rainfields. In a later paper (Kumar and
Foufoula-Georgiou, 1996a), the authors established empirical connections between
wavelet-based statistical characteristics of the rainfall data and physical storm
characteristics. The results from this work were then used to develop a model for
the disaggregation of spatial rainfall based on a coupling between its meteorological
and scaling descriptions (Kumar and Foufoula-Georgiou, 1996b). Other studies in
this area include those by Gollmer et al (1995) who used Daubechies D8 wavelets
to analyse liquid water path (L WP) data in a study of marine stratocumulus cloud
inhomogeniety, Tian et al (1999) who employed wavelet packets in a neural network
study of cloud classification, and Chapa et al (1998) who used Morlet wavelet
transforms to interrogate cold cloud index data in order to detect periodicities in
convective activity in South America.
7.5 Other areas
7.5.1 Astronomy
There has been a recent surge in the use of wavelet transforms for the analysis of
astronomical data, both time series and spatial data. Aschwanden et al (1998) have
performed a multiresolution analysis of hard x-ray time series of solar flares. Their
analysis allowed them to determine the shortest timescales associated with strong,
smoothly varying, and weak flares. Furthermore, the fastest significant time struc-
tures (in strong flares) were found to be related to physical parameters of propagation
and collision processes. Morlet wavelets were used by Townsend (1999) to analyse the
line-profile variations of rapidly rotating stars exhibiting non-radial pulsation.
Hughes et al (1998) used Morlet wavelets to analyse over 20 years of radio flux
data from the astronomical object BL Lac object OJ 287. They found modulation
of the total flux over the whole time period. However, this modulation changed its
period midway through the time series. This behaviour is explained through a
'shock in jet' model by the authors. Morlet wavelets have also been used by Lucek
and Balogh (1998) in a study of Alfvenic fluctuations in solar wind data. Bedding
et al (1998) have reported on a wavelet-based investigation of the switching between
two pulsation modes of a star. The switching of the pulsation modes between 332 and
175 days is clearly seen in their transform plots associated with the light curve time
series. Bijaoui et al (1996) used discrete wavelet transforms to provide a new statistical
indicator of cosmological distributions according to scale. The spatial structure in
cosmic microwave background (CMB) maps has been assessed by Tenorio et al
(1999) using spherical Haar wavelets. They went on to use the soft-thresholding of
planar Daubechies wavelets coefficients to denoise the map: both to remove local
point sources and to suppress (nonlocal) instrument noise. Sanz et al (1999) used a
variety of two-dimensional transforms together with wavelet thresholding to denoise
CMB anisotropy maps with added noise at different signal to noise ratios and then
compared their results to other methods of filtering. Pagliaro et al (1999) have
Copyright @ 2002 lOP Publishing Ltd.

analysed the substructure of clusters of galaxies using three-dimensional cubic
B-spline wavelets. Fang et al (1998) considered the problem of galaxy clustering.
They used Daubechies D4 wavelets to investigate the bias associated with scale
dependence of the clustering associated with galaxies. Pando and Fang (1998) have
developed a discrete wavelet spectrum estimator which they have used to analyse
the power spectrum of the spatial distribution of Lya clouds. Meiksin (2000)
employed Daubechies D20 wavelets to provide a statistical characterization of the
absorption properties of the Lya forest. The method was applied to the Keck
HIRES spectrum of Q1937-1009 and proposed as an easily automated procedure
for basing a comparison between measured and predicted properties of the Lya
forest. Rabadi and Myler (1998) used an image of Saturn to illustrate their fast wave-
let-based algorithm for the image reconstruction from Fourier domain information.
Finally, Aussem and Murtagh (1997) have employed neural networks to predict
sunspot time series using wavelet decomposition.
7.5.2 Chemistry and chemical engineering
A good place to begin the search for information concerning the use of wavelet
analysis in chemistry is with the review paper by Leung et al (1998). This com-
prehensive review covers the application of wavelet transform techniques in chemical
analysis from 1989 to 1997 and contains 130 references to other articles in the field.
The authors cover a variety of topics in chemistry including flow injection analysis,
chromatography, infrared spectroscopy, mass spectrometry, nuclear magnetic
resonance and ultraviolet-visible spectroscopy, voltammetry, quantum chemistry
and chemical physics. The paper by Alsberg et al (1997) provides a comprehensive
overview of the wavelet transform targeted at chemometricians. It gives a brief history
of the wavelet transform before tackling both the continuous and discrete wavelet
transform, including wavelet packets, and their applications. Depczynski et al
(1997) provide a brief but clear introduction to the discrete wavelet transform (and
associated multiresolution analysis) and its use in the analysis of chemical signals.
See also the paper by Shao and Cai (1998). In a more general review of chemometrics,
Wold and Sjostrom (1998) have highlighted the wavelet transform as an emerging
tool for dealing with large sets of very similar variables (from spectra to chromato-
grams). More specific studies concerning wavelet transforms in chemistry and
chemical engineering include: the use of spline wavelets by Zheng et al (1999b) in
the study of electrochemical signals; the analysis of flow regime in gas-liquid
bubble columns by Bakshi et al (1995) using wavelet packets; wavelet-based methods
for process monitoring developed by Shao et al (1999) and Tsuge et al (2000); an
integrated wavelet and neural network-based framework for process monitoring
and diagnosis by Zhao et al (1998), Chen et al (1998, 1999a), Wang et al (1999)
and Yang et al (2000); a brief mention of the role of wavelet transforms in the
treatment of noisy electrochemical data by Cottis et al (1998); the analysis of magnetic
resonance spectroscopy data using continuous wavelets by Ding and McDowwell
(2000) and Serrai et al (2000); the application of wavelet-based neural networks
to hybrid modelling and optimization of a chemical process by Safavi et al (1999);
the wavelet-based error control and adaptation strategy within the simulation of
Copyright @ 2002 lOP Publishing Ltd.

