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Transmission
of Information
by Orthogonal Functions
Henning F.Harmuth
With 110 Figures
5
Springer-Verlag New York Inc. 1969
DR. HENNING F. HARMUTH
Consulting Engineer
D-7501 Leopoldshafen / Western Germany
All rights reserved
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag
© by Springer-Verlag, Berlin/Heidelberg 1969. Printed in Germany
Library of Congress Catalog Card Number 79-79651
Title-No. 1590
To my Teacher
Eugen Skudrzyk
Preface
The
orthogonality
communications
extensive
work
since
use
However,
Ten
little
results
retical
tional
its
very
in
this
before
the
have been published
functions
appear
to
sults.
It
again
the
produced
the
logy
was
that
of orthogonal
In
this
ample
tions
as
a
emphasis
well
as
on
years ago.
functions
block
first
block
of
independently.
state
opera-
the only
that
time
and this
way to derive
of
new,
made
the
re-
on
the
available
and
example
functions.
Walsh
on
techno-
useful
Walsh
functions,
sine-cosine
pulses
were
known
semiconductor
really
pulses
functions
However,
at
pulses,
placed
is
number
based on orthogonal
advance
is
theoretical
solid
of
the little-known
literature
in
and
circuits.
complicated
functions:
book
since
be
in
rather
arrival
many
and
theory
exploited
Conscious
considerable
field
examples of orthogonal
sine-cosine
a
integrated
and
A theory of communication
could
been
use could be made of the theo-
experimental
amplifiers
useful
has
beginning.
later
years
working
were
people
functions
was made of it by KOTEL'NIKOV
1947.
in
of
func-—
derived
from
them.
There
tions
are two
are
of
major
reasons
practical
a number of mathematical
are required,
and
shift
cosine
mission,
of
interest
orthogonal
in communications.
func-—
First,
features
other than orthogonality
such as completeness
or 'good' multiplication
theorems.
usefulness
why so few
One
quickly
multiplication
functions
whenever
and
learns
shift
and
to
appreciate
theorems
for
multiplexing
mobile
one
tries to duplicate these
of
radio
the
sine-
trans-
applications
PREFACE
pe
of
actually
later
This
from
andof
frequency
spectrum
phase
this
generalization
equipment
variety
economic
using
but
is
potential.
Walsh
mately
Itis,
should
not
cosine
functions
The
be
as
work
has been sponsored
der
Verteidigung
this
SCHULZE
E.SCHLICKE
encourage
in
and
attenuation
and
possible
de-
to
on
based
stimulate
of
the
to see
why
problems
particular
Walsh
filters
orthogonal
Co.
on
is
digital
functions
as
continued
was
sine-
networks.
functions
he
wants
Prof.F.A.FISCHER,
the
case
inti-
Bundesministerium
Deutschland;
their
the
previously
time-invariant
of
been
about
artinbinary
digital
thank
work
has
to
favor
the
new
this
A considerable
extend
years by the
Allen-Bradley
some
and
functions
to
In
area
for
only
competitiveness
Republik
to
not
controversy
tend
for
the
many
Dr.M.SCHOLZ
of
of
concept
equipment
due to some
state
opportunity
and
much
forlinear,
der
the
competitive.
difficult
for
functions.
equipment
new
methods.
important
are
author's
it
orthogonal
always
and
e.g.,
of
offer
to
the economic
connected to the
circuits.
take
Thisis
equipment
of
made
must
still
which
functions,
should be treated
orthogonal
multiplex
lead
equipment
there
ortho-
frequency.
of
must
compatibility,
introduced
of
but
the
derived from it as frequency
must be economically
of
developed,
of
and
filters
why
of
response
theory in engineering
understanding,
of
functions
Walsh
practical
raised
functions
concepts
frequency
Sign
Any
be
transmis-—
But sooner
generalization
the
such
or
The
shift.
to
systems
other
ledto
question
power
had
sine and cosine
gonal systemof
differently
presence
signalsinthe
question
the
the
of noise.
with
attention in connection
attracted
feature
orthogonality
the
mainly
was
is
1960
to
sion of digital
or
be
produced.
Prior
that
can
more
or
think
to
tries
a million
of which
systems of functions
one
if
comprehended
readily
is
quirement
re-
second
this
of
severity
The
produce.
to
easy
be
must
reason is that the functions
The second
functions.
by other
among
Dr.E.
support.
the
engineering
to
first
Dr.
to
applica-
PREFACE
tions
to
era
of
Walsh
him.
Help
functions;
has
been
well as administrative
Prof.
F.H.
LANGE
Innsbruck
N.EILERS
ruhe
of
Prof.
Rostock
University,
(FTZ-FI
SANDEN
of
the
Thanks
are
nische
Hochschule
greatly
in
indebted
scientific
Prof.
G. LOCHS
and
the
of
Dr.H.HUBNER
Dipl.Phys.
Dr.E.KETTEL
of
AEG-Tele-
Prof.J.FISCHER of Karlsof
Technische
Hochschule
Technische
Hochschule
Aachen
University
of
Southern
Darmstadt
studyofthe
who
showed great
applications
and
California.
particularly due to Prof.K.KUPFMULLER
encouraged
as
gentlemen:
Darmstadt),
and
Prof.G.ULRICH
of
further
GmbH, the late
Prof.H.LUEG
J. KANE
is
Dipl.Ing.W.EBENAU
Prof.K.VON
University,
Iimenau,
rendered
Bundespost
Bosch
AG,
author
problems by the following
University,
of the Deutsche
funken
of
the
of Tech-
interest
described
and
inthis
book.
De. fF. PLCHUER,
Mannheim
were
of
great
of the-book.
and
devoted
to
time
to
as
book
the
were
Ubertragung;
for
the
College,
editing
of
University
mathematical
Telephone
sections
Mrs.J.OLSON
and Telegraph
of
the
task.
published
Cu.
manuskript,
Many of the
inthe
Archiv
a
picder
Mr.F.RUHMANN of S.Hirzel-Verlag
their use.
Mrs.F.HAASE
1969
St.Olaf
first
Dr.L.TIRKSCHLEIT
of Innsbruck
the
indispensable
permitted
E.HARMUTH-HOENE
January
of
International
well
this
courteously
due
of
as
elektrischen
are
University,
helpinimproving
much
thankless
in
Linz
Prof.D.OLSON
Mr.J.LEE
tures
oF
University and Dr.P.WEISS
for
the
Last but not least,
typing
and
to
my
thanks
wife
Dr.
proof-reading.
Henning
F.
Harmuth
Table of Contents
ere catei Slerevelalel sielahers wrere¥ebeteterers coe
INTRODUCTION.....-.....4 MATHEMATICAL
FOUNDATIONS
des
FUNCTIONS
ORTHOGONAL
1.11
Orthogonality
1.12
Series
1.13
Invariance
of Orthogonality to Fourier
formation
THE
6
and
Expansion
FOURIER
TRANSFORM
from
Linear
Independence......
by Orthogonal
AND
ITS
Fourier
Functions...
Trans-—
GENERALIZATION
1.21
Transition
Transform
1.22
Generalized
1.2%
Invariance
of Orthogonality
lized Fourier Transform.<...
1.24
Examples
of the Generalized
Fourier Transform
1.25
Fast
1.26
Generalized
Fourier
Walsh-Fourier
GENERALIZED
Laplace
Series
to
Fourier
the
Genera-
Transform
to
Transform
Transform......
FREQUENCY
1.31
Physical
Interpretation
Frequency
ioe
Power Spectrum, Amplitude Spectrum, Filtering
of Signals coer eee eee ee
1.33
Examples of Walsh
Spectra
Fourier
of
the
Generalized
Transforms
al
and Power
oi
TABLE
OF
2.DIRECT
2g
CONTENTS
1x
TRANSMISSION
OF
ORTHOGONAL DIVISI
AS GENERALIZ
ON
ATION
FREQUENCY DIVISION
Ere eRnCDL Open caudol
ee
2.14
ete
2.15
2.21
Cree
(OT
by
COMMUNICATION
Sine
ee
and
oe
FILTERS
BASED
ell COMMUnMa
Ceai
WALSH
Wsuachoevsis:
TRANSMISSION
OF
Walger
71
U2
81
86
OmeChanmed:
«ce dehee ees
O41
FUNCTIONS
NOW Pas Se Lal GCI S sy. «6s. ers) cuss sre: siereve «6
Luts PeOUUCICy
AMPLITUDE
ON
60
64
CHANNELS
Eose pois acame
TCL Gls cide sels
OuUciicy.
POPUL
4.11
OF
pUMABaAe
beri ety LOM Olm
Zo De Meee
Diwt
AND
Frequency Response of Attenuation and Phase
onatG of a Communicapion, Caanmel.......s.s6
SEQUENCY
%.CARRIER
TIME
Gre o1onal Se. pare
Transmission of Digital Signals
Cer ie Be WSC cere Sacehi c= east ee
Wye
Zee
OF
eee
ce eee
(SLOMAN Ssuis e = aie erehe cess ciecee iecichnew as
Amplitude Sampling
and
Orthogonal DecompoSSLOD orecae eee are! Grcbsie cic as«10k & Geeetaehes ole oo eerteakare
ctreuats) for Orinogonal Division. ..tss....-.
eam DUES
CHARACTERIZATION
ee
SIGNALS
tao
Gab om
oOo
94
SG
97
HPLLGOrE ss «sss ss oe soles oa ees
104
SIGNALS
MODULATION(
AM )
Modulation
Bile genet
Wes
J, NOUVEAU
and
Synchronous
ELE
of
Demodulation....
oye
h CMs . gates 6 ss 6 0a ole es s6ne eco 50 8
LERING see
Sideband
ae
3.14
Methods
Correction
of
Time Differences
in SynchroTAOS
SMO GUE enWall Ollilcetete swell olarate) eters stich els: slersiene:clele) 6 447
TIME
BASE,
POSITION
Pee
imesBase
2022
Mate
Modulatiom
pee
COC en OCP abiOm
Position
AND
CODE
154
MODULATION
(TEM) 0. 626 case cee se es
Modulation
CCM)
Modulation......
114
4.15
TIME
Single
ewe cs ete ws sees eevee
106
155
ees ay
(TEM). s...ssecs
es cee c caste e ee secee ses
159
TABLE OF CONTENTS
3.31
of Walsh
Radiation
3.32
Propagation,
3.43
4.1
SINGLE
Definitions......
4.12
Density Function, Functionof a Random Variable, Mathematical Expectation. .....sccesecc
Moments
and
OF
Ae)
Addition
4,22
Joint
Aways, aiaral oeraga Ce
CharacteristLG
137
188
Huncutome
sc clerete tom
VARIABLES
of
Independent
Variables.
ss... =0<6
194
Variables
198
DEPENDENCE
WS
Coviartance
wands
4. 52
Cross—
=AUutbOCOrrelaulonm
SERIES
re ea
Distributions
of Independent
STATISTICAL
“and
OF
comre basromc = al rere mie we cients
ORTHOGONAL
EXPANSION
OF
TO
21
214
STATISTICAL
FUNCTIONS
Thermal
5-12
Statistical Independence
of the Components
of an-Orcuhogonel
Expeneicne...e<see
en sea ee ere
ADDITIVE
Nowsegec.
YunGulole «see.
FUNCTIONS
STOCHASTIC
.
Sell
2-21
5 cee eee eietetreeereerereneeenerere ee
ee
DISTURBANCES
Least
Mean
pample
5.3
167
173
Doppler
VARIABLES
5.APPLICATION
PROBLEMS
Bee
Effect......
Recognition..........
VARIABLES
COMBINATION
Del
Shape
160
Dipole
4.11
4 15
4.3
by a Hertzian
Waves
Antennas,
Interferometry,
4. STATISTICAL
WAVES
ELECTROMAGNETIC
Dis NONSINUSOIDAL
Square
Pirie
Doce
Kramplesiof
poo
Matened
».24
Compandors
eo
ee eo ere
ee ee
from
een
CireuiGe. va ceed cieenie
cee Risieteiees
«Pa lteraeus
MULTIPLICATIVE
Deviationof a Signal
ciieichce sua
for
an eae cea
Sequency
ere
C24
ey,
ee ucie ensnere
rene 240
Signals.....s.c...+.
299
DISTURBANCES
yeo|
Interference
2-32
Diversity
Fading...
Transmission
a. «sc «0 6 Ke Wee ersete Outer LOO
Using
Many
Copies...
243
TABLE
OF
CONTENTS
6.SIGNAL
6.1
DESIGN
IMPROVED
RELIABILITY
CAPACITY
GOwl|
Measures
ote 6.aie eta eee -
245
6.12
Transmission Capacity
of Communication ChanMEV. cscs Stetststencls
elepe s/s ela s/c.steisiores sjait sucia ayateresa e
Zon
ERROR
of
Bandwidth..:..
Delay
and
PROBABILITY
OF
Sipnal,
a evecaler
Distortions.
...
SIGNALS
ree
weaker
Ower
Tami Fed
Se — iy Mem DATS
OVenalS. sc. ssc
ces e0 se ss
268
Us Dal COS sekere ele sclalere tens) sis sts ele ATA
CODING
limeruencttany meh ILeMmerdGiGtels > cycle crelsi ete leleteners
ZFS
6.32
Orthogonal,
Transorthogonal
and Biorthogonal
MOE S 5 ood
6 Ob OOED AGH OR DODOOKO GODS COGUE
want
280
6.55
loding,
2s.
288
Go.
+-“lernary-Combinatiom
Alphabets. 220.5...
cee
ASNS)
form
Error—lree
Transmission,
GIS mC OMpi
me palOnurhulapita betris
wou Oredieie
7...
21eed)ctslers eres! «
Pe
BA RVG Bos OG Dishbd) BYatOLG L.LOND <6 «6s s06 « 1s «60s
MAP ewetees se eo ois s/wiwie| tajels = cr = oles
numbered
are
Equations
chapters.
6
chapter
front
260
Error Probability of Simple Signals
due
to
eee MeN ODCyclecate a eters atererert eiahareie’ s srete cos s oie oie ene 262
Gremlne © Gantt
the
6.
6.21
Gee Ome
6.3
FOR
TRANSMISSION
Cupeoienadl
6.2
XI
of
is
made
the
in'chapter
4.
by
number
writing
of
the
to
an
the
«
502
cies ese cies ¥ oss «9.010 ® o's B78)
consecutively
Reference
5 650
299
within
equation
number
of
of
the
each
one
of
a different
chapter
(4.25)
e.g. n,
equatio
for
in
(25)
Introduction
Sine
and
cosine
munications.
defined
The
functions
concept
play
of
by the parameter
a unique
frequency,
role
in
com-
based on them,
f inthe functions
Vsin
is
(enft+a )
and Vcos (2nft+a).
There
are
many
to
possible
hardly
reasons
of
communications.
made
it
possible
forms
wave
almost
the
any
further
delay
and
were the most
coils
of linear,
ticular
fine
desirable
sinusoidal
has brought
functions.
cosine
on
Sinusoidal
functions
ortho-
and
the-
The
the
e.g.,
is
par-
no
analyzing
be
that
semi-
of
advent
functions
Walsh
socalled
out
of
capacitors
There
should
turns
It
the
based
signal,
of aradar
a tre-
demonstrates
change.
filter,
whyadigital
reason
structure
unchanged.
had
systems
The
functions.
aradical
based
the
sine
on
filters
digital
are
A
the
attenuate
only
remain
networks
invariant
was
elements.
circuit
that
economically.
resistors,
as
was
it
But
circuits
functions
complete
other
long
time
of
advantages
conductors
and
over
as
functions,
gonal
non-sinusoidal
circuits
cosine
and
sine
was
early
transistors
functions
the shape and frequency
advantage
mendous
ory
them,
the systemof
Hence,
It
the
voltages.
produced
be
invariant
time
in
integrated
sinusoidal
favoring
role.
and
simple
ramp
the
could
functions
linear
that
fact
of
arrival
factor
tubes
such
or
pulses
block
as
before
unique
functions
Electron
produce
to
this
other
produce
days
not
for
simpler
and
ASuCIe.
are
less
pagation of electromagnetic
waves
conductors.
the
BERT
and
tion
show,
mitted
the
dominance
1
solutionof
general
that
dipole
can
of sinusoidal
partially
explained
Harmuth, Transmission of Information
classof
or can
radiate
waves
by the
free
wave
solutionof
a large
distortion-free
a Hertzian
be
The
important
in
be
the
for
space
equation
can
regenerated.
radio
proalong
by d'ALEM-
telegrapher's
functions
non-sinusoidal
in
the
or
be
equatrans-
Similarly,
waves.
The
communication
can
invariance
of their
ortho-
INTRODUCTION
2
have
functions
comeback
as
is that
functions
munications
the
almost
be
can
mathematical
taken
often
engineer,
sine
tool.
functions
quency
is
that
he
cosine
and
crophone
The
of a time
sentation
function by sine
is
only
many
of
orthogonal
one
sions that
into
to
tions.
output
of
will
be
nals
and
not
the
for
riers
that
to
in
used
are much
correspon-
systems
of
of
functions,
book
for
orthogonal
sine
the
func-
Legendre
etc.
in
include
carriers,
used for theoretical
multiplex
and
methods
of
the
and
that
functions
of
sig-
the application of ortho-
since
sine
analysis,
radio
cosine
in-
lines and networks.
systems.
exist for them,
frequency
will be shown
and
functions
representation
characterization
as
modulation
it
of
systemof
theory must
functions
amplitude,
more,
the
expan-
For instance,
superposition
a
cylinder
systems
special
usedinthis
only
see
systems
a microphone.
of
complete
A consistent
gonal
may
many
of
mi-
repre-
series
transforms
for
is
functions
Complete
series.
also
transform
parabolic
voltage
General
stead
one
Hence,
polynomials,
are
ofa
the
cosine
functions
Bessel
of
voltage
permit
generally
There
Fourier
the
and
is
This
superposition
ones.
to the Fourier
series
communications.
ding
possible
functions
correspond
expansions
in
among
a
Actually,
transmitter.
of
communications
output
the
com-
time to fre-
analysis.
the
sees
instinctively
in
by
in
analysis
from
this
of
granted
functions
orateletype
Fourier
transition
for
cosine
superposition
a
which
a result
much
so
by
represented
for
used
functions
time
all
functions,
sine and cosine
a
made
and
sine
of
features
important
most
recently
cables.
digital
One of the
ampli-
as
relays
have
they
and
lines,
such
were
fiers,
electromechanical
using
century,
49th
lines of the
The telegraph
existed.
always
sinusoidal
transmit
not,
need
nor
not,
could
that
lines
wire
open
or
Cables
delays.
time
varying
under
gonality
phase
antennas
and
but
It
will
which
modulation.
canbe
cosine
also
as
be
are
carshown
correspond
Further-
designed
that
3
INTRODUCTION
radiate
non-sinusoidal
The
tions
transition
to
general
the
as
theory
the
systems
simplifications
matical
waves
from
of
of
the
a limited
finite
The
been
the
complications
far
theory
which
that
an
avoid
infinite
substituting
signal
the
a parameter
interpreted
concept
of
frequency
theoretical
may
with
Sign
100
changes
of
aS
also
sine
and
number
crossings
is
mensionally.
100
of
cycles
unit
functions
per
One
per
unit
cycle
number
of zero
to
cover
is
introduced
hasno
frequency
and
The
term
wavelength
response
sequency
X =
v/f
per
sequency
as
A sine
for
unit
of
makes
for
are
of
and
in
sine
definitions:
ave-
"sequency"
and
Thus
cosine
power
spectrum
or
spec-—
power
attenuation.
+t =
oscillation
frequency
wing
in
order
frequency.
frequency.
cted
with
conne
n
of sequency
tutio
di-
useful
the
time"
by sequency
of attenuation
period
zero
it possible
to replace
frequency
of
of
is
half
Thenewterm
of
func-
functions
It
"one
the
crossings
numerically
meaning.
identical
response
of
general
number
concept
» for
more
the
obvious
are
concepts
concepts
half
generalization
sequency
and frequency
The
zero
functions.
forthis
important
200
second
crossings
non-periodic
functions.
time".
of
out,
half
of
crossings
are defined
introduce
the more general
"one
has
second
second.
as
of
functions.
cosine
interpreted
per
cycles
Zero
which
the term
be
per
per
a
has
result
MANN [1], STUMPERS [2] and VOELCKER [4] pointed
frequency
a
composed
occupies
functions
of communication
based on orthogonal
crossings
trum
by
mathe-
e.g.,
occupies
satisfying
cycles
such
the
may,
time-limited
of
most
tion
rage
to
brings
time-function-domain.
number
of zero
1s
One
orthogonal
of
this
of
the
is
canbe
time.
Any
generalization
so
Frequency
to
as
func-
functions
time-frequency-domain
number
section
sine-cosine
orthogonal
communication.
time-function-domain.
or
of
troubles
fact that ome
any signal
section
of
of
well
etiicienticy.
system
f leads
to
the
1/f
and
Substifollo-
INTRODUCTION
4
multiplied
crossings
zero
the
of
in time
paration
se-
(average
T = 1/p
period of oscillation
average
by 2)
they
nology.
pears
if
sine
and
simple
are
known
semiconductor
tech-
system
of
Walsh
quency
and
than
faster
sine
on
of
strictly inthe
ly have
active
radiation
ning
Walsh
geometric
optics.
waves
the
been
Walsh
can
optics
On
theoretical
only,
other
since
hand,
wave
electromagnetic
field for basic
research.
radio
inthe
to
the
waves
implies
region of visible
question
of
why
The
that
light,
white
be
there
non-Sinusoidal
dal
are
on Walsh
practical
waves
problems
in
is
are
a
that
challenging
canbe
non-sinusoi-
leads
should
of
wave
of
this
concerterms
doubt
such
and
for
a sine
little
waves
waves
recent-
practical
generation
light
progress
very
answered
optics
ra-
circuits.
be
is
They
functions
Only
to
the
among
ahead.
Their
Most
or
Digital
electromagnetic
found
functions.
presently
as carriers
years
stage.
se-
frequency
integrated
scale
Very
functions
considerable
non-sinusoidal
antennas
of
over
functions.
will require
of large
in the development
the
based
cosine
and
however,
Applications
functions
equipment
for
when
Walsh
applications.
certain
multiplex
coils.
experimental
an
Walsh
applications
and
application,
in
and
these
that has advantages
systems
digital
Simpler
on
of
system
time-invariant
linear,
Furthermore,
circuits,
the
as
capacitors
systemusing
developed
promising
the
for
based
filters
developed.
multiplex
filters
ther
is
functions
multiplex
most
components,
ap-
functions
time-variable
linear,
for
based onresistors,
been
time
are
known
digital
binary
cosine
been
are
applications
to
little
sequency
have
has
such
The
circuits,
prac-
tied
all
on
its
Several
are
to be as ideal
based
are
intimately
applications.
tical
and
crossing)
theory in engineering
any
of
test
acid
(whee
4
by
multiplied
isthe velocity of propagation of a zero
v
in
separation
(average
v/®
crossings
zero
space of the
The
A =
wavelength
average
be
generated
ultimately
decomposed
1.11
ORTHOGONALITY
into
sinusoidal
The
Walsh
5
functions.
functions,
emphasized
sentl
the y
most important
tions
by
in communications.
communication
more
for
subsystem
a
are
this
complete
found
system
around
conductors
practice
into
them
of
1900
in
Walsh
the
to
1923
the
of
functions
when
[6],[7],
were
recently
aware
func-
for
century.
The
been
have
[9]
introduced
engineers
common
of
standard
was
scheme
this
which
used
transposition
J.L.WALSH
of
used
conduc-
[4],
to
seems
The
known
been
were
19th
Communications
not
are pre-
of
functions
BARRETT's
mathematics.
have
functions,
end
Walsh
they
transposition
by J.A.BARRETT!.
according
thematicians
very
the
towards
purpose
the
Rademacher
lines.
of
book,
functions
are hardly
although
for
years
wire
tors inopen
These
engineers
60
than
inthis
exam
of non-sinusoi
pledal
and
ma-
until
usage
[8].
1. Mathematical Foundations
1.1 Orthogonal Functions
1.11 Orthogonality and Linear Independence
A
system
Wentening
in
gonal
dition
{f(j,x)}
tunetions,
the
holds
of
real
£(0,%),
x,
interval
and
almost
£0¢1,x),....
= x
= x,
if
everywhere
is
the
talled
non-
ortho-
following
con-
true:
x1
J fC5,e)f£Ck,x)dx = Xj)
Ge
Xo
Sik =
7 lores
=
ik,
Pes
Orton]
ut.
by FOWLE [5] in 1905
is mentioned
1'JOHN A. BARRETT
the transposition of conductors according
or
inventof
Walsh
functions;
see
particularly
page
a
(OAS merowie ae
as
to
FOUNDATIONS
41. MATHEMATICAL
6
are called orthogonal and normalized if
functions
usually
are
constant X,; is equal 1. The two terms
The
the
al
malized.
orm
or orthonor
dterm orthon
the single
touce
red
may
functions
orthogonal
of
A non-normalized system
the system
For instance,
ys
be normalized.
alwa
is normalized, if X; of (1) is not equal 1.
orthogonal
functions
system
A
linearly
called
is
Systems
of
s
of system
of lin-
cases
special
functions.
independent
early
functions
are
{x;fts3=)}
m
of
{f(j,x)}
if
dependent,
equation
the
m-
2s e(j)f(g,x)
is
satisfied
c(j)
being
=°0
for
orthogonal
if
inthe
comstamy
A system
can
(2)
values
is
systemare
multiplication
each,
all
of
zero.
The functions
independent,
products
(2)
always
interval
linearly
f(j,x)
x,
are
satisfied.
always
(2) by
without
=x
= x,
all
constants
called
linearly
Functions
of
independent,
and
since
integration
yields
c(j)
an
of the
=
0 for
oC).
{eCa; =v} of m linearly
be
orthogonal
of
not
x
f(j,x)
transformed
functions.
One
into
may
a
independent functions
system
write
the
{f£(5,x)]
following
of
m
equa-
tions:
ECO,
)=76,0
2 00,2)
£(1,%)
=
£(2,x)
="0,,
(3)
eee)
¢,, 200, x)
€C0,x)
S cl etlyxy
4 Cy, B(2,%)
etc.
Substitution
of the f(j,x)
equations
for
xy
tr
Hy aeercOy
se asc
determination
into
of
the
(1)
yields just enough
constants
x,
(4)
Xo
xy
Ay
J 2° (1, x)ax= %), (eC
Xo
x)
c
ei
aren,
XQ
2
xy
xy
J f° (2,x)dx=X,,
[£(0,x)f(2,x)dx
=0,
f£(1,x)£(2,x)ax=0
Xo
Xo
Xo
etc.
1.11
ORTHOGONALITY
9
Thevcoet ficients
for
normalized
tually
yields
system
{e(3,x)}
satisfy
(4)
Figs.1
Xz,
Xj,.2.
systems.
values
of
are
arbitrary.
ip follows
for
the
linearly
from=(2)
coefficients
independent
They are
that
Cpq
(4)
as
1
“ac=
only
functions
a
could
identically.
to
3 show
The
independent
The
functions
examples
of
orthogonal
variable
is the normalized
of
Fig.
-% = 0 = 4; they will
1 are
orthonormal
be referred
to
functions.
time
@
inthe
as
sine
=
"tis
interval
and
cosine
elements.
odd
One may divide them into even functions f, ‘Gates
functions f,(i,@) and the constant 1 or wal(0,86):
© Ci,0-)
=
FoCi,o)
=
"Ve cos
2719
=
f, (1,8)
= V2 sin
2nié
-#
=
0Face
Ger)
=wal(0,6) = 1
= undefined
6 <
ees
Bt
So
eo
So
-8,
9
>+8
ial, Ns
ra
ew 000
sal(1,9) KJ
| 000
a6) 700000
——
B12) ee re
0
cal(2,8)
—+——_—_-
sa(36) EF
—_f —
4 0100
soo
cal (3,6) —f—L-A—_L_
FH~_s 66 td
LN
Pain
(46) FLSA Ess at
a to
=&
SS
PITT
SPP
Weos4ma — sall5,0)_ 4+ FH EF -LFSCS:g (100
F
10 1010
WH HE:
call 5,4) -—H7A-
eon
re
taro
aa CS
tos
Fso2 .: 1100
LAL
cal (6,6) ELA
sal(7,) HALF-ALLLF-XLFL_s£“BS 101
FAL
HE
EFL: 4s 110
cal(7,6) -ALFLA
acm
iat
fener
sal(8,6) AEE
FLL
ft
=f
ec
fl A = ik
0
@=t/T——
85 111
Ee
1/2
e lements.
i
and cosine
1
sine
Orthogonal
c
left)
ILS 5
Orthogonal Walsh elements. The poe
gee oe
a ce
the right give jin decimal and binary ee
-1,
tion wale j.0) 12 used. Wwal@et.6 = cal(i,6), wal(2i
sine
cosine
elements
other hand, yields the periodic
It is easy
gonality
is
to
see,
that
satisfied
1/2
J Wesinenie
for
and
f 1V2cos2mie
interval.
Fig.1
of
on the
functions.
(1) for
cosine
1/2
de =
“Wa
1/2
condition
sine
that
= O yields
sine and cosine
the
unde-
is
continuation,
periodic
pulses;
cosine
and
a finite
-4 = 6 = #% by f(j,6)
interval
of the
outside
outside
and
sine
the
of
Continuation
the
zero
identical
is
a function
and
only
used to emphasize
is
'pulse'
The term
outside.
fined
interval
finite
a
in
defined
is
tion
a func-
that
emphasize
to
used
is
'element'
term
The
FOUNDATIONS
MATHEMATICAL
1.
8
ortho-
elements:
dé = 0
iz
2
J V2 sin Qnie-V2 sin 2nk@ dé = [2 cos 2mié-V2 cos 2nk® dé s6;,
-1/2
-1/2
1/2
J V2 sin anis-V2 cos 2nké dé
-1/2
2
f VIERA
-1/2
Fig.2
-
Colle od
shows
more
sal(i,9).
It
-1.
is
orthonormal
-
Walsh
functions
These
functions
Consider
-1
the
following
+%.
0
1/2
odd
and forth
first
two
-# s @ < O and
of
these
or
of a constant
functions
between
+1
functions.
+1
in the
products
has
= 4+4
= 0
0
product
of
the
in the intervals
the intervals -
-1/4
back
integral
+ f (+1)(41)de
-1/2
of
and
of the
interval
The
functions
value:
J 6+1)(-1)d0
The
jump
Walsh
consisting
cal(i,@)
product
in the
0 = 6 <
systemof
elements,
even
equal
interval
the
the
exactly
wal(0,@),
and
= O
these
second
and
-- = 6 < -2
= 6 <0
and
products
again
third
element
yields
+1
and 0 = 6 < +4, and -1 in
++ 5 6 < +#. The integral
yields
zero:
0
J (-1)(-1)d0 + [(-1)(+1)ae + (41) (41 )a8 . (41) (-1 ae at
a
= 16
0
We
1.11
ORTHOGONALITY
One
may
easily
duct
of any two
tiplied
9
with
verify
that
functions
itself
is
the
equal
yields
the
integral
zero.
Fig.3
nal
are
functions
shows
functions.
functions
vanish
tions
pro-
A function
mul-
C44) Crd
ior
the value 1 in the
integralis’1.
The
orthonormal.
thus
a particularly
simple
Evidently,
product
the
system
of
between
orthogo-
any
vanishes
and the integrals
of the products
too.
For normalization
must
the
two
must
amplitudes
of the func-
be V5.
f(0,8) pal sea| Seis
Co
the
products
(-1)(-1). Hence, these products have
whole interval -$ = @ = +# and their
Walsh
of
S 22%: it f (Ov)
a
f(1v)
(2,8) penseteeweet ste! SES
f(2,v)
i
ee
eae
OR
(PN Sl
F, (0) Se ee
ee )
ee Fy (v)
+++ — 94/7
;
Fig.3
5
ae
Orthogonal
block
Bernoulli
polynomials
Gj 6) and. £5,Vv)
Fig.4
Cop
v=fT
pulses
eich).
Fig.5
Legendre
polynomials
(right).
An
nal
of a linearly
example
of
system
functions
but
independent
not
polynomials
Bernoulli's
are
orthogo-
B, (x) [4], (51:
B, (x)
Bai x)
=
11, Be
2
=
4x?
=
x.-
$, B(x)
Se
$x,
Ace)
=
=
x4
x?
-
x
ox
3
+ +
att x2
=
1
Sie(g)By(x) =
I
condition
The
j.
order
-f
polynomial
a
is
B j(x)
FOUNDATIONS
MATHEMATICAL
4.
10
(e)
j=0
highest
term
applied
to
the
in
it.
One
may
see
in
proves
from
the
Fig.4
are
interval
orreCysse) in
DN
and
sum
This
polynomials
lization
then
the
values of x only if c(m)x™ is
is the
;
= 0. Now c(m-1)R_.(x)
the
the
same
reasoning
linear
can
be
independence
of
polynomials.
Bernoulli
Bernoulli
e(m)
implies
This
zero.
all
for
satisfied
be
can
without
calculation
not orthogonal.
-1
= x
= +1
one
that
the
For
orthogona-
may
substitute
Coy.
Blow Asie).
Eid
ere
hy ek eee eg
PX)
Using
the
constants
Xj
=
eee
2/(2j+1)
one
obtains
from
(4):
frax =X, =2
J [oy + oy (x-t)
ax = x, = 2, flere +0;, (x-b) lax = 0
The
coefficients
tained.
The
lowing
form:
enn eed] =
"l,
c,,
Py Gz)
P, (x) = $(5x?
=
with
shows
the
P(x)
- 3x),
X
=
first
=
Pj (x)
—(3x?
BP, (x) = $(35x*
(j
five
= 1, etc. are
polynomials
E85
These
are the Legendre
-1/2
plied
= #, ¢,,
orthogonal
»
polynomials.
2
4)
for
=
readily
ob-—
assume
the fol-
1)
-30x?
Pj(x)
4 3)
must
be multi-
normielization.
Eee
ee
polynomials.
1.12 Series Expansion by Orthogonal Functions
Let
a
orthonormal
function
F(x)
system
B(x) = S' a(g)t(3,x)
j=0
be
igh
expanded
in
a
series
of
the
eS
(6)
1.12
SERIES
The
value
EXPANSION
of
the
14
coefficients
a(j)
may
be
obtained
multiplying
(6) by f(k,x)
and integrating the
interval of orthogonality x, sx s x,:
in the
by
products
x
f F(x)£(e,x)ax = a(x)
(7)
XQ
How well is F(x)
are
represented,
if the coefficients a(j)
by (7)? Let us assume a series > b(j)f£(3,x)
determined
having
m terms
terion
for
ation
Q of
a better
yields
shall
be the least
'better'
its
from
F(x)
mean
cri-
The
representation.
square
devi-
representation:
m-1
- [e260 Seep
tGisxpl dx
ee
“| (odaae~ 25,05)
j=0
Using
(7)
yields
and
Q in
the
the
The
Xan
orthogonality
following
x]
Q=
ae) (Jyxedax + | a ShGec. ar dx
Xo
Ne AGREE
X9
j=0
last
term vanishes
for b(j)
The
assumes
socalled
its
of the
functions
£(j,x)
form:
f F’(x)dx -
deviation
eice
a Lo(g) = aid}
=
a(j)
(8)
andthe mean
square
minimum.
Bessel
inequality
m-]
eo
xX
j=0
j=0
Xo
follows
from
(8):
Sia2(j) = Slat(j) s f Fe (x)ax
The
upper
since
the
holdetor
The
zero
of
summation
integral
any
system
complete,
and
to
limit
with
quadratically
value
does
of
the
may
not
be co insteadof
depend
m
and
is
called
orthogonal,
mean
square
deviation
increasing
integrable
m
for
in
any
the
function
interval
vim 65f'UR(x) - j=0S!a(g)e(3,x)]?ax = 0
Megat
on
m-
must
1,
thus
m.
{f(j,x)}
if
(9)
x,
normalized
Q
converges
F(x)
= x
is
that
= X,:
(10)
FOUNDATIONS
41. MATHEMATICAL
2
equality
in-
Bessel
the
in
case
this
in
holds
sign
equality
The
(9):
(11)
a?(j) = Kaf Fe (x)ax
Ss
j=0
resistance.
unit
the
is
integral
The
(11),
to
sum EGG
the
same
function
The
is
function
no
the
quadratically
the
of
energy
voltage is described
the
or its
lee
F(x)
system
represents
This energy
it differently,
Putting
Tee
whether
sumof the
series
is
integrable
function
a
then
equals,
terms
the
in
energy
by the
expansion.
said to be closea',
Let
across
time
of
F*(x)
of
energy dissipated in the resistor.
according
the
meaning
as
a voltage
represent
F(x)
physical
Its
theorem.
val's
follows:
as
is
Parse-
or
theorem
is known as completeness
(11)
Equation
time
if there
F(x),
[ P2(x)ax < oo
(12)
Xo
for
which
the
equality
xy
{ FG SC), x
dsetanO
(43)
Xo
is
for
satisfied
Incomplete
mit
a
convergent
integrable
practical
an
functions.
orthogonal
an
expansion
instance,
lowpass
expansion
system
of
j.
functions
of
Nevertheless,
For
frequency
by
of
of orthogonal
series
interest.
ideal
exactly
values
all
systems
—
in
series
are
output
may
of
per-
quadratically
they
the
filter
a
all
do not
be
the
of
great
voltage
of
represented
incomplete
functions.