multicomponent mixture processes by Briesen and Marquardt (2000); the use of
discrete wavelets by Stephanopoulos et al (1997) in the mining of fermentation data-
bases; the scale and magnitude thresholding (which they call smoothing and denoising
respectively) of experimental spectra using wavelet and other transforms by Barclay
et al (1997); and the numerical procedure based on wavelet collocation suggested by
Liu et al (2000a) for the solution of models for packed-bed chemical reactors and
chromatograph columns.
7.5.3 Plasmas
There are many papers concerning the use of wavelet transforms in this area. Some
recent ones include: the studies by Dose et al (1997) on fusion plasma transients;
Santoso et al (1997) on nonstationary plasma fluctuations; van Milligan (1997) on
the nature of plasma edge turbulence; Dong et al (1998) on coherent structures
within tokamak plasma turbulence; Bruskin et al (1998) on the reconstruction of
the plasma density profile using wavelet analysis of microwave reflectometer signals;
Bruskin et al (1999) on plasma fluctuations using wavelet spectra and cross spectra;
Heller et al (1999) on scrape-off layer intermittency in the Castor tokamak using
wavelet bicoherence; and Jakubowski et al (1999) on the analysis of plasma fluctua-
tion measurements using beam emission spectroscopy. It is interesting to note that
most of these studies employed the Morlet wavelet.
7.5.4 Electrical systems
The use of wavelets in the detection of transmission line transients in power cables has
been investigated by a number of researchers. A brief introductory tutorial intended
for power system engineers is provided by Kim and Aggarwal (2000). Goswami
(1998) has used semi-orthogonal wavelets to evaluate the reflection coefficient for
open/short transmission lines. A wavelet transform approach to the solution of
multiconductor transmission line equations has been developed by Raugi (1999).
Huang et al (1999b) employed a Morlet wavelet transform approach to detect various
simulated power system disturbances, including voltage sag, voltage swell, momen-
tary interruption and oscillatory transients. Zhao et al (2000) have also proposed a
wavelet-based approach to fault detection and classification in power cable systems.
Their system, however, uses a multiresolution algorithm based on discrete ortho-
normal wavelets such as Daubechies, Coiflets and Symmlets. Both wavelet packets
and continuous Morlet wavelet transforms have been used by Pham and Wong
(1999) to perform harmonic analysis of power system waveforms. They validated
their approach on both synthesized waveforms and power system waveforms
measured in the W estern Australia system. A discrete wavelet transform-based
feature extraction technique for the discrimination between internal faults and
inrush currents in power transformers has been developed by Mao and Aggarwal
(2000). A method for the reliable analysis of ultrawide-band detected partial
discharge (PD) currents has been developed by Angrisani et al (2000) based on the
Daubechies D16 wavelet. They illustrated their method on experimental PD signals.
Morency and Lemay (1999) have described a method based on the Daubechies D4
Copyright @ 2002 lOP Publishing Ltd.