"A complete orthonormal systemis always closed. The inverse
of this statement holds true,
if the integrals of
this section are Lebesgue rather than Riemann integrals.
The Riemann integral suffices forthe major part of this
book. Hence,
‘integrable' will
mean
Riemann integrable
stated.
unless otherwise
1.15
INVARIANCE
Whether
series
be
a
OF
ORTHOGONALITY
certain
function
of aparticular
told
nuity
from
or
such
Ne)
F(x)
orthogonal
simple
boundedness'
-
be
system
features
[5]
can
of
expanded
{f(5,k)}
F(x)
as
in
a
cannot
its
conti-
[7].
1.13 Invariance of Orthogonality to Fourier Transformation
A time
tain
function
conditions
of
means
the
f(j,@)
by two
may
be
represented
functions
a(j,v)
under
and
b(j,v)
cer-
by
transform:
Fourier
(5,0) = f [aCj,v) cos 2nve + b(j,v) sin2mveJav
(14)
ag.
= bP.
C459
ny
e=
5,6) cos 2rve
0)
sin anve qe,
It is advantageous
the
two functions
e(j,v)
=
a(j,v)
aero ltows
Hurceion
a(j,v)
=
a(j,v)
of
vy)
and
a more
fT
applications
to replace
and b(j,v)
by a single
function’:
(16)
that
aC(j,v)
b(j,v)
(17)
=
is
an
even
and
bj,Vv)
may be
-b(j,-v)
yield
+ b(j,-v)
for
C17)
g(j,-v):
(18)
= aCij.v) - bCij,v)
regained
from
g(j,v)
by means
(18):
FLeCi.y)
ar e(j,-v)]J
#[e(g,v)
=
g on
in
functi
Usthe
in
certain
v=
Vv:
and
b(j,v)
and
(16)
a(j,v)
Cy
(16)
6 = t/T,
bigs)
(15)
= a(j,-v)
g(j,-v)
of
+
a(j,-v),
Equations
for
a(j,v)
trom
aieodd
dé
symmetric
(19)
e(j,-v)]
g(j,v)
one
may
write
(14)
and
(15)
form:
'For instance, the Fourier series of a continuous function does not have to converge in every point.
A theorem
due
to BANACH states, that there
are
arbitrarily many
orthogonal systems withthe feature,
that the orthogonal
series of a continuously differentiable function diverges almost everywhere.
2Real notation is used for the Fourier transform to facilitate comparison with the formulas
of the generalized
Fourier transform derived later on.
MATHEMATICAL
4.
44.
FOUNDATIONS
£@),0)=
i e(j,v)( cos 2nvé
+
sin 2mvé)dv
(20)
calgictiyy
fer eae
cos Onyvé
+
sin 2mve )dé
(21)
(20)
vanish
since
OLAS
sLOia Cae
We
Let
{f£(j,6)}
-40 = 0 = +4@
isaneven
a(j,v)
be
a system
and
zero
nite.
The-functious
f(3,0)
Their
orthogonality
integral,
f£(j,0)f(K,0)d0
and
in the
@ may be
are
Fourier
is
b(j,v)
orthonormal
outside.
sin2nvé
b(j,v)
and
a(j,v)cos2nvé
of
integrals
The
an
in
odd
interval
finite
or infitransformable’.
= by
(22)
—oo
may be rewritten’?
using
(20):
cos emve
00
[ e(k,v)(
J £03,0
+
sin amve )dv]de
= 6,
fek, vt f £(5,8)( cos anve
+
sin 2mvé )d6]dv
= 5%
f eCi,we(k,v)dv
= 6,
(23)
—co
Hence,
the
{£(§j,9)}
Fourier
yields
Substitution
transform
an orthonormal
of
an
orthonormal
system
{g(j,v)}.
g(k,v)
= alkyy)
system
of
BCI,Ve) = ee
er vs
into (23) yields it in
terms
of
the
+ bUk,Y)
notation
a(j,v),
Dea yes
feGi,welk,vav = fla(j,v) + dCj,v)]LaGk,v)
+bCk, vay
—oo
a faCg,vja(k,v) + b(j,v)b(k,
vay
= 6),
—co
'Orthonormality
implies
transform
inverse
The
and
the
integrations
grands
are
may
absolutely
be
the
existence
transform
interchanged,
integrable.
of
the
(Plancherel
since
Fourier
theorem).
the
inte-
1.15
INVARIANCE
OF
ORTHOGONALITY
15
Fig.6 Fourier transforms
g(j,v) of sine
ses
according’
to
“Nigw..\.a) \wal(O,es
Cel
o~os ond...
Fig.6
Sine
and
d) Veo sin 470,
shows
as
cosine
pulses.
elements
of
Side
interval
the
an
Fig.1
by
and cosine pulb) Vesin ene,
“e) {2cos 4r6..
example
the
These
Fourier
transforms
pulses
are derived
continuing
them
identical
from
zero
of
the
out-
-% = 6 = +4:
1/2
200.)
= if 1( cos
ap:
2mv@
+
sin 2nvé )de
1/2
gc(i,v)
= [| Vecos 2mié( cos 2nvé
+
ee
CoH)
sin 2nvé)de
-1/2
i
ayo (Sia8
Tess
TO
Vai
Saat
T
1/2
= i V2 sin 27ié( cos 2nvé
Gelisv)
.
V=1
+
sin
_
sin
vei)
Cvtl
ve )dé
Biz
a
ay 2( sint(v-i)
z
Fig.7
rived
by
outside
2(0,V)
shows
the
continuing
the
interval
142
-= f wal(0,@)(
“1/2
TT
Ov=n
Fourier
the
Tr
(vti
transforms
of Walsh
elements
of
)
Aye
Fig.2
pulses
identical
-4 = 60 = +#:
cos 2nv@
+°
;
sin 2mvd )dé@ =
sinty
a
dezero
1.
16
2
= [ sal(1,8)(cos anvé + sin arvé )dé = ee
“1/2
;
ee
in?
gs(1,v)
go(1,v)
One
=
sinénv
may
functions
time
physical
meaning.
oscillation
Fourier
FPig.2.
d) -sal(2,6),
Fig.8
g.(2svie=
from
into
of
the
The
with
these
even
transform
values
to
see
transform
Negative
Fig.7
ne
readily
functions
sine
FOUNDATIONS
MATHEMATICAL
examples
frequency
into
odd
frequency
oscillationof
reference
transforms
2a) wal(O,0),
g(j,v)
e). cal(2,6).
Fourier transforms
to
=
O,
g(j,v)
if
time
and odd
functions.
a perfectly
frequency
@
even
functions
valid
y is
the
a
co-
Fourier
of Walsh pulses according
bd) sal(1,0),
£(2,8) and £(3,6) of Fig.3.
that
frequency
have
nv/4
cos a
of the block
¢)
-cal(1.4),
pulses
f(1.9
eo?
1.13
INVARIANCE
transform
has
value
Fig.8
ORTHOGONALITY
the
oscillation,
imme
OF
if
same
the
but
value
Fourier
Opposite
shows the
pulses
of Fig.3.
V7
for
+v
and
transform
sien
for
+V
are
no
the
as
a) sine
same
abso-
and\—\v.
Fourier transforms
They
—VeebGe
has
longer
g(j,v)
of three
block
either
even
odd!.
or
£(6,8)=-V2cos(616+m/4)
F(5,8)=V2sin(61.8+m1/4)
£(4,8)=-V2cos (418+11/4)
1/4)
sin
(418+
£(3,6)=V2
f(2,8)=-V2 cos(218+n/4)
F(1,8)=\/Zsin (21 8+n/4)
f(0,8) = constant
eT
2
'
Es
4
2
ney
a
2
Q=t/T —~—
Fig.9
Orthogonal
jumps
of
equal
Fig.9
shows
ses.
so
They
that
6 =
-
shown
are
and
at
of
sine
and
6 = -% and
cosine
shifted
functions
6 =
+#.
compared
have
Their
jumps
Fourier
pulses
6 = +#.
a system of orthogonal
time
all
in
system
hight
sine
and
with
those
of
equal
transforms
having
cosine
of
pul-
Fig.1,
magnitude
g(j,v)
at
are
Fig.10:
e(j,v) = StS
(25)
, xk =-45 for even j
k = 1(j+1)
for
odd
jg.
‘The Fourier transforms of the various block pulses are
different
but
their frequency power spectra
are
equal.
The power spectrum is the Fourier transform of the auto@orrelation function
of a function, and not the Fourier
transform of the function itself (Wiener-Chintchin theorem).
The
connection
between
Fourier
transform,
power
spectrum and amplitude spectrum
is discussed in section
1.32.
2
See
also
ta].
Harmuth, Transmission of Information
(4,8)
(2,8)
FOUNDATIONS
MATHEMATICAL
4.
4S
£(0,8) £(1,8) £(3,9)
sa!
ff \
Fig.10
Fourier
pulses
of
shape
LCi
0)
a
cy
¥, (8),
=
Gai
ee
g(0,v)
=
ya: Gawd,
2d
Ss eee
¥
(4rv)
(26)
g(2i+1,v)
ese
=
(-1)' Woies (Ary)
ere
SORT
-o5}
Fig.11
The
rabolic
functions
4,
cylinder.
-4x’
¥,)(x) =__e
=
ite
=
same
[5]:
g(2i,v)
se
in
shown
the
have
e(j,v)
transforms
Fourier
their
and
Fig.11
cosine
and
sine
cylinder
the parabolic
of
4508)
functions
The
of the
g(j,v)
transforms
Fig.9.
OCOR
¥, (8)
Amys
=
O=t/T
4;(8)
or
y; =;
yi
He, j (x); ;
en
decreases
—>
th
=
v=2fT
f
on
oe
4x?
d\i
$x?
= e&* 3 (- S)ie
ax
eda
for large
(4n
el
He, j (x)
ae
eon
V2
ee i
absolute
ee
values
of
6 propor-
tionally to 6! exp(-#6*) and ¥, (4tv) decreases for large
absolute values of v proportionally to (4nv)! exp[-4(4ny)?}.
Pulses
with
quire
a
the
shap
of parabolic
e
particularly
small
part
cylinder
of
the
functions
re-
time-frequency-
1.14
WALSH
domain
FUNCTIONS
for
19
transmission
of
a certain
percentage
of
their
energy!.
1.14 Walsh Functions
The
Walsh
functions
are of considerable
a close
connection
between
as
between
cal
and
in
sal
cal
were
while
and
the
For
letters
to
use
times
the
milar
duality
single
three
sine
wal(j,9)
re
he
cal(i,6),
functions
difference
Wol(O,@)
be
sal(i,@)
may
be
for
s
name
con-
at
other
convenient.
defined
and
Walsh.
more
while
Walsh
andc
connection,
the
sometimes
more
is
as well
A si-
functions.
instead
of
A
the
cal(i,@):
= sal(¢i,¢e)
defined
(27)
by the
following
equation”’’:
wal(2j+p,6)
Om
is
There
letters
this
functions,
wal(2i-1,6)
wal(j,@)
The
from
is
exists
may
cal(i,o)
functions,
indicate
it
function
notation
wal(0,6),
sine
derived
cosine
function
he
to
and
communications2.
and
purposes
and
sal(i,9)
functions.
are
functions
=
Peet
'al'
of
in
sal
chosen
exponential
walC27,0)
The
cosine
computational
venient
wal(0,@),
interest
1s
= (-1)
Hs)
fae,
= 9O for
wal[j,2(e+t)]
1, -c,4.-45
+(-1 walt j,2¢0-4))}
wallO,e)
=
for
412
6 < =—4, 0 > +#.
0 <4;
(28)
1Pulses of the shape of parabolic cylinder functions use
the time-frequency-domain theoretically
'best'. This good
use has not been
of much practical value
so
far, since
sine-cosine pulses
and pulses
derived
from sine-cosine
pulses
are
almost
as
good, but much easier to generate
and detect.
2The probably oldest use of Walsh functions in communica-
tions is for the transposition of conductors [18].
3Walsh functions are usually defined by products of RadeThis definitionhas many advantages but
macher functions.
does not yield the Walsh functions ordered by the number
of sign changes as does the difference equation. This r=
of frequency in
der is important for the generalization
functions
the
are
functions
Rademacher
1.31.
section
-sal(1,9),
sal(3,6),
sal(7,0),..inFig.2.
may also be defined by Hadamard matrices
4(j/2] means the largest integer smaller
9*
Walsh functions
Pass.
or equal
;
#j.
wal(j,26)
obtained by shifting
terval 0 =6<+#.
j = 2, p = 1.
GME
the
Using
values
= 0
[0/2]
and
p=1
j =0,
cases
in-
the
into
right
the
to
the
consider
example,
an
As
is
< 0, and wal[j,2(@-z)]
-# <6
interval
the
left into
to the
wal(j,20)
shifting
by
is obtained
wal[j,2(0+4)]
< +4.
209
-+
interval
the
into
squeezed
is
but
shape,
has the same
wal(j,29)
The function
wal(j,6).
function
the
consider
equation
difference
this
of
explanation
For
FOUNDATIONS
MATHEMATICAL
4.
20
and
[2/72]=7
Gloigeulias) y
wal(1,0)
(-1)°*
{wal [0,2(0+4)]
+ (-1)°*! wal[0,2(8-4)]}
wal(5,8)
(-1)'" fwal[2,2(0+4)]
+ (-1)?*! wall2,2@ +4)]}
Tt may be verified
obtained
from
multiplying
-1,
and
the
=
cal(1,6)
to
is
The
to
by
function
function
sal(3,@)
to
the
right
product
of two
is
is
to
by
+1
half
to
its
to
the
and
the
the
ia
width,
the
left
right
by
squeezing
multiplying
by
= sal(1,6)
shifted
shifted
obtained
left
the
wal(1,0)
squeezingit
that
is
that
that
half its width,
shifted
shifted
Fig.2
wal(0,@)
the
wal(5,0)
from
by
+1.
wal(2,8@)
function
function
=
that
that
is
by -‘.
Walsh functions
yields
another
Walsh
TUNE
TLOM:
wal(h,@)wal(k,@)
This
relation
rence
may
equation
them
with
= wal(r,9)
readily be proved
for
each
wal(h,@)wal(k,9)
same
form
The
as
is
the
sign
wal(k,@),
turns
satisfies
a
of the
somewhat
modulo
wal(h,§)wal(k,@)
The
and
It
writing
out
diffe-
and multiplying
that
difference
the
the
product
equation
of
the
(28).
determination
equation
equals
wal(h,@)
other.
by
® stands
written as binary
value
of
cumbersome.
2 sum
of
h and
r
The
fromthe
result
difference
is
that
= wal(h@k,@)
for
an
numbers
(29)
addition
and
0@®@1=+=1002=1,060=1
r
k:
added
©%
modulo
2.
k and
h are
according to the
rules
= 0 (nocarry).
Addition
1.14
WALSH
modulo
FUNCTIONS
2 is
what
puters.As
wal(6,@)
one
an
and
eas
a half
adder
example,
doe
in binary
s
consider
the
wal(12,6
Using ).
binary
obtains
10
for
the
sum
com-
multiplication
numbers
for
of
6 and
12
6 @12:
ClO pen ets
Open?
Seid
digital
co
VOD Ores
ces eh
be verified
It may
equals
wal(12,6)
a
of
that
the
wal(6,6)x
product
yields
itself
(41)(41)
products
the
with
function
Walsh
only
since
wal(j,@)wal(j,@)
eo
Fig.2
wal(10,0).
product
The
MaubeO.G),
Occur.
from
and
(-1)(—1)
= wal(0,@)
C50).
=O
The
product
of wal(j,@)
with
wal(0,@)
leaves
wal(j,é)
unchanged:
wal(j,@)wal(0,@)
Ae
= wal(j,¢e)
(31)
=
Since
the
plication
addition
of
Walsh
modulo
2 is
associative,
the multi-
functions
must
be
associative
too:
(wal(h,@)wal(j,0)]wal(k,@
)=wal(h,9 )[wal(j,0)wal(k,6 J (32)
Walsh
functions
cation.
tions
is
Equation
yields
defined
the
unit
by
law
functions
is
and
is
an
modulo
2.
to
(32).
Walsh
the
consider
and
dyadic
number
what
both
h are
inverse
element
the
element
itself;
to (31); the
The
commutative
be
func-
the
according
by
of
discrete
k and
two
may
the
are
of
(31)
phic
k and h, that
product
or
the group
speaking,
subgroups,
the
Abelian
tically
determine
multipli-
(30)
in
To
to
equal
hold
factors
to
(29),
to
respect
function;
is wal(0,8@)
shown
with
that
a Walsh
(50)
is
group
(29) shows
again
element
ciative
forma
group
group,
assoWalsh
since
commuted.
functions
of
the
Mathemais
isomor-
group.
of
elementsina
group
and
its
if two numbers
ers
can occur,
numb
smaller
written
or
as
equal
binary
2° - 1, are added
numbers:
ice
eree
Daas
:
herpes oie
R=
O64
ae
esa
Dy t* Psy?
h@®j=
Cae
the
h =
ke
number
occurs,
of
factors
Diet
for
thus
functions
are
all
the
(29)
of
This
to
the
role
Walsh
cal(i,@)cal(k,9)
a
functions
cal({ (i-1)@(k-1),9]
is
sine
and
required
is
for
i
requires
is
number
system
inthe whole
functions
interval
of
an
in
the
the
integer
and
thus
Walsh
interval
series
system
whole
and
and
This
expansion.
{ sin2mv8,
interval
thus
theorem
-co<
denumerable,
cos 2mié
is
The
the
Fou-
cos eT v8 }
6 <
+a,
while
vy
non-denumerable.
functions
-co<
sin 2mni9
-# s ®@ = +%#.
a Fourier
orthogonal
that
a real
cosine
in the
transform
The
2
= wal(0,é@)
orthogonal
Note
of
(35)
sal{[i®(k-1)]+1,6}
which
ele-
sal{[k@®(i-1)]4+1,6}
Il
rier
These
2'
follows:
sal(i,@)sal(k,@)
system
s.
powers
multiplication
i]
The
sr <
functions.
cal(i,@)sal(k,é@)
are
0
to
contain
= cal(iek,é)
sal(i,@)cal(k,@)
cal(0,@)
bi-
wal(0,8)
contains
Evidently,
the
as
in
and vice versa.
Subgroups
subgroup
Walsh
rewrite
if all
that
functions
cosets.
for
may
h has ones
wal(2'-1,8),
= 2°"
one
means,
functions.
Since
for
obtained
number,
Walsh
2°
is
front
in
os
where
the
of
(34)
factors
number
© j. This
zeros
wal(0,@)
(27)
amet
number is obtained,
resulting
(2°*-1)
subgroups.
2°/2'
=
CUE Fn
the
all
largest
the
h =
important
Using
ee
+'(p,®
if
zero.
contains
it has
an
1;
a total
the
play
Heoess
The
j has
wal(25-1,9),
ments
q,2'+
nae ee ee Pap Se ei
notation
A group
0
Pye = 2
]
is obtained
nary
y
2 are
O.
are
Ca
s-]
1
pee
k yields:
h and
of
equals
FOUNDATIONS
oe Digital
® ae
powers
j and
these
Sotan tapegpnn
BIL
2 sum
smallest
The
eligi Seige
Gagne
module
The
of
MATHEMATICAL
4.
ee
8 <©
orthogonal
is denoted
and
complete
by {sal(u,@),
1.14
WALSH
FUNCTIONS
Cal(,9)},
later
on,
sal(i,®)
can
where
be
that
is
a
number.
Rademacher
ema
by
stretching
functions
the
sin2mnié
be
shown
by 'stretching'
and
cos 2mié.
from
the
periodi-
and
cal(1,6).
From them
Walsh
functions
known
We aerate 6, ats Se =GO<
yp be
of
the
An-
[9]:
[8],
functions
starts
sal(1,@)
subset
58) '= cal(1,2*6)\
now
It will
may be obtained
definition
due to PICHLER'
one may define
Popes
real
system
and cal(i,é@) just asthe syst
{sin
em
2mv6, cos 2rv0}
cally continued
Let
»
this
obtained
other
25
. sal(2",6). =) sal@ ,2"6)
(36)
aU I<) 14.00%
written
as
binary
number;
co
u
==)! ieee
UU.
sum
be
=
s=-00
1S
either
has
at
1
or
a finite
most
of the
fined
SN
es vie
O.
44h
uw is
number
a finite
binary
as
TW
called
of
oe
of
cal(u,®)
a)
dyadic
terms.
number
point.
2s
This
binary
and
or
olele
rational,
means,
digits
the
must
the
right
to
sal(yu,@)
if
there
are
then
de-
follows:
co
cal(u,0)
= |] cal(u,2*,0),
-ao<
8 < +00
(37)
s=-0CO
—Ca LO, ose =Oo
se <50
sal(u,0)
={
Bal(e,0)
= caliee
™ 6 )sal(2e 0),
SEE
g=even
cal(u,@)
and
sO)
Decne
number;
sal(u,@)
are
<.co
u=
dyadic
-co<16
irrational
<c,
u = (g+1)/2%
= dyadic
showninFigs.12
and
rational
13 for
the
'The non-denumerable
system of Walsh
functions required
for the Walsh-Fourier transform is due to FINE [12], who
also pointed out first the existence of such a transform.
The correct mathematical theory of the Walsh-Fourier transwhich are somewhat diffeform using sal and cal functions,
rent from the system used by FINE, is due to PICHLER [9].
as well
or Pichler transform appears fair
A term like Fine
as shorter than the cumbersome term Walsh-Fourier transWalshthe
this term, because
use
form. Mathematicians
caseof the general Fourier
Fourier transform isaspecial
on topologic groups, published by VILENKIN two
transforms
years after FINE's paper [22].
intervals
value
the
a line
of
sal(u,8)
a
versa,
cal(u,8)
of
value
a certain
y for
of
function
as
sal(u,@)
or
values
the
shows
Vice
ui.
of
value
certain
or
cal(u,9)
obtains
one
a
y-axis
the
to
parallel
line
6 for
of
function
as
6+axis
the
to
parallel
indicate
By drawing
-1.
value
the
areas
white
+1,
areas
Black
-3<90<+3.
and
-4<u<+4
FOUNDATIONS
MATHEMATICAL
4.
24
6.
4
4
3
w
2
2
1
40
ne
H
r
4
-]
2
-2
3
3
-4
#
Oe
oO
0
-1
-2
=3
1
3
2
Q—e
Fig.12
(left)
The functions
cal(y,®)
—~—3 < 0 < +3, -4'< yp < £4.
A function,
is
obtained
§-axis.
by
drawing
cal(1.5,9)
is
a
line
+1
at
where
yw
=
this
in the
interval
¢.@.
<al(1.5.6),
1.5
parallel
line
runs
to
the
through
a
black area and -1 where it runs through a white area.
At
borders between black and white areas use the value holding forthe absolutely larger yw. The function cal(y,1.5)
is obtained by drawing a line at 8 = 1.5 parallel
to the
U-axis and proceeding accordingly.
Fig.13
(right)
The
functions
sal(u,@)
in
the
interval
“3 <9 < +3, -4 < yw < +4. The
values
+1
and
-1
of
the
functions
are
obtained by drawing lines as explained in
the caption of Fig.12. At borders between black
and white
areas use the value holding for the absolutely smaller u
or
9.
The
There
are
no
following
computations
wal€u,9)
with
functions
additional
Walsh
= wal(0,6),
Ceun.e.) = calen, ey
clGiyoy) = selGi,6),
sal(0,®@)
formulas
or
are
sal(u,0).
important
for
functions:
O-sie<i
sl
a epg hip
“isiesraie
Abo |
a
(38)
—% = 0 < +4
1.14
WALSH
FUNCTIONS
cal(u,oe0')
eal (yu; o8joal(w,o")
sal(u,9@6')
Since
extend
=
sal(u,6)sal(u,é
@ and
the
6@' may
ges
or
addition
negative
modulo
one
2 to
has
cal(u,@)
may
to
negative
(40)
= -(a.@
(-b)
equal
to one
This
positive
of
=a@b
a @
b =
of
1.
(39)
ty)
-b:
(-a) @ (-b)
u is
be
definition
,-a and
numbers
ae
25
half
or
bd)
the
sal(u,é@)
easily
be
average
number
of sign
inatime
chan-
interval
of duration
veryfied forthe
periodic
functions
cai2,9)
and sal(i,@). by counting the-sign changes in
Pis.2. calGu,9) and sal(u,9) are not periodic;
ifwis not
dyadic
the
rational,
average
duration
Ef
1 still
an
known,
yielded
the value
-% = 6 <
yu as
one
an
by
of
that
respect.
+1
a
everywhere.
saying
interval
of
sine
function
is
feature
is
from
This
sinusoidal
Walsh
Assume
foraWalsh
+#. It follows
yield
additional
-1;
the
interval
surement
Hintox
acti)
measurement
value
of
# = yp <
yields,
functions
functions
that
and
uw is
the
A®
interval
undetermined.
thus
-1
forthe
The
W;5
required
Au
since
tained
The
more
within
product
with
transmission
information
increasing
interval
restricted
interval
this
this
O = u<1.
# = 6 <
to
the
rate
about
of
<
meaand
yy to. the
of
the
successively
y remains
the sequency
remains
1
smal-
1.5
A doubling
measurement
which
A@A4y
1 39
restricts
3 4 <°0.75.
for
as the uncertainty
interpreted
tions.
the
inter-
13 that
1 according
to Fig.12. Afurther
e.g.
smellerssovervals
halfs
in
are
a measurement
functioninthe
Figs.12
thestmpervalid. > = 0 <2;
interval
half
time
be a functioncal(u,9)
with wy inthe interval
Let
be
section
known
inthis
val
time
small
expressed
of
per
true.
is
has
ler
changes
information
at the rate zero.
different
must
interpretation
sign
holds
function
frequently
quite
the
of
arbitrarily
the
transmit
but
number
and
constant
may
for Walsh
relation
func-
not
zero,
information
the
observation
exact
value
interval
is
of
AQ.
u
is
ob-
FOUNDATIONS
41. MATHEMATICAL
26
inclined
words may be added for the mathematically
A few
reader about the connection between the systems {wal(0,6),
and {1,V/2sin 2ni9,V2cos 2nié}. Both
cal(i,@), sal(i,9)}
are
orthonormal
may
base
rier
is
bothof
them
series
andthe
Fourier
that
from
may
both
of
system
the
group
topologic
the
functions
dyadic
rived
from
most
of
Walsh
of
the
-
numbers
real
is
the
of
Walsh
the
group
de-
real
the
func-—
and discontinuity
different
by the
of
of
the
between
functions
dyadic
of
group
topologic
difference
derived
group
system
representations
caused
and
the
is
character
The
The
groups.
character
the
groupis
striking
functions
the
numbers.
from
binary
of
this
character
is
continuity of circular
-
tions
real
dyadic
set
the
The
numbers.
the
Fou-
reasonfor
The
{ cos kx, sin kx}
which
derived
be
group;
},
of
group
may
from
functions
er
theories of the
transform.
derived
be
circular
similar
very
on
one
and
L,(0,1)
space
in Hilbert
systems
topology
[8,11,12,20].
group
1.2 The Fourier Transform and its Generalization
1.21 Transition from Fourier Series to Fourier Transform
The
of
Fourier
every
Fourier
series
facilitate
from
transform
communication
is
shown
here
understanding
orthogonal
Consider
cosine
belongs
engineer.
the
elemants,
series
of
to
to
the
Its
derivation
in
a
the
more
special
orthogonal
orthonormal
system
the first few of which
elements
fs5(i,8)
and
the
constant
knowledge
way
general
from
that
the
will
transition
transforms’.
{f(j,6)}
are
The elements f(j,8) are divided into even
odd
basic
of
shown
elements
sine
in
and
Fig.1.
f,(i,8),
f(0,9):
'The transition from
the
Fourier series
to the Fourier
transform has mainly tutorial value. A mathematical
correct transition without
an additional assumption is not
possible,
since the Fourier series uses
a system of denumerable functions but the Fourier transform one of nondenumerable functions. A corresponding remark applies to
the transition from orthogonal series to the generalized
Fourier transforms in section 1.22.
1.21
FOURIER
TRANSFORM
£COV00
Ge)
=
ey
Si= wal(0;0)
= 4
Detcdey
V2 cos 2mie
f,(i,@)
= V2sin 2nie
undefined
39° 1,
Sine
cosine
outside
sine
the
elements
interval
f£(0,6)
8:)
DC.
=
Polio)
Periodic
is
a
way
of
to
the
An
triangular
is
obtain
periodic
-00< 8 < +00
(42)
the
If conditions
in
a finite
interval
defined inthe
series
of
sine
function
and
cosine
is
one
continue the sine
must
two of
continued
the
possible
and
ways
Periodic
continuation of the
periodic
continuation
Hence,
the
panded
in
tions.
If,
continued
has
to
which
Let
be
are
periodic
a series
on
of
expanded
zero
be
of
its
If
the
the
a
outside
expanded
inthe
cosine
sine
requires
elements.
Fig.14a
and
triangular
and
same
important:
function
and
series of sine
that
triangular
elements
sine
periodic
hand,
the
particularly
triangular
the
of
satis-
interval of definition,
cosine
are
top
are
orthonorsame interval as
is expanded into
O outside the interval
in
on
of the
triangular functionof
other
by F(@) =
F(@)
elements:
the
-$ #90<#.
shown
elements.
outside
interval
definition.
required for convergence
function
way;
of
interval
fied, one may expand F(@) into a series
mal system fe Gioe)} being definedinthe
F(6). The triangular function of Fig.14a
a
periodically
the
a function
extend
F(@)
example
continued
to
see sin-omk8
Consider
a function
Fig.14a.
be
9 <+#
1
continuation
special
0 >+4
V2 cos anid
sat eed
=
may
-% =
(41)
functions:
cosine
and
6 < -#,
-2,. So.
Week7
and
af A Ree
is
cosine
ex-
func-
function
is
-# =9
< #, it
cosine
pulses,
interval.
in a series
of
sine
and
cosine
FOUNDATIONS
41. MATHEMATICAL
28
cosamie + a,(i) sin 2ni8]
B(@) = a(0)£(0,0) + V2 Sifa,(i)
1/2
a(O) = fF(@)£(0,8)de
a
=
JF(@)de
-1/2
-1/2
anu) aay [PC ) cos amie de
(43)
-1/2
a.(i) = V2 [F(8) sin2nie ae
“1/2
The
coefficients
angular
a,(i)
a(O)
function
are
of
zero,
and
a,(i)
Fig.14a
since
the
are
plotted
in Fig.15a.
triangular
for
All
the
tri-
coefficients
function
is
an
even
sbGuaNe
NGALON
Let
the
replaced
variable
by the
Cea 7.5.)
new
® on the
right
variable
@':
hand
side
of
(44)
substitution
"stretches"
the elements
V2cos 2mi8 and f(0,8) by a factor §. The new
orthogonality
is
-#§
= 6<#8.
The
\2sin 2mié,
interval
of
system
of
orthogonal
the stretched elements V2 sin2miée', V2cos 2mié'
not
normalized,
amplitude
The
to
be
multiplied
-#§
nuation
and
over
the
these
is
not
by
§”
to
stretched,
of
= 98 < -# and
# #6
the
Y2sin2nié
and
is
shown
the
is
§-times
stretched
the
same
as
wide.
functions
functions
have
normalization.
continued
< #8 by F(6)
stretching
for
andf(O,6')
have
the stretched
retain
but
of
F(®)
functions
elements but are
square
§ rather than ‘1. Hence,
F(9)
val
since
as the original
integral
yields
be
Ses
This
is
(43)
of
§ = 2 and
into the inter-
= O.
This
conti-
£(0,8),V2cos
2mié
& = 4 in Figs.14b
elialel (ec
The
ments
expansion
has
the
of
F(§)
following
inaseries
of the
stretched
ele-
form:
P(8) = 7p{a($,0)£(0,8') + V2 S'La,(,i)
cosamie! +
+ a,(8,i) sin 2nie']}
(45)
1.21
FOURIER
TRANSFORM
29
wal (0,8)
ae
——
Ces
oN
rn
x
_
Se
Se=e
Ne
ee
DOS
<i
N,
Nea
ae
TaN
Seoe
/2sin2x8
/2cos 210
ta
/2sin&n8
Nae ~ 2 cos428
=
nm
wal (0,8’)
V2 sin 2x8’
—<
nen
cates
V2 cos 2x6"
2 sin 420"
/2 cos 418"
/2 sin 616"
><> V2 cos 218
~
ae
aa Ns
Zz
/2 cos 618"
=
ee, SRO
LASSE
N
—
= V2sin 3x8
AS4
ff?08308
See oe Se
V2 sin 816"
2 cos 88"
.
b
N
Lf (sind
SAS
[2008 408
Per pares
=
-4
wal (0,0')
=,
—
va
4/2
B'—=
=
ph
= Eee
4
-- F08)
wal (0,8)
/2sin6/2
\2 sin 2n0' = Se
/2 cos 276"
/2
ae
Se eae
sin4n0' oe
a
\/2cos
16/2
eee
ae
/2sinx 0
= \2cosn8
/2 cos4n0"
/2 sin6x6'
2 cos6x8"
<x /2sin3n6/2
ee
ee
See
/2sin 6x0
:
Lf
=
==
ht
}
——
c
Expansion
cosine
elements
a) -
= 6 < $,
Ceo
aA CEs
Pees
<1,
2
of
t
a function
having various
n
iH
F(@)
i o—=
2
Ca
(_—=
3
in a series
intervals
/2cos 3116/2
of
sine-
of orthogonality.
V2 Coe emi0, Ve sin 27i6}
cos On(#i)8 V2 sin 2n(4i)6}
40 ),V2 cos an(4i)8,V2 sin On(£i 6}
{wal(0O, 8)
#6me
yal o.t
wal(0O
pied
‘
+
-A/2
Fig.14
atl
FOUNDATIONS
41. MATHEMATICAL
50
The
but
This
2nié'.
of
is
used
be
may
it
generalization
=
cos
2mi(6/g)
=
cos 2n(i/e)e
=
sin 2ni(@/é)
=
sin 2n(i/é)e
£ C059
=
notation
72)
=
f£(0/€,6)
is
trivial
the
with
for
sine
of dea point
as
transform:
Fourier
the
the
TO.)
combined
be
may
9@'
in
for
cos 27i0'
sin 2nie'
The
argument
functions
cosine
parture
contained
the
i in
factor
and
ae
factor
(46)
78)
£t07E
strictly
formal
andis
of no con-
sequence.
The
series
expansionof
F(@) assumes
the following
form:
V2 >!fa,(€,i oosengorads,i)sinande}}
F(a )=zela(s,0)#(G,9
i /&= VE
2) E/2
Ca Garey
ae
:
|e J F(e) cos ange de
-E/2
;
=
ag(8,1)
le
2
|
g/2
é
2
ale
sin 2ns0
F(@)
(47)
dé
&/2
a(g,0)
= rE f F(@)ae
-§/2
Introduction
ac(#) =/é
of
new
eas
constants,
a s(#) =Vé
Rl Ey ty
a(2) =Vé§ a(@,0),(48)
yields
nae.
ie
cnt
tet ein
ates
FCG) Se
=sae)
iC B72 diLac(g)oos
engo+as
2190 ]} (49)
0
F
aly) and
and
c;
the
sine
Let
lim
i,8 00
age
fy iP
ac(#)
they
and
are plotted
for € =
hold forthe
cosine
expansion
elements
of
§ increaseto infinity;
ac(#) = a,(v),
ae
2 and
of
Fig.14b
i/€
& = 4 in Fig.15b
F(@)
shall
in
and
a
of
c.
remain
Lin as(#) = ag(v)
i,k —co
series
constant:
(50)
1.21
FOURIER
TRANSFORM
a
05
04
04
i
be
3 02
eS 0.
0
a
>
=
02
s 0.
he?
eye
Chan
a
0.5
04
|0.4
4
i/I—~
to
i may
Fig.14.
be
any
merable.
of
on
tions,
the
denotes
i
and
thus
aan
transform
curve
as
be
be
for
the
i/&
are
allowed
to
denu-
be
any
non-denumerable,
would
denumerably
Fourier
limit
well
must
integrals
contains
See
ee
~-a,
as
hand,
number
‘
the
§
number.
other
0
2
—~
expansion
of the triangular
sine and cosine elements ac-
a factor
following
series
but
the
real
the
Fourier
by
integer
v,
non-negative
some
a-(v)
stretched
5 0.3
0.2
0.4
0
d
Fig.15 Coefficients
of the
function F(@) in a seriesof
elements
wie
1/4
Os
uf03
S02
SOA
0
b
cording
sh
0.6
be
many
zero.
Hence,
orthogonal
contains
or
the
func-—
non-denumerably
many.
The
end
limits
a,(v)
and
a,(v)
follow
readily
from
(46)
(47):
a,(v) =
&/2
3
co
lim/2 f F(@) cos ange do = 2 [ F(e)cos 2nve
E—oo
-F/2
de
—00
a)
E/2
Pec
order
consider
F(8,) as
plotted
to
along
Fog.16.
to
1/€.
by
dg
=y2
gh
iN F(@) sin 2nve
The
Hence,
(49)
find
the
an
integral
numbers
distance
the
is
dé
i
representation
for
a certain value
® = 0,. Equation
(49)
a sum of denumerably many terms, which
in
given
co
asl 2 J F(6 ) sin engé
sats
In
i
sum
axis atthe
points
between the plotted
of
the
terms
i/§
multiplied
equal to the areaunder
the
step
yields
may be
as
terms
F(@),
shown
is equal
by
A AZ eis)
smbnakenrmmeyal
FOUNDATIONS
41. MATHEMATICAL
oe
series
Fig.16 Transition from Fourier
to Fourier transform.
x(0)
(0) = a(P)t(F,%)
Xa/72)
=
a (E2 cos ange
+
+
as(2 V2 sin ange
0 Wf 2G 3K Sif
of Fig.16.
Using
(49),
bitrarily
close
following
integral:
FCO)
= V2
for
The
lower
limit
limit
of the
of
butes
able
the
in
sum
(52)
and
integral
an
odd
not
function
(49)
may
be
of
be
shows
of
approximate
v.
large
this
area
ar-
values
of § by the
a,(v) sin 2nve]dv
(52)
integral
is
zero,
because
approaches
zero.
neglected,
for large
assume
only
+
in
little
not
(51)
following
F(@)
(49)
must
could
Equation
the
sum
arbitrarily
v
numbers
the
of
may
sufficiently
flactv) cos 2mve
0
wer
term
one
denumerably
interpreted
that
a,(v)
Hence,
(52)
since
values
the values
of
of all
many
as
it
§.
real
of
may
be
and
lo-
first
contri-
The
vari-
positive
them,
a Riemann
isaneven
the
The
or
the
integral.
a,(v)
rewritten
is
into
form:
= (Lay) cos Onvé
A(v) = #V¥2a,(v),
+ B(v) sin 2nvéjdv
(53)
BCv) = 4V¥2a,(v)
a,(v) is identically zero for the triangular function of
Fig.14; a,.(v) is plotted in Fig.15d according to the following
ac(v)
formula:
:
3/8
8
= 22 ! (1 - 39) cos emve
dé
sin
i
34nv/8 \2
= ale (Saas )
1.22
GENERALIZED
FOURIER
TRANSFORM
DO.
1.22 Generalized Fourier Transform’
Conasid
system of er
functions
orthonormalized
script
an
c
odd
in
indicates
function.
results
the
will
interval
of
an
even
© may
be
{f£(0,0),f.(i,@),f,(i,8@)}
interval
be
-40
finite
or
orthogonality,
such
be
to
in the
defined
f,(i,@)
Hence,
s
the
The
of the
non-nega-
be
f,(i,@)
cross
from
do not
functions
A function
is
-$@ = 9 < 4@
interval
sub-
havi
an infinite
ng
differentiable.
or
continuous
‘The
subscript
as the functions
@ = O.
at
values
to positive
negative
have
the
infinite.
functions
all
let
and
6 = 0,
for
and
applicable
to functions
all functions
Let
cylinder.
par
abo
lic
tive
s 6 < 4@.
function
expanded
F(9)
in
a
series:
B(@) = a(0)£(0,8) +S? Lag(4)f_ (4,8) + ac(4)P(4,9)]
a,.(i)
=
0/2
if F(0 )fc(i,8
-0/2
de
(54)
@/2
= J F(O)f,(i,6
)d0
as(i)
-9/2
@/2
a(O) = [ F(e)f£(0,6)de
-0/2
@ is replaced’
and
(4
Gea
/ vy,
The
tions
by 9'
in the
functions
£(0,0),
y=
y(t) > 1,
expansionof
is
obtained
pene)
F(@)ina
in
= OO
series
of the
analogy
to
(55)
stretched
;
other
generalizations
au Rel Soa
see
func-—
(45):
(56)
B(@) = gofa(s,0)2(0,8") + Dilac(S AF eCi,8") A
'For
£,(i,8)
8%
ea
Soe
|}
[1,2].
systems of
of
2Thre method used applies to a large class
proofs can be obtained withmathematical s.
tion
Exact
func
individual
for
requirements
mathematical
out excessive
the results of
of functions only. For instance,
systems
i
this section seem to apply for dyadic rational values of
in reality
of Walsh functions;
i/gé = uy only in the case
i/€.
of
values
real
all
to
they apply
3
Harmuth, Transmission of Information
1/y
factor
(de/é )
oni
@' may be written onthe right hand side of (56).
had been replaced trivially by 2n(i/§)@ in (46);
connected
8 are not necessarily
and
and £,(i,@)
formal
should
be
until
proved
CO;
0")
The
=
Pour
=
Ci.
series
et
yo) =
o7 9)
rather
i
in f,(i,@)
particular,
than
i/§
a fraction:
CLs sae
Peala/e
=
In
otherwise.
a symbol
considered
foro 0.) = fore 7)
cipal leony
since
ons
must be considered
following substituti
the
purely
product
as
of
6 instead
that
with i so
is combined
the
into
continued by F(@) = O
40 = 6 < #y®.
and
-ky® = 0 < #y0. F(6) is
intervals -#y® = 9 < -4@
The
interval
in the
orthonormal
are
functions
stretched
The
FOUNDATIONS
MATHEMATICAL
4.
34
(57)
ey
tUO/e, 33
expansion
of F(@)
assumes
the following form:
PCO )=pofa(s,0)£(2,0)+ >; Lac(S,4)F,(F,0
4a,(6,4)2, (F011
i/E =1/F
(2,4)
a.(§,i)
;
a
;
(=,6)de
= many 6 )f,)f. (B18)
teva
ae
=
eo
(8,4)
= 7bwok? (8)
Go)
F.(=,8
7 ye/2
(,0) m== 75 a
eee Om
New
Y _y@/2
(58)
)dée
=e
(8) )dé
F(@
coefficients
are
introduced:
‘
ac(g)1 = Vyac(6si),‘ ag(#)1 = Vyas(8,i),
aP) = Vya(s,0)(59)
In
order to make (58) and (59) more
one
must
demand
either
the
same
as
= y
is
1/6
limit
for
'The
hand
left
limit
large
that
the
value
hand
limit
differ.
a formal
coefficients
for
constant,
values
than
all
or
of
values
that
i and
shall
be
notation,
ac(#) or
as(z) have
of
§,
they
i and
converge’!
as
long
toward
a
&:
taken,
if
left
and
right
1.22
GENERALIZED
;
i
=
algae
Again,
one has
number
and
FOURIER
TRANSFORM
:
el eer
to
al,
a
Sean ss p=
postulate
thusis
aD.
that
uy is
non-denumerable,
(60)
a non-negative
while
i or
i/€
real
is
de-
numerable.
The limits (60) exist,
limit functions
follows'!:
and
f,(u,8@)
yo/e2
lim
if fo (3,8) and £5(z,9) approach
x
i F(8 )£¢ (3,8 a8
eee
= lim
ea /2
yo/2
wine
i,§ co
Se
are
that
f,(u,8)
defined
yo/2
f[ F(O)f.(u,6)ae
ey
as
(61)
(2
yo/2
F(@)f5 (7,0 )ae = pam) Gea.
(1,0)d8
~yo/2
oe
a2
y = y(§)
The
functions
fc (%,8)
and
f£.(F,9)
val
-#y® = 6 < #y®
to the limit
fs(u,8). This type of convergence
gence’
i
from
=
lim
F(6)
interval.
to
;
(62)
-y@/2
+
ye/e
‘i B(8 )f, (F589 )d8
-yo/e2
be
a function
Equations
(62)
that
reduce
vanishes
to
the
outside
a finite
following
simpli-
form:
In order
B*
functions
f,(y,8) and
is called 'weak conver-
(61):
ac(u) = fPC@)fe(u,9)d8,
'The
oe
F(6
inter-
lim , f F(8)£, (7,9 )d0
= —co
fied
(51)
yo /2
aoQi).2
Let
in the
[4].
follows
Aetu)
converge
to
find
an
alu) = f FCe)f.(u,e)ae (63)
integral
representation
for
F(6),
integrals shall represent Cauchy's principal value.
must hold forall quadratically integrable functions
a
consider
a
as
COE
plotted
along
instead
of
given
by 1/y as
a step
gral,
function.
This
€ and thus
if
by (58)
area
y(€)
is
may
grow
to
equal
the
area
represented
be
beyond
all
the
multi-
terms
the
of
sum
the
Hence,
1/y.
is
terms
plotted
plied
between
distance
The
Fig.16.
in
as
i/&€
be
i/y = i/y(8)
points
axis atthe
the numbers
may
which
terms
many
denumerably
of
sum
yields
(58)
Equation
8,-
6 =
value
certain
FOUNDATIONS
MATHEMATICAL
4.
36
under
by an inte-
bounds:
£5
F(@) = ff Cac(ufc(u,8) + ag(u)
)1a0
(use
(64)
0
ac(u)
and
transform
ag(u)
of F(8)
are
for
called
the
Equation (64) is anintegral
generalized inverse
grals
the
actually
functions
riable
wp
cannot
and
the
Fourier
-
normalized
and
-
real
tions
negative
f
uae)
=
f,(u,9)
numbers
£.(-n,9),
is
an
ag(u)
into
form
the
FOO) =
co
of
y
is
One
or
these
without
its
inte-
specifying
closely.
variable
v
The
va-
in
the
called
a generalized
may
extend
integers
for non-
the
defini-
numbers:
function
=
-f,(-p,6)
of
function
of
and
show
(63)
function
9
@ as
as
well
well
that
of uy. Hence,
(65)
as
a,(u)
(64)
as
uw,
and
is aneven
and
of
of
u.
may be brought
(534):
Jf [AC)£e(u,8)
£e.(u),
the
of F(@)
Whether
more
Fourier
f,(u,6).
are defined for positive
and fs5(u,6) are defined
+ Bludfs(u,lay
8
—oo
ACy)
as
Hence,
fein, 9)
even
is an odd
role
uw only.
real
f5(u,0) is an odd
Equations (62)
stated
and
frequency.
f,(i,8)
f,.(u,@)
negative
be
fs5(u,8)
same
transform.
f£,(i,8) and
i only. Hence,
to
representation
exist
usual
generalized
f¢(y,8)
Fourier transform.
f,.(u,9)
plays
the
functions
Boi) = $eccu)
(66)
1.23
INVARIANCE
OF
ORTHOGONALITY
37
1.23 Invariance of Orthogonality to the Generalized Fourier Transform
Consider
the
G(u) = VatACu)
function
+ BCu)]
Since
A(u)
G(-y)
= VetAC-u)
G(y):
= BV2Lac(u)
is even and
B(u)
+ B(-y)]
is
+ as(u)]
(67)
odd,
one obtains
= V2tACu)
for
G(-y):
- BCu)]
A(u) and Blu) may be regained from G(y):
ACu) = 4V2[G(u) + G(-u)],
Using
Or
s¢20)
G(u)
and
one
may
By)
rewrite
= 2V2G(u)
(63)
and
- G(-u)]
(64)
(68)
into
the form
€21):
(8) = V2 f GCu)Lfc(u,e) + fo(u,8) Iau
(69)
G(u) = 4V2 f FC@to(u,0) + £5(u,e)]a0
(70)
—Co
se
tei made
Ataf. Cu,0)
in
and
Consider
that
a
(70)
of
thesfact,
BCw)f,.(u.9)system
vanish-outside
{f(j,8)}
a finite
that
the
integrals
of
vanish.
of
orthonormal
functions
interval:
co
J HOS, 0 DF (&, 806
Let
g(j,u)
HCO).
e(j,u)
=
Six
the
denote
Olt follows
C749
Fourier
generalized
transform
fromsC70)%
(72)
OVE (Unt) + fe Cu 0) de
= aVe2 i Poss
Equation
(71)
may
of
be
transformed
as
follows:
F ej, )fav2 feCkwlte(use) + f5(u.8 )]ausas
ie
—oCo
F e(eyu)tave [Cd
te(us8) + £(u,8)]4e}dy
[o.e)
ll
mo
=
Je weCd
wdy = 5%
—oo
Moe
felg,uor-
system
orthogonal
an
into
transform
Fourier
generalized
the
by
transformed
is
interval
nite
fiae
outsid
vanishes
that
{f(j,9)}
system
An orthogonal
FOUNDATIONS
MATHEMATICAL
4.
38
1.24 Examples of the Generalized Fourier Transform
function
iM)
15
The
=
of
substituted
and
B(x)
4).
transformations
following
-1
8
2 = <
+414:
x
=
26
are
(74)
=
ae
2s
made:
= Py (ee)
Pe Gi
b-
etc.
the
Gly)
Cage
yy ea te)
(-1)' (ai - 19B,, (20)
oo
a er
The
system
the
{f(0,6),
interval
positive
for
sitive
first
-(3x?-
is
= PoCOy = Cad
in
=
polynomials
orthogonality
Pea)
ch
Legendre
for
Fig.17
2) (Cpe
interval
£(07,8)
of
trif
Transformothe
Fourier
generalized
the
Consider
angular
-§ = 6 s +#.
@ = 0,
and
differential
few
PCO,8)
polynomials
=r,
P.(i,6),
FeCl, o
all
All
= ey ee
as
is
orthonormal
functions
functions
quotient.
read
P,(i,9)}
P,(i,6)
P,(i,@)
Written
have
are
a po-
explicitely,
the
follows:
P.(1,8)
=
-8V5(120"
-
1)
7s)
Ps (2,8)
=-¥7(208" -38), P. (2,8) = 79(5608' - 12087 + 3)
The
coefficients
a.(i)
readily
computed:
a, (i)
f P(e), (4,8 ae
!
- 1/2
a,(i)
3/8
I
aioe
for
Fig.17a
may
$8 )Pc (i,8 ae
be
(76)
0
[ F(@)P.(i,0)ae
1/2
ag(i)
and
3/8
0, a0)
8
= 2 f (1 - $0)ae
- 1/2
0
a (i) and a(O) are plotted in Fig.18a.
Let @ in(75)be
replace
byd9' = @/y, where y = y(e&)=
ace sec Fa. Ue eae: P.(i,®@) are stretche
over doubledthe
interval
as
shown
in
place
byd
the streched
Fig.17b.
functions
The
functions
P,(i/2,8)
(75)
and
are
re-—
P,(i/2,é6):
1.24
EXAMPLES
=
——
_—
Py (28)
—
OF
TRANSFORMS
See
eo
c-—
ae
Be)
|
|
P, (28)
ee
Ga
i
See
a
& FIG)
wal(0,8)
ch
|
ESS
al
ee
ae
7
pean
PB, (2,0)
ee
ie (2,8)
A
a0)
)
eg
Ps (26')
Ps
See
SSS
a
=
P(8/28)
(26’) ee
—Py (26') ===!
|
ie(2/28)
=
3
P,(3/2,8)
8)
4
0
4 O— = 2
L_ P.(14,8)
—P, (26') ———
Se
Po (1/4,8)
a
ie
P, (20°) <P
Ps (28’)
<i P, (2/4,8)
eT
(2/8)
L_ P,(3/4,8)
A
mea
—P, (28") a
ee
oa
/48)
Ph
mS
Py (20°) 2S
Pe (5/4,8)
YS
pee
ee
Cc
Fig.17
dre
Expansion
polyno mials
a?
WA
4IA
TIA
ec ete
1/4
of afunction
having
1/4 B"
0
various
Sia &, {waltO,9),
9 S Alc fwal(0,9),
9 <—es {wal(0,@),
F(@)
pecs
1/2
of Legen-
in a series
intervals
Pe
P,(4/48)
0")
of orthogonality.
Pett,
2e.0),
)s
Pe (a7e,9.)}
FOUNDATIONS
MATHEMATICAL
4.
40
(77)
P.(1/2,8) = Ps(1,9/2) = ava(ee)
- 11
P.(1/2,8) = Po(1,8/2) = -BV5(12(88)
P,(2/2,8) = Ps(2,0/2) = -V7[20(#8)- 3(#8)]
Po (2/2,6) = Pp(2,0/2) = sV9[560(#8)’ - 120(88)°+ 3]
The
coefficients
a (4/2)
a,(i/2)
= FRC@)P, (i/2,8)ae
have
the
following
value:
-2f (1 - $0 )P, (i/2,8)de
(78)
=f
Values
of
aecise)
exactly the
since,
as the
P,.(2/2,0)
the
Jee f=
by
(75)
the
4 as
in
Fig.18b.
coefficients
is*mot,
functions
interval
= VCE
plotted
same values
¢.g.,
Let
the
are
equal
be
a2)
of Fig.18a
P.(1,8).
stretched
substitution
shownin
They do not have
6'
over
=
four
6/y,
times
where
Big .17Gz
P.(1/4,8) = Pg(1,0/4) = ay3(48)
P.(1/4,0) = Po(1,0/4) = -#V5012(46)?- 1]
P(2/4,0) = Po(2,0/4) = -7[20(40)°- 3(40)]
P.(2/4,8)
Some
y =
(79)
= Po (2,8/4) = 4V9[560(40
)* - 120(30)?+ 3]
coefficients
a.(i/4)
are
plotted
in
Fig.18c:
a,(i/4) =] F(@)P,(4/4,8)ae = 2 wich $0)P,(i/4,0)a8 (80)
23
In
of
orderto
i and
lues
0
of
§, one
the
needs
polynomials
of
j and
Ppa
a eae
for
hans
EAs
small
(74)
one
Pe (4,8 /8 =e
obtains:
cos uze
of
is
a,(i/é)
for
= P,(i,6/€)
6/€.
values
of
cos”
[Cj+e)cos
large
that
values
forlarge va-
An asymptotic
known
ath
7qzsinl (j+k
= ae
Using
limit
P,(i/€,8)
i and small values
Legendre
values
compute
holds
series for
for
large
x:
x + dn)+
x + 4n)}
(81)
1.24
EXAMPLES
OF
TRANSFORMS
44
|oS7
0.3
=
02
0.
1 03
=
S 0.2
= oO
So
i
0
a
e
0
ead
parr
aay
ed
Ou
0
1/2,4>——
jee
d
= 03
3
502
=
= 0
s
0
c
Se hiet
He
hashear i
eae
Fig.18 Coefficients
of the expansion
of the triangular
function F(§) in a series of Legendre polynomials according to Fig.17. a,(u) is the limit curve for the polynomials stretched by a factor § +a,
iiew
imi be tumetson-P.(1,9)-
and
a,(u)
follow
fom.2.>.co-:
FC,0 a= fq 008 40
(82)
co
A
3/8
8
ac(u) = f F(@)P¢(u,8)49 = rf (1 - $6) cos 4yue ae
5
: we
a
(u)
angular
is
is
the
in
coefficients
a.(v)
for
tion
in
scale
of
of
Fig.15
Legendre
equation
lues
of
j to
to
for
c.
a,(i/2)
and
One
a,(u)
may
Cet
readily
fromthe
that
Tre
differential
may
see
are
is
re gra
values
how
of
the
a,(u).
equal
except
equa-
generally
Oris
x
It
to
differential
this
tri-
polynomials.
a,(i/4+) converge
in Fig.18
see
for small
reduces
the
transformof
the
Legendre
One
and
polynomials
ee eee
This
Fourier
Fig.17
Fig.18a
a,(i),
factors.
(Cee)
(83)
generalized
function
plotted
2
SSS)
and
2. ca.
C84)
large
and
f
sine
equationo
so:
va-
cosine
functions:
ES
a
Ae
Oe
Cee)
SSS
ipa eae
a
cei
ae
Eee
Hes ia
oes
ee
as
te a
eee
a
Ra
Se
FOUNDATIONS
MATHEMATICAL
4.
42
ealPca 8
| arn
ae
el
$318)
lg eo
a at
ge
pp
eC
aed
aa
ee wt
Se gee ee PAF
We LST
cay SATs
a
USL
PALS woeLL
PLLA
ts
a
-1
t Sooo
|
|
=i
el ge Es
ee
a
eee
Oy)
Salil
e (=e See,
eee
oe
eT
eee
es
ne)
s0l(98) Ee
7s
=
sal(6,0)
“EFL
sal(4,8') =25, ofan
sal(6,0) =o
=
aaa
al (48)
sal
2
sal (1/2,8)
Poe
ed
ate
sai00)
WeDo erzeasee eSt
}
Lt FL
a
!
Sones
a
=o Sd
ets
sal (6/2,8)
sal (3,8)
LE sal(7/2,8)
FLFLFS—
sal (4,8)
_
sal (8,0
b
F (8)
sal(1,8") =
= sal (1/4,8)
sal(2,8') -
r~ sal (1/2,8)
sal(3,8’)
sal(4,8) =
sal (5,8’)
ae a
sal (7,8
sal (8,8
sal (9,6
a cays
sal(11,6') --
_ sal (3/4,8)
P sal(1,6)
_ sal (5/4,8)
== gal (3/2,8)
sal (7/4,8)
== gal (2,8)
= gal (9/4,8)
= sal (5/2,8)
- sal (11/4,8)
.
aa aE
7
== gal (3,8)
al 8).
Je
Pea
Ll
r t
ig eT! (gy VRE ge FO eyYN sr EN peg ET pe PR) es Nealpen Pg
sal(15, 6) Se
i
ae
eer
sal(16,0) =e
c
a
a
4/2
—1/4
|
o
AM cy EY ty
eh
ms
ier ee oe
0.
r
1 o6——
7
0
|
1k B'—=—
4/2
e
Sa
gall(4,8)
Fig.19 Expansion of a function F(@) in a series of Walsh
elements having various intervals of orthogonality.
a) —- s0 <4,
{wel(O,0),>Gel(i,6))
seits, Go)
bh ~1
@) -2
= 6 < 1,9
s6 < 2,
(wellO,e),
{wall0,$),
cal(i/2, 8),
cal, ay
sal(i/2, 8 )}
sal(i/4,e)}
1.24
EXAMPLES
Hence,
TRANSFORMS
interest
not
for
defined
reduced
by
systems
by
such
Fourier
functions
Walsh
rather
thanadifferential
amore rewarding
of
Fourier
distinguish
Walsh
for
theory
PILCHLER
the
functions
Walsh
to
functions.
This
not
did
distinc-
Walsh-Fourier
of
mathematically
into
rigorous
-
odd
and
even
functions
sal
Walsh-
the
FINE
separated
and
expected
polynomials.
due
is
to
f£(0,6),
f,(i,9)
and
f,(i,8)
repre-
functions:
6( O76 j)=wel 0,0),
triangular
2c(i,8 )=cal(i,o),
£,(1,0)=sal(i,6 ) (86)
function
of Fig.19a
1/2
aero
yields
the coefficients
3/8
a(0) ar F(@)wal(0,8)a@
Bead)
be
may
However,
The
func-
a difference
[L2).
Let
The
cal
is
cosine
by
they
applications
the
functions
that
-
functions
sent
for
and
transform
FINE.
communications.
to
analysis
sine
equation,
due to
that
equations,
which are
than lt
resu
Legendre
and odd
even
between
important
is
tion
is
transform
functions,
defined
are
of the Fourier
generalization
The
transform is main-
orthogonal
stretching
to the one
Since
yield
of
differential
tions.
to
44
the generalization of the
ly of
are
OF
= 2 J tive 30 de
30 cal (i,8)as,
eG)
=O
oO
Fig.20a
With
shows
y(2)
=
sal(i/é,6)
=
cal(i,9)
@ ome
continued
original
values
of
obtains
sal(i,@/eé).
cal(2i/2,8)
that
the
some
and
a(O)
and
a,(i).
cal(i/g,8)
=
cal(i,0/é)
of Fig.19a
Inspection
cal(4i/4,6)
periodically
are equal
over
of definition.
interval
to
to the function
or four
double
This
result
may
be inferred readily from the difference equation
Hence, it holds in the interval -$ = 9 <#:
Bal(ise ) = can (2i1.6/€)
Cpt
er l= obey
sees
3 Gal(Zi/e,0)
—$) S40) <4.
and
c shows
times
also
(28).
FOUNDATIONS
41. MATHEMATICAL
44
Inspection
in the
hold
ing relations
Cal(i,6)
toc
Fig.19a
of
follow-
the
that
further
shows
-% = 96 < B:
interval
(87)
= cal (21/2,0) = calf (2it1)/e,0)
= cal[(4i+2)/4,8)
= cal[ (4i+1 )/4,8]
= cal(4i/4,8é)
= calf (4i+3)/4,8]
= uw,
(€i4+n)/E
Substituting
= é€al(i,6)5
i, \ASoy
CalQu,6)
=
ese\A yn <4
Corresponding
Sali,
4260
relations
ca ceLuL,
fal
<a
obtains:
one
< 141,
2 Su
calG,0)
Wal CO,e },
O51, --5E= 1; See
Ye
= cal[(gi+n)/é,9]
i
Fees
are
obtained
SY
Ss
for
(88)
sal(u,8):
ot ee yeywe
(89)
<¢
The
limit
derived
functions
here
in
an
cal(yu,8)
heuristic
and
sal(u,8)
manner
for
the
-% = 6 < #. PICHLER has obtained
cal(u,8)
and
a
for
whole
mathematically
-0co<8<co,
but
rigorous
his
of mathematics.
proofs
Fig.12
presentation of the
[2].
way
and
the
require
13
functions
show
a
a
cal(u,§8)
very
very
and
have
been
interval
sal(u,8)
in
interval
good
command
ingenious
sal(u,9)
re-
found
by am
Functions
that are identical
inthe
yield the same expansion coefficients
obtains
for
a,(u)
ac (uy) =
a, (i)
and
interval
for
-4
F(8).
Hence,
fFCe )cal(i,e ae
i
i+1
IIA i=
A
< y
IIA
H-
IIA
A
28
-1/2
=
1/2
(Ce )se1(a,0
“1/2
ag(i)
Il
ac(u) =
a(O)
1/2
fFCe ae
i}
-1/2
=
S0.<
43 130],
one
a,(u):
V2
ey
= 6 <4
See
a8
i=)
0
i
=
(90)
1.25
FAST
WALSH-FOURIER
TRANSFORM
0.4
04
0.3
0.3
1 02
=
=}
0.2
Als
504
er
;
0
a
os
0
i
2
0.4
3
ee
4
c
:
0
04
2
3
4
ere
a
(4i+9)/4 ——
ter
| 0.3
alg
A5
3
=}
0.2
JO)
—
04
STO
Ss
0
b
ees
Fig.20
to
F(@)
Fig.19.
chedy
by
a.(u)
ea factor
Fig.20b
The
to
for
Lo
expansion
of
of
Walsh
the
triangular
elements
according
elements
stret-—
& i co,
for
the
of
Walsh
Cdeiticients
the
series
a [(4i+n)/4]
d
computation
simple
a
Pe
isthe limit curve forthe
ao {(2i+n)/2),
in
of
into
OC
d
—=—
Coefficients
function
0
ee
(2i+n)/2
the
functions
functions,
a(0),
and the limit a,(u) are shown
triangular
a,-(i)
occ
rOumOm
70
A net COMM
Obtain
a-(u)
and
a.(u)
fee | < 147 or 3-1 <4
a,(u)
since
and
function
a;(i)
emt
the
to
is
very
compute
and
Ol
Fig.19.
a.(u)
has
only
Omit).
in
and
one
of
plot
eieOM
the
thesé
Le GOmeL—
intervals
O24
lato
<1,
= i.
1.25 Fast Walsh-Fourier Transform
The
be
time
required
drastically
Fourier
to
transform.
A
transform was found
by
[2,3].
fast
KANE,
of
pictures
forsignal
according
between
to
Consider
the
[1]
transform
[4].
a
even
number
function
Fourier
of
and
of
by
and
odd
GUINN
in
functions
changes
some
WELCH
a two-dimensional
and
as
of infor-
have
The form presented
sign
F(@)
Walsh- Fourier
generalized
used
may
known as fast
for the compression
WHELCHEL
[5].
fast
and
have
transform
a method
corresponding
GREEN
classification
tinguishes
the
by means
ANDREWS and PRATT
Walsh-Fourier
mation
obtain
reduced
in
interval.
used
it
here
dis-
lists
them
Fig.e.
Let
this
The
intervals.
of
average
transforms
Walsh-Fourier
and
from
the
= 8
O2n
s
C+D ERereGeH
air
SA=B=C—Dth+ F+GtH
-A-B+CyD+E+P-G-8
= alGjee= aclu),
= acta) = a. tus
=
a. (Cl) =Backins
‘is
+AeB-—C-DtEti-G-H
=
ae(2)
ast),
4 <<
s e
eh B=OFDLE—E=G4
—A+B+C-D+E-F-G+H
= acter
= apis
2 Ss
=o
= a,(3)
= ag(u),
ei
a Ss
=
0 <a.=°%
a <
—A+B-C+D+E-F+G-H
= ac(3)
= acy),
4s4
<4
+A-B+C-D+E-F+G-H
= a,(4)
= a.(u),
oo
eee
There
tions
are
2°(23-
necessary
ag(u).
The
ditions
to
fast
only.
1) = 56 or
obtain
Note
that
generally
the
Walsh-Fourier
the
2"
Walsh-Fourier
which
the
case
the
transform!.
For
an
refer
to
A,
...,
B,
Column
1
samples,
and
1.
lists
again
neral
4
the
sums
with
and
are
of
and
a more
notation
Bre
shows
column
the
are
1
are
of
shown
each
added
Walsh-Fourier
or
in
of
case,
does
consuming
samples
notation
two
Sho
each
of
notation.
column
column
subtracted.
coefficients
2,
2.
which
in
transform
8 amplitude
general
differences
in
ad-
Walsh-Fourier
the
differences
column
and
2"n
transform
time
together
with amore general
sums
yields
fast
O lists
shows
previous
column
of
Column
H together
differences
column
the
explanation
Table
a,(u)
requires
multiplications,
Fourier
1) addi-
coefficients
require
fast
2"(2"-
transform
not
of
ee
Bo
of
help
the
ac(u)
obtained
be
may
H with
...,
B,
A,
values
for this nunm-
F(@)
of
fit
square
func-
byastep
represented
is thus
functions
step
these
of
a.(u)
s
are denoted
8 interval
in the
F(@)
mean
least
isa
that
tion
F(@)
H.
...,
by A, B,
per
values
average
The
of
be discussed.
2* = 8 will
case
special
the
illustration,
For
vals.
y
subinter
all
wide
equ
2"
into
be divided
interval
FOUNDATIONS
MATHEMATICAL
4.
46
:
the
Sums
while
The
ge-
terms
of
The
third
a(0),
a. Ga
'A fast Haar-Fourier transform may be derived forthe complete orthogonal
systemof Haar functions
[6]. This trans-—
form may be even better suited for digital computations
than the fast Walsh-Fourier transform
(personal communi-
cation
from
H.C. ANDREWS
USCLA).
1.25
FAST
WALSH-FOURIER
Table
1.
Fast
42
TRANSFORM
Walsh-Fourier
Par(soges0g)
17 ransform
0,0
tse
+A+B
HO)
0,0
+A+B+C+D
0,0
0,0
SE pet 1,0
=+A+B+C+D+E+F+G+
OO
0,0
SUS SnD)
)
0,0
So'3 ter 5 ee
0,0
0,0
CS
0 Al
So 3
—A-B+C+D
=-A-B-C-—D+E+F+G+H
onion 00
Mas Goh Se
1 iy OTA
0‘ =
la ok 0
aly Spe
—A+B+C-D
0,1
A-B+C+D+E+F-G-H
01
-( Sy) haere
+A-B+C-D
0 Se C854
0,0
0,0
a3;
=+4+A-B-—C+D+E-F-G+H
+H+F+G+D
yh).
1hOPee
Ose
8)
Ae
£
Oiier.
6.4
=+(5,')-8,',
)
4
33 Oe
3
0,
1,1
1
heyy eo
=—A+B-C+D+E-—F+G-H
—H+F+G-D
0
0
ita
3, ies!
-(8)/, +83',) So’ ACS Ghee
2cii.).
The
fast
Walsh-Fourier
formula
from
the
Walsh
functions
j/2]+
rela
2).
Tayi
Altea fai
that
ee
of
ee
[j/2]
=
largest
Ontors)
=
number
As
of
difference
integer
evenyix
0) In
PeeROn eee.
o"=
or
transform
a recurrence
cee
yal
heey)
=+A-—B+C—D+E-F+G-H
+E-F+G—-D
end
ee.
=-—A+B+C—D+E-F-G+H
—E-F+G+D
=
10
Same
=s1l-
for
consider
be represented
by
that
follows
(29):
j/ 2),x
sfi7)]
or
j.=
=P Onl san yy
amplitude
an-example
smaller
can
equation
(91)
equal
#j
odd
pe=
Oy or
14.9
=
0.2 .m;
samples
the
term
for
j=3,
p=1,
k=0,
Br
= (1) lac,
This
is
of
ol) Syd
right
corner
x may be producedina
binary
ber.
Division
number
The
number
to
right
j= 25 210111
The
left
the
the
binary
point
by one
of
the
binary
point
is
x.
Example:
45 211-59 =1011.1,
computation
-.2'-1.
is
starts
It follows
from
ay , k = 0....2"'-1,
terms
inthe
[3/2]
= [1/2]
quire terms
The
second
and
can
column
sj while
fast
the
2"
be
only terms
2”'
terms
inverse
computing
the
the
ane
with
recursion
g lil2h*
1, both
coefficients
formula
difference
7 £(-4) ih?)
j/2],x
7
Jue
gli/2)*
[j/2e]
are
the
terms
with
yielding
transform
A,
B,...,
This
for
the
of
Bo
;
git! )
and
2"-'
terms
zero
or 1,
[j/2]
is
H from
the
done
ere
One
-
=
0.
obtained
may be
he
re-
available.
may be
ite
coef-
by inobtains
the following
j/2
k,m
j,0
j,1
veg ek)
be
written
=
(4 piri?) Gone $ (4)
F glk,m )
1;
x
k+p,m-1
0 or
Further
x
values
k,m
may
These
terms
formulas:
k+l, m-1
Both
=_ Oa.
the
x = O are
,» since
Walsh-Fourier
recursion
fromthe
sum and
two
1.
aa
ais jac
= 0 that
the computationof
the
PiCienvesstO) se aULeaws coe as(4).
verting
terms
computed.
Table
the
[j/2],
=41410T1,)
= [0/2]
of
Bet permit
j may thus be O or
The
Fife
place.
= O cannot be computed,
since this would
terms
aye andthe
with
[j/2]
num-
bya binary
represented
j be
Let
2 shifts
by
the
to
and
[j/2]
quantities
er
as follows:
comput
p=
lower
terminthe
the
with
identical
Table.
The
by
Ae
= Wyeaee=
14/2)
=
(3/2)
with
Tbe rollows
n=3o°
FOUNDATIONS
MATHEMATICAL
4.
48
= largest
=
together
0 for
j =
integer
in
6veny
smaller
one
formula:
S0er7
or
(924
fore)
equal
4).
=
ooae
1.31
GENERALIZED
FREQUENCY
49
1.26 Generalized Laplace Transform
The
and
Laplace
its
transform
inverse
GS)
Vimo
may
be
X(o,v)
written
| ECO Jer2? ci *™".
of
as
a time
function
F(@)
follows:
ag
3)
0
F(@)
=
It
be
ef
is
apparent
considered
factor
Baplace
mark
to
that
be
Laplace
F(@)
functions
in
transform
and
(93)
transform
transform
quadratically
not
from
the
(94)
a Fourier
makes
e799
are
that
X(o,v)e2™"G9
may
F(6@)e-?9 . The
Fourier
transformable
integrable.
generalized
The
follows
notation
real
of F(6)
of
from
re-
this
(94):
ag(o,v) = f F(e)e?* £6(v,9)a9
(95)
0
Bevo,Vv) =
f Fede
# oCy,0
G0
0
co
HCA}
=m
“eo?? J pe
Co4 Vs AN)
a,(o,v)f.(v,9)]dv(96)
—co
The
do
integrals
the
since
integrals
the
vanish
The
(95)
of
factor
not
the
e-%?
sufficiently
usual
do
the
generalized
might
fast
assumption
have
make
for
F(@)
=
lower
limit
Fourier
-co
as
transform,
them divergent.
F(9) must
large
O for
negative
6 < O is
values
used
of
@.
here.
1.3 Generalized Frequency
1.31 Physical Interpretation of the Generalized Frequency
is
Frequency
vidual
Its
per
a parameter
physical
unit
of
Gerpreted
time".
as
interpretation
is
normalized
The
“number
of
cycles
in
"number
indi-
cycles
of
v=fT
is
in-
interval
of
du-
frequency
a time
the
or { sin 2nft}.
{-cos 2nft}
of the systems
functions
usual
distinguishes
that
igeysiiewaieg hike
The
4
generalized
Harmuth, Transmission
frequency
of Information
may
be
interpreted
as
"ave-
divided
u is
interpreted
by 2". The generalized
time interval of duration 1 divided
frequency has the dimension [s"]:
The
(97)
/ Tae Lo
een
ie
eH
definitionof
the
generalized
sen so that it coincides
to
sine
lation
200
and
the
of
per
as
sine
to
spaced
It
is
useful
need
frequency
oscillations;
sal
waves
cy
as
in
three
well
as
is
a
for
use
one
Consider
Fig.2.
may
the
i equals
the
interval
the
periodically
new
One
reason
is
Walsh
one
-§ = 6 < #4 and
continued
number
zero
cros-
spaced
but
it appli-
not
equally
used
'sequency'
is
in
that
for
the
term
connection
with
are
transvera frequen-
The
there
measure
per
i/f
sign
is
will
i/€
and
of
functions.
are stretched by a factor § they
gesinthe interval -#@ = 6 < 42;
sequency
is
'zps'.
cal(i,@)
number
» =
of
second divided by 2",
abbreviation
the
are
term
that
functions
half
The
or
zero
space which have
crossings
the
of
periodic.
the
already
oscil-
same
frequency makes
be
sequency.
zero
the
are equally
even
dimensional
"average number of
which
whichis
crossings
another
sine
cycles per second
zero
m.
cho-
if applied
a
frequency'.
introduce
frequency
frequency,
100
100,
the
not
been
One half the number
functions
whose
to
generalized
damped
has
of the generalized
which
generalized
Hz
of
of
has
For instance,
second.
cosine
frequency
that
equals
that
and
functions
and
100
per
second
definition
cable
the
frequency
dimension
Sings
with
functions.
crossings
crossings
and
cosine
with
zero
per
crossings
number of zero
"average
as
frequency
generalized
The normalized,
by 2".
time
of
unit
per
changes
sign
of
number
"average
as
or
by 2"
divided
time
of
unit
per
crossings
zero
of
number
rage
FOUNDATIONS
MATHEMATICAL
1.