wavelet for the detection of erroneous data points (spikes) present in a signal sampled
from an analogue-to-digital convertor.
7.5.5 Sound and speech
Early papers by Grossman et al (1987) and Kronland-Martinet (1988) discuss the use
of the Morlet wavelet in the analysis of speech and music sounds. Guillemain and
Kronland-Martinet (1996) have presented techniques for the estimation of relevant
parameters from sound signals such as frequency and amplitude modulation
corresponding to each spectral component of the sound. Pielemeier et al (1996)
provide a review of time-frequency analysis of musical signals which includes wavelet
transforms among a variety of other techniques and, more recently, Fujinami (1998)
has described a wavelet analysis of a sound image localization transfer function in
headphone listening which reveals important features for spatial hearing. The
compression of high quality audio signals using wavelet packets is described by
Sablatash and Cooklev (1996) within a more general treatise on the subject.
A Daubechies wavelet-based method for detecting transient underwater signals
from within a changing background sound environment has been developed by
Bailey et al (1998). They applied their method to the detection of dolphin
sounds-clicks and whistles-from within noisy sound data segments. Quyen et al
(1998) have developed a method based on the Daubechies D4 wavelet for the
classification of underwater mammals (whales and porpoises). Speech recognition
is tackled with Haar wavelet transforms by Uchaipichat and Parnichkun (2000) in
work which combines wavelet decomposition and neural network classification.
Weiss and Dixon (1997) have outlined a wavelet-based denoising method for the
removal of unwanted backscatter from high-frequency underwater acoustic signals.
Unoki and Akagi (1999) have addressed the problem of extracting a desired acoustic
signal from a noisy signal using wavelet analysis. Kermit and Eide (2000) have given
details of an audio signal identification method in which Haar wavelets are used to
preprocess audio signals (speech and music) prior to identification using a neural
network. Obaidat et al (1999) have described a wavelet-based method for the
detection of pitch period, an important parameter in the design of automatic
speaker recognition systems (see also Obaidat et aI, 1998). A text-to-speech synthesis
system is described by Kobayashi et al (1998) which incorporates discrete wavelet
analysis in the determination of pitch period. Sarikaya and Hansen (2000) have
investigated the impact of stress on monophone speech recognition accuracy and
proposed a new set of acoustic parameters based on high resolution wavelet
analysis. A method for the encoding and enhancement of wideband speech signals
using wavelet packets has been described by Carnero and Drygajlo (1999). Mikhael
and Ramaswamy (1995) used speech signals to illustrate a method of signal
representation using mixed transforms and Singh et al (1997) have selectively
enhanced acoustic events in speech signals using wavelet subtransform domains.
Pinter (1996) has given a summary account of many of the types of wavelets that
have been employed in the representation of speech signals including the Morlet,
Haar, Le Marie-Meyer and Malvar wavelets, together with wavelets derived from
biophysical models of the inner ear and wavelets with psychoacoustical features.
Copyright @ 2002 lOP Publishing Ltd.

Pinter then described the development of perceptual wavelets and their application
to speech enhancement.
7.5.6 Miscellaneous
In an introductory review of patterns that occur in extended dissipative systems,
including sand waves in the desert, the stripe patterns of fish, Rayleigh-Benard
convection in fluids and magnetic domains, Bowman and Newell (1998) have
suggested the continuous wavelet transform as a useful tool for the extraction of
order parameters from these patterns. Li et al (1996) have provided a review of the
application of wavelet transforms to optics and Gharbi and Barchiesi (2000) have
used Daubechies wavelets in a method to characterize the local spatial frequency
separation in near-field microscopy. Wen et al (1996) and Deng et al (1999) have
detailed wavelet-based approaches to the verification of handwritten signatures.
Other applications of wavelets include the digital watermarking for tamper-proofing
of images (Kundur and Hatzinakos, 1999; Tsai et aI, 2000); the use of discrete wavelet
analysis for pattern generation in architectural design (Sariyildiz et aI, 1998); the
automatic detection of incidents on motorways (Cohen and Jing, 1995); the study
of intermittency in discrete dynamical systems, namely the logistic map (Figliola
and Schuschny, 1995); the analysis of the three-dimensional topography of a
clamshell surface (Toubin et aI, 1999; Diou et aI, 1999); the tracking of moving targets
using multiresolution analysis (Hong, 1999) and matching pursuits (Chen and Ling,
1999); the construction of a Lorentz-covariant superposition of light waves (Han et aI,
1995); lip reading using both Coiflet wavelets and Fourier methods (Yu et aI, 1999)
and automatic human face recognition and expression analysis using both wavelets
(Wiskott, 1999; Kondo and Yan, 1999; Lien et aI, 2000) and wavelet packets
(Garcia et aI, 2000).
Copyright @ 2002 lOP Publishing Ltd.