50
=
the
If
the
have
2i
wy will
sal(i,6)
in
changes
in
sequency
of
functions
sign
chan-
be one half
'The number of sign changes per unit of time has been used
to define
an instantaneous frequency of frequency modulated sinusoidal-oscillations
[7,2.5).
1.42
POWER
the
SPECTRUM,
average!
number
of
Sign
i
changes
in
an
interval
of du-
Ae
WEG
Consider
nued
as
Legendre
They
have
interval
number
by
a factor
€ and
i/é
normalized
by
the
=
periodically
& makes
the
uu becomes
conti—
duration
one
half
of
interval
time
per
changes
sign
the
the
Pp. Cie rand P.(@1,%.) of Fig. 17an
changes in the interval -k = 9 < +4.
sign
them
replaced
example
polynomials
equal
of
Let
a further
2i
Stretching
iG =
FILTERING
variables
y
non-normalized
and
the
this
average
duration
8 in
variables
of
sin2mvé
f
=
v/T
1.
be
and
BAUS
sin 2nvé
The
=
time
tain
base
the
angle.
sin on (£1 )e =
T drops
three
This
functions,
is
sin 2nft
out.
(98)
Sine
and cosine
parameters
amplitude,
not
complete
which
so
do
for
not
have
functions
frequency
systems
sequency
and
of
con-
and
phase
orthogonal
time
base
con-
nected
by multiplication. Walsh functions sal(y,98) or Legendre polynomials P,(u,8@) have a comma between yw and 0.
Hence,
the
sal(u,6)
These
=
substitutions
sacbCpl! .b/'h)-,
functions
rameters
»
Vsal (p?, = S22),
p/T
and
Ps (u,9)
containin
amplitude
=
their
=
t =
0T
yield:
PP, (ol,t/T)
general
(99)
formthe
V, sequency
a, delay t,, and
four
pa-
time baseT:
VPs (pT ,t-tHD).
1.32 Power Spectrum, Amplitude Spectrum, Filtering of Signals
One
from
may
the
interpret
may
from
derive
Fourier
it
as
the
frequency
transforms
frequency
the sequency
interpret
the
generalized
function
power
+
Gy
and
In analogy,
spectrum.
one
function
Fourier
aé(v)
(51)
a,(v)
and
as(v)
aé(y)
transforms
of
+ ag(u)
He Cy)
end
derived
ba. (uy
I'The sequency of a periodic function equals one half the
a nonncy
number of sign changes per period. The sequeof
chansign
of
number
the
s
half
one
equal
function
periodic
ges per unit of time, if this limit exists.
4*
FOUNDATIONS
41. MATHEMATICAL
De
notation
the
using
integrated
and
coefficients:
the
for
(59)
of
squared
be
(58)
Let
spectrum.
power
a sequency
as
(63)
(62)-and
of
-/0 9)
Oy
a F*(e)d8 = —>dioee te a(+)f(S,
e -Ff {a@r,
Ferea
+
ai
i
See
"i ee Pt,
,
oe
+ ag(h)te(F,e)1} a8
The
integrals
of the cross-products
of different
vanish
sreliseotet
ya
functions
due to the orthogonality
of the functions.
(075,64),
yuela
£2¢2 6,0
The
inte-
rand 2 oie, 82 multiplied
by
1:
( F2(ayae = ytpazQ)5 + j/kSt
ca2cd)
+ 02(4)I]
=1/
eS
apa
—oo
The
be
sum
has
the
interpreted
sum
may
be
same
as
form
the
replaced
as
area
by
an
that
of
under
a
(58).
step
integral
for
Hence,
function
large
it
may
and
the
values
of
§&
and y = y(&):
a[ F?(o)a0 = f Caz(u) + a2(y)lau=#f[az(u) + a2 (u)]au (100)
—-co
Using
non-normalized
co
notation
one
obtains:
[o.e)
J F’(t/T)at = Tf Car(pP) + a2(@T)]d(@T)
TLaté(u)
+ az(u)]du
ac(ulfc(u,9)
to
+
as
as
atu)
the
has
sequency
if
energy
the
the
energy
of
ac(utdufc(u+du,e)
as(u+du)fs(u+du,9),
preted
is
the
of
dimension
power
integral
the
of
spectrum
signal
power
or
(101)
and
of
the
acg(u)fs(u,8)
F*(t/T)
F(8).
and
sequency
components
may
power
is
Hence,
to
inter-
ena ery) +
be interpreted
density
spec-—
Tye UIT
Using
G(v)
the
= ACv)
function
+ BCv)
one may rewrite
into
the
G(v),
= $V2[ac(v)
the frequency
followings
form:
+ Bel VIN
power
spectrum
a2(y)
+ az (Cv)
1.32
POWER
ey)
SPECTRUM,
+ poy
Use
has
power
e=et
been
FILTERING
hnCy)
made
spectrum
eB Cy)
of
may
(16),
be
DD
= Cy)
(19)
rewritten
and
als
eo G2
(52).
The
square
not
root
power
sequency
cosine
sequency
+ G2(-y)
(103)
Lac (vy) & EelGus may be interpreted
amplitude
spectrum.
possible forthe
and
The
follows:
ai(u) + ag(u) = 4[A*(u) + B2Qu)] = Gu)
frequency
C102)
square
spectrum,
functions
Such
root
interpretation
[a2(u)
a
since
is
an
for
is
+ eeu) 2a toratie
of sine
feature
specific
required
as
it!.
Using
the
re-
lation
Mein«
one
+ Bcosx
may
=
rewrite
(A°
=i A
+ B*)'*cos (x - tg
(52)
as
=)
(104)
follows:
oo
Mies
V2)
lacy)
+ 22(v)i!cos [emve
Pac(v)
+ as(v)]"?
= te = = Jdv (105)
0
Die
stactor
quency
tude
to
amplitude
of
the
that
the
phase
do
permit
spectrum,
oscillation
angle
Um
have
an
not
this
Of
a,(v)
the
the
function
Filters
put
signal
scribed
connexion
@
ee
Vary).
ac(u)
generally,
into
an
operators.
linear
with
denote
and
spectra
the
ampliregard
of
like
the
square
as(u)
are
fre-
without
Systems
theorem
of
vy
as
functions
(104)
root
just
do
not
[az(u)
like
+
a,(v)
of the
even
and
odd part
systems
that
change an in-
F(é).
more
F(@)
by
describing
Let
or,
interpreted
frequency
addition
amplitude
be
since it represents
with
interpretation
fs aero sie - However,
and
may
The
systems
complete
an
output
operator
signal
concept
is
of
systems
and
of
F,(@)
linear
particular
of
be
de-
operators
importance
orthogonal
{f(j,9)}
may
in
functions.
a complete
system
cosine are required
and
sine
of
1'The addition theorems
in
theorem
chin
Wiener-Chint
the
of
derivation
the
for
Hence, other systems of functions have no
real notation.
Walsh
theorem.
Wiener-Chintchin
direct analogue to the
the dyadic
on
functions have an abstract analogue based
correlation
function
[F(8 )G(eer de.
1.
54
f(j,6)
signal
input
or
function
to a particular
of
Application
functions.
orthogonal
of
FOUNDATIONS
MATHEMATICAL
sig-
an output
generates
nal g(j,6):
the
system
functions
all
for
hold
= a(jar(,¢)
Q Sa(402G5,8)
=
j=0
proportionality
S* na(g)2(5,9)
law
superposition
(107)
law
(108)
j =0
Q may
be afunction
operator
and
variable;
the
of
of
j and
system
otherwise
example
plitude
it
they
a linear,
modulator.
Let
h(k,6)
may be,
e.g.,
carrier
wal(k,9).
carrier
yields:
6.
If
describes
are
linear
Q depends
are
and
time-variable
an
input
ted
by the sum Da(j)f(j,9)
asine
carrier
and
is
F(8@) be
andthe carrier
Amplitude
linear
¢,
2sin
modulation
j=0
(5,9)
It
=
bc,
is best
h(k,9)
is
the
time
time-invariant.
system
signal
on
the
am-
represen-
by h(k,@)
2nké@
with
or
=
Q.
a Walsh
suppressed
F(a )h(k,8) = aF(8) = a S'a(gj)£(5,8)
= S'a(gde(3,8)
if
of
(t£(J.0
7):
Qa(g)f(j,8)
An
law
superposition
the
and
law
proportionality
the
linearif
called
9 is
operator
The
(106)
cy
ACH
=
yo.
(109)
j=0
ef Cn.6)
to use
a
Walsh
Walsh
functions
carrier
wal(j,@)
wal(k,6).
One
for
f(j,8)
obtains
for
gCg,8):
e(j,8)
If
= wal(k,@)wal(j,0)
h(k,§8)
functions
tem
isasine
= wal(k@j,0)
carrier
Ve sin 2nk§
f(0,@),
{f(j,8)}.
one
should
use
V2sin2mie and V2cos 2mi8 for the
The functions g(j,8@) are then
g(0,8) = V2 sin Snke
g(2i,0) = cos 2n(k-i)@ g(2i-1,9) = sin 2n(k-i)®
+
fezeo, 21, Stade eee
9
cos 2n(k+i)9
sin 2n(k+i)e
eee
the
sys-
1.32
POWER
The
SPECTRUM,
definition
lopment
of
differential
to
time
invariable
The
by
If
a
(106)
OEC3
to
invariable
for
described
the
of a filter.
=
The steady.
nal
F(8)
Curpus
el Geb)
ey
sponse
are
output.
have
signal
It
to
has
time;
its
is
often
credited
be
been
widespread
to
a
a linear
of
operator
form
9 and
functions
in this
of
and
the
an
hand
side
{f(3,6 )},
9.
Equation
case:
eigenfunction
of
function
are
(110)
and
phase
state
of
has
9 even
to. be re-
f[j,6@-6(j)].
described
shirt.
Vcosenft
a,(f)
is
the
by
voltage
electrical
the
This
applied
frequency
description
to
the
input
V,(f)cos[enft+a,(f)]
=
and
a-(v) are called frequency re-
phase
shift.
Fourier transforms
F (9)
if
The frequency functions -2logV.(f
)/V
attenuation
follows
from
Let
a,(v)
an
and
input
sig-
a,(v).
The
(52):
ye [ {a,(v)Kg(v)coslanve 4. a, (v))+
a ale)
0
+ ag(v)Ke(v)sin[2nve + a,(v)]}dv
w= VoCV)/V3-
The
for
of
ope-
Q
long
f(j,98)
of filters
—2logV,.(v)/V
sponse
require
eigenfunctions
shifted
a voltage
appears at the
then
differential
not
theory of communication
of attenuation
that
restricted
C1110)
right
time
characteristics
assumes
deve-
Ch, 6)
frequency
response
was
operator.
system.
following
baie
by the
Inthe
a
by
the
It is convenient to-call
placed
does
communications
choose
Se DCL
ou
it
during
coefficients,
necessarily
definition
the systemof
assumes
Poi
changed
First
[1].
choose
8)
not
time
into
systemis
may
has
theory.
mathematicians
omeviesiree
one
or
by WUNSCH
book
linearity
but
present
introduction
DD
operators with constant
a differential
used
of
communication
to
rators.
FILTERING
description
of
8 = £0,
of
attenuation
telephony
usually
filters.
described
9 = t/T.
filters
and
phase
Matched
by
means
means
by
shift
of
frequency
is eminently
filters,
the
of
re-
suited
on the other hand,
pulse
response.
A
is
voltage
output
the
of
shape
the
and
input
the
to
applied
§(8)
function
Dirac
the
of
shape
of the
pulse
voltage
FOUNDATIONS
MATHEMATICAL
4.
56
reference to sine and cosine funcD(@)
tions is required. Which system of functions is used for
description of afilteris strictly a matter of convenience.
No
is determined.
Let
the
the
input
voltages
of
Vfc(y,9)
a filter
f.(u,8)
and
fs(u,9)
ralized
Fourier
and
instead
are
the
transform
of
same
(63).
attenuation.
term
tions
other
than
between
input
sisting
of
are
sine
filters
Walsh
coils
and
that
resistors
sine
and
and
tions.
Such
tions
than
Let
forms
a
cosine
but
a,(u)
are
sine-cosine
F(8)
and
the
Let
and delaybe -2 log Vo(u)/V,
The
output
By Ce)
=
signal
fla Gu
Ku
and
one
attenuate
sine
and
described
con-
fo4a.e0
may
design
storages,
and
delay
cosine
func-
Walsh
func-
by
functions.
have
a.(u).
relations
integrators,
will
func-
filters
F.Gi,6))
distort
delay,
to
simple
for
However,
better
called
applied
These
If
which
will
are
be
exist
functions.
filters
by
cosine.
voltage
multipliers,
switches,
signal
@5(u)
cannot
capaciters
contain
functions,
[u,8-85(u)]
and
and
output
and
gene-
be
shift'
steady
in the
-2log Vc(u)/V and -2 log V.(u)/V
@,.(u)
'phase
The
The functions
at
Let
the
occur
occur
the
Since
onft.
to
voltages
V5(u)fs
called
Vcos
that
applied
shall
and
output.
be
state
Ve(u)f.[u,8-8c(u)]
filter
Vfs5(u,8)
follows
generalized
the
steady
state
-2log Vs(u)/V
from
)fCu,8-86(u
Fourier
and
trans-
attenuation
@c(u),
& (ur
0-06(u)]
dau
(64):
]+a(uKs(u
flu,
0
KeCu) = VeCu)/V,
Comparison
Gov)
would
sine
put
of
Kg(u)
(111)
= V5 Cu) /V.
and
(112)
occurs in (711), Gut Met
occurif
and
shows
kec,)
that
end
frequency filters would
cosine
voltage
RAVES
Vsin
functions
onft
of
would
tage V.(f) sin [2nft+a,(f)]
the
then
same
only
a g(v).
K,(v)
Such
distinguish
frequency.
produce
rather than VC
the
and
terms
between
The
output
invol-
oa [enftt+a(f)].
1.343
EXAMPLES
Such
a distinction
OF POWER
time-variable
frequency
Filters
the
between
circuit
filters
based
sine
given
later
sine
sine
and
are
and
and
a7
sine
element
which
on
periodic
tween
SPECTRA
and
An
cosine
can
linear
cosine
cosine
cosine.
and
requires
thus
and
not
some
occur
in
time—invariant.
pulses
rather
functions
than
distinguish
exaof
sucha filter
mpl
e
on
be-
will
be
on.
1.33 Examples of Walsh Fourier Transforms and Power Spectra
Fig.21
shows
transforms
epeceraas
time
G(u),
(a)
functions
a,(u),
F(@),
as(u)
their
and their
Walsh-Fourier
sequency
power
enae(u):
G(u) = 472 f F(e)Lcal(u,8) + sal(u,eiae
ac(u) = aV2[GCu)
pacer
act
+ Gl-u)],
as(u) = aVetGCu) - GC-y)]
= G (a) + G°C=1)
F(8)
{waOe)
(113)
: G(u)
Loa
ac (j1)
a
a5 (Ww)
ad
(u)+a2
(1)
accel prsRit a
2
2 2waigze)—L_,_L_ ee
ata ee
eee
fae
—{ft—
—1 _
6 cal(,8) -—f>+ Lp en — 1A
———__
—n.n—
7ol(26) FPA
—_____
_n__
4 sal(18) -—f—+- —{fH—_-
a
sh Serer
V2
Fig.21
a1
4
V2 -4
—
0
i_it—}
4 -4
i
0
4
4-4
F(@), their
Some time functions
forms
ag(u)
eee)!
may
by apower'
as(u)
ac(u),
G(u),
One
—_-__-.
yp
nt
0
|
on
8 sal (3,8) oF-HEL+
ak
a
eee fe
=
see
of
CC)
that
2 in
andtheir
4
+—_1
do
tt
4 4
0
4
-4
Walsh-Fourier
sequency
power
0
4
trans-
spectra
eG (a).
compression
the
of
time-domain
the
ames,
produces
lolleyele
jouilksie
a proportional
is
yu <@.
-©<
interval
whole
the
in
value
stant
a con-
has
G(y)
transform
Its
limit.
in the
obtained
6(8)
function
delta
The
G(u).
of the transform
stretching
FOUNDATIONS
MATHEMATICAL
1.
58
G(y) of the
further see, that the transform
‘sequency-limited'.
pulsesinlines 1,4,5,-.,8 are
One may
Walsh
This
is
that
analysis,
a time-limited
quency-limited
in
shown
to
Fourier
Fig.6
Fig.1
for
known
sine
the
The
cannot
Walsh-Fourier
time-bandwidth
infinite
transform
sequency
idth
inthe
case of Walsh-Fourier
bandw
A class
limited
of
may
cal(i,®@)
time
be
and
functions
inferred
sal(i,@)
-—3 = 90 = 4%. Their
that
from
vanish
of
Walsh
PCOS) =o
=
Let
have
F(6)
the
=
course
to
transform.
and
sequency-
Walsh
time
pulses
interval
transforms
vanish outside
and
sequency-limited:
sal(i,@)
=
vant spe
O for
|@|
transform
aee oe (114)
>
G(u).
It
holds:
(115)
> cle
The
orthogonality
generalized
catly,
The
the
of
the
= yp s +(i+1) o0r-i = yp = +i.
consisting of a finite nun-
Walsh-Fourier
the
Walsh-Fourier
avoids
of
|e| > 4
= Ooms!)
iH
time
time
outside
Sie G Jeet G6)
izl
cal(i,@)
F(@) = O for
to
is
0 wal (OCG
wal(0,9)
GCs
pulses
are
Fig.21.
Walsh-Fourier
the sequency intervals -(i+1)
Hence,
any time function F(@)
ber
refers
bandwidth
according
products
Fourier
fre-
transforms
pulses
ordinary
analysis;
Fourier
havea
Fourier
cosine
and
of
result
function
transform.
go onto infinity.
troublesome
the
well
the
to
contrast
in
of
a
systemof
Fourier
transform.
functions
is invariant
transform
and that includes
the
Hence,
one may write
G(y)
expli-
af
the coefficients alO), a,(4)
and
a.(i) Gr the
(114) are known.
Let g(0,u), eli.) and 2.4 sa
denote the Walsh-Fourier transforms of wal(0,@), cal(i,é@)
and sal(i,®@). One obtains the transform Gu) of F(A):
expansion
{
GCu) = aCO)g(O,u) + Difa, (idecli,u) + as(ides(igu)]
i=0
(176)
1.33
ecm
EXAMPLES
OF
POWER
Uncuionere(O,u)5
shown
may
in
Fig.21,
readily
larger
59
2201s).
second
infer
values
SPECTRA
the
6c Cin), sees B.S,
column,
shape
lines
of
Vey
g,(i,y)
and
eo
WZ
wal (0,0)
|wal(0,8)
Wz
12
(2 sin 2n6
¥2 sin 270
WwW?
Wy2
z
cos 2n0
W2 cos 210
1NZ
uz
\2 sin 416
V2 sin 410
W2
v
V2 cos 416
ee
-15
eS
-10
a5
yipetor
i.
of
WZ
SNe
eare
sey Oe Oe
\2 cos 410
EEE
0
5
10
15
-F
-10
3
0
5
10
15
Fig.22 (left) Walsh-Fourier transforms
G(y)
of the sine
and -cosine pulses
derived
from
the
elements
of Fig.‘1.
Fig.23 (right) coefficients
of the expansion of the periodically continued sine and cosine elements of Fig.1 in a
series
of periodic Walsh functions cal(i,®) and sal(i,é@).
Fig.22
cosine
One
pulses
may
formed
the
that
readily
functions
and
cosiné
functions.
The
spectra.
Fourier
The
Walsh-Fourier
vanish
see
is
ag(i)
functions
band
of
the
in
transforms
the
Fig.2%
the
of
shows
of the expansion
a
series
spectraofFig.22
to
interval
orthogonality
preserved.
analogy
series
outside
how
and
a;(i)
a(O),
cients
Sine
shows
Fourier
of
are
transform
corresponding
of
sine
-%
\A
=
the
the
and
9 < %.
transcoeffi-
of periodic
periodic
Walsh
by line
replaced
of
periodic
a pulse
function
and
is
evident.
power
s
the frequency
show
ee G-(-\) storie first five
Fig.24
Poca
spectra ae(v) + a6(v)
and cosine pulses
sine
of
TRANSMISSION
2.DIRECT
60
continuing
obtained
by
interval
v < 0.
Walsh-Fourier
the
Civicne
continuation
This
spectra
power
for
even
as
them
transform
are
into
the
is of muchless
G(y),
interest
G(v)
transform
since
they
the
0 <©o
functions
Fourier
the
for
than
-co<
interval
whole
in the
curves
The
signals.
under
of
energy
the
represents
T
by
multiplied
curves
the
area
The
Fig.45.
of
pulses
block
the
and
Fig.9
are
or
always
ia CivOiaiss
2. Direct Transmission of Signals
2.1 Orthogonal Division as Generalization of Time and
Frequency Division
2.11 Representation of Signals
Consider
ber
of
having
sets
of
atelegraphy
characters.
32
characters.
5 coefficients
character
1:
character
2:
In
An
general,
alphabet
example
It
is
with
value
+17
+17
+1
+1
+1
+1
+1
+17
+1
+1
-1
may
afinite
the
teletype
to
represent
usual
+1
the characters
containing
is
+1
or
nun-
alphabet
them
by
-1;
etc.
consist
of
sets
of
m
coef-
2:
6?)
+62(-)
—=
4
766
V—_
=
33d
7
ot
40fF[Hz)-~
468
Pig.24 Frequency power spectra a2(v)+a2(v) = G2(v)+G2(-yv)
of the sine and cosine pulses of Fig.9.
a) £(0,8); b)
£(1,6),£02,9)3 co) $C,0),PC4
6) Comverd te the frequency
power spectrum of the block pulses of Fig.4 if they have
five times the energy of the block
pulse
of Fig.9. The
frequency scale in Hertz holds for T = 150 ms.
2.11
REPRESENTATION
ficients
having
fues*+1
or
SIGNALS
arbitrary
61
values
rather
than
just the va-
-1.
The
following
omer
O),
Ay(1),---ay(§j),---a,(m-1)
notation
is
appropriate
character
The
representation
another
tions
important
f(j,9).
of
characters
representation.
Let
by time
m
time
be multiplied
functions
and
the
products
of
the
character
representation
C7)
functions
Consider
the
a,(j)
efficients
added.
be
func-
co-
by the
obtains
One
func-
time
by the
y
m-)
Pr(0)=>, a, (i)£(5,8)
The
from
coefficients
functions
The
are
malizedinthe
obtained
grating
¥Y(6)
a,(j)
may
orthogonal.
product;
f(k,0)'
2
be
regained
et
em
eequal
eye)
oe tlie
Fig.3.
Fy(@)
teletype
The
=
F,(6)
at
sampling
time
The
frequency
j=0
1/2
5; Let
functions
as
proper
a@y(0),
F,(8)
£0),0.°are
voltage
of
times.
also
by
time
of
Fig.4
f(j,v).
= ay(k)
equal
sblock
of
+1
shown
pulses
the
(3)
and
in
of
usual
may
transmitted by the
obtained
by amplitude
terms
transmission
the
division
or
process:
shape
current
a,(j)
be
Hence,
pulses
functions
'correlating
time.
coefficients
may
and
this
the
the
or
nor-
inte-
a,(3)
then
and
the
f(k,@)
for
ay(2),
if
a,(k)
expression
used
has
linearly
coefficient
J £(5,8)fCk,8)d8
function
of Fig.3
multiplex
block
V2
is
simple
orthogonal
The
with
shorter
Siay(j)
represents
signals
F,y(9)
m-1
vequali—1.
be
-$ = 6 < 4.
the
values of the
signal
Let them
individually
{f(j,9)}
particularly
is generally
-1/2
aeiwy
is
by multiplying
the
with
process
interval
J Fy(e)£(k,0)d0
by
(2)
F,(6), if the system of functions
independent.
Mir
is
Pee):
TLOM
is
in
case:
this
the
OF
also
are
be
used.
interpreted
as
The character
y is then repre-
If
FY(v) is applied
sented by the frequency function
Fy(v).
to
usual
terms
for
Recovery
in time
is
not
or
by
means
of
frequency
of
Theoretical
racters
these
in
is
always
this
that
is
more
freedom
represent
How
m-dimensional,
is
of
this
by
chavector
orthogo-
rectangular
vectors
of
of
time
application.
representation
equals
the integral
for
or
The
number
than
frequently
space.
case.
the
larger
pos-
division
a particular
having the unit
functions
£7(3,0)ae
the
is
orthogonal
there
for
signal
to
Consider
vectors
orthonormal
1/2
in a
related
coordinates
computation
is much
Hence,
sampling
functions’.
division
system
by vectors
functions?
tesian
of
best
by
further
appropriate
functions
are
of orthogonal
terms
investigations
representation
nal
are
division.
the
without
orthogonality
The
orthogonal
systems
for
their
of these
division
coefficients
systems
(4).
multiplex
advantage
choice
domain
of
to
voltages
transmission.
of
transmitted
for most
according
useful
type
frequency
orthogonal
or
the
possible
Recovery
sible
of
this
recover
frequency
or
multiplex
Frequency
filters.
output
the
sampling
by
coefficients
the
one may
filters,
bandpass
frequency
5 suitable
TRANSMISSION
DIRECT
2.
62
e;.
the
The
square
car-
length
of
the
f(j,86):
=e,6,
i
(4)
ae
-1/2
The
scalar
vanishes
connection
sentation
since
product
they
between
may
thus
of
are
two
vectors
e;
and
perpendicular
to
each
orthogonal
be
function
expressed
by
the
and
e,,
j # xk,
other.
vector
The
repre-
orthogonality
re-
lation:
1/2
J £C3,0)f(c,8)ae@
=e.0,
= 8,
(5)
-1/2
A character
y is represented by the vector
F, in
signal
'More than one amplitude
sample is then needed to compute
the coefficients.
Such a processis, however,
a method to
compute the integral (4) and thisis not what is generally
understood as time or frequency division.
2.11
REPRESENTATION
OF
SIGNALS
69
space:
m-1
=
2 ay (J) ej
Instead
early
(6)
of morthogonal
independent
tained,
if
linearly
As
the
one
This
functions
may also use
representation
f(j,9)
are
not
m
lin-
is
ob-
orthogonal
but
independent.
a practical
F,(8)
vectors
vectors.
composed
example
of 5 sine
consider
and
a
cosine
teletype
elements
character
OC
Oe Cua
oaT@
calle aaed OF
myo)
=
a,(0)r(0,8)
+
a, (1/2 sin 400+
+ a,y(3)Ve2 sin 676
—eeeei<oe,
T-equals
is
150
9
=
150
ms,
a,(j) are
an on-off
of
if
val
= 6 <
at
the
output
=tes
amuch-used
to
#.
The
the
time
a
output
standard.
of
voltages
dueto
traces
a negative
the
of
shows
32
(-1).
This
system,
a&
+1
signal
and
those
2sin2n8
of
symmetry
ending
added
of
at
to
-1
the
tion. The elements V2sin 2n@
imi? » fomethis ‘reason.
F,(6)
with
voltages
five
inter-
integra-
a,(j)
the
16
between
is
(+1)
indicates
a negative
instead
O for
of
of
the
interval
each
of the
characters
of the teletype
system.
In an on-off
lack
the
reach apositive
value
value
or
receiver
output
tracesvior
value
apparent
+1
the
oscillograms
sume
The
and
at
during
teletype
O
coefficients
coefficients
52 different
the
character
during
the time
5 integrators
5 output
16
of
Fig.25
the
are
the
The
The
voltages
values
s.1 [here
16
teletype
to f2cos6n6.
6 = 4.
voltages
alphabets.
(7)
5 multipliers
which multiply
f(0,6@)
00<a
and
of
+1 or -1 forabalanced system,
system. Let F,(@) be applied
represent
(7)
duration
5 multipliers
are integrated
-§
tors
the
whichis
5 functions
the
+ ay(4)f2 cos 4n8
+
t/'D.
ms
simultaneously
the
ay(2)V2cos4n8
caused
traces
value
the
by
for
=
4%
a balanced
would
at
traces
an
9
9
as= #.
ending
additional
for synchronizacharacters
and
V2cos ené
do not
appear
2.
64
Fig.25
ents
DIRECT
Detection
+1 and -1
by
of 42 different
composed of sine
Duration of the
All
t he
three
discussed
coefficients
Ss entation
of
functions
f(j,8)
F y(9). Some
p 1one,
m
vectors
coefficients
the
signals,
are usually
presentation
The
by
by
e;
one
representation
as
coefficients
time
will
signals
pulses.
450 eis.
of signals
contain
vector
one
the
Fy,
time
voltage
of
functions.
be
coeffici-
teletype
«
ine
permit
by
such as the output
available
the
cross-—correlation
and
representations
a,(j).
of
TRANSMISSION
discussed
1
the
time
function
a microTheir
in
re-
2.134.
2.12 Examples of Signals
Fig.c6a
Signal
shows
space.
functions
for
two
characters
The
same
the
block
characters
pulse
Ore
= 4ECO58),
FYCe)
are
f(0,9)
Seul(C4) sis) ye
Bee)
F,=e
= =2.0,89
and
shown
or
the
F,=
below
Walsh
-e,
in
as time
pulse
2.12
EXAMPLES
OF
rae)
SIGNALS
Cm
65
wal (0,6)
£0.87 eal
-1/2 6 24/2
Fig.2e6
Characters
dimensional
signal
Figs.26d, e andf
tors
the
Fo =
e,
and
e,,
characters
@,
+ e,,
or
(4.8)
6 V2
represented by points in one and
spaces and by time functions.
two-
show
characters
vec-
from
two
of
Fig.26d
F,=
@,-
have
0,,
constructed
functions.
the
F,=
following
-e,
from
Writtenindetail,
+0,
,
form:
F,=
-e,- e,
or
Cop asni Or
+e.o yey G0)e= 100,68)
BRCG)
= =000, Oo ret 10) oF, (e er-f00,8)
or
ie Cope a= wallGO
6) e=ssalC1
F,(@)
5
=-wal(0,0)
-
=1f(15.0)
= £(1,0)
8);
Fy Ce) = wal(0,6)
+ sal(1,6)
sal(1,@),
F3(8@) =-wal(0,6@)
+ sal(1,86)
Harmuth, Transmission of Information
functions
these
of
posed
the
Fig.26d;
below
shown
£(1,6),
£(0,9),
functions
Te
uc)
to
com-
F,(8)
them.
above
shown
are
F,(@)
are
sal(1,0)
and
wal(0,@)
characters
TRANSMISSION
DIRECT
2.
66
Be es
te} P-saca
-i72
6 V2-1/20 W2
Fig.27 Characters represented by points in a two-dimensional signal space and by time functions.
Nn
The
plied
terms
to
vectors
assume
like
a
binary,
the
or
2,
characters
functions
3 or
"binary
character
Fig.27a
sentation
are
character"
the
of
and
quarternary
Fig.26,
since
multiplied
by
values.
are
more
not
one
be
shows
terms
if
function.
characters
of
socalled
trans-—
The
characters
read invector
repre-
follows:
a
that
that
applicable,
vectoror
ap-
individual
coefficients
Fig.27
generally
than
the
may
three
alphabet.
as
of
4 different
consists
shows
orthogonal
ternary
2.12
EXAMPLES
OF
SIGNALS
67
Fo= #¥3e, + #e,, F, = ORT
Cty a SO
a
e, is multiplied by one of the three coefficients V3, 0
or -gV3, e, by one of the two coefficients -mor t=).
rit
the vectors
e, and e, are rotated relative to the signal
points,
representa
are
obtained
tions
ferent
coefficients
efficients
for
the
functions
are
shown
for
e, and
f(0,@)
Rpegoe= eV 570056)
shows
alphabet:
Fo,=e,,
Ee);
than
The
tain
two
Signal
gonal
The
Paco
look
very
if
or
All
CHig
vectors
similar
the
27a)
equals
UF,
a
If
than
a
Il
alee TEC
F, y
i
(v3
the
by
shown
5*
characters
one
vectors
integrals
e,)
of
by: (5).
the
of
a
socalled
bior-
to those of Fig.26d.
The
characters
are
composed
of
functions.
show
have
cars—.h,
distances
points
the same
signal
points
and
of
between
the
transortho-—
distance
0 to
cer-
1,
F,-F, . The
from
1 to
each
2 and
square
of
3:
(F,
- F))’ = (-8¥3e,
- $e,)
zi
of
-sal(1,0)
F,= -e,-
signal
from
Fr,— Fos
length
composed
and
F,(@) = +sal(1,0)
characters
linesinFig.27
points.
eres
their
four
vectors
alphabet
other.
co-
- #sal(1,6)
disappears,
dashed
dif-
model:
F,= -e,;
characters
more
Signals
e,.
orwal(0,@)
- $sal(1,8),
the
thogonal
similarity
for
three
different
+ $£(1,80)
= -#/3wal(0,8)
These
have
ortwo
+ ££(1,9), oF, ce) =ieC4, ey,
F,(0) = #¥3wal(0,0)
Fig.27b
that
vector,
£(1,@)
vector
F,(@) = -BV3f(0,6)
F,(8)
three
and
the
below
each
=-f+h-=3
ar eae
I
I
2
are
must
products
It follows:
represented
replace
of
the
by functions
scalar
products
respective
rather
by
functions
the
as
TRANSMISSION
DIRECT
2.
68
fer, (e) - #,(e)}?ae = ft-#V3e(0,0) - 32(1,8)7 ae
fte,(e) - F,\Ce)}ae = ‘ft av3e(o,e) - 32(1,0)?ae
(tp,(e) -F,(e)]?ae = f [¥ar(07,0)7* de "=93
2
|
\N
-1/2
-1/2
I
WN
-1/2
1/2
-1/2
-/2
to
obtain
must be added to the
the character F, (0).
1/2
i [P,(9)
- F,(8)]’ 48
is the
energy
required
to transform
F,(@)
character
Ea
character
V2
[FY (6 de
=1/2
of the
sents
order
in
F,(@)
character
into
that
function
is the
- F,(@)
F,(@)
}
:
if
F,(@),
is the energy
of the character
distance
the
of
energy
a
of
signal
that
point
F,(6).
from
7
integral
the
the
The
square
Mar
origin
repre-
character.
Fig.28 Characters represented by points in a threedimensional
signal space.
Fig.28
spheres
shows
represent
vectors
ferent
tances
c are
depend
characters
represent
the
@,),
from
distances
@,
and
e,
composed
of three
signal
points.
between
are
between
the
adjacent
chosen
points
in
The values
of the coefficients
orientation
stance
the ,
four characters
of
of
vectors.
The
between
them
rods
points.
Normalization
is
signal
The
adjacent
shown.
Figs.26
and 27. It
equal.
on
the
the
the
unit
so
that
No
is
dif-
the
dis-—
Figs.28a,
invector
vectors
unit
b
and
space
e;.
For in-
transorthogonal
alpha-
2.12
EXAMPLES
bet
of
Fig.28
OF
may
Cogs Ay
ee
my
-
Oe,
SIGNALS
be
69
written
-
Wayge,
F, = -Fe, + Ve,
= 5 2) 28)
F,=
0€,
+ Z72\3e,
all
four
The
energy
Boe
of
follows:
Pay,
iV3e,
0@, +
as
characters
is
equal:
Fi = FRR Fj 8
The
distances
between
the
four
signal
points
are
also
equal:
eh
ge
2
lead
2
seem
erate
coy
Sipe
7=sic3.8)
Co
.
ie
ee
jae
Se
a OO
eel
HE
—o
|
a tse
AEs La) rere alee 19005
£28)
Fig.29
1
Characters
of Fig.28
Fig.29a
by three
two
pulses
£(0,0),
case
the
is
reasonable
opposed
the
of
to
signal
simple
£(1,9)
wal(0,8),
Walsh pulses
In
following
represented
by time functions.
shows arepresentation
of these
block
as by three
it
cal(1,8)
points
vector
the
are
four
alphabet
coordinate
located
representation
on
characters
f(2,0)
-sal(1,8)
biorthogonal
orient
and
as well
~cal(1,0)<
and
Fig.28b,
of
system
each
results:
so
axis.
that
The
These
characters
Walsh
functions
be writmay
Fig.28c
co-
the
of
if the axes
form,
of the cube at their
the surfaces
intersect
system
ordinate
of
alphabet
simple
a particularly
in
ten
in Fig.29b.
the
of
characters
The
shown
three
or
pulses
block
three
of
composed
are
F5= -@
Fre*=03, 0 Fp="—€;,,
-Fi> @) see Fz = e7>
Fo= 6p,
TRANSMISSION
DIRECT
2.
70
centers:
Fy
=@)
+e,
+@
F, = -e9+e,
+;
+ 6;
—'4,
F,
-@,
EC oneOy
a es
Fe eet Ogae, Fa Ss
f=6, -e,
—'e,
F, =-e,-6, -@,
Fo =e,
Fig.29c
shows
pulses
if
and
The
perspicuity
characters
four
or
m+1
of
a)
m+1
are
b)
The
qual.
mination
There
ber
of
ditional
for
are
the
This
One
alphabet
between
(m-1)
m+1
0.
al-
alphabets
al-
may
compose
from
m
he
func-
coef-
foliosaes
computation:
characters
is
the
+
m+1
equal.
(m-2)
characters
This
+....+
can
are
O-
1 = -m(m-1)
characters.
equat
are
ions
available
m(m+1)
conditions.
m+1
e.g.,
binary
by m(m+1)
y = 0...
their
true,
conditions.
m +
coefficients
is
andthe
computation.
m1,
lost,
readilybe specified
Transorthogonal
for
all
m
between
of #m(m+3)
of
may
is
The
biorthogonal
c.
available
of
block
vectors.
characters
are specified
distances
distances
A total
and
yields
three
three
functions.
the
J = Os...
energy
statement
of
representation
of atransorthogonal
a,(j)}
The
or
of
considerable
These
conditions
vector
alphabets
Figs.29b
characters
fielents
the
vectors
require
tions.
of
characters
phabets
composed
pulses.
consist
of more than
some
more
the
ready
of
characters
Walsh
the
characters
for
these
three
=-e,+e,
coefficients.
be
chosen
for
the
deter-
A considerable
freely
or
fixed
numby
ad-
2.13
AMPLITUDE
SAMPLING
al
2.13 Amplitude Sampling and Orthogonal Decomposition
The
sampling
a Signal
and
cosine
cies
[1..6].
Hertz
by KLUVANEC
[7].
tions
that
a
and
= Ap
per
in
is
if
limited
eH ee
into the
signals
cussed
later
evident
on.
without
turns
frequency
-4 s v = 4%.
Legendre
polynomials,
phase
system
angle
I!
to
of
v
=
fT
etc.
are
sine-cosine
$n being
introduced
withno
> % may
decom-
for
1(6+3).,
(649
:
.sequency
will
be
that
disit
de
1,
transform
of
be
pulses,
suitable
to
is
be
the
in
the
pulses,
functions.
will
ha-
expanded
outside
Walsh
simplify
= V2 sin (2riv+én)
eetmdie
components
vanish
2(2i-1,v)
series
Tt
simple
= f2cos (aniv+ar )
21,
a
its
sin
used,
The
the
result:
1
ewe
in
so
pulses
a= 0 FOr eveeeer and Vo <= — es
ded
to
result
fed@2kerat)
Fourier
a frequency
of
system
that
Sine-cosine
ZC
The
be
sam-
amplitude
It will be shown
functions
F(8)
functions
g(0,v)
g(j,v)
out
signal
orthogonal
interval
following
Walsh
interval
calculation.
limited
anormalized
of
It
of
states
functions
of
inthe
equivalent
orthogonal
Corresponding
composed
A frequency
series
The
zps.
func-
theorem
by 2Ap
in
in
generalized
been
sampling
amplitude
incomplete
wetinnto aes oo
limited
a
measured
determined
of orthogonal
sequencies
signal
is mathematically
position
ving
with
determined
is
that
section,
this
Ap
has
frequen-
measured
a superposition
of
completely
second
sine
is
sampling
KLUVANEC's
that
with
Af
if
systems
complete
f£,(pt,t/T)
2nft
states,
periodic
completely
is
theorem
sampling
other
of
cos
second
per
consisting
signal
O =
ples
This
analysis
and
0 = f = Af
Inessence,
fe(ol,t/T)
2nft
samples
for
Fourier
a superposition
sin
interval
amplitude
of
of
functions
the
in
2df
by
theorem
consisting
these
poe
(8)
< ate
G(yv)
of
pulses:
a signal
F(§)
is
expan-
V2sin(enive+dn )+a(2i V2cos(ariv+ar)]
7 (a (2i-1
G(v)=a(0)+
TRANSMISSION
DIRECT
2.
Ye
iz]
1/2
av
2cos(anivedm
= f° Gv)
etait
pile
= Le
aCOy
=e
1/2
(9)
= di G(v)V2sin( eniv+gn )dv
aC2t=1)
-1/2
The
inverse
Fourier
transform
co
F(8)
= [ [G(v)C cos anvé
+
yields
F(8):
sin 2nve Jdv
—oo
The
sum
G(v)
(9)
is
is zero
substituted
for
G(v).
outside the interval
Keepinginmind
-% = v = % one
that
obtains:
F(@ )=a(0) seul
Sin.78 >)[a(2i~1 SER PE) 1a (21 SER OEE) (10)
A frequency
by
limited
a series
functions.
of
It
the
signal
F(@)
incomplete
follows
from
are
orthogonal.
One may
ating
the
integral
©
Sinn(@+k)
sin
1(0+;
ee
Kgs
£25
a
os
The
OF
ae [lc
obtained
at
the
functions
a(0),
by sampling
times
@ =
prove
represented
system
1.13
it
be
of ais
that these
directly
by
func-
evalu-
= 6,
(11)
eee
coefficients
be
de
thus
orthogonal
section
tions
(Saas
may
t/T
ee
a(2i-1)
the
= QO, 41,
and
and
amplitude
t2,
&..
aa
a(2i)
of
of the
FOr
are
(10)
signal
anstence,
zero
for
may
F(0)
bi
8
=
0
and $4509 is 1. Hence, it holds F(O) = a(0).
It follows
and
tion
a(2i-1)
of
F(8)
from
may
(11)
also
by ae
be
that the coefficients
a(2i)
decomposi-
functions.
a(O)-=
f p(o Sin ged) ae =)
a(0),
obtained by orthogonal
FCO)
fer
jg =
“ately = Rai? for y=
a(2i—1)
= FU)
«fore 4
ey
<7
2.14
CIRCUITS
The
FOR
equivalence
decomposition
A, Sinemy,8
and
pling
these
would
a(2i-1).
amplitude
and
with
additional.
orthogonal
limited sigoscillations
discrete
of
number
B, cos 2nv,é6
again
73
sampling
Vi > @ Deradded to (6).
filter with cut-off
frequency
oscillations,
v = % would
and amplitude
sam-
yield
the coefficients a(0), a(2i) and
decomposition of the new signal F(9) +
Orthogonal
since
yield
contribution:
the
sinn
(6+
]—== o43
and
(21°
B, cos ary,6
and
A, sin 2mv,8
functions
:
JCFC@ )+A,sinery
6+B,cos2ny,9
a(0);
yields
also
cos 2my,e)
(eh, cllenv,@.+0By
a(2i-1),
no
DIVISION
restricted to frequency
finite
ideal lowpass
suppress
of
is not
a
Let
nals.
An
ORTHOGONAL
ae:
ge
sinn(0+
nora)
Oo
-Co
veos tt, Thood, fp o> 4/22je85=°t/T.
It
remains
lations
do
to
not
tion
D(@)
be
with
frequency
be
yield
added
|v|
then be zero inthe
the
Fourier
outside
time
any
to
that
which
interval
-#
interval.
The two
each
bands of oscil-
either.
Let a func-
contains no oscillation
> #. The Fourier transformof
the
to
continuous
contribution
F(@),
transformof
this
orthogonal
shown
(13)
other
and
= v = %. Onthe
same
hand,
1S)
TCO+ 9
Fourier
must
other
sint(84+
functions
the
D(@)
ZErO
transforms
are thus
must
for
hold
the
functions:
f v(o
Se
Sin nttsDag =O
TC
(14)
O+ 9
2.14 Circuits for Orthogonal Division’
Fig.30
shows
FG
transmitter,
4 £41.
ted
block
a,(j)
5 coefficients
generator
a
generates
which
are
diagram
by orthogonal
5 functions
orthogonal
The five coefficients
by voltages,
See [1] -[11]
which
for amore
for
have
detailed
A function
division.
£(0,0)...f£(4,8)
in the
interval
ay(O)...ay(4)
a
constant
of
transmission
the
are
value
discussion
of
at the
-T
=
represenduring
the
circuits.
74
2.
DIRECT
TRANSMISSION
s
are multiplied
tion
f(j,@)
interval -#T = t < #1. The func
The
M.
multipliers.
the
in
a,(j)
by the coefficients
R
are added by the resistors
five products a,(j)f(j,9)
and
RA.
It
is
enters
and
transmitted
then
f£(j,8) used in the
cients.
with
each
M.
FG
signal
is
5 functions
for
the
coeffi-
and retransmitter
the
in
The
the
one of
transmitter
as carriers
generators
Function
gh
throu
the amplifier
5:multipliers
to
applied
simultaneously
multiplied
receiver
the
signal is
resulting
The
TA.
amplifier
operational
the
ceiver
must
the received
be synchronized.
The 5 product
of s
signal
with
the
functions
I during
integrators
the
at the integrator
tages
a,(O)
to
ay(4)
Another
set
at
the
of
five
a,(4) is transmitted
functions f(0,@) to
inthe
transmitter
these
functions
representing
-3T
s t < #7
in
are
integrated
-#T
= t < #T.
outputs
time
the
and
represent
terval
$#T = t < 31.
Transmitter
the
vol-
the coefficients
represent
t = #7.
coefficients
denoted
receiver
are required
periodic
coefficients
T = $7
in
The
by
a,(0)
to
during
the interval 4#T = t < 317. The
f£(4,8) of the function generator FG
and
are
the
f(j,9)
interval
with
period
a,(j)
transmitter
are
the coefficients
The
integrators
again.
Hence,
T.
The
voltages
during
the
interval
changed
ay(j)
in
the
suddenly
during
at
the in-
receiver
are
Receiver
Acca aah @) Block diagram for signal transmission by orthogonal functions f(j,
FG0).
function generator, Mmultiplier,
Pigeon
TA transmitter amplifier,
RA receiver ampliier.
2.14
CIRCUITS
reset
at
FOR
t = 4T
ORTHOGONAL
and
start
DIVISION
1
integrating
the
voltages
vered from the multipliers during the interval
For
be
practical
augmented
modems
are
delivered
by
use
a
the
to
at the
time
diagram
the
4T < t < 37.
Fig.30
circuit.
has
coefficients
into
the
required
the
at
obtained
desired
to
Furthermore,
the
coefficients
#T into
of
transform
transmitter
also to transform
ver
block
synchronization
required
to
the
deli-
a,(j)
form
the
and
recei-
form.
+ ——
wal (0,8)
sal (1,8)
cal (1,8)
gal (2,8)
cal (2,6)
sal (3,6)
Fig.31
for perio-
cats.)
ave weaken nuney tons. By banea—
ry counter,
X multiplier
=
half adder, z input for trig-
Generator
sal (4,8)
Wa
Cea
ger
pulses,
sal(5,8)
set
pulses.
n
input
for
re-
cal (5,8)
sal (6,8)
cal (6,8)
sal (7,0)
cal (7,8)
sal (8,8)
Fig.31
shows
repeated
cally
sal(i,é@).
This
a
for
circuit
Walsh
functions
circuit
is
the
on
the
periodi-
cal(i,@)
or
wal(j,6)
based
of
generation
and
multiplication
functions wal(j,6) as given by (1.29). Binary counters B1 to B4 produce the functions wel 15600 =
= sal(4,@) and
= sal(2,6@), wal(7,9)
sal(1,9), wal(3,0)
in Fig.41
shown
= sal(8,@). The multipliers
wal(15,9)
theorem of the
produce
tem
The
of
from
Walsh
function
these
Rademacher
functions
wal(0,9)
functions
sal(1,0),
isaconstant
the complete
cal(1,0),..,
positive
sys-
sal(8,9).
voltage.
The
2,
only.
Comparison
adder
shows
i poses
no
functions
circuit for
the
of
tion
-V. The generaliza-
a negative voltage
1 for
output
an
and
difficulties.
fay and
Consider
counters
a
Walsh
rather
depend
are
the
counters.
There
are
a frequency
discrete
sine
switching
sently
compare
between
sequency
of
“1018
=
zps
of
the
10
frequencies
of
at
100
of
having
20
or
aging
the
MHz
the
such
will
binary
It
is
a
million
representative
zps
time.
were
a
a generator
circuits
10®
GHz
of
the
sequency
This restricts
10
of
problems.
of
digital
present
to
one
delivering
from
binary
A total
driving
On the other hand,
10ns.
(b).
can be obtained.
their
simplicity
functions
Gzps
adder
any
generator
foramulfunctions
half
Fig.31.
produce
fastest
100 ps and
Walsh
in
synthesizer
functions.
times
to
drift
the
a
functions
accuracy
no
to
of
shown
pulse
to
for
generator
Walsh
The
trigger
worthwhile
that
as
required
functions.
on
4
different
adders
possible
function
than
048 576
19 half
of
values
higher
with
Table 2. Truthtables
tiplier fortwo
Walsh
pe
+V
voltage
a positive
for
stands
O
output
an
if
adders,
half
be
may
Fig.31
in
multipliers
the
that
with that of the half
table
truth
this
of
in
+1 0r -1
values
the
assume
functions
Walsh
since
Table
shown
as
table
truth
a
having
gates
are
multipliers
TRANSMISSION
DIRECT
2.
76
are
pre-
the highest
=
100
Sine
Mzps
waves
produced
to
with
decades
ago.
Fig.32
phase
of
shows
stable
Fig.1and9.
demacher
a
sine
The
functions,
function
and
binary
from
fundamental
sinusoidal
three
the
times
In practical
generator
cosine
oscillations
counters
which
functions.
frequency of the
applications
it
for
is
B1
the
and
generation
for
the
of
pulses
B2
produce
Ra-
filters
extract
the
harmonic
has
The
first
fundamental
better
to
oscillation.
leave
out
the
2.14
CIRCUITS
FOR
ORTHOGONAL
DIVISION
rae
cos478
sin4n8
Fig.32
iGrous.
Generator for phase locked
8 binary counter, F filter,
pulses.
x andy are
filters
and
functions
to
by
a
complementary
produce
outputs
a better
superposition
sine and
z input
of
cosine funcfor
trigger
the
counters.
approximationof
the
of
Rademacher
sine
functions.
g
Vy=-\j cal(i,8)
cal (i,8)
\,cal (1,8)
-cal (i,8)
sal(1,8)
\V,sal(1,8)
-sal (i,8)
cal(k,8)
V,cal (k,8)
—cal (k,8)
b
Fig.33 Multipliers
by Walsh
function
multiplication
b) multiple
There
multiplies
for the multiplication of an arbitrary
a single multiplication,
functions.
are
three
two
basic
voltages
(e.g.
types
that
of
can
filter
bank).
The
first
values
only,
multipliers.
assume
two
circuits.
by logic
V,
arbitrary
having
sume
a few
type.
Voltage
output
any
V,
terminal
field
emission
equal
-V,
ground
inverting
nal
must
The
bitrary
mented
sistors
ally
FETis
of
at
be
basic
voltages.
by
Hall
and
type
In
effect
quad
circuit.
It
where
of
this
only.
V,
The
may
have
(-)
also
non-
this
multipliers,
elements.
multiplier.
to
V,
multiplies
type
can
field
etc.
A
tran-
are
due
Fairly
ar-
imple-
emission
shows
usual
V;.
two
be
devices
applications
the
termi-
equal
Fig.433b.
in
These
Fig.34
from
inverting
the
multiplier
must
The
and
shown
V3
terminal
V,
is
the
if
conducting.
requires
am-
non-inverting
grounded,
at
of
deviates
The
is
input
temperature
drift, price,
diode
-1
non-conducting.
principle,
logarithmic
or
be
This
V,.
canas-
FET
multiplier
this
-V;,
fully
inverting
Let
that
rang
ofe
the operational
unsatisfactory
for practical
the
tive
the
bring
voltage
example
+1
amplifier
the
terminalisthen
third
impedance,
is
to
or
an
wor
as ks
follows:
transistor
potential.
shows
values
voltage
of
(+)
also
variation
the
The circuit
input
to
the
assumes
within
A.
plifier
Fig.43a
equals either+V,
voltage
value
only.
values
V,
withavoltage
values
a
multiplies
type
second
The
implemented
is
of multiplier
type
This
V.
-1
V and
+1
say
TRANSMISSION
DIRECT
2.
VAS.
usu-
to
low
suitable
a representa-
one
by not
using
transformers.
Fig.34
n
The
voltage
Multiplier
V,
in
Fig.33a
using
diode
assumes
the
quad.
values
+1
or
-1
2.14
CIRCUITS
only
and
ry
Four
binary
16 values.
field
ORTHOGONAL
may be considered
digit.
assume
FOR
emission
complicated
P.SCHMID.
one
voltage
Fig.45
the
is
be
digits
transistors
and amore
to
to
represent
network.
available
in
amplifier
yields
accuracy.
charging
this
the
switch
The
switch
capacitor.
is
usually
The
by
of
the
form.
feedback
of
voltage
that
input
voltage
with
the integrator
by dis-
practical
a field
but
output
the
s resets
four
Suchamultiplier
capacitive
an
can
one in Fig. 44a
results,
digital
The
bina-
that
requires
the
excellent
proportional
to the integral
great
than
yields
anintegrator.
operational
multiplier
rather
by one
avoltage
resistor
It
be
must
shows
Ves,
represented
Acorresponding
is
due
DIVISION
implementation
emission
of
transistor.
s
R
v,(t)
principle
circuits
This
the
for
circuit
pulses
integrator
for
the
sine
makes
sin2nit/T
following
output
Ce).
function
of
of
the
generator
function.
functions.
pulses
fact,
v,(t)
suffice
Superior
Fig.36
shows
according
to Fig.‘.
that
cos 2nit/T
are
sine
and
cosine
eigenfunctions
of
equation:
y(-sT)-yQT),
v,(t)
any
of
y'C-8T)=y"CeT)
amplifier
A;
fv, Goldee~ (R05) fv
voltage
pce) = CRC,
use
cosine
and
voltage
= CRO,
output
The
and
differential
ae tty = 0,
pire tnt
The
and
detection
are available
for special
a detector
Integrator
mac ret
Multiplier,
in
Fig.45
t
= -v2(t)
of Fig.36
0b ae.
(15)
is
(16)
of A, equals:
Sh ry Celet
I
(R,C,R,C,y' [fv Cb! dtdt'+(R,C,R,0,)"
I
-v,(t)
(17)
Ce" dtat'
ffv,
yields:
terms
the
reordering
and
twice
Differentiating
TRANSMISSION
DIRECT
2.
80
(18)
va Ch) = = Ce, Brey yee)
wiley 2 (RC RG
vp (t)=-v, (t)
Fig.36
Detector
for sine
cos 2mit/T
R, = TiR,;
Choosing R,C,R,C,
of
(15)
and
cosine
pulses
and
(18)
= (T/2mi)
identical.
makes
The
the
shape
terest,
+27,
of v, (t)
since
the
v, (1/2)
outside
and.v,
Fig.37
oscillograms
= 1.
Fig.38
120;
of
this
a
with
shows
voltages
28,
The
729
losses
that
out
at
use
of
circuit
the
100
regeneration.
plication
lies between
The
lower
the
frequency
limit
is
of
resonators.
of
The
is
to
tuned
and
t
Ry =.7iks;
i=
f=
i = 128
and
128,
for
129
to
its
input.
Fig.36
are
comparable
readily
frequency
of
by
1Hz
pulses
and
leakage,
operational
for
to
thou-
obtained
range
and
detection
cosine
fed
=
= (-1)'V,.
that
are
fractions
of
for
in-
at
Q-factors
of several
Hz
determined
response
i?¢k end
v,(1/2)
v,(t)
cycles
are
of no
closed
k equal
circuit
128
cycles
a frequency
of
the
of
is
sand v,(%)(for
of
with
orm 140
those
of mechanical
sand
of v,(t)
oscillograms
pulse
for
= 0 and
V, cos 2nkt/T
means
cosine
s, are
zero
v,(1/2)
term v,(t)
for -#T = +t = #7.
interval
s, and
(2/2), are
yields
shows
of this
switches
V,; sin 27it/T
input
and
left hand side
inhomogeneous
is equal to V, cos 2nkt/T or V, sin 2nkt/T
The
sinemit/T
according to Fig.1. R,C, =7T/2mi, R,C, =T/emi,
s; and sz, are closed at t = +2/2.
with-
its
ap-
about
100 kHz.
the
amplifiers
upper
[6].
by
2-15
SINE
AND
COSINE
PULSES
81
ig-5/ (lett) Typical voltages of the circuit of Fig.36.
Peron pusevOlbage v,(t) = Vsin2nt/T;
Brand C: resulting
vyolbages v,(t)
and
v,(t);
D:
input
volta
v, (Gas
aev-cos -1u/1; Hand
F:-resulting
om meZompclleescalker e> mls airy.
voltages
Te Capomes Coals
Pics 5o (right) Typical voltages of the circuit of Fig 46.
Circuit is tuned for the detectionof
sine and cosine pul
ses with i = 128 cycles.
Output voltages v,;(t) shown are
caused
cycles
v=
by input voltages v, (t)
(wand JoOecrvoles (C).
7o9me.
(Courtesy
Allen-Bradley
with 128 cycles
Duratvion.ol the
P.SCHMID,
R-DURISCH
and
(A),
129
traces, 1s
D.NOVAK
of
Co.)
2.15 Transmission of Digital Signals by Sine and Cosine Pulses’
One
and
block
Hertz
Tiss
they
amplitude
limit
The
tSee
-
6
[1]
of
an
same
[11]
orthogonal
Harmuth, Transmission
FPig.4
canbe
idealized
fom
sampling
fluence.
using
pulse
through
devectiom
without
more
the
rate
of
per
lowpass
block
second
filter.
pulses
intersymbol
by
in-
holds forthe
"raised
examples
of transmission
systems
functions.
of Information
of
correction
transmission
for
transmitted
frequency
cosine
pulses"
Fig.39.
Those
be
influence.
intersymbol
0
T
27
t—=
per
pulses
second
a
and
in practice.
of
ideal
any
way
to
Sine
[12,12].
and
which
of
cosine
The
more
than
limit
approached
complex
of
one
equipment.
three
spectrumis
shown
on
the
red for transmission
zfés
fg =
second
reasonable
tion
wal(0,@)
sine
pulse
4 block
any
not
if
deviation
large
seem
to
be
half
the
Nyquist
rate
least
not
without
pay-
and
pulses
of
two
This
or
9 permit
per
second
pulses
per
close
by
may
be
T only.
right.
The
1/T.
and
to
One block
seen
-
Hertz
second
using
The first
Its
pulse
transmis-—
and
from
and
more
Fig.40
consists
frequency
frequency
shall be defined
of Fig.1
a
Fig.1
pulse
systems of functions.
duration
per
be used
occur,
arbitrarily
does
at
at
of
of
is
to
arbitrarily
pulse
It
lead
There
used,
pulses
than
Nyquist
be
shows
mitted
cannot
can
transmitted;
may
a block
-asO
pulses
2 pulses
penalty.
higher
can
is
transmit
amplitudes
is
faster
to
these
pulses.
sampling
and
rates
Hertz
the
one
large
pulses
transmit
a power
Sion
such
between
amplitude
ing
However,
synchronization
crosstalk
if
Hertz.
Arbitrarily
a sequence
from
permit
cos ant/T,
1+
Fig.39 Raised cosine pulsesintime domain:
4 4+ cos 2n(t-T)/® and 1 + cos 2n(t—20)/7.
The
requi-
functions
sample
with
the
correct
to
circuits
res
may
crosscorrelation
by
tection
orthogonal but linearly indedetected by amplitude sampling. De-
pulses are not
They
pendent.
in
shown
are
them
of
Some
domain.
in time
TRANSMISSION
DIRECT
2.
82
band
power
requi-
somewhat
arbitrary
can
be
then
trans-
Hertz.
identify
and
cosine
block
to transmit
pulse
duration
the
of
T each.
a
duration
The
pulse
as
block
pulse,
func-
5
47
instead
of
power
spectra
of
2.15
SINE
AND
COSINE
the
pulses
for
transmissionis
1.5
pulses
are
PULSES
shown
are
in
83
Fig.40.
reduced
to
transmitted
The
0
=f
per
bandwidth
s to
second
and
required
2/ 51.
Hence,
Hertz.
0
Fig.40
systems
Comparison
of
of functions.
the
Considerafurther
ries
mit
of
cosine
is
to
means
pulses
0
that
bandwidth
step.
of
one
=f
Instead
5T.
T
2
sine
required
3/5T
a
se-
pulses
and
frequency
band
Fig.40.
second
frequency
band for the simultaneous
trans-
are
according
2
to
pulses
=
transmitting
each,
one may trans-
pulse,
The
by various
per
1.67
3 3f,/5
required
of
duration
block
pulses
of duration
reduced
This
and
5 block
simultaneously
fy=/T
transmitted
Hertz.
The
required
mission
of one block
duration
(214+1)T
pulse,
equals
mission
rate
equals
Hertz’.
This
rate
i sine
0 = f-s
i cosine
pulses
of
(i4+1)/(2i+1)T.
The trans-
(2i+1)/(i+1)
approaches
and
2 for
pulses
large
per
second
values
of
and
i.
Wsee Wi4 | for, a detailed discussion
of the fraction of
energy outside this band. This paper
also
discusses the
application
of KRETZSCHMER's principle of partial response to signals consisting
of sums of sine and cosine pulses.
6*
84
2.
Table
shapes
pulse
transmitted
for
and
of
of the
equipment
different
proaches
2.
Table
Number
pulse
of
shapes
and
rapidly
different
may
One
Hertz.
and
21i+1
TRANSMISSION
different
of
of pulses
(2i+1)/(i+1)
number
the
increases
21+1
number
the
for
second
per
number
4.
values
4 shows
DIRECT
thus
see
that
complexity
the
(2i+1)/(i+1)
as
pulse
shapes
ber
(214+1)/(i+1)
of pulses transmitted
per
Hertz for a transmission system using
sine
the
and
ap-
nun-
second and
and
cosine
pulses.
Table
4.
Utilization
Transmission
rate
is
of
a 120
6.67
Hz
wide
teletype
characters
per second;
channel.
duration
of a character is
150 ms. First column lists the pulse,
second the frequency of the function from which it is ga-
ted,
third
the subchannel
(su.)
and
digit
(di.)
the pulse is used. carr.
stands for carrier
tion, sync. for character synchronization.
for
which
synchroniza-
[pulse
[ flHe]
feu.
ai.[[ sinpulse
| fla)
[au. ai |
wal(0,@)
O
carr.
1870
60
c
sin
cos
sin
cos
sin
cos
Sin
cos
sin
cos
Sin
cos
sin
cos
sin
cos
278
2T6
416
46
676
6TA
819
816
1076
10780
1276
1276
1476
1416
1676
1678
Guo
6.67
5359
Boose
20
20
26/567
26.67
44.55
D515
40
40
46.67
46.67
Baer
DiBin BIS
:
SS
FUP
AW
EW
FWD
PO
PO
PD
INN
NIN
AASA
cos
1819
Sin
cos
Sin
cos
Sin
cos
Sin
cos
Sin
cos
Sin
cos
sin
cos
sin
cos
2076
2079
2276
2279
2476
249
2678
2678
2876
2876
3078}
4078]
32m760|
32m6|
4479@|
44m6|
ew
60
66.67
66.67
1339
75.59
80
80
86.67
86.67
94.44
94.43
100
100
106.67
106.67
113.33
114.43
EEO
EEE
DANDNN
WMEWNM
EWS
SU
EWN
UP
AU
2.15
SINE
Table
AND
COSINE
4 lists
of the periodic
in
not
‘start-stop'
ving
sine
multichannel
but
they
are
abeonhenreceiver,
teletype
A teletype
to
200
large
number
Tests
be
can
other
that
in
sible
power
wer
one
sion
is
There
of one
channels
out
that
works
over
and
be
One
able
long
more,
at
were
loading
rates
but
for
high
least
synchronous
one
For
2400
transmission
Error
depending on how
fast
rates
lost
or
comparison,
the
data
permis-—
the
load
should
very
telephone
may
po-
requires
power
It
is
chan-
transmis-
that
the
bits/s.
occur in switched
distances.
trans-
coding
Exceeding
but
and
between
than
24 teletype
system
channel
im-
10”
No
speed
such
permissible
exceeds
3.
de-
With
of
the
used.
some
becomes
condition
afactor
transmit
which
operate
channel,
exchangers.
channel,
usual
is
crossing
characters
channel.
accomodates
telephony
to
jumps
tive to phase
can
quite
systems.
pointed
methods
telephony
zero
a
4 can
error
severe
system
usual
characsin2nt/T
exceeding
telephone
TELEX
bandwidth
eight
ious
two
the
holds for transmission
more
loading by about
loading
the
figure
of
signals
function
a telephony
power
without
amuch
error-reducing
the widely used
of
shown,
between
nels
in
have
This
into
end of the
telephony
of
obtained
from
signal.
question
loading.
mission
channels
arri-
buffer
teletype
has
Hz
and
is
transmitter
fed
the
according to Table
the
subscribers,
be
of
Gs66
/le=
the
periodic
A
quality of the
less
two
ms.
teletype
portant.
power
150
system
pending onthe
a
may
duration
synchronization
as
used
end=is
they
The
mands
signals
a
must be added tothe
pulses
that
and their
Transmission
through
to
frequencies
gated,
Teletype
fed
slope at beginning
negative
100
so
Ouns
be
the
are
system.
synchronously
to be
Wit
with
must
equipment.
is assumed
ters
pulses,
they
teletype
stop
and
which
synchronous.
fed
Start
Fig.30.
85
cosine
from
asynchronously
which
and
waves
use
a
PULSES
be
sensinet-
increase
to
synchronization
reestablished.
reason,
why sine and cosine
transmission,
is that
pulses
telephony
yield
channels
very
reli-
are designed
2.
86
periodic sine and coofsion
transmis
containing very
pulses
cosine
and
Sine
free
distortion
for
sine
functions.
many
cycles
fer
little
son
is
orthogonal
rence,
shapes
pulse
however,
affects
cau-
noise.
noise
affects
is
used
interfe-
than
more
pulses
sampling
mainly
Pulse-type
equally.
block
particul
if arly
amplitude
rea-
thermal
thermal
5 that
chapter
in
shown
be
will
It
than
rather
interference
Another
are
channels
telephone
in
sed by pulse-type
all
distortions.
ayion
or attenuat
del
errors
s
and suffunction
periodic
the
to
close
come
that
TRANSMISSION
DIRECT
others,
detection.
for
9.2 Characterization of Communication Channels
2.21 Frequency Response of Attenuation and Phase Shift of a
Communication Channel
Communication
attenuation
and
channels
phase
are
shift
usually
of
harmonic
function
of their
frequency.
A
to
the
state
the
Vc-(w)
=
input
cos
ac(w)
as
and
omitted,
sine
wt,.(w) =b,.(w)
of
if
are
and
only,
steady
the
attenuation
equal.
Since
for
line
and
the
The
and
phase
it
is
applied
output
why
with
are
constant
such
time
its
and
periodic
zero
into
the trans-
ends.
equation
The
differential
described
coefficients,
invariant
Vcoswt
that
be
channels.
at
equation.
be
shift
may
functions
differential
will
c
those
equation
s,
if
its dimensions
it
lg V/V,(w)
communication
chann
be divided
el
by ordinary
In particular,
of
known
as
is
information
at rate
of
the
the
parameter
shift
well
circuitry
by apartial
described
ponents
w.
characterization
difference-differential
equation
at
attenuation
and phase
cosine functions
transmit
described
ferential
are
frequency
Let the communication
is
Vcoswt
voltage
itis interesting
to investigate
used
mission
voltage
by
oscillations
w[t-t.(w)]
is measured. The quantities
function
Vsinwt
are
and
specified
items
The
line
is
or
a partial
terminal
CLTCULtLY.
or
difference-dif-
are
not
by
a
if
the
as
too
large.
differential
circuit
coils,
com-
capaci-
2.21
FREQUENCY
tors
and
anput
RESPONSE
resistors.
of
sucha
and
sinusoidal
shape
se
of
other
phase
the
frequency
phase
instance
but
more
02 w
Ox2
L,
32 w
=
©,
and
(4) —
LO sry
=
(LA
R and
A are
free,
case
if LA
as
w(x,t)
is
ee
f(x-ct)
ow
equal
RC.
[£(x-ct)
and
=
-
is
by
per-
shape
the
by the
of
telegra-
O
(19)
capacity,
The
line
general
resistivity
is
distortion
solution
is
in this
«G = 1/ VC,
and
and
a
distortion-free
According
to
boundary
Au
if
ee
=
example
are
ductors
L = 2.01x10°3
during
if
functions
they
are
The
determined
only
transmission
feature
also
change
is
holds
condition
x
is
holds
an
for
‘electrically
[1] a line
of length
following
wires
Henry/km,
atnot
short'.
electri-
for
x
aie
following
Ohm/km,
One obtains
fLAs— RC =O:
conditions.
This
[2]:
(21)
consider
copper
The
cm.
the
Z=\
x< ge - =
an
lines,
K.W.WAGNER
short,
(20)
g(x-ct) are arbitrary
initial
4.95
Hence,
respon-
+ g(x+ct)]
delay.
2g) U
RAw
Its
functions
18
-
length.
suffered
by these
of
since
described
inductivity,
unit
tenuation
As
functions
an
the
A characterization
Walsh
line
se
the
per
bey eic.
cally
as
output;
follows:
= 6°
by the
the
[3].
ae RC
conductivity
state
the
preserved.
complicated
Consider
a transmission
puem'emequation
steady
to
changed.
is
functions
applied
at
are
shift.
and
for
possible,
the
by the frequency
characterized
-
fectly
voltage
in
voltage
attenuation
functions
these
appears
shifted
and
be
may
circuit
A sinusoidal
circuit
attenuated
the
87
an
of
open
wire
3 mm
diameter
typical
values
© = 5.9x10°*
line.
The
at
two con-
a distance
apply:
Farad/km
<A * 0
Z = 540 Ohm
and
22/R = 225 km.
This
line
is
regenerative
Inserting
km.
225
smaller
than
at shorter
amplifiers
dis-
over
any
cable
be-
distortion-free
signals
one may transmit
tances,
distances
for
line
distortion-free
a
like
TRANSMISSION
DIRECT
2.
88
distance.
tween
insulated
typical
subscriber.
and
exchange
copper
values
of
wires
R
This
lowe
line
lew
following
apply:
The
will be electrically
sow
50
2Z/R
will
coaxial
and
100
then
cables
Ohm.
be
Let
of
However,
one
must
keep
equation
does
not
allow
tionof
the
skin
cosine
seems
Despite
to
the
be
One
solution
variant
some
subscribers
in
partial
coefficients.
= us
practical
described
and
mind
for
impedance
20
km.
that
skin
Thisis
between
the
Z
10
be-
Ohm/kn.
the
order
amplifiers.
telegrapher's
effect.
An investiga-
functions
other
than
sine
cosine
sine
and
lacking’.
important
of
a wave
resistivity
be
forthe
role
in
reason
and
the
is
theory
Assume
equation
that
of
a
w(x,t)
space
functions
of
BERNOULLI's
differential
represented as the product
a time variable v(t):
Wore:
for
distances
results,
a distinguished
lines.
10
usual
effect
these
have
the
between
of magnitude
play
short
euLIL.
usual
tween
Wk
The
diameter.
paper-
7x10 Henry/km, C = 3.3x10° Farad/km,
70 Ohm/km, Z = 145 Ohm, 2Z/R = 4 km
L
was
mm
are
conductors
The
0.8
telephone
a
consider
example,
a further
As
transmission
method
for the
with
in
time
(19)
variable
in-
may
u(x)
rCh)
be
and
(22)
distortion-free
by MEACHAM
do
[3].
line
using
semiconductors
Superconductive
cables
are
almost distortion-free and transmit switching transients
in the nanosecond region [4,5]. Such superconductive
cables could have great practical potentif
organic comial,
pounds can be developed that are superconducting at room
as some physicists believe to be possible.
temperature,
2.21
FREQUENCY
Substitution
of
tal
SP
Oh
-
89
u(x)v(t)
into
RA)u
yields
two
ordinary
=)
(23)
2
LC Sey + (LA + RC)
4 v = 0
Their
are
eigenfunctions
e?! | where
(19)
equations:
differential
ax
RESPONSE
A-RA)x,
sin \(A-RA)x
and
follows:
as
defined
y is
cosy
y = Hee (CHS) oh J
BERNOULLI's
solutions
the
of
boundary
lution
that
riable
lines
the
by
sine
would
dimensional
with
w(x,t)
= f(x-ct)
solution
a
and
asa
R=
It
special
A =
0.
has
the
674
same
form
term
like
distortion-free
have
be
to
rent
rent
sons
but
ting
sine
waves
frequency
sequency
bands.
and
be
factor.
waves
obtained
of
general
is
for
the
de-
one-
telegra-
solution
There
are
is:
Radio
that
aradio
link
waves
sine
do
not
functions.
operate
in
diffe-
instead
operate
in
diffe-
the
transmitters
by
except
to
excellent
channels
mainly
(20),
Hence,
line.
have
may
radio
as
described
not
they
are
reasons
receivers
radio
is Missing.
do
bands;
for allocating
these
or
transmitters
of
(25)
DiemeGuenuation
Different
sova-
separation
a
case
Its
of
Time
+ g(x+ct)
behaves
a
method
dependent
is
par-
initial
certain
the
permit
space
equation.
propagation
equation
not
ifTine saayse
functions.
electromagnetic
wave
paemis
This
of
satisfy
itis
for
and of other
equation
cosine
course
of
importance
that
and
a time
into
the
great
However,
conditions.
propagation
scribed
of
equations
favours
solution
The
is
telegrapher's
editierential
tee
and
method
en
practical
rea-
according
to frequency,
simplicity
rather
than
of
implemen-
laws
of
na-
radio
commu-
possible with Walsh
nication is indeed theoretically
waves.
ture.