Appendix
Useful books, papers and websites
This appendix aims to provide the reader with a shortlist of useful books, papers and
websites. They have been selected by the author for their extensive content and/or
clarity of presentation.
1 Useful books and papers
There are a large number of wavelet books in the literature, from those aimed at
a mathematical audience to those which deal with a specific scientific discipline.
The mathematical and statistical literature is greatly weighted towards the discrete
orthonormal wavelet transform and associated transforms, e.g. the nondecimated
discrete wavelet transform, due in part to its nice mathematical properties.
However, the applied scientific literature is more balanced between the continuous
and discrete wavelet transforms. There are many good texts available covering the
background theory or specific applications. However, this section is restricted to
three texts.
· The World According to Wavelets. In her book Hubbard (1996) provides an
excellent account of the history and use of the wavelet transform. As it subtitle
says, the book is 'the story of a mathematical technique in the making'. Much
of the text is written using no mathematics at all, and where mathematical
explanations of the concepts are employed to convey some of the concepts, the
treatment is minimal.
· Ten Lectures on Wavelets. By Ingrid Daubechies (1992), this is one of the first
wavelet texts and has become a standard in the field.
· A Wavelet Tour of Signal Processing. A more recent text by Mallat (1998), this
book provides much of the mathematical detail underlying useful wavelet
transform techniques used in practice.
There are a number of papers in the literature which transcend their target audience to
provide a lucid explanation of some aspect or aspects of wavelet transform theory
which can be accessed by a wider audience. Some of these papers together with a
brief description of their contents are listed below:
Copyright @ 2002 lOP Publishing Ltd.

· Kumar and Foufoula-Georgiou (1994): This is the introductory paper in the book
edited by Kumar and Foufoula-Georgiou. It provides a concise account of both
the continuous and discrete wavelet transforms.
· Alsberg et al (1997): Targeted at chemometricians, this comprehensive paper
contains some historical background, some basic theory of the continuous and
discrete wavelet transforms and wavelet packets before ending with applications
including denoising, baseline removal and compression. The appendix contains
some useful information too, including details of the HYBRID and VISU thresh-
olding methods.
· Labat et al (2000): This is a well illustrated, comprehensive account of the use of a
variety of continuous and discrete wavelet methods (e.g. wavelet spectra and
cross-spectral analysis, wavelet coherence, hard thresholding, multiresolution
covariances) in the analysis of rainfall-runoff relationships.
· Wong and Chen (2001): This paper concerns the analysis of multi degree- of- free-
dom systems exhibiting nonlinear and chaotic behaviour using the Morlet wavelet
transform. The paper uses a large number of clearly presented examples of the
modulus and phase plots for the Morlet wavelet transform of both test signals
and the nonlinear signals under consideration.
· Jawreth and Sweldens (1994): This overview paper is packed full of information
concerning the continuous and discrete wavelet transform.
· Williams and Armatunga (1994): A good place to find out more practical infor-
mation on the implementation of the multiresolution algorithm. The paper
makes explicit the mathematics involved with signal decomposition and recon-
struction.
· Mallat (1989): The original multiresolution paper. Very insightful.
· Meneveau (1991a): Although this paper is aimed at the fluid mechanic commu-
nity, it contains a lot of useful information concerning the discrete orthonormal
wavelet transform.
· Abramovich et al (2000): This is a good place to begin a literature search on the
statistical application of wavelet transforms. Among other things, it contains a
concise overview of the various methods of wavelet thresholding.
2 Useful websites
Typing 'wavelet' into a search engine should produce a very large number of sites
containing wavelet material. A brief list of useful websites is given below with a
short note on the contents of each site. This is not a comprehensive list, but rather
has been compiled to give the reader some good places to begin a search. Most of
them contain a large number of hyperlinks to other useful sites. These sites were all
active at the time of writing.
· http://www.mathsoft.com/wavelets.html
This site contains (or rather is) a very large list of other websites which contain papers
for download covering introductory material, applications and theory. The list is fairly
comprehensive and it is a very good site from which to begin a search for information.
Copyright @ 2002 lOP Publishing Ltd.

· http://www.wavelet.org/wavelet/index.html
This is the website for Wavelet Digest, the e-mail periodical which contains news and
views from the wavelet community including details of new books and papers, recent
theses, software, courses and conferences, jobs and not least a section devoted to
questions and answers from the subscribers. It contains a lot of useful information
for both novice and expert. You can sign up to Wavelet Digest from this site. The
news group is open to all and the questions posed range from the very basic to the
thought provoking.
· http://paos.colorado.edu/research/wavelets/
Christopher Compo and Gilbert Torrence's site. This is a very nicely presented site. It
has an interactive bit where you can submit your own data for wavelet analysis. The
answers to frequently asked questions and software are also available.
· http://www.mame.syr.edu/faculty/lewalle/wavelets.html
Jacques Lewalle's site-the continuous wavelet transform is explained through many
illustrations.
· http://www.public.iastate.edu/ rvrpolikar /W A VELETS/waveletindex.html
Robi Polkar's site-lots of well illustrated introductory information.
· http://www.amara.com/current/wavelet.html
Another good place to begin.
· http://www.stat.stanford.edu/rvwavelab/
Site containing W A VELAB software-a long list of routines for MA TLAB@.
Copyright @ 2002 lOP Publishing Ltd.

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