It
will
be
shown
later
on
that
mobile
achannel
of
features
that
are
There
have
been somany
py the
transmitted
shapes
proposed and used
signals.
telephony
for
telephony
signals
functions.
Hence,
is
that
signals
there
the
output
will
mation
of
a
independent
cularly
by apartial
so
for
like
p,
t
described
by
This isnot
sounds
is
time-variable
with
excitation
why one
position
of
consider
ons of some
sine
other
Experimental
MAILE
and
considered
LUKE
and
using
of
4000
functions
to
MAILE
filters
up
and
has
be
have
that
zps
to
such
to
a
there
no
sounds
to
systemof
built
a
telephony
permit
Walsh
pass
through
a frequency
of
rather
4000
parti-
sinusoi-
ofa
not
Hz.
the
equation
no
of
reason
super-
functi-
functions.
TASTO,
Walsh
may
LUKE,
indeed
functions.
multiplex
functions
with
producing
Signals
of
sy-
excitation
orthogonal
superposition
a
equation
consist
voice
sy-
activa-
Such
is
and
by
vo-
The
andis
BOESSWETTER,
that
and
approxi-
good
particular
functions
shown
sine
differential
is
by KLEIN,
of
consonants,
system
and
There
cosine
telephony
sinusoidal
The
partial
a
complete
work
others
k.
coefficients
function.
should
differential
voiceless
or
why
function.
anda
for
difficul-
oscillations.
a sine
coefficients
sounds
with
time-invariant
is
cosine
sustained
A long
consisting
with
cords
described
is
function.
vowel
and
represented
signals
sinusoidal
of afew
vocal
reason
a microphone.
of
the
stem
cy
voice
a voltage
sum
producing
time
dal
as
hold
regard
The
superpositions
Consider
not
preeminent
appear
overwhelming
no
sine
of
channels.
regarded
voltage
produce
by the
ted
be
should
wel
stem
is
functions.
cosine
be
functions
these
characterization of telephony
the
ty
superposition
use-
to
practice
general
is
It
a
as
it
only
does
This
a channel.
characterizing
for
one
ful
the
as
one
pulse
that
transmission
digital
for
a particular
claim
to
hard
be
would
distinguished
by functions
signals.
the
describe
+o
It is reasonable
angle.
another
from
nels
chan-
of communication
characterization
Let us consider
TRANSMISSION
DIRECT
2.
90
system
up
to a sequen-
than
sine-cosine
There
is
no
dis-
2.22
CROSSTALK
cernible
an
that
tions
and
synthesizer
has
these
formants;
A theoretical
argument
contains
"sequency
work
on
built
16.fil=
speech
these
Walsh
used
Has
analysis
as
formants"
functions
are
investigations
explaining
OANDY
5.14.
tiesecuLen
retical
functions
by sine-cosi
decompos
edne
as voice
just
has
using?
sequency of Walsh funcaccording
to the frequency
of sine funcshown fora few examples, that voice de-
by Walsh
frequency
BOESSWETTER
fora vocoder
according
to the
than
KLEIN
composed
a
filter
rather
tions.
oA
differen
of performan
ce
ce.
analyzer
ters
PARAMETERS
contains
continuing.
results
is
functions
early
as
given
“in théo—
1962
[6].
2.22 Characterization of a Communication Channel by
Crosstalk Parameters
Having
to
be
shown
question,
how
theory
of
quires
a
all
or
least
the
a
Let
functions
a
(1,0)
channel.
that
the
tained
systems
the
of
a,(i)
fiom
since
sine
An
is
orthogonal
functions.
characterization
channel
capacity
system
of
re-
apply
in
to
As
will
sim-
section
6.1.
orthogonal
functions
and
The
of
For
time
the
state
the
being,
This
{f(j,9)}
ldne.
only.
voltage
the
hold
is
to
case
is
of
is
class
obof
of a distortionlg V/V-(i) =
the
communica-—
generalized
applicable
produces
large
or
= K.(i)
the
vol-
the input
channel
be such
fora
attenuation
= b.(i)
Vf,(i,®)
let
inthe
V.(i)/V
shift
f(0,0).
applied
V.(i)f. [i,@-6-<(i)]
will
generalized
functions
constant
V. coswt ig
voltage
oupput.
phase
input
of
will
instead
term
the
of
general
functions
that
£,(i1,8)
§ @.(i)
channel.
the
f,(i,9),
freeitransmission
=
based
on orthogonal
complete
functions
the
have
them be divided
into even functions
steady
ao
not
raises
may be characterized.
A consistent
systems
more
do
functions
characterization
many
this
Consider
tetemvte
of
channels
sine-cosine
they
discussion
{f£(j,9)}.
of
communication
by
communication
effect
plify
else
method
at
aside
odd
that
characterized
the
to
sine
output
delay,
and
co-
voltage
Vf(0,9)
The ‘constant
a(O)
=
lg V/V(O)
=9beCe
6<(4)
and
= a,(1)
1g W¥s(i)
V¥.(i)/V = K(i),
by
defined
are
delay
and
Attenuation
Vo(i)f,[i,e-95(i)]-
TRANSMISSION
DIRECT
2.
92
-61, VCO) ZNSE COD
(0)
yields V¥(O)£[0,6
= bo).
6(0)
and
Let the functions of the system {£(0,0),f,.(i,@),f,(i,9)}
substitution
by the
stretched
be
&€ increase
is obtained according to section1.22.
be (i)
ag(i),
lar,
one
and
b,(i)
obtains
for
{f,(u,8),f, (u,9)}
the
Kk. (i),
Im particu—
f, (u,9)
functions
special
a,(i),
Ks(i),
K.(u)---b,Q@u).
become
i and
Let
=u.
i/§
i-~
The system
beyond all bounds.
=
= /2cos 2mud =V2coswt and f,(y,9) = 2 sin 2mud =V2 sinwt
the frequency functions K,(w)...b,(w). The indices c and
s may
be
omitted,
frequency
tions
are
of
special
case.
In general,
the
delayed
be
expanded
and
functions
but
Vgc (1,0) is obtained
@c(i,8)
are
Vf,.(i,9)
into
a
of
Hence,
by
shift
phase
the
delayed;
channels
distorted.
instead
of the
functions
and
obtained.
communication
attenuation
response
and
of
cosine
and
attenuated
b(w)
and
a(w)
K(w),
racterization
nuated
if sine
equally
are
the
the
is
not
included
only
The new output
of
the
cha-
frequency
system
as
atte-
function
V,(i)f-[1,6-0.
(i)
series
same
func-—
}-mee
{f[0,9-
6c(i),fefk,0-9c(i)], f,[k,8-6c¢(i)]}.
The value of the delay
8,(i) will be defined later on. The variable is now k,
Wiibesigs
Ss a COMSLENG \(=) we ey ea Ce en
co
Bela e) = Keer, O)fp0yeseu(a4 Dd, (RCed, ck )f, Dc, 8-6.
ae
k=]
+. ECeL, Sk) fuCk,6=05
Gj),
K(ci,0) = fe.(i,e)£[0,0-0¢
(i)Jag
K(ci,ck) = fec(i,e)f,[k,0-9
(4),]a9
E(ei,sk) = [g-(i,o)f,(k,6-8,(i)
Jae
(26)
2.22 CROSSTALK PARAMETERS
Consider
depends
the
on
integral
6,<(i).
Let
assume
its absolute
s
= b,(i)
and
6,(i)
®.(i)
the
distortions.
value
for
coeificients
communication
of
the
KG
the
i = k.
so
two
K(ci,ck)
attenuation
has
variables
with
de-
K(ci,sk)
are
variable
k,
i and
that
it ap-
fora distorting
one
Kici,O),
vector.
a
value
K(ci sez )
delay Ce)
line
and
K.(i)
the
Its
that
distortion-free
K,.(i)
by
by a matrix
presented
the
channel.
have
for
chosen
The generalized
is then defined so,
K(ci,0),
Moyebeesreprescnted
K(ci,sk)
be
maximum.
creasing
rhe
K(ci,ck)
= b,(i)
proaches
generalizations
93
and
i
and
KCci,;ck)
and
may
be
re-
K(ci):
med. CO isk1c 1 ak Ce4/,s1 PyKCcd.c2 akK(e14s2)2.
UGC. 0 eK Cee, cl) K(e2, 4) |) Reco. CZOK (Cease.
i C500) Kies, 61) KhCc5),s1) Kles,¢c2) Klesjs2)..
een
(27)
Me
output
voltages
Ve,(i,6)
are
obtained,
if Vif, (i,9)
baaGesusotevi- 41,0) 18 applied tothe input. Coefficients
K(si,O), K(si,ck) and K(si,sk) are obtained
in analogy to
(26).
Duo
The
CL
16
matrix
replaced-by
Transmission
cients
as
K(0,0),
line
K(si)
Has
the
form
of
the
bined
Ke
K(O,ck)
and
yields
Vg(0,6)
K(0O,sk)
which
andthe
may
the
be
coeffiwritten
matrix:
three
ante:
matrices
K(ci),
K(si)
and
K(O)
Boi)
may
be
com-
one:
RCO Oe
K(s, 0)
(oe Oume
CeO
Res 0)
The
(27),
81%
of Vf(0,@)
K(0) = (K(0,0) K(0,c1) K(0,s1) K(0,c2) CHEE
The
matrix
terms
functions
(CO, ot) ek
O,c1)
K(st, 51) K(s1, C11 )
Ce leet) Kcl,c1)
ee etn (seact)
K(c2,61) Kice,e1)
outside
f(j,@)
of the
are
not
KUO, 82)
K(s1, 82
Kicl.e2)
K(s2,82)
K(ce,s2)
main
KC0,¢2)..
OS, Co als
Klel1,¢2)..
)K(s2,ce).~ | (28)
Klcz2,c2)..
diagonal
distorted,
of
The
if
K vanish,
terms in the
and
= K(ci,ci)”
K.(i)
= K(O,0),
K(O)
coefficients
tion
attenua-
of
set
one-dimensional
the
become
diagonal
main
TRANSMISSION
DIRECT
2.
O4
8 Csa,.5a5).
The delay times 9,(i) of (26) and the corresponding
of the
delay times 9,(i) and 6(0) for the transmission
and Vf(0,8) may also be written as
functions Vf,(i,@)
ReaGie
uetasest
sxe
8(0)
O
O
OMT
e)
Ce)
O
Oe
eta)
Om
O
2S)
O
6
The
tion
0
two
for
Distortions
the
in
a
transmission.
One
correction
this
an
of
appropriate
K and
Sans
system
of
channel
a
K,
the
functions
cause
crosstalk.
for
ie :
® characterize
application
term
ic
er
Meta
matrices
channel
ss
of
matrix
Hence,
while
{f(j,9)¥-
crosstalk
the
communicain
crosstalk
® may
be
multiplex
K is
for
the
matrix
called
is
the
de-
lay matrix.
2.3 Sequency Filters Based on Walsh Functions
2.31 Sequency Lowpass Filters
It
can
has
be
f5(u,8)}
Equation
signal
applied
shown
instead
of
(1.112)
had
F,(8)
to
functions
P,(8) =
been
characterized
at the
the
in
by
sine
section
a
system
and
been
filter
1.32
of
cosine
functions
functions
obtained,
output,
how
that
if the
a
filter
Pie tehsowy
[i
—
yer
represents
signal
input.
the
F(@)
Substitution
of the system of
{cal(u,8),sal(u,9)}
into (7.4172) yields:
is
Walsh
[acu dKe(u)oalfy 50-0 (41 +
(30)
+ as(u)Ks(u)sal(u,9-85(u)]
tau
The
derive
following
filters
relations
from
(30):
of
section
1.24
are
needed
to
2.31
SEQUENCY
LOWPASS
eal(u,@)
= wal(0,0)
calcined)
f= calG
eal Gu,e)
=
—s
5,
=
0-<
Let
6i<
a
Lae
Oy
ang
i Sty
se Be
(31)
Snes)
ily
es
25
eee
G(6)
be
divided
6 < #,.....F(9)
into
denotes
is
on
@ae(U)
signal
required
(40)
ei
the
canbe
may
from
G(@),
which
derived.
be
The
computed
a
sections
not
place
beginning
F(@)
with
42
e=se(0)
<= [FCO )wai(0,6 ae
= ac(i)
Ale
V2
= | F(6)cal(i,e
)de
any re-
synchronization
and
coefficients
for
-$ =
section
in the im-—
does
but
the
time
the
-# = 9 < %. Suchadivision
intervals
of
ls
signal
strictions
gnal
56)
sali,6)
2, f=
terval
FILTERS
end
a,(u)
the
help
si-
of
the
anda,(u)
of
(31):
Onsen
(22)
an Dey ne ee igs
= /i2
(2
ase(u) = a(i)
= [F(8@)sal(i,é)de
i-1 <p
IIA
H-
Si f74
The
specific
possible
ted
a
to
features
transform
function
F(§)
of
the
by
an
the
Walsh
functions
representation
integral
into
of
make
a time
it
limi-
arepresentation
by
sum:
fe [a,(u)cal(u,é)
F(@)
+ a, (u)sal(u,é@)jds
eee)
0
Co
a(O)wal(0,0)
+
ys a
Ci cal Gare)
is anGh sala, 60]
iz]
as
The
attenuation
the
delays
ter.
They
may
§<(u)
be
Sical
realization.
to
able
be
than
an
to
coefficients
and
chosen
The
represent
6,(u)
freely
K.(u)
in
(30)
within
following
the
and
determine
the
choice
output
Ks(u)
limits
is
signal
made
by asum
as
well
the
fil-
of
phy-
in
order
rather
integral:
Ke (u)
=
K(O),
ee)
=
6(0)
p=
oa
4
Gy)
=
Kew
O@c(u)
=
Gy
Pas
ele<
tet
ec)
=
EC
Dee
=
85 (i)
ye
De
Se
eS
(34)
DIRECT
2.
WG
Equation
form:
following
the
assumes
(30)
TRANSMISSION
F, (@)=a(0)K(0)wal
lO, 0-9(0) 4 'fa,(4)Ke (i calli, 0-0¢(i)]+
iz]
+a.
Let
us
consider
(i )K, Cisaltas O83
a filter
for
which
the
(359
following
holds:
K(O) = 1, 8(0) = 1; K.(i) = Ks(i) = 0
Gs)
follows
F,(@) =
pass
ter
of
filter
or
in
(30)
to
(35):
fa¢(u,@-1)an = a(0)wal(0, 9-1)
0
form
The
from
order
lowpass
integral
the
-
more
to
filter.
andatime
suggests
precisely
distinguish
Its
diagram
block
in
it
An
diagram,
circuit,
Sequency
lowpass
c) practical
A operational
filter.
circuit.
amplifier
(37)
calling
-asequency
from
diagram
Fig.41b.
F(8) =F (t/T)
Werk.
-—. != :
ie
Pies
(36)
is
the
shown
a
lowpass
usual
integrator
fig
H
this
in
lowfil-
frequency
Fig.4‘a,
I determines
-F,(8)
:
a) block diagram, b) time
L intvesrater.
11H holding
2.32 SEQUENCY BANDPASS FILTERS
a(O)
according
the
interval
S,-
The
the
section
to
-$
(32).
a(0O)
integrator
is
then
G(9)
inthe
by integrating
G(6)
during
is
version
ter
of
numerical
with
4 kHz
this
to
The
output
will
have
Cee
cut-off
is
made
here
analysis,
cy lowpass
equals
of
which
per
A signal
the
of
lowpass
rate,
1/2T
=
lowpass
at
the
fil-
output
analysis.
filter
of Fig.41
if
second.
sampling
[8] -
a practical
Fourier
it
also
Hence,
the
= 125
4000
theorem
happens
to be trivially
filters
»
IIA rwlK
wo
amplitudes
per second
T = 1/8000
» =
6 <
# = 6
and
frequency
theorem
mistbe
sequency
-4 \=
interval
filteris
shown in Fig.41c.
information
amplitudes
rie. 41>
(37)
to
a(O)wal (0, 6-1)
Hence,
in, Fie.41a,
signal
of the sequency
same
according
1)
-
independent
sampling
for
4# = 6 < 3 is obtained
interval, etc. a(0)
frequency.
8000
the
the
independent
Ca
has
a(0O)
time
considera
cut-off
filter
according
values
s,.
interval
the ng
duri
of this sequency
lowpass
For
of
outp
by ut
switch
switch
in the
shown
H is
by
value’.
stored
and
circuit
eho leine
that
obtained
a(O)
9=+%
at
sampled
sample
at d
the end
interval
with
isaconstant
voltage
the
be
reset
wal(0,9
by
multiplied
be
wal(0,9)
can
= 6 < #4 at the integrator
of
must
97
has
8000
steps
of
us
long;
the
zps=4
kzps.
Use
of
Walsh-Fourier
simple
for sequen-
[10].
2.32 Sequency Bandpass Filters
Let
1.174
are
sequency
s
theorem
of Walsh
multiplication
tion
simple
derive
us
bandpass
needed:
= cal(i¢k,0)
= sal{(k®(i-1)]+1,96}
sal(i,6)sal(k,@)
= cal[(i-1)®(k-1),6]
The
of
stands
7
= wal(0,9)]
and
on
(38)
I
multiplication
sine
The
in secderived
functions
Gal(i,é@)cal(k,6)
sal(i,9@)cal(k,9)
fealCO,8)
filters.
the
theorems
cosine
right
Harmuth, Transmission of Information
(38) are
functions,
hand
side
except
very
those
toar
simil
that
one
term
only
for the
two terms
of d
instea
is
sideband modulation. This makes
sequency bandpass filters by a
yields a single (sequency)
implement
to
it possible
band-
frequency
for
used
little
but
known
well
principle
signal
a
by
carrier
Walsh
a
of
modulation
the
that
A consequence
frequencies.
the
of
difference
the
and
sum
TRANSMISSION
DIRECT
2,
98
filters.
pass
the
Let
F(9)cal(k,8)
= a(O)cal(k,8)
"sequency
shifted"
one
obtains:
(48)
Using
cal(k,9).
with
multiplication
be
(33)
of
F(@)
signal
+ S* (a¢(4)eal (48k, 0) +
i#k
by
(39)
+ ac(k)wal (0,6) +
+ a,(i)sal{[k®(i-1)]41,0})
+ a,(k)sal{[k®(k-1)]+1,6}
Passing
by
(46)
this
signal
through
a lowpass
filter
yields in analogy to (47) the output
described
signal
F,, (8):
Py, (@)=a,(k)wal(0,0-1)= f a, (u)oal(u,e)cal(k,e-1)du (40)
cal(k,é-1)
=
cal(k,6)
Multiplication
tered
signal
to
of
its
F,,(9)
by
original
cal(k,9-1)
shifts the fil-
positioninthe
sequency
do-
main:
F, (o)=F,, (@)cal(k,6-1)=a, (k)ceal(k,6-1)=
wal (0,6-1)cal(k,9-1)=cal(k,9-1),
The
filter.
last
For
MUltaplaer
integral
its
in
suggests
practical
front
of
cal’ (k,9-1)=1
the
name
after
tiplication
The
the
(41).
sequency
Fig.42
sequency
lowpass
shows
one
filter
a
bandpass
must
lowpass
(49).
such
(41)
sequency
implementation
the
Fig.41 to perform the multiplication
plier
Fac(udeal(u,e Jadu
put
filter
A second
a
of
multi-
performs
the mul-
bandpass
filter.
same function cal(k,6)
is fed to both multipliers
since cal(k,8) hasthe period 1 andisthus identical aes
cal(k,6-1). Suitable multipliers are shown in Fig.34. Note
2.32
SEQUENCY
BANDPASS
F(8)
FILTERS
oo)
Fo(8) Fig.42 Sequency bandpass filter. M
g
multiplier for Walsh functions, LP
sequency lowpass filter.
that
multiplication
Cevion
by
Signal
unchanged,
+1
by
a
Walsh
function
means
by
multiplication
-1
multiplileaves
+1
by
multiplication
only;
—1
or
reverses
a
its
am-—
of
se-
1,
and
plitude.
Fig.43
quency
for
several
are
shows
a
attenuation
sequency
bandpass
shown
to be 1.
The
midieabewtnas
through
whey
tor
the
the
as
filter
function
with
coefficients
values
of
i
cal(i,®@)
for
or
cross-hatched
K(0)
=
Ke (i) and K.,(i)
which
areas
at the band
function
filter;
dOeaole
The
hatched
the
delay
lowpass
filters.
ZEro,,excepy
and
they
limits
U=
sal(i,@)
areas
are
i
passes
indicate,
that
pass
K(0)=4
K(0)=4
\
WEY
oe
sa
0
K,(1)=4
K,(4)=4
Z
S
2
‘Ss
es
—
K,(2)=4
ai
j
4
s—
—EE
0
2
3
SOK
(2)=4
a
|
0
EE
4000
T=05 ss
_
8000 12000s”
(3—S>
Fig.43
The
all)
of
width/(lower
The
7%
|
2
a
0
ae
|
=
K.(2)=4
following
band
0 ——eEEEEE——EE—|
0
i
2
3
0
0
0
bandwidth
relations
|
4000
T=125 ps
Bet
delay
8000 12000s*
of
0
=
Au/u,
may
be
denote
as
Seen
2
|
4000
T2513
800057
ae
filters.
sequency
equals
1 for
quotient
band-
u,-u,=i,-1,=4u
us
i
ji—>
|
Cae
and
limit)
K,(2)=4
a
_—E
4000 8000s"
Fig.44.
wo
Peyote
jee
|
T=125 1s
Attenuation
filters
| 125ns
a
(ae
normalized
Ks(1)=4
oe
ao
—EE
0
K.(4)=4
jes
the
relative
to
hold,
bandwidth.
for
AuU/u;
from
Fig.43:
YO Mic Ee 4
fore macy
achieved
with
multiplexing
are
square
between
is
and
-60
The
hold
an
the non-normalized
u as well as
signal
at
filter
the
107
and
filters
10°
of
The
is
steep
used
to
sequency
filter
filters.
This
think
terms
in
filters
quantization
edges
of
use
the
result
of
is
square
means
the
between
-40
frequency
switches
signal.
that
Keeping
would
interval
sections
in the
Attenuation
and
pass-band
inherently
no
stop-bands
energy
is
bands.
This
-~<
were
-# =
defined
Real
is
to
or
for
shown
mind,
surprising.
used
filters
Fig.43.
delay
filters
a
functions
rather
than
6 < #,
dela
of sequency
y
isnot
delay
68 < +0
according
attenuation
passed.
in
disappear,
if the Walsh
interval
to
filters.
this
discontinuity
whole
also
introduce
The
the
lies
startling
change
of attenuation
is not
the
root
shown
in Fig.43
discontinuous
im
the,
» which
Fig.43
shown
voltage
mean
the
the
assume
delay
output
present.
and
stop-bands
may
second.
attenuation
the
in
use
would
one
that
per
in Fig.44.
plotted
are
us
In practical
of
between
the
practical
engineer
the
be
can
only
1
dB.
However,
of
band-
relative
equal
ler
smalor
of
0.0017
thus
in
sequency
the
into
the
determine
idealized.
infinitely
for
time
fed
amplitudes
response
and
attenuation
sal(k,®)
signals,if each
deviation
0.071
deviation
omm
channels
the
telephony
sequency
Fig.44
mean
eerek. (ae
T = 125
show
independent
The
—~
circuit.
for
»
of
values
8000
=
sequency
» = u/T
sequency
The
this
normalized
The
Kee
bandwidths
Relative
width.
1ce
Fig.42
of
filter
bandpass
=I
or
cal(k,@)
functions
The
Kee
torek (k=
eee 1
AU/Uy
Vor
=
Ko C2)
for
4/2
AM/iy
in
TRANSMISSION
DIRECT
2.
4100
are
Hence,
distortions.
ideal
pass
by dashed
filters,
energy
lines
in
in
constant
there
are
Delay
in
since
16
the
stop-
Pig .43.
2.32
SEQUENCY
BANDPASS
cael
Fig.44
Walsh
(left)
FILTERS
aae
=
|
Approximation
functions.
A:
04
of
sinusoidal
sinusoidal
functions
wave, ae
AO)
by
Malba
ceamrerce
wale,o
(Cs.
2.(4 sal(1,¢); 0:
a.(1)cal(1, 8);
ees
ce salve.
es fois the sum of B and C; Cys he cu or
DCE)
ei
SmuMom SUM OieD, C. Diand Hee lime: base I=
=ipms;
Gorizonval
scale O.5 ms/div.
Fig.45 (right) Walsh-Fourier transforms
of sinusoidal waves. A:
sinusoidal
waves, frequency 1 kHz, various phases;
horizontal
scale
transtorms
ac
O.1
gl)
ms/div.
and
a,(¢r)
B
and
of A;
C
show
time
Walsh-Fourier
base
T = 1.6
ms;
horizontal
scale 625 zps/div.(Both oscillograms courtesy
C.BOESSWETTER and W.KLEIN of Technische Hochschule DarmStadt
Fig.44
sequency
the
shows
a sine
filters
output
of
a
and
wave
the
sequency
(A)
at
the
resulting
lowpass
input
output
filter
of
several
signals.
with
K,(0)
B is
bandpass filters with K,(1)
ofuts
the outp
and K,(2) = 1. F, G and Hare) thevontputs
C, D and E are
= 1, K.(1) =1
d
in paralconnecte
filters
bandpass
ined
several
obtafrom
em eee
él: (0) = KyC4) = 4 CP); KeCO) =e
ReCOvee
Ke(1)
Fig.45
functions
phases.
B, the
The
(A)
>= ee
sequency
of
equal
amplitude
amplitude
oscillograms
output
and
=k C1)
shows
The
the
and
traces
spectra
of
sinusoidal
frequency
and amplitude but various
spectra
spectra
B
ae
amplitude
B
a.(u)
=
a,(gTI)
are
shown by
a,(u)
= a.,(yT)
are
shown
C were
obtained
by
sampling
voltages
of abank of 16
adding
TRANSMISSION
DIRECT
2.
102
and
sequency
C yields
filters.
the
by
C.
the
Squaring
sequency
power
spectra.
10
Q=t/T—
Fig.46
filter,
Sequen
widecy
band
H holding
i
ee Stet s Ginafilter.
LP
sequency
lowpass
2.32
SEQUENCY
Bandpass
filters
bandwidths
bandpass
his
BANDPASS
Au/u,
for
alowpass
T.
A further
output
chosen
of
There
pass
tive
may
of
(6)
Signel®F’
is another
filters
of
bandwidth.
pass
the
Pyeceti.¢.)
important
Fig.46
The
of
b
LP?
and
LFP1
in
is
and
LP1
of
SP. The differ-
output
the
yields
between the band-
besides the different
rela-
sal(u,6) as well as cal(y,8@)
Fig.46inthe
calG@i,6)
The
shown
are
filter.
difference
42
functions
filter
Or
and
..2
1/4,
or
of
of
over
circuit
and
bandpass
wide
this
intervals
delay times
holding
a
integrates
filters
period
Fig.46a
of
voltages
the
shows
aes
time
LP2
°T/3,
or
Fig.46
according to Fig.41,
filter
lowpass
relative
LN
over
different
The
T/2.
permit
only.
LP1
1/2,
two
compensated by the
are
ence
to
equal
...
signals
integration
the
b;
and
Fig.46a
Fig.42
bandwidths
lowpass
the
of
voltages
to
filter
duration
Gimewinbervals-of
AOS
1/3,
relative
integrates the input
duration
LP2
according
= 1, 1/2,
filter
circuit uses
which
FILTERS
may
pass
pass—band,
a’
filter
while
on-
according
to
Fig.4e.
Fig.47
Sequency
highpass
oi and
(b)
LP
sequency
sequency
bandstop filter
lowpass filter,
BP
variety
A great
sic
a
ters
Parallel
according
sal(u,@)
AUy/iie=
as
We
filter
bandstop
filters.
well
Aes
filters
as
1/5,
derived
yields
cal(u,@)
os;
On
filter.
the
ba-
highpass
and
from
may be derived
connection
Fig.42
to
bandpass
shows asequency
Fig.47
discussed.
types
sequency
pass
of
sequency
from
lowpass
of several
filters
other
and band-
bandpass
that
and have relative
the
filter
hand,
let
fil-
pass
bandwidths
one
may ob-
Fig.44F-H.
of
oscillograms
the
by
shown
as
only
cal(u,@)
or
sal(u,0)
pass
let
that
filters
bandpass
wide
tain
TRANSMISSION
DIRECT
2,
104
2.33 Digital Sequency Filters
transformation
of
while
only,
subtractions
and
additions
requires
signals
see
filters
digital
are
Walsh-Fourier
numerical
that
is
reason
functions
Walsh
on
based
ters
fil-
of sequency
applications
promising
most
One of the
numerical Fourier transformation requires multiplications.
Fig.48 Block diagram of adigital sequency filter.
LP sequency lowpass filter of Fig.41;
AD analog/digital
converter, ST digital storage, AU arithmetic unit performing
additions and subtractions,
DA digital/analog converter.
For
an
explanation
sider the block
through
tion
a
of
diagram
sequency
of
F,(6@)
at
its
output
of
the
steps
converter
delivered
the
AD
at the
digital
digital
Fig.48.
lowpass
amplitudes
tal
a
A signal
filter
have
are
sequency
LP.
of
storage
racters
be
Fourier
transform
denoted
ST1.
ty A,
of
16
the
Let aset
B,
step
steps of duration
per
of
16
T/16.
1.25
may
The
time
which are
unit
T to
consecutive
cha-
...,8,...,P.
section
func-—
analog/digi-
characters,
characters
con-
F(6) is passed
Let
converted by the
into binary digital
rate
filter
The
be
Tease
used
Weta
to
obtain
from
these 16 characters the 16 coefficients a(0),a,(1),
ac(1),---,45(8). Additions and subtractions only have to
be
performed by the
veral or all of these
ac(5)
alone
ficients
is
-a,(5),
arithmetic
unit
coefficients.
computed
+a¢(5),
as
shown
+ac(5),
AU
to
Assume
in
obtain
the
Fig.48.
-a,(5),..
one,
se-
coefficient
The
16
coef-
with the signs
corresponding to the signs of cal(5,6) in Fig.2 are transferred into the digital storage ST2. Reading these 16 coefficients out serially through a digital/analog conver-
ter
DA yields the analog
output
signal
F,(8).
The
connec-
2.33
DIGITAL
SEQUENCY
tion
between
input signal F(@)
(32) and (41):
follows
from
Be uD)
Pe
2
FILTERS
ee
(05
and
output
ei)
S
signal
eG)
an
(42)
Po(8) = ac(5)cal(5,0-1) = f ac(uoat(u,e)ae
5
Let
F,(8)
quency
minal
value
Fig.42.
according
Fig.48
10°°
If
and
the
if
negligible
error,
+
2).
loge
(On
The
usual
of
one
eropaene,
For
analog
which
tions
Colour
of
Stands
filters
or
two
for
three
tive
is
basic
can be programmed
effort
filters
and
ANDREWS
even
a
of
cross-
of
and
filter
the
enormous
sequency
an
and
pictures,
The
func-
ofami-
grayness
two
third
space
are
time.
of
va-
func-
variable
the
that
Digital
functions
of
two
functions
are fed into
of
one
variable.
filters
usedit
a
with
compared
PRATT,
advantage.
digital
for
The
rela-
inherent
the
and
important
aswas
of
such
programmed
of two variables
functions
filter
pictures
or
dB.
-72
voltage
time.
a
obtains
one
to
output
Dunctions
becomes
of still
information
The
Fig.48
a function
though
becomes
have
deviation
loge 2) +
capable
the
colours
to
of digital
simplicity
frequency
KANE
are
as
variables
Shestilcersscriallyslaike
computing
a
to
introduces
square
in
used
funetion
picture
variables
in
compute
to
used
voltage.
pictures
or television
three
attenuF,(8)
Let
referred
are
—-(6G0 + 40
such
a
space
the
of
filters
is
10°
unit
i #4 5,
crosstalk
filter.
a mean
as
16
variable
a black-and-white
riables.
=
2"
a,(i),
conversion
a
The
voltage.
unit
a
is thus 10 log 10 °/2" = -(60+
autenuabion
Spcrosstalk
tions
dB
in
no-
F,(8)
of
to
referred
talk attenuation
its
deviation
obtains
one
from
distribution
with mean
to
analog/digital
40f 107° /2"
ae C5)
se-
digital
samples
an
voltage.
SRO)
the
Amuchhigher
a
square
mean
the
have
into
deviate
coefficients
the
by
obtained
be
cal(5,@)
F,(9)
referred
of
-60 dB.
is then
a,(i)
Let
to aGaussian
attenuation
agion,can
unit
obtain
by feeding
ed
of
deviation
square
erosstalk
and
be
filter
filter
for
of
the reduction
pointed
out
in
1.25.
3. Carrier Transmission of Signals
3.1 Amplitude Modulation (AM)
3.11 Modulation and Synchronous Demodulation
The
previous
chapter.
The
F,(@)
by means
of
or
be discussed
will
output
denotes
to
emphasize
of
coefficients
functions
time
a,(j),
a
contain
cal-
the
number
finite
signals:
teletype
as
be
e.g.
F,(@) is used
notation
The
that
such
function,
time
any
will
$(k,6@)
functions
These
{#(k,6)}
functions
time
of
a system
of amicrophone.
voltage
F(@)
of time functions
transmission
now.
F(9)
carriers.
led
the
in
discussed
been
has
{f£(j,e)}
functions
thogonal
or-
of
byasystem
ax(j)
constants
of
transmission
m-
By(8) = S ay(4)£(4,08)
j=0
The
carriers
functions
at
mathematical
could
not
lines,
wave
pulses
are
form
Multiplex
systems
of
with
exact
meaning
lized
sense
of
why
for
links,
well
etc.
using
The
with
respect
as
a
more
this
is best
general
and
other
explained
a
functions
via
wire
trains
to
of
from
the
modulation
of
such
do
meaning
terms
modulation.
not
sideband
by an
multipli-
carriers
sideband
carriers
single
other
Periodic
Amplitude
such
term
cosine
neither
as carriers
in cables.
suited
view.
and
transmission
extent
a group
sine
however,
inheren
atly
single
filters.
here
is,
reason
holds
radio
point
yields
used
This
particularly
functions
There
usedto some
that
mathematical
predominantly
physical
guides,
Functions
sideband
a
used.
are
are
present.
nor
be
block
cation
#(k,@)
the
need
single
modulation
than
used
example.
in
usual.
a
is
The
genera-—
3.11
SYNCHRONOUS
Consider
a Signal
DEMODULATION
amplitude
F(@).
Let
modulation
F(8)
be
of
a cosine function by
into a Fourier series
expanded
-# = 6 < #:
interval
in the
107
;
oo
F(a) = a(O) + V2 Di La,(i)
cos 2ind + as(i)
sin 2ind]
(1)
iz
Let
all
F(6)
terms
be
pass
of
implemented
tion
The
2.31,
limit
carrier
sum
very
but
filtered
upper
through
the
alowpass
with
much
index
like
of
F'(8)
the
sum
J2cosQ,9
(1).
i > k.
is
has
of
suppresses
Such
filters can
filters
in sec-
no
importance
i = k rather
Amplitude
by F'(@)
that
the sequency
implementation
signal
filter
than
here.
i =o
modulation
of
as
the
yields:
(2)
FT(@)V2cos
2,8 =
3 Ca. (i)cos(Q,-ani )6-a,(i)sin(Q, -2ri)e]
iz!
k
,8+ >, Ca, (4)cos(A) +2mi )e+a,(i)sin(a,+2ni )e]
+a(O0)V2 cos
i=)
The
first
sum
lows
the term
is
produced
represents
Let
SCS)
represents
with
by
lower
sideband.
frequency
2,
of
DC
component
of
F'(9).
the
the
the
the
upper
the
It
carrier,
The
folwhich
second
sum
sideband.
F(8§)
be
expanded
into
a Walsh
=
alO)
+
S' Lac(i)eal(i,e)
=
a(O)
+
opel ape
it
series:
~eaeta
ereeD
eal(i.6) |
(3)
+ 6201 )Wwal@2i-1,9)]
—-s6<@
Let
F(8)
suppresses
enal
Ft(9)
has
i =k
expansion
series
number
terms
all
of
wal(j,0)
a
through
pass
of
with
ae
sequency
index
upper
Ft(é6)
terms.
Amplitude
by Ft(9)
yields:
and
i >
limit
FT(9)
modulation
lowpass
filter
that
k.
The
filtered
si-
of
the
sum
(3).
The
have
of
thus
a Walsh
the
same
carrier
+
= a(O)wal(j,9)
Ft(e )wal(j,9)
TRANSMISSION
CARRIER
3.
108
+
fac (i)wal(2iej,9)
er
£ a, (i wall (21-106
7,00 39
On
places.
binary
1 lowest
DigOe \e=tii pou)
C2i 51)
tM
Gede=
relations
following
The
nota-
os
the
zerat
gj Have
let
and
2k
than
larger
j be
Let
tion.
‘lo-
Consider,
in binary
n digits
2k having
a number
example,
for
in
sideband'.
Lower
partly
upper,
‘partly
or
wer’
j,
of
value
the
Depending
(4).
be',
an 'upper
descri
may
sum
this
only
sum
one
is, however,
There
rier.
car-
unchanged
the
by
cases
in both
is transmitted
a(O)
component
DC
the
that
shows
(2)
and
(4)
of
Comparison
(4)
hold:
a oy, 1soe Sgn ee
+
ly
4
oe
(5)
see e
ee
Ce ns
es tera Oa)
All
ger
indices
than
choice
2i®j
the
of
and
index
j.
This
(2i-1)@j
j
of
the
of
the
sum
carrier
corresponds
to
an
(4)
are
wal(j,®)
for
upper
larthis
sideband
modu-
lation.
As
a further
binary
places.
example,
One
CeCe
at)
eta
16S 5g ie
Now the
all
ee,
corresponds
The
lues
Jj, 1f
west
than
to
numbers
of
Week.
the
i larger
and
neither
zeros
binary
places.
This
Why
lower
does
sideband
amplitude
riers
yield
Walsh
carriers
two
the
n lowest
(6)
for
only
in
the
carrier
sum
(4)
wal(j,6).
are
This
modulation.
will
other
nor
be
for
values
ones
corresponds
to
only
a
certain
smaller
onthe
partly
vathan
n loupper,
modulation.
modulation
Sidebands,
only
the
(2i-1)®j
than
j and
j has
partly
(2i-1)®j
of
sideband
and
at
case:
ee me
i
eee
index
a lower
ones
this
eee
a
2i®j
2i®j
j have
in
es ea
i
indices
smaller
let
obtains
one
but
sideband?
of
sine
amplitude
Forthe
and
cosine
car-—
modulation
answer
of
consider
3.11
SYNCHRONOUS
the
DEMODULATION
multiplication
theorems
AO?
of
sine
and
cosine:
2 cos
1@
cos
k@
= +cos(i-k)@
+ cos(i+k)e
eesim
710 cos
kK@
=
+Ssin(i-k)6
+ sin(i+k)e
2 cos
¥@
sin
k@
= -sin(i-k)@
+ sin(i+k)e
2 sin~16
sin
k0
i
PhewesiS
right
be
is
a
us
Sides
of
cal(i,6@)
-
sime
cos(i+k)6
or
cosine
equations.
and
cosi@é
amplitude
or
sini@é
Fourier
modulated
once
more
the
=
=
=
=
is
only
(8):
bet
or
and
the
one
[xeonG
cal(k,@)
function
or
Walsh
in
a
for
on
the
sal(kj6)
components
the
be
= 110)
right
hand
carriers
of
a
carriers.
sideband
amplitude
1-4
signal
There
but
and
one
modulation
that
is
not
function
of
Walsh
modulation.
modulation
of
a Walsh
carrier
Fig.49a.
cosine
carrier
demodulated
PECGiay
encase
Mne
term
a filter.
single
—e1) 1)"34
Ce ="4)
(Ge 74] veOne
inl
tel
Imi
Is}
Walsh
lower
(8)
Pe
reason
why amplitude
yields
A circuit
filtering.
theorems
=io@k
cal(r,6)
sal(r,ée)
sal(r,@)
cal(r,9@)
sal(i,@)
Thisis
modulated
theorems
multiplication
'side-function'
firs:
Hence,
functions:
one
be
a
(7).
upper
may
produced.
multiplication
one
shown
of
carriers.
the
onto
The
the
sinké@
carriers
modulated
is
those
is
amplitude
only.
on
or
components
onto
are
functions
cosk@
modulation
of sine and cosine
of
consider
functions
Let
side-oscillation
sideband
eobereo
cal k.6)
Sai(1,6)cal(k,6)
€al(i,é)sal(k,@)
sal(i,¢@)sal(k,6)
There
two
are
consequence
Walsh
of
andalower
double
Let
of
that
upper
the
sum
sides
of these
carriers
Signal
An
a
hand
+cos(i-k)@
C2)
signal.
Half
This
the
9,9
modulated
with
by multiplication
o,0u0y2
om
V2 cos
cos
0,8 = F°(6)(1
the
right
The
second
power
power
is
loss
hand
term
lost
is
side
+ cos
F'(6) in(@)
cos
the
be
9,80:
20,9)
represents
must
on
by
V2
(9)
the
de-
suppressed
average
by
by
this
if the product
unimportant,
TRANSMISSION
CARRIER
3.
410
multiplication
can be amplified before
Ft(@)f2cosQ,8
with 2 cos 0,8.
1,0.
Let a signal Dt(@) be transmitted by a carrier Vecos
yields:
by Yecos,9
demodulation
Synchronous
[Dt(@ V2 cos 2, 8172 cos 2)9=D*(8 )[cos(Ny-2, )e+cos(0,+2, 6]
(10)
Let
-VvV2Zy
2m
eo)
the
signal
auxiliary
be modulated
of the carriers
Q,8
(11)
Vq-
&
v
OS
gq?
F'(@)V2cos
carrier
conditions:
satisfy the following
and f2cos,é@
without
will be received
F*(6)
D*t(@), if the frequencies
from
interference
V2cosa,@
an
vg.
frequency
cut-off
through
pass
signals
demodulated
the
with
filter
frequency
a
let
and
only
Vg
0 = v 8
interval
the
in
frequencies
with
oscillations
contain
Dt(@)
and
F'(@)
signals
the
Let
be
J2cosQ,8
and
by multiplication
with
first
let
multiplied
the
2cos
product
by
then
(Q)-0,)8:
{LFtC8 V2 cos 298]V2 cos 2,932 cos (M)-0y)8
=
(12)
= F'(9@)[14+ cos 2(0)-M, )6+ cos 20,0+ cos 20,8]
The
desired
terms
on
Let
the
term
right
a signal
synchronous
F'(8)
is
hand
side
D'(@)V2cos
modulation
obtained.
must
The
be
[D*(@ W2 cos (n,-20, )e]~2cosa,d
to
signal
(13)
may
be
desired
be
received.
Direct
(9) yields:
=
(13)
= D'(8)[ cos 20,8
The
not
filtered.
(0,-20,)8
according
three
filtered,
if
+
the
0]
cos 2(a,-0,
frequency
band-
width of Dt(6) is sufficiently small. Hence, there is no
interference between Ft(@) andthis image signal. This is
not so, if the signal D'(@ V2 cos (a,-20,)8 is first multiplied
by an
lated
by multiplication
auxiliary
carrier
with
2cos
Q,8
and
then
2 cos (M)-,)8:
{[D*(@
V2 cos (a,-20, )e]V2 cos 0,9}2 cos (259-0,)8
=)
aE
60:)[ cos 20,0
demodu-
+ cos 2(29-22,)8
+14
=
(14)
cos 2(0,-2,
6]
3.11
SYNCHRONOUS
The
term
FT(@)
is
from
(7)
DEMODULATION
D‘'(@)
appears
affected
and
(14)
of
cosine.
would
if
there
were
the
one
no
Amplitude
us
term
rather
used
instead
of
the
of
two
of
by
image
on
the
of
(14),
sine
and
signals,
right
hand
(b)
for
is
to
that,
lev
a
signal
filter;
if
carriers.
anyway
Dt(9)
be
Let
means
in
carriers
the
signal
= Fe)
very
carriers.
little,
systems
multichannel
transmitted
ley)
contrary
filtered,
fromdifferent
be
Walsh
sine-cosine
of
usually
signals
to
term
M multiplier
by wal(j,®):
= Ft(e)wal(0,e)
sequency
required
separate
order
demodulator
processes,
sine-cosine
difference
this
and
lowpass
(4) be multiplied
demodulation
synchronous
filtering
Lem
high
no
is
ever,
than
(a)
same
(Ft(6)wal(j,9)Jwal(j,6)
j @j = 0
to
side
cal (j,8)
modulator
consider
Ft(e@)wal(1,6)
There
hand
theorems
interference
Walsh carriers.
LP sequency
for Walsh functions.
are
right
multiplication
be
b
Let
the
W7.)s
Sidenote
Fig.49
on
by the image signal Dt(@). One may see
that the reception of image signals is
a consequence
There
aa
Howsince
in
.
To show
channels
by
a
second
car-
wale, 0):
Dec e.)
bC8)
+ SoCeeCH
i=l
=
s o(r)wal(r, 8)
r=0
calcite)
Orci
Ssalta, ed
(16)
wal(j,6)
by
pig )wal(1,0)
of
demodulation
Synchronous
TRANSMISSION
CARRIER
3.
We
yields:
(17)
i] D*(e)wal(1@j,6)
=
[pt(e@ wal(1,0)]wal(j,6)
2k
>; c(r)wal (18jér ,6 )
r=0
and
wal(r,@)
with
s
be filtered
signal
will
is
pass
ee Or
Only
2
j and
function
demodulated
the
with
cut-
of
wal(1®jé@r,6)
(17)
filter
r,
the following
filteif
condition
1 be
larger
than
binary
Swe
The
condition
ree
> 2k
be
modulo
=k
zeros
at
its
eabisty
eke
have
will
zeros
Let
only
eee
at
(19)
one. 6 ee
ere
in
order
sides
of
for
(20)
(18)
to
hold.
Adding
ek
(21)
has
n lowest
1 modulo
j
yields:
4S
that
Adding
them
(18)
digits.
obtains:
a eel
Seal
transformation
fact
One
n binary
(20)
2 on both
last
the
2
ways to satisfy
have
andlet
gl ete
!satisfied,
tee Pi Os
2k
2k
places.
lv Goj,
(18)
possible
the number
lt
must
ec ae ere
many
Let
y lowest
ata)
jee
lowpass
this
two
of the
Oi
wo
by a sequency
ogt Te
discussed.
the
The
functions
satisfied:
SD
be
through
Walsh
y,
or
onl
further
No Walsh
p=k.
sequency
off
wal(0,@),
Let
only.
= 2k
r
i =k
with
sal(i,@)
cal(i,9)
functions
Walsh
contain
and Dt(@)
Ft(e)
Let
uses
the
n binary
binary
2 in
(20)
relation
places
j © j =
only,
while
O and
j
has
places.
yields
a
second
possibility
(18):
=
Conditions
Pr
+7]
like
(23)
(21) and (22) divide the sequency
spec-
3.11
SYNCHRONOUS
trum
into
trum
is
divided
into
ment
of
certain
frequency
more
general
Walsh
DEMODULATION
sequency
carriers
{CF
for
based
an auxiliary
as
the
group
with
the
the
sequency
theory
wal(h,8)
channels
is
no
are
interference
then
=
D*(6)wal(1,6)
be
it
(23)
= Ft(e)
by
image
this,
let
multiplied
first
by wal(h,®)
To
later.
demodulate
show
used.
A
for
F*(@)wal(j,6) first
= [F*(@)wal(jéh,8)]wal(jan,8)
There
require-
wal(jéh,é@):
8 )wal(j,6)Jwal(h,9)}wal(j@h,9)
carriers
spec-—
channels.
will
be given
and
a carrier
by
for
multiply asignal
carrier
frequency
channels
bandwidths
allocating
on
to
by multiplication
just
frequency
method
It is possible
with
channels
AAD
signals,
a
if
received
Walsh
signal
and then by
wal(j&h,9):
{([DT(6)wal(1,@)Jwal(h,
9) }wal(jéh,
@)
=
Con)
Dt(@)wal(1@h®j@,8)
= Dt(e)wal(1ej,9)
2k
>, ¢(r)wal (1¢j@r, 9)
I
r=0
Dt(@)wal(1®j,8)
through
=k,
absence
of
a
as
only
sequency
lowpass
the
image
signals
Walsh
function
of
Walsh
Amplitude
orthogonal
Most
but
of
theorems
Of
8
Waléh
can
may
(8)
the
that
have
be
well
better
known
produce
not
sime-cosine
Harmuth, Transmission of Information
of
be
systems
may
on
could
cut-off
is
traced
the
and
with
(18)
be
diagram
modulation
this
filter
that
pass
sequency
satisfied.
The
to the occurrence
right
hand
side
of
the
(1.29).
forthe
synchronous
demo-
carriers.
systems
other
or
theorems
shows
a block
component
condition
of
dulation
far,
as
one
Fig.49b
No
no
long
multiplication
way.
contains
functions
of
other
discussed
in very
shown
due
to
practical
have
or
functions,
two
terms
the
advantages
insufficient
an
functions
one
complete,
much
same
so
effort.
multiplication
asinthe
but aninfinite
case
series
of
or Walsh
strongly from that of sine-cosine
differs
dulation
mo-
amplitude
Their
one.
a complete
not
but
system
gonal
m
an orthofor
pulses
block
Carriers of periodic
terms.
TRANSMISSION
CARRIER
%.
144
functions.
3.12 Multiplex Systems
through
sed
modulated
a
via
after
plex
one
are
to
several
signals
such
Single
modulators
the
There
link.
at the
method
it
carriers
as
well.
m
modulators
The
the
receiver
phase
must
This
at
be
means
the
from
m lowpass
multiplex
same
synchronized
the
frequency
The
are
inserted
m
is
signal is multi-
carriers
The
that
were
carriers
to
those
in
must
be
right
suppress
multi-
non-sinusoidal
demodulated
filters which
signals of wrong channels.
the
transmitted
demodulation
to
transmitter.
difference very small.
through
applied
by the
principle,
be
frequency
Synchronous
be
mo-
sideband.
to separate
received
used for multiplexing
mitter.
can
amplitude
filters
one
suppress
receiver.
and
plied
in
sideband
methods
In
then
may
and
added
are
carriers
common
carriers.
sine orcosine
m
onto
dulated
are pas-
Signals
and
filters
lowpass
frequency
telephony
m
telephony.
for
system
tiplex
mul-
a frequency
Consider
in multiplexing.
is
modulation
amplitude
of
applications
important
most
the
of
One
the
in
transand
signals
the
pass
the contributions
A practical
frequency
mul-
tiplex system differs of course fromthis principle,
since
specific features of sine and cosine functions are utilized
in
practical
tures,
which
complete
The
of
modulation
carriers
are
amplitude
lation.
cosine
and
methods
Two
equal
Here
sine
systems
two
sideband
systems.
of
Two
of
known
in
frequency
by
bandwidth
are
is
on
share
those
fea-
with
other
and
single
functions.
quadrature
are
equal
signals
emphasis
functions
orthogonal
modulated
frequency
the
two
inthe
produced,
modulation
frequency
but
90°
multiplexing.
phase
independent
case
of
difference
Signals
quadrature
eachof which
has
of
modu-
twice
3.12
MULTIPLEX
the
SYSTEMS
bandwidth
than
in
the
Signals
of
a
of
occupy
one
of
sideband
a common
the
[11].
two
i.
either
cal(i,8@)
quency
sideband
quadrature
by means
voltages
and
a'.
periodic
The
modulated
at
Walsh
the
used
be fed directly
g*
of
of
available
the
the
sequency
may
denote
modulation
and
are
too.
sal(i,9)
is
single
carriers
means
that
sideband
to
appear
at
two
d'
lowpass
to
The
transmitter
that
LP,
step
The
at
a
delay
another
125
c'.
the
multiplied
Walsh
same
voltages
ap-
are
fed
equal
are
at
to
the
They
may
lowpass
fil-
b'.
The
and
and
voltages
b and
headset.
produce
produce
is
two
c
S
M by the
b'.
modulate
of the multiplier
g'
are equal tothose
into atelephone
in
a
fil-
b and
points
added
voltage
This
filters
transmitter.
lowpass
outputs
are
out-
points
amplitude
multipliers
g and
The
to
and
transmitter.
outputs
multiple-
Fig.50.
applied
applied
d and
for
modulation.
their
M
secar-
the
apart
of sequency
are
One
but
through two sequency
obtained.
receiver
inde-
modulated.
case,
refer
multipliers
the
by two
twice aswide
principle
carriers
in
at
h'
h and
outputs
those
e
the
in
one
modulation
spaced
microphones
carriers
sequency
through
ters
the
used
functions
pearing
a Signal
in Single
for
modulated
only
for
the
voltages
receiver
the
those
be
as
of
two
voltage
terms,
sideband
sideband
to
of Walsh
fedto
two
exist
generated ineither
is
have
of
Step
are
output
both
carrier
by
methods
cal(i,8)
sal(i,®@)
They are passing
LP.
better
amplitude
Single
explanation
These
modulated
methods
single
are
modulation
put
ters
one
is suppressed
several
functions
or
sequencies
xing
since
Only
for
Quadrature modulation means that cal(i,@)
signals.
pendent
and
sal(i,@)
as
For
Lacking
Walsh
sequency
well
at
modulation
quadrature
are
rier
bandwidth
suppression.
by
each
as
are
more
Signal,
band.
sidebands
There
No
per
frequency
generated
Corresponding
There
signals.
occupied
frequency is amplitude
multiplexing
them
is
modulation.
this
for
original
baseband
certain
and
the
415
of
us
125
us
delay.
and
The
these
delays.
Multiplexing of analog sigis no problem in modifying
transmission inone direction.
nals will be discussed. There
input and output circuits for
a
for
that
principle
the
of
two-wire
hybrid
frequency
line
coming
circuit
into
branch.
A signal
on
one
the
inputs
of
1024
sequency lowpass
filter
The
the
switches
of
a
the
from
a
the
and
a
than
transmitter.
filters
by
applied
It
sequency
are
split
receiving
branchis
with cut-off
lowpass
im-
more
subscriberis
transmitting
of
no
multiplexing.
transmitting
LP
are
multiplexing
sequency
or
time
omitted
are
details
they
since
of
digital
transmission
the
for
Such
signal.
discussion,
further
portant
The
telephony
PCM
the
from
channels
Seven
signals.
of ausual
are required
binary
of
transmission
for
channels
inputs of the
the
to
applied
are
only
-V
For
types of signals.
other
or
+V
voltages
the
instance,
for
Channels
1024
with
system
shows amultiplex
Fig.51
for
indicate
Fig.50
of
diagram
time
the
of
sections
dashed
TRANSMISSION
CARRIER
3.
AAG
to
passes
a
of 4 kzps.
driven
by
pulses
of the timing generator SG. The input signal F(§@) is transformed into a step function F%(6);
F(6) and Ft(9) are
shown
F(e)
in
After
to
Fig.52
filtering,
one
of
42
pliers
M.
The
are
Walsh
without
thogonality
the
the
delay
of
125
signal is amplitude
carriers
first
shown in Fig.52.
nal
us
between
four
171
142
carriers
Duration
interval
to
T and
coincides
modulated
inone
wal(0,6)
to
position
of
with
the
on-
of the multi-
wal(4,8)
their
stepsofthe
orsig-
F *(9).
32 modulated
S32
into
nels
is
but
and Fit(9).
one
each
are
chosen
as
choosing
size
carriers
group.
As
obtained
example
of
groups
are
combined by the
a result,
as
shown
32
in
groups
with
Fig.51.
only;
principles
and
supergroups
adders
The
for
42
81
to
chan-
figure
32
judiciously
will be discussed
later.
The
output
voltages
of
the
adders
are
amplitude
modu-
3-12
MULTIPLEX
SYSTEMS
,
AV?
1
'
oc’
fe)
a Ss
ee
t—_
Fig.50
quency
channel No.
Principle of a
lowpass filter,
Transmitter
sequency multiplex system.
M multiplier,
S adder.
Receiver
32 groups
32 channels each
T33-T64
LP
se-
channel No.
WMS T3Z
Fig.51 Sequency
multiplex
system with
1024
telephony
channels for transmission in one direction.
LP
sequency
lowpass filter; Mmultiplier;
S adder; TG, FG, SG trigger,
function and timing generator.
118
carriiers
of
i
Generation
5.
Table
3.
CARRIER
4
to
TRANSMISSION
multi-
the
by
T42
ce
wal(k,6)wal(1,@) = wal(j,@) and oe
ee
41,
)wal(
l(1l,
,@)wa
wal(k
tion
plica
multi
143 to T64 by the
ee
REMPBRRSSRRSAR
SORIAMARWNHO
SNH
eonnwne
COM>!
CHNORADID
wo pe
lated
ers
onto
M.
voltage
the
the
Adder
is
steps
Walsh
8343
carriers
adds
the
obtained
is
at
equal
to
tude of this
output
independent
amplitudes
the
sequency
nal
may
may
be
At
be
to
signals
LP,
O
which
with
pass
are
the
to
output
bey x125
per
sine
the
the
then
equal
The
to
or
Walsh
T™1
the
those
The
ns.
A step
width
The
ampli-
occupies
receiver
sig-
or
it
in
32
then
in
carrier.
multiplied
to
to
of
= 8 192 000
signal
the
134%
carriers
multipli-
This multiplex
first
carriers
to
* 122
Mzps.
is
through
833.
8000x(42)*
second.
signal
inthe
32 voltages.
us
directly
a
T64
of
assumes
= » = 4.096
modulate
M with
32 multipliers
ters
9
signal
receiver
multipliers
lated
the
transmitted
used
the
band
134
resulting
T64
T3432.
sequency
in the
and
The
demodu-
lowpass
transmitter.
fil-
3.12
MULTIPLEX
SYSTEMS
119
F(@)
F*(@)
OE
ees
een goer
i As ce! aes Sg aos
meee
eases a eee
wl Oe
wi.) EEE
tt
ee
ES
pal0 oe
8
SSE
ae
a
fer
afi fa ea
eS
0
eS ee Ea ee
125
ee Ee
ee ee ee ee ee
250
eS eee
375
us
500
t——_
Fig.52
Time
diagram
Table 6. The 2x42
tem of Fig.51.
for
the
carriers
multiplex
of the
sequency
function
calli,6), salli,@)
system
multiples
calli,8), salli, @)
RWHR
RHA
HAHAHA
BHCOIMBRA
Pee
eee
NNrFrR
OS
Om
OO
m
ol
cel
ce
|
ce)
ia
ian
sys-
CHONANRwWNHRO
Ree
Bee
eee
NOQOrwnro
pDnNnnwnnree
Sow
Rone
ror
boCoane
dec,|
binary
000000000
| 0000000000
16
000010000
2
| 0000100000
32
000100000
| 0001000000
48
000110000
0001100000
64
001000000
0010000000
80
001010000
| 0010100000
96
001100000
0011000000
001110000
0011100000
010000000
0100000000
010010000
0100100000
010100000
0101000000
010110000
2
0101100000
011000000
0110000000
011010000
0110100000
011100000
0111000000
011110000
0111100000
100090000
1000000000
100010000
1000100000
100100000
1001000000
ecoqgaononceccooaaonasoo
100110000
1001100000
101000000
1010000000
101010000
1010100000
101100000
1011000000
101110000
1011100000
110000000
1100000000
110010000
1100100000
110100000
1101000000
110110000
1101100000
111000000
1110000000
111010000
1110100000
111100000
1111000000
111110000
1111100000
46066004660666
-
oo
tooa
wal(j,@)
een
|dec. | binary
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
Fig.51.
function
wal(j,@)
ti
eee
ae
of
0
o)ra
o
@eaenreonmeeunsoaw®omaonaoanmnegogauagunoawnnwnwnonmaonxaonon
the
the
multiplication
-wal(1,0),
functions
by binary
ted
functions
digits
of
the
ry number
is
be
between
fitted
available
lost
always
sequency
the
Walsh
2'°
purely
elements.
this
to
To4.
of
the
The
Walsh
carriers
are
functions
tion
...,
wal(0,6),
cation
means
coset.
One
by
the
T32
may
To4.
The
are
progress
the
as
of
by
reordering
wal(0,@)
of
the
with
143
modulation
All
134% to
functions
6 with
a
cosets
wal (0,6) .n<
cosets.
wal(31,6).
are
32
carriers
the
carriers
of Table
...,
=
functions
32
The
with
wal(31,9)
2!'9/25
each one of the
wal(992,9)
no
be-
formagroup
number
of the
usable
multiply
to
to
there
to
to
are
generated
elements
T64
wal(1,6),
can
just
the
a
used
wal(0,@)
to
only
used,
last
channels.
be
There
functions
obtained by multiplying
wal(32,0),
may
the
as a bina-
T1
T4343
carriers
wal(1023,8)
133
wal(j,@)
that
way of chosing the carriers.
functions
Thisis
see
wal(j,6)
written
carriers
32
adjacent
to
2° elements.
subgroup.
--wal(31,8)
ble
empirical
may
completely
theory
wal(0,@)
The
with
is
between
group
functions
subgroup
of
of
The
two of the
band
bands
Concepts
yond
zero.
any
notation
sequency
normalized
sequency
One
sal(i,@).
cal(i,®@),
as
as well
five
inthe
table
Walsh
carriers.
these
Ta-
wasted.
bandwidthis
of
choice
cross-
so that no
must be chosen
showninthis
are
Fig.31.
in
sequency
no
and
genera-
be
can
functions
shown
T64
possible
a
6 shows
ble
as
to
1443
produced
is
talk
Rademacher
counters
carriers
The
wal(7,0),-..wal(2"-1,6)..
wal(3,6),
5.
Table
by
shown
is
Rademacher
the
from
(1.29)
theorem
of
by means
generation
Their
132.
to
™
carriers
for
used
wal(0,@) to wal(31,6) are best
functions
The Walsh
used.
carriers
in the
only
differ
methods
two
The
modulation.
sideband
single
for
as
aswell
lation
modu-
holds for quadrature
Fig.51
of
diagram
block
The
TRANSMISSION
CARRIER
3.
420
any
possi-
T64
are
wal(0,6),
one
func-
Sucha multiplielements
one of the
tions wal(0,@), wal(1,6), ..., wal(31,@).
multiply wal(42,6 ) with these 32 functions,
of
each
32 func-
One may further
then wal(64,¢@),
3.12
MULTIPLEX
SYSTEMS
ja
etc.
The
are re
atotal of 323? = 2'®8
such products, which
means there are 2'®8° possible choices of carriers 13% to
Te4,
none
duce
crosstalk.
of
Fig.52
which
shows
carriers
wal(0,@)
riers
11
to
be
a more
of
lated
132
sequency
bandwidth
sine and cosine
carriers
besides the Walsh
to
in
wal(3,@).
Fig.51.
The
this
having
car-=
have
The
32
to
to
modu-
pass
a
modulation
sequen-
and
frequency
between
as
would
case.
without
or pro-
them
type of quadrature
connection
close
may use
multipliers
in
added
This
filter.
One
type
could
be
sideband
the
shows
waste
complicated
carriers
single
would
cy multiplexing.
One
May
functions
Mee O or
Signals
readily
that
Lie
is
carriers
F(6)
are
doesnot
hold
have
multiplication
or-Walsh
Fig.51
1024
nization
can
Ofepne
sequency
134
similar
used
for
output
voltage
channels.
tion
tudes.
of
The
the
of
be
input
filters.
Té4.
They
those
of
must
sine-
can
a
synchrowhich
is
vol-
constant
of
Tracking
with
relative
statistical
a
form
times
averaging
long
by
the
shifts.
built
functions
suppressed
of
of
orthogonality
to
be
between
more
a
if
The
invariant
Walsh
be
line
or
wal(2"-1,0),
2”.
channel
them
onto
one
filters.
Let
Fig.53.
of
of
is
signal
the
is
is
rise
v,(t)
multiplex
steps
completely
v,(t)
and
signal
the
a telephony
width
signal
If the
to
transmitted
ation
ents
for synchroniz
Requirem
from
lowpass
to
can
if. the
transmission
function
is
the
to
can
tracking
inferred
system
interval
synchronization
functionsis
and
background
any
Fig.51
Actually,
A Walsh
modulated
The
ease.
be
lock
that
filters
by
of
carriers
extra
function,
Rademacher
the
the
that
a finite
(T52
theorems
an
applied
is
in
to
receiver.
signal.
a Rademacher
tage
Fig.52
functions.
and
channels
7]
for
shows
transmitter
from
filtered
This
eosine
see
orthogonal
122
represent
with
system
ns.
contained
transmitted,
may
times
The
in
the
1024
informa-
its
it
be
ampli-
suffices
—
the
absence
steps,
in
in
the
nization.
exactly at the
rise
time
In
error.
must
is
be
thus
than
Amplitude
ns,
rise
general,
less
presence
points
122
122
time
is
of noise.
However,
to v,(t)
tegral
over
taken
Crone
tOnv «Ceol
circuit
va(t),
cal
which
can
shown
of
do
this.
ry
the
due
to
emission
Consider
modulation
and
the
ter
are
in
plus
synchronization
no
detection
one may
error
method
readily
see
by integration,
e.g.,
Fig.543
without
The
within
is
maximum
The
...
synchronization
is
from,
frequency
can be solved
field
ZT,
sampling
by
T
in
that
since
to
Of
v,(t)
the
is
the
in-
propor-
evel
theoretically
problem
in Fig.53.
there
a poor
can be reconverted
The
O, T, 2T,
if
so
ns.
sampling
v,(t)
regained
be
may
voltage
step
original
va (t)
v,(3T),
to
v.(2T)
from
The
be
v,(2T),
to
by va(t)
shown
as
etc.
v,(T)
from
change
to
ns
122
that it takes
slow
synchro-
time
rise
the
Let
time.
Consider the rise
the
for
interval
tolerance
thisisthe
and
vals,
inter-
long
ns
422
the
in
anywhere
done
be
may
sampling
The
information.
the
all
obtain
to
order
of
amplitudes
the
sample
to
-
noise
of
TRANSMISSION
CARRIER
3.
Wee
use
wider
of
a
transistor
for
time
in
Walsh
functions.
signal
F*(@)
filtered
on
top;
the
Itisa
with
into
classi-
approximate
a filter
which this problem
variable
the
The
by
element
of
Single
original
a sequency
time
theo-
the
filter.
discussion
of
shown
to
v.(t)
framework of sequency
F -
a
transform
any ringing.
theory
simplicity
the
Fig.54
will
shift
sideband
signal
F(é)
lowpass
fil-
between
F(98)
and
F%(@)
is omitted. The Walsh carriers wal(0,@), wal(2,6),
wal(4,6), wal(6,8),.......
are shown. Their time base is
250 us, whichistwice the durationof the steps of Ft(9).
The
filtered
expression
F¥(6)
signal
in
the
interval
= c(O)wal(0,6)
8 = (t-t,)/T;
t+
F"(@)
is
O
represented by the
= t <250
following
Us :
+ c(1)wal(1,6)
= 125 us,
T = 250 us;
(25)
+ s9<
}
3.12
MULTIPLEX
SYSTEMS
Vel2T)
sipleaelll”
123
wat)
F(Q) ——
Valt)
:
VelT)
_ vet)
velaT)
ny
FM)
Vel4T)
volt) el
wal (0,) | Sa
L___
ca ee)
ee es)
hE
a
ee
|,
a eee
wall6,8) ALL
FER ESF
T
0
el
-wal (2,8) ———_+—_
+ }__+_
20
ees
i208
Fig.53
(left)
Finite
rise time
of astep
perio
the conversion.
of) va(t)
also-reconverts v,(t) into v,(t).
Fig.54
(right)
multiplex
tools
mo lmen
Amplitude
of Fig.54
are
functions
occupy
Fig.54
250
us.
also
quadrature
is
the
just those
in
the
shows
single
between
Fig.54.
whole
sine
Walsh
function
sequency
filters
Ts
is
P5Ouus -
carriers
Walsh
function
carriers
Hence,
functions
the
wal(2j,6)
with
modulation.
functions
sine
of
on
a Walsh
the
as
(25),
Ft(6)
of
wal(1,@),
which
does
a
by F%(@)
sideband
not
(26)
wal(2j+1,0)
modulated
sequency
and
signal
The
geries.
Te
fil-
filter
+ ¢(1)wal(2j+1,0)
available
modulation is based
first
lowpass
the
Their amplitude modulation
a (frequency)
dence
of
= c(O)wal(2j,6)
out
sideband
the
Carriers
one
and
The
yields:
produces
left
for
anne
of
function
v(t).
of asingle
base
DosemrOr
by Ft(e)
Modulation
diagram
Time
modulation
Pf (6 )wal(2j,6)
that
Time
system.
into
time
The
fact,
well
as
however,
band.
base
doesnot
in
Walsh
of
yield
correspon-
the
case
of
that
wal(0,6)
of
a Fourier
contains
the
to
a
Fourier
Fig.54
be
increa-
belong
series.
Let
the
time
base of the
carriers
in
special
case
of
of
time
the
the
single
class
the
base
of
sequency
which
can
with
signals
be
frequency,
whichis
signals,
times
separable.
herently
to
lead
to
those
need
Since
marked
to
are
other
some
by
their
frequency
The
yet
as
the
sideband
systems,
the
is
to
of
or
time
is
signals
Time
various
marking
in
signals
sequency,
one
are
that
are
fretheir
mul-
unknown
order
will
with
that
times.
having
networks
radio
by
multiplex
multiplex
systems
ease
marked
delay
hand,
the
reason
inherently
additional
sequency
not
equal
networks
independent
communication
for
systems
separated.
onthe
have
appears
of frequency multiplex
communication
signals
is
of
filters.
multiplex
and
multiplex
tiplex
delay
in
combined
quency
feature
time
single
of
carriers
lowpass
A characteristic
compared
the
drawbacks
and
modulation
T =
for
only
systems
sideband
Quadrature
investigated.
where
base
possible
many
these
been
Advantages
......
2,
1,
0,
j =
us}
are per-
wal(4j,9)
wal(8j,9)
and
us,
T = 500
for
only
mitted
carriers
The
side-
single
sequency
of
examples
systems are obtained.
band
4000
more
Two
Ft(@).
signal
changing the
without
us
1000
or
us
500
to
us
250
from
sed
TRANSMISSION
CARRIER
3.
der
to
also
expect
very
be
in-
them
similar
signals.
supergroup B, 64 channels
ae
u®)
cane®)
supergroup A, 32 channels
es
a
256 = 320
Fig.55 Occupation
wide base bands.
Fig.55
er
je
3h~SSAB KapsSIZ
of sequence
bands by multiplexi
y Muitiplexing
y
q
shows a possible
sequency
allocation
for
4
kzps
groups
3.12
MULTIPLEX
and
supergroups
tion
B
is
are
so
of
band
64 to
128
= cess
Khe
group
band
from
192
to
A in
Modulation
wal(128,6@)
wl2ekzps.
of
the
Se,
sieval
100
of
a
band
in
kzps
and
from
func-
Single
assumed.
Oto4kzps.
the
are
sequency
wal(32+2j,0);
wal(96,8)
the
interval
shifts
42
256.
are
channels
ty
a
from
128
into
the
agroup
wal(192,8)
by
256
kzps;
the
are
G2,
384
A into
64
Table
band
are
marked
shifted
fromthe
“10°
is
of
the
shows
by the
are
the
384
denoted
by
shifted.
The
16
wal(j,9),
anstance,
“onto
is
from
position
for
modulated
A
carrier
which
carriers
baseband.
occupies
the
band
to
are
wal(50,6)
signal
a supergroup
channels
7
channels
sequency
the
to
supergroup
resulting
channel
The
into
These
from
Fig.55.
of agroup
in
band
carrier
a
2450e8>.,
wal(50,6).
or
carrier
wal(64,8)
Walsh
us
occupies
carriers
the
kzps.
the
sequency
channels
jue
256
The
B
128
alloca-
supergroup
Fig.55.
shifts
supergroup
of
basebands
that
The
of
the carrier
Shifts
it into
Go
kzps.
the sequency
kzps;
supergroup
occupy
modulation
shifts
192
32,
This
A or
subgroup
than
make
a group
Od
network.
supergroup
and atim
base
e of 250
channels
from
Amplitude
group,
mathematical
modulation
basebands
a)
et
communication
wit
j smaller
h
individual
Sixteen
a
W2D
that
the
wal(j,@)
sideband
The
in
chosen
cosets
tions
to
SYSTEMS
the
equal
band
the
carrier
to
from
2x50
100
=
to
104 kzps. The carrier wal(50,6) becomes wal(82,6)
by multiplication with wal(96,6), or wal(114,6) by multiplication
the
with
band
2x114
=
wal(64,9)
from
228
wal(210,0)
cupies
to
the
of
292
kzps,
2x244
488
kzps.
and
a
group
supergroup
164
the
Finally,
in
of
=
556
a
Channel
kzps
the
or
10 occupies
the
carrier
B.
bands
kzps,
transmitted.
The
with
B,
from
wal(50,¢)
Channel
a
Signal
10 oc-
lower
= 420
2x210
supergroup
band
wal(178,4),
wal(146,@),
supergroup
wide
4 kzps
case
A.
‘168
carriers
2x178.
being
to
kzps.
the
eyqiGu=
Consider
=
wal(244,@)
or
one
232
of
one
becomes
in
2x82
limit
or
kzps
supergroup
occupies
A
the
Bi
eo pear
= OV) ae
512
kzps
the
carrier
256
<@
the
band
<
0 <
» <512
band
(j = 192,
<»
< 256 kzps
the
extract
wal(64,6)
which
the
supergroup
(j = 64,
<@
for
band
of
a
<
128
a
B be
484
the
band
190) is transposed
into
that
the
126).
...,
sequency
bandeset
the
into
A sequéency
128
of
can
kzps
A multiplication
kzps.
the
to
from
Bis multiplied
shows
254) is transposed
cut-off
it
shift
=
from
band
the
2. .9eG2)5
0,
...,
a
O
band
can
is
Let
having
filter
7
...,
(j =
keps
kzps
128
lowpass
2! x4
band
the
Supergroup
Table
(j = 128,
©» <128
equal
in
in
channels
wal(128,0).
< 384 kzps
to
transmission
the
channels
32
32
shall be extracted.
to
by
is
filtering
of
is
sequency
example
the
the
or
kzps
484
to
Either
B.
supergroup
256
kind
extract the group
wanes
further
a
as
Consider
simple
cut-off
the
if
only
possible
sequency
This
A.
supergroup
the
and
cut-off
kzps
256
with
ter
will
fil-
A lowpass
kzps.
128
of
sequency
cut-off
with
filter
lowpass
sequency
a
of
by means
group
the
extract
may
One
Big. 5d5¢
to
according
kzps
512
to
64
from
band
sequency
TRANSMISSION
CARRIER
3.
‘N2e
band
< » <
128
by
kuapa;
256
A.
supergroup
multiplied
by
wal(192,8)
instead
of by wal(128,0). The band 256 < » < 384 kzps (j = 128,
---, 190) is transposed into the band 128 < » < 256 kzps
(j = 64,
2-2,
...,
126), the band
254) into the
A sequency
128
lowpass
kzps,
group
which
filter
now
individual
be
542
< 128kzps(j
can
contains
sequency.
any
position
with
the
dual
channels
The
in
proper
and
can
by
extract
the
other
the
band
be
or
lowpass
filtered
the
kzps
(j
= 0,
...,
the
band
channels
Walsh
groups
in
time
of
all
may
spectrum
carrier.
The
2j
<9
with
filter
signal
sequency
remodulate
done
in
multiplication
by a sequency
off
dulate
channel
extracted
filtering
what
0 <9
< » <
=
0
192,
62).
<®
<
of super-
B.
Any
can
band
384
<
2ejt+4
kzps
wal(j,9),
and
having
then
by
4 kzps
be
shifted
to
multiplying
it
extraction
channels
without
channels
multiplexing.
is
of
indivi-
need to demo-
very
It
cut-
may
similar
to
be
to
used
3.12
MULTIPLEX
Table
of a
SYSTEMS
dey
2 Transpositof
ion
the carriers wal(32,0)..wal(62,9)
group to the carriers wal(64,6)..wal(126,0)
of a
Supergroup
A
and
a Supergroup B.
Zeek epoe Ore tos
the
The
25),
carriers
sequency
its.
of
wal(128,0)..wal(256,6)
the
carriers
is
equal
of
to
supergroup A: 2 groups
group:
16 channels
carrier wal(96,8)
96=1 100 000
carrier wal(64,6)
64=1 000 000
7
Zz
2
3
4
5
6
7
8
32
100
100
100
100
101
101
101
101
110
110
110
110
111
111
111
111
000
010
100
110
000
010
100
110
000
010
100
110
000
010
100
110
66
68
70
72
74
76
78
80
82
84
86
88
90
92
supergroup
1 000 010
1000 100
1000 110
1001 000
1001 010
1001 100
1001 110
1010 000
1010 010
1010 100
1010 110
1011 000
1011 010
1011 100
1011110
|
|
|
|
|
|
|
B;: 2 supergroups
A
carrier
wal(192,8)
192=11 000 000
10 000 000
10 000 010
10000100
10 000 110
10 001 000
10 001 010
10 001 100
10 001 110
10010000
10 010 010
10 010 100
10 010 110
10011000
10 011 010
10 011 100
10011110
route
tion
It
|
|
;
channels
that
11 000 000
11 000 010
11 000 100
11 000 110
11 001 000
11 001 010
11001100
11 001 110
11 010 000
11 010 010
11 010 100
11010110
11 011 000
11011010
11011100
11 011 110
202
2
y
2
|
}
2
22%
through
3
|
p
2
y
y
|
|
2
switched
a
edthe channels
so farthat
assum
supergroup
holds
exchange
case
10 100 000
10 100 010
10 100 100
10100110
10 101 000
10 101 010
10 101 100
10 101 110
10 110 000
10 110 010
10 110 100
10 110 110
10 111 000
10111010
10 111 100
10111110
11 100 000
11 100 010
11 100 100
11 100 110
11 101 000
11 101 010
11 101 100
11101110
11 110 000
11 110 010
11 110 100
11 110 110
11111 000
11 111 010
11111 100
11111 110
communica-
[14].
into
assumption
the
|
been
has
combined
same
|
individual
network
carrier
wal(128,6)
128=10 000 000
|
|
|
1100 000
1100 010
1100 100
1100 110
1101 000
1101 010
1101 100
1101110
1110 000
1110 010
1110 100
1110 110
1111 000
1111010
1111 100
1111110
98
100
102
104
106
108
110
112
114
116
118
120
122
124
A
if all
true
into
groups
channels
are
or
B are
channels
and
groups
This
synchronized.
are
combined
supergroups.
combined
and
into
Now
in
the
consider
at differgroups
are
synchronized.
groups
same
groups
these
these
that
the
have
j
|
j
wal (27,6).
LJ
LI LI
cannot
assume,
|
that
assume
iu es) is O) ANSE
|
j
|
We
j
|
wear
Th
wath Ohm (le Fw Le
foe ee
wal (96,4)
ey
LJ L—J
however,
base
time
wal(34,0) |
wal (63,8) |
into
may,
One
super-
combined
One
exchange.
level
ahigher
at
groups
are
groups
these
and
exchanges
ent
TRANSMISSION
CARRIER
3.
128
1
LJ
grup4 ay 1 T 2 ROSS
5 1 6 S888
group2 a 1] 2 ROSSA
5 T 6 SYSS8y
a
eel
a
ee
es Se
ee
grup! cA
4 72 RSS
5 T 6 BSS
group2 cL 4 | 2 K3NSN 5 T 6 SSS
group! f SSNS
7 T 2 BSS
5 TO
group2 FMT 4 TP2 BSR
5 TT 6 B87
wiiter=
lowpass
=e
Lee
P2222 LLANELLI
LBS
SS
Se
a
0
0
4/32
7.8425
ee
4/46 @ —= 3/32
45.625
23.4375
T= 250 ps
SS
4/8
34.25
pos
=
Fig.56 Principle for the combination of two non-synchronized groups into a supergroup A according to Fig.55.
The
time
base
figure
and
combination
of
unsynchronized
will be discussed
shows
on
wal(127,6)
top
in
the
the
with
groups
reference
Rademacher
interval
with
to
equal
Fig.56. This
functions
wal(31,0)
O = @ < 4. The
multipli-
and
These
cations
wal(127,6)wal(63,@)
= wal(64,6)
wal(127,9)wal(31,8@)
= wal(96,0)
yield
the
to
the
functions
carriers
one
supergroup
ference
Lines
for
a of
ding
functions
to
for
wal(96,@).
transposition
A according
to
of two
Fig.55.
are
groups
in-
They are the
re-
synchronization.
Fig.56
non-synchronous
the
wal(64,8)
required
show
groups.
wal(32,@),
Figs.54
and
55;
symbolically
These
signals
the
wal(43,0),...,
the
amplitude
signals of two
consist
of
wal(63,8)
of
these
sums
of
accor-
functions
Fe(ee ULL
EPLEX
SYSTEMS
Wes,
depend
ons
the particul
signal transmitted
ar . Fig.57 shows
that a signal containing the functions wal(32,6),wal(33,6),
---,
wal(63,9)isastep
odd
with steps 1/64 wide. The
wal(35,8),..., wal(64,9) are not
wal(33,8),
functions
in Fig.57,
shown
function
since
they
Say Gi skeveGOm
only
differ
4)
cua
the interval -3 = 8 < Ofrom the even functions wal(32,8),
wal(34,0), ..:, wal(62,0). The signals in the Vines a 6f
Fig.56 are divided into intervals 1/64 wide. Their amplituaes
these
in nt
consta
are
Seivcue
Fig.57
sare
signal
a
that
val
-2 +2
the
amplitudes
Sign
must
< -4 +
-$ 26
in
2k
any
two
fl
8
of
Fig.56
value
of the signal
The
amplitudes
Ta
Ghestupervals
show
such
eal
Fig.57
9
Walsh
Harmuth, Transmission
generally:
and
opposite
1 amd
Cae
intervals
alternatively
absolute
2,
5 and
of
equal
absolute
hatched
and not hatched.
value
and opposite
4,
4
2.<_<
(mas
holds
oF Ga’ '* < en ten
have equal
a
inthe inter-
-v
value
al ll eGRies
Meecaecumi.
Bines
wal(32,9),
v in the interval
result
absolute
in-
intervals
2k
ce
This
same
the
have
functions
amplitude
the
< -# +2.
£86
the
individual
from
see r
furthe
may
amplitude.
the
have
The
One
“2.
containing
having
and
WoL (65,0)
wey
intervals.
2,
1,
by
aenoved
5 and
6,
——E——————
sign
etc.
os
————— ee
a
a
ga
ieee iia
functions
pies
of Information
a
ee
Pr
Snes
a
eek ae (hued
9
Walicqeon
see
rea
ness
ry
wl?
C21 ;0)) =
seawall
Gl.
cal Gi,d).
trigger
by the
riers
sufficient.
yet
not
the
O with
absolute
equal
which
for
ling
this
and
so.
1 has
circuit
to
that
groups
correctly
but
f and
reference
to
arbitrary
shift
groups
since
each
by
other
any
not.
line
wal(96,8).
from
that
shifted
with
1/32.
of
of
Such
an
channels
in
division.
Written
symbolically,
following
(group
yields
1)wal(64,9)
the
the
signal
of
supergroup
of
+ (group
supergroup
following
two
2)wal(96,6)
A
by
wal(64,@)
or
wal(96,80)
signals:
[(group
1)wal(64,8)
[(group
= (group 1) + (group 2)wal(32,6)
1)wal(64,6) + (group 2)wal(96,8)]wal(96,6)
=
One
of
pressed,
the
in
It
is
(group
2)
quence,
+
(group
terms
order
on
to
easier
than
(group
the
to
are
2)wal(96,9)]wal(64,6)
2) +
the
(group
right
obtain
others.
Thisisof
(group
obtained,
sides
1 or
the
terms
no
may
be
[(group 2)wal(32,@)]wal(32,0)
1)wal(32,0)
wal(32,6)]=
= (group 2)
(group
1)
be
sup-
2 separate-
(group
practical
wal(42,6):
[(group
must
group
2)wal(32,0)
which
=
1)wal(32,6)
hand
group
suppress
sinc
the e
terms
wal(32,0)
A has
form:
Demodulation
ly.
f.
2 is
and
aninterchange
cause
would
multiple
samp—
2 from
differs
2 maybe
a
lines
group
wal(64,6)
1 and
of
group
problem of synchronization
the
by
signals
the
takenfromline
be
of time division,
the
yields
the
have
intervals
two
with
1 begins
Shifting
modulation
of the carriers
for
time
not
is now synchronized
1
Group
c
Group
is
holding
Group
Note
value.
is
6 =
time
amplitudes
2 in whichthe
1 and
at
c begins
line
2 in
Group
intervals
synchronization
This
wal(96,@).
and
wal(64,6)
are
They
c.
lines
of
signals
car-
the
with
synchronized
the
yields
1/64
of- duration
interval
an
during
voltages
sampled
the
holding
and
indicated
times
the
a at
lines
als
signof
b
line
pulse of
the
Sampling
TRANSMISSION
CARRIER
3.
130
and
1)
or
conse—
(group
1)x
demodulated
by
3.12
MULTIPLEX
The
a
terms
sequency
Vetoeeo
tion
SYSTEMS
(group
lowpass
lvone
1) and
filter
68 = 1752501752
(group
that
<
duration
channel
telephony
tend
over
two
4 in
line
f,
equal
absolute
gration
value
yields
zero
and
and
wal
4 and
4,
Pores
wal(42,6)
52,0)
makes
etc.
equal.
Fhe
asienalis!
pass
the
signals
ecules
siens
wine
(sroup
the
Fig.56.
us’
¢c andof
1024
the
ex-
group
have
signals
the
inte-
suppressed.
teandeeroup
Aim
intervals
2)wal(42;0)
Hie. 56
Their
for-a
intervals
cancellation
sequency
dintegra-
Hence,
are
in “the
Thereisno
through
of
sign.
opposite
Muka pLrcabloniOneroup
Booy
of
line
2 in
amplitudes
the
These
integration
group
of
the
which
line
by
over
the inter-
etc.
to°7s6125
These
system.
intervals
in
last
equal
ig
suppressed
integrates
0 = 2/32)
inter
arevals
shown inthe
Hhon-normalized
2) canbe
by
sors
lowpass
Oscillograms
Pine
1 and
2;
integra-
-(eroup
1)
filter.
of
sequency
filver:
nC: carrier
wal(>,o
3D:
first modulation
Ft(0O)wal(5,6);
E: carrier
wal(9,@);
F: second
modulation Ft(6@ )wal(5,0 )wal(9,6 )
=Fit(¢@ )wal(12,8); horizontal scale
50 us/div.;(courtesy H.LUKE
and
ReMALLE.
of
AEG-Telefunkeéen AG)*
Fig.58
system
is
g*
developed
shown
for
oscillograms
some
shows
by
LUKE
clarity
and
MAILE.
instead
of
of asequency
The
one
carrier
of
the
multiplex
wal(5,98)
carriers
—55
about
to
and
dropped
was
transmitted
by
a
of Fig.55.
-5%
dB
with
the
telephony
tracking
would be high enough to meet
compandors
are
used. However,
mainly of interest
at
present,
the
than
enough
system
Post
is
peak
-54
in this
power
case.
by
digital
of
the
is
more
multiplex
sequency
HUBNER
is
signals
attenuation
advanced
An
if signal
multiplexing
limited
crosstalk
dB
developed
being
Office
for
and
attenuation
standards
sequency
signal
extracted
and
This
filter.
used
was
line
signals
telephony
was
equipment
this
in
if the synchronization
about
function
Walsh
or wal(64,6)
synchronization
extra
an’
if
dB»
modulation
obtained
attenuation
crosstalk
The
wal(96,@)
than the carrier
rather
second
the
for
shown
is
s wal(9,8)
(orsuauaulya
iglalew
Bor tie salic™rearon,
Im Figs50°
Wal (62,6)
s00))
Wal(42,8),
TRANSMISSION
CARRIER
3.
Age
West
German
Department.
3.13 Digital Multiplexing
It
has
been
based
on
gital
filters.
pointed
Walsh
Since
shifting of signals
expect
that
telephony
before
can
be
bandpass
just
sequency
easilyby digital
two
out
functions
as
equipment.
sequency
filters
multiplex
multiplex
signals
that
implemented
require
systems
systems
Consider
according
to
filters
easily as disequency
do, one will
can be implemented
the
multiplexing
Fig.50
for
of
illustra-
WAL Owl,
Two
signals
These
signals
Fig.50.
The
interval
125
form
an
by
digital
F,(6)
are
and
amplituedes
us<
F,(@)
represented
t <
ina
250
analog/digital
representation
are
by
to
the
particular
be
+110110
for
a
and
interval,
us, are transformed
converter.
multiplexed.
curves
Table
FY(8)
a'
say
in
the
into digital
8
and
lists
the
-011010
for FM). Multiplexing of these two values will be discussed with reference to Table 8. It is assumed that F,(06)
and F,(@) are signals
of an 8-channel multiplex system.
The
8 Walsh
riers.
Only
functions
two
responds to an
of
the
activity
wal(0,8) to sal(4,8)
channels
factor
carry
of
0.25,
are used as car-—
Signals.
This
cor-
which is represen-
3.13
DIGITAL
Table
8.
MULTIPLEXING
Digital
sequency
FY(8)
and
to
respective
the
135
F(@)
multiplexing
according
line
of
two
signals
to Fiet50.hc,bcly. seeirefer
in Fig.50.
F(@)
stands
forthe
sum
@fer™(o )eak(4,0) ——
+
——
2 pleco
Te
sal(1,9)|sal(3,0@)
-F(6@ )
-011010
+011010
+011010
-011010
+011010|
—011010
-011010
+011010
|+0011100
|+1010000
|+1010000
|40011100
-0011100
|-1010000
|-1010000
|-O011100
eerste
sal(1,9)
sal(3,6)
+0011100
+1010000
+1010000
+0011100
+0011100
+1010000
+1010000
+0011100
+110110000
+110110
tative
for telephony
multiplex
channels
-sal(1,@)
and
during
peak
traf-
Bie
The
two
sented
by
anudec
sot
carriers
8
digits
Table’)
+1
co.
or
The
-1
-sal(3,6)
as
shown
negative
sign»
of no importance
here. The carriers
amplitude
numbers
shown
of
modulated
+110110
inthe
column
Same
sline
presented
by F7(9)
and
columns
e
is
d and
obtained
tinicodumnrc
by numbers
and
-011010
by
Fi(9)
The
the
The
one
ithe
columns
c
carriersis
yield
by
two
or
digit
the
-1
signal
numbers
multiplex
more
8 times
+1
multiplex
adding
¢'.
having
of)
the
-sal(1,9)
and -sal(3,8)
multiplied
d'.
and
in
can be repre-
as
-F(9)
of
the
signal
is re-
than
Ff(@)
or
BECO).
The
8X8
-F(9@)
may
could
use
this
xing
would
case,
and
be
channels,
digits
of
-
digits
be
block
sequency
time
one
each
with
pulses
division
for
division
same
the
if
in many
transmitted
64
the
including
number
parity
channel.
.as
check
Such
ways.
of
-
sign
For
amplitudes
would
be
used
transmission.
in
time
digit
a
signal
one
-1.
In
+1
or
for
multiple-
The
64
pulses
multiplexing
of
8
to
the
7
were
check
the
instance,
added
digit
would
permit
correction.
error
no
but
detection
error
single
TRANSMISSION
CARRIER
3.
134
Demodulation of -F(9) is done by multiplying -F(9) with
-sal(1,9) and -sal(3,6). The resulting binary numbers are
shown in columns g and g'. Integrationof F(e )sal(1,6) and
F(@ )sal(3,9) means adding the 8 numbers in columns g and
8 yields the original
tical
to
of
the
principle.
The
signal
value
and
-F(@)
0011100
two
of
active
and
each
to
correct
errors
tage
over
useful
time
be
1010000;
can
errors,
corrected.
time
division.
information
if
the
be
numbers.
two
can
contains
Hence,
it
unchanged
would
is
activity
obscure
only
numbers
signs.
if
one
This
number
corrected
In
with
in
There
The
many
is
cases
and
two
to
the
also possible
more
than
that
fourths
of
time
two
advan-
reasonis
three
0.25
due
with
a definite
underlying
is
signs
for
changed
itis
thus
transmitted
factor
negative
comparison
cases
this
absolute
typical
is
by
most
and
is
of
explanation
the
two of eachhave
positive
channels.
interference
three
method
Inclusion
-F(6).
to
section
of
transformation
signal
be
course
of
would
numbers
these
multiplex
the
time-consuming
less
values
by
pracs
of rite) and P(e) = The
Walsh-Fourier
fast
the
apply
1.25
obtain
to
way
Division
-011010000.
and
+110110000
yields
which
g',
no
the
division
is
used.
A considerable
number of variations
quency
multiplexing
scheme
gated.
However,
possible
great,
thatno
about
their
the
definite
relative
of
Table
of
the
8 have
number
of
conclusions
digital
been
investi-
variations
have
been
se-
is
reached
so
yet
merits.
3.14 Methods of Single Sideband Modulation
Amplitude
a double
modulation
sideband
theorems
of these
for
the
elimination
very
well
functions.
of
by orthogonal
Consider
two
of
sineorcosine
modulation
one
due
There
to
the
are
sideband
carriers
a numberof
that
yields
multiplication
canbe
methods
analyzed
functions.
transmitters,
both
radiating
sinusoidal
4.14
METHODS
functions
of
gm.
OF
of
SSM
frequency
The carriers,
Ft(@)
and
D*(@)
Ve sin Q,9-
be
a
Dt(@),
reproduced
phase
Q,),
but
having
amplitude
shall
It
have
is
exactly
form
Ft(6)
assumed
that
the
the
a
receiver,
between
and
V2singn,@
a phase
difference
modulated by time
the
at
difference
V2cos 9,
frequency
but
the
that
Q,can
there
received
locally
the
functions
Ve cos y,0 7and
is
carriers
carriers
produced
V2 sin (Q,8+a). Multiplication of a received
V2 cos (099+a),
Signal S(6),
S€e)
“WS
= F'(6) Y2cosn,0
by V2cos (Q,8+a)
+ Dt(e) y2sina,6,
and [2 sin(0,9+a)
S(6) V2cos (N,@+a)
= F*(9) cosa
+ F'(Oicoe
S(o Ve sin (0,e+000=
C27.)
yields:
+ Dt(0) sina +
(2iigesa ) +
—F'(oy.sina
(28)
D (8) sin (Q,8+a)
+ DiC) cosa
+
(29)
+ F'(@) sin (20,6+a) - D*(8) cos (20,6+a )
The
termsonthe
tiplied
by cos(2n,@+a)
frequency
components
filters.
only
The
if
riers
the
of
cy
phase
two
they
(28) and (29)
mul-
contain
high
shall
be
Ft(@)
as
Hence,
difference
signals
Ft(9@)
and
into two phase
and
cosine
car-
#n may
Dt(6)
without
each frequenchannels
channel.
utilizationof
by
Dt(@)
two
phase
sine
permits
or
a
Putting it differently,
here
very
suppressed
then
a vanishes.
but
subdivided
demodulation
of
contain
difference
frequency
canbe
sides
sin(20,@+a)
independent
will be denoted
nous
or
only;
interference.
channel
hand
right hand sides
equal
transmit
mutual
right
both
which
Synchrochan-
phase
nels.
through
the
channel
in
for
order
instance,
quire
low
sine
a
that
certain
to
replace
Ft(é@)
and
them
make
in
Ft(@)
D*(@)
frequency.
modulated by the
but
channel,
be
through
never
transmitted
One
may,
and
re-
no energy
practically
be-
be
de-
(29)
signal
circuit of Fig.59.
cosine
the
distinguishable.
have
The
always
may
function
time
A certain
by 1 + F*(@)
S(8@)
The
may
signal
then
[1+MF‘(®)]x
%.
(14+ cos 2298)is obtained at output
1, and
4s
au
Obtained
Single
To
show
this,
of
sine
and
let
cosine
Dt(6)(1+ cos Q,8)
2.
to
according
pulses
channel.
into aseries
expanded
be
F(9)
practical
cosine
and
sine
through
a signal
excellent
an
is
modulation
transmitting
for
means
outvpuu
sideband
TRANSMISSION
CARRIER
136
Fig.1:
(30)
B(g)=a(0)£(0,9)4V2 5’ Lac(4) cos 2mig+a,(i) sin 21ie]
jz
ap SS 5
Fourier
transforms
pulses
shown
are
g(0,v),
given
by (1.24).
@c(i,v)
The
and
first
g,(i,v)
five
of
these
transforms
are
in Fig.6.
feedback loop
Sine.
(Mra ins0,e—L-!2sin(248+cx)
Ty
TM output
[I+MF*(@)}+c0s20,8)
Fig.59 Correction of the phase difference between received
and local carrier
V2cosQ,8
and V2cos (Q)84+a). Mmultiplier; LP frequency lowpass filter; HFO high frequency
oscillator; PS
variable
phase
shifter; PD
fixed phase
shifter for 90°
Let
us*denote
V2sinQ)8,
cos Mg8 and
dois (8),
these
products
dsc (8) and dgjs (0).
boc(v) =
are
denoted
8V2lg(0,v-v,)
hejic (Vv) =
= Q,/2n
The
signs
2nie
sinNyd,
The
2sin
2niex
dog (8), duc (8),
Fourier
by hg <(v),..-.,
&
£(0,8)x
transforms
of
hss (v):
g(0,v4y,)]
Coy
#V2lg.(i,v-v9)& Sold svtVed
=) 8V2le,(i,v-v,)
Vo
£(0,9)V2cosQ,9,
2cos
2sin 2ni@ sin ® by doc(8),
products
hsic(v)
the
2cos 2718 cos N98,
in
parenthesis
+ gs (i, viv,)]
hold
for
the
Fourier
transforms
3-14
hoy
METHODS
(v2
The
OF
Daeg
SSM
(v)
Fourier
ons
F(@)V2cos
and
(41):
oy
and
re
(v).
transforms
2,9
G,(v)
and
G.(v) of the functiF(@ V2 sin gQ,6 are obtained from (20)
and
Ge(v) = a(O)hoe(v) +91 Lac(d)bene(v) + asi thei (v)1(Z2)
1=]
Gs(v)
a(O )hy
-
:
at as@iy) bey)
(v)
oF S fea. Ki bt ic
.(v)
=1
Considersthescase
efficients
of
Ge(v)
the
equal
ait)
and-all
zero to get anunderstanding
The
G,(v).
and
= as€t)-=4
resulting
co-
of the shape
transforms of
Fourier
functions
Vecos 2n@
V2cosn,6,
V2cos ane
VerimManoe
2 cos),
Vo sin 2n0*V2 sin 150,
ere
Shown
in-the
Vecose2né
equal
and
zero
The
first
four
V2 sin2né
outside
following
PeOlmuMemu
bans
the
are
V2 sinn,§,
lines
of
Fig.60.
cosine
and
sine
interval
single
Orisouine
sideband
bch
rlrsh
signals
four
See. onl 610.0
+9
Sinem
£-¢(8)
=
cos 2n8 cosN,@
-
sin er sin 099
fsc (9)
=
cos 270 sin)?
-—
sin2nd
Peso
me
ecOs TO cos (0,6 +
Fourier
transforms
almost
all
of
pulses,
that
be
on
(43)
cos 049
of these
their
functions
functions
energy
in
are
£4,(6)
the
sideband
rier
sinQ,@
all
and
vy <
four
cosine
the
implementation
of
SSB
called
modulation
[2]:
Both
(44)
phase
contain
carrier
single
of
ding to (33) isusually
method
Q,/en.
signals
cos,@.
sideband
second
energy
sine
The
F(6)
in
are
car-
practi-
modulation
method
A signal
sideband
channels
the
in
sand: f,.(8)
the
since
shown
upper
feccenpemdeice (Op havermost of their)
lower
Fie s60:
COs 0,0
Weegee
used,
derived
Sil cn? sin {),0
Linesapestoposotetig.e0.eThehave
that
may
lanes
=
The
Note
-% = 6 = ¢.
eC
cal
other:
or phase
accorshift
is modulated
AD
single
yields
a filter.
of
by means
sideband
one
suppressing
by
result
same
the
obtains
modulation
the car-—
onto
modulated
is
all
with
signal
same
the
sumordifference of the modulated carriers
of SSB
sideband signals. The first method
9,6;
cos
rier
shifted
phase
90°
oscillations
and
0,0,
sin
carrier
the
onto
TRANSMISSION
CARRIER
3.
ee)
2cos 21t/T ° cos Gol
~
LS)
2cos 2Tt/T
‘sin Wot
wy
2sin 217 t/T ‘cos@ot
2sin2T7t/T-sin@ot
wn
cos
n
cos 2M t/T- cos aot - sin 2M t/T-sinwot
$44
Lhd
b4§
@
©
-foT
cos 21 t/T: Sin@ot - Sin 2TTt/T- cos Wot
Fig.60 Fourier transforms
plitude modulated by sine
Line
9 in
Fig.60
be
disregarded.
of
lines
this
Single
if
same
the
used.
this
is
for
not
a
number
two
The
way
is
phase
positive
single
and
of
channels
usually
handicapped
- sin 2Tt/T-cos
21M t/T- cos Qt
of
referred
by high
looks
values
sideband
double
channels
exploitation
-
cos
of sine and cosine
and cosine pulse.
transform
6
sideband
+ Sin2Tt/T-sin apt
shows why negative
This
5 and
cos 21 t/T-cos@ot
Oot
st ~
fT —_
the
+ Sin 2Tt/T - cos ar
meee
+ LSPPLLESBR
N
less,
21 t/T-Sin@ot
frequencies
like
of
of
a
to
as
crosstalk
cannot
transforms
fT;
neverthe-
modulation
certain
each
double
the
ve
am-
signal.
sideband
in
carriers
frequency
frequency
sideband
quadrature
inthe
case
permit
band,
channel
modulation
are
in
modulation-—
of
telephony
3.14
METHODS
OF
transmission.
hand
causes
Sionif
ter
SSB
more
as
wide.
Thermal
of
for
double
sideband
signal
from
phase
sine
through
one
the
sine
the
band
pro-
used
is
would
noise
as
half
filtering
but
the
side-
equally,
methods
otherwise
channel
or
asingle
receive
both
from
channels.
The
Sine
signals;
one
transmit-
the
of afrequency
both
other
sideband
band;
phase-sensitive
the
transmis-
energy
all
on
signal
through
frequency
channel
that
the
phase
A double
influences
noise
course
digital
either
transmits
cosine
modulation,
in
used.
energy
of acertain
the
vided
are
all
transmitter
well
sideband
distortions
filters
transmits
band
149
Single
cosine’ channel
as
SSM
investigation
and
.cosine
functions
are
plot.
results
The
tions.
not
Consider
Everuneoi one
wal(2i-1,v)
functions
wl2ek,6)
wC2k+1,0)
of
pulses
suffers
frequency
are
the
Wall@.))
modulation
from
limited
simpler
Walsh
insteadof
are
amplitude
to
the
of
with
Fig.2
,ecalia,
vy) =wal(2i,V)
obtained
= ig wal(2k,v)
by
cos
functions.
a Fourier
that
of
these
and are cumbersome
obtain
functions
time
by means
fact
The
Walsh
as
iand
to
func-
frequen-
alia ,Vv)=
following
time
transformation:
2nvedy
(44)
= iy walecked,+) sinenvedy
9 Kes
OF
1)
"2..."
—oo
The
or
2k+1,
functions
have
the
w(j,@)cos,9
Fourier
following
2fw(2k, 8 )cost,8cosenveds
and
Il
w(j,6)sinn,§9,
j =
2k
transforms:
wal(2k,v-v,)
+ wal(2k,v+v,)
wal(2k,v-V,)
- wal(2k,v+v, )
-0o
afw(2k,8 )sind,@sindnvede
co
2k+1, V-¥,)+wal (2k+1,v+¥)
2fw(2k+1,8)cosQ,@sinanvede
= wal
2fw(2k+1,@ sind, ecos2nvedd
=-wal (2k+1, v-y,)+wal (2k+1,v+y)
-oo
Or)
3. CARRIER TRANSMISSION
440
or
tions
the
from
may be derived
nds
only
sideba
lower
the
upper
time
func-
in
energy
all
having
signals
following
The
54):
w(0,@ )cosM@
+ w(1,0)sinQ,d@,
w(0,0)cosNes - w(1,0)sinQ,6
w(0,6)sinMge
- w(1,0)cos,8,
w(0,0)sin se + w(1,0)cosQ,6
w(2,8 )cosf,ge@
+ w(3,0)sinQ,@,
w(2,0)cosQ @ - w(3,0)sinQ9@
w(2,0)sing,@
- w(3,0)cosQ,@,
w(2,0)sinQ,6
+ w(3,0)cosQ,6
(36)
Four
Fourier
in Fig.61.
solute
The
value
increases.
for
the
transforms
of the functions
arrows
The
direction
upper
of
sidebands
Wo(0,8)cos $2,8+w (1,8) sin Q,8
jel
indicate
of the frequency
ee
ee
in which
of
the
andis
(436)
direction
wal(2k,v)
arrows
reversed
are
and
shown
the
ab-
wal(2k+1,v)
remains
forthe
unchanged
lower
ones.
lt
w(0,8)sin 22,8-w(1,8)cos228°
2
eee
D
Fig.61
W(08)cos
22,8-w(1,8) sin 298
es Fare pe
Fourier
transforms
of
frequency limited single
Signals; v, = Q,/ent.
some
sideband
el
w(0,8)sin
2298+w(1,8)cos298
A block
band
modulation
signal
Two
diagram
F(@)
signals
components
wise
sideband
fed
The
a
through
at
second
phase
The
the
two
their
method
phase
cos 2ny,9
sum
of
of
The
the
difference
of
shifting
whose
90°
and
lower
sidelimited
networks.
oscillation
but
are
sin2nv,9
products
a
single
frequency
outputs,
difference
carriers
modulated.
signal,
the
shown
in Fig.62a.
appear
have
equal.
plitude
is
is
for
other-
are
am-
yields
an upper
sideband
signal.
4.14
METHODS
OF
SSM
444
g(B)cos 21r(vy+1/2)8+
+h(B)sin 2(vq+1/2)8
Fig.62
(left)
Outphasing
method
of single
method
sideband
(a)
and
modulation
(b)
SARAGA's
fourth
of acarrier
with
frequency
vy, byafrequency limited signal F(9). PS phase
shifting
network, M multiplier,
S adder, BP bandpass fil-
ter,
F'(8)
single
sideband
signal.
Fig.64 (right)
WEAVER's third method of single sideband
modulation of acarrier with frequency v)+% by a frequency
limited signal F(@). OS oscillator, M multiplier,
LP lowpass falter.
A very
riers
ted
similar
cos
env@
signals
plied
An
these
A further
WEAVER
[4].
riers
of the
pass
sin
oor
band
shows
F(8)
a
-and
cos
md
Trequency,
The
filtered
signals
carriers:
sin
[4].
the
two
baseband
Asimple
block
no
multi-
and
in
around
bandfilter
method
diagram
energy
with
of
frequency
the
for
carrier
sup-
cos
OT
the
v4 +8) 0.
imple-
the
band
onto the carf,T
=
the
middie
Vg=
and
its
to
in
pass
onto
due
v,=
is
carriers
modulated
is
outside
3 +] ismodulated
modulated
am(y ,+8)8
are
carshif-
signal
is generated;
cut-off-frequencies
are
The
phase
sums
modulation
with
= fT
with.
2v,.
sideband
2)
used.
filters.
The
to
signals.
1
lhe
SARAGA
added
sideband
frequency
single
76
jol.
to
are
producedinthe
undesirable
A signal
f) 20477
ioe
are
Fig.63
mentation.
Oes
is due
env
upper
signals
double
the carrier
presses
sin
according
to Fig.62b.
together.
addition,
method
and
through
f£,T =
high
The
#.
# or
low-
The
frequency
sum
yields
nce
,
the differe
signal
sideband
upper
an
CARRIER
%.
442
TRANSMISSION
a lower
sideband
signal.
w (0,8)
-w (1,8)
1
AEs
-2w(18) cos 18
2
2w(0,8)sin x8
-2w (1,8)
sin 18
3
299(8)=9o1 (8)+9q2 (8)
2g,(8)= 91 ()+9n (8)
Fig.64
2)
forms
2h,(8)=hy,(B)+ hyp(6)
tO
eR
thod
Fourier
trans-
of the third
of
single
let
the
me-
side-
:
ae
ees
9478)
Yoo8)
hg,(8)
hy (8(8)
3
ho (8)
hy(8)
Eee
24g (8) cos 2m (vy+1/2)8
2q)(8)cos 2x (ve 1/2) 8
+--
10 f=
2hg(Q)sin 2m (vq+1/2)8
--+--
2hy(8)
sin Zr Vq+/2)8
Go (B)cos 2reWvq+I/2)B+hg(B)sin 2ry+1/28 g,(B)cos 2rr(vq+1/2)8+ h,()sin 2ne(v,+1/2)8
__..
__ Ss) 49
+-- EE
L
i
4 1
22
TL
as
i
oa
ji
4
J
ET
=
x
te
4
Ls
RO
Ne
weed
ial
cy
limited
input
functions
transform
F(6)
=
It
(34)
>, [a(2k)w(2k,
k=0
(37)
ee
2
through
F(8§)
and
from
>
be
method
expanded
w(2k+1,9)
the
8) +
one
4
ee
WEAVER's
signal
sufficesto trace
series
of
w(2k,8)
rier
et
1
For an explanation
the
+
Walsh
frequen-
intoaseries
derived
by the
functions.
a(2k+1 )w(2k+1, 6) ]
even
the circuit
and
of
Fou-
one
odd
of Fig.63
(37)
function
of
the
rather
than F(6).
The simplest functions, w(0,9) and -w(1,6),
are used. Their
Fourier transforms wal(0,v)
and -sal(1,v)
are
shown in
Fig.64, line 7. The arrows point in the direction of in—
creasing
absolute
values
of v.
Modulation
of
cos7mé@
shifts
4.14
METHODS
the
Fourier
the
left
OF
443
transforms
of line 1 by # to the right and to
(line 2). The transform shifted
to the left is
hatc
for
clarity. The two shifted transforms are
hed
superimposed where they overlap and have equal signs.
Shown
shown
Modulation
right
andthe
(line
4).
the
SSM
h,(98)
which
etsoushowm
transform
signals
g),(9),
yield
the transforms
yiel
d
Lines
g,,(9) andh,,(6),
g,(8)
and
h,(4);
by3@ )) and
t-h)(6)),
sof
transforms
)odd.)
4and5).
(lines
of
whe
the
left
outside
to
which
to
The resulting
4and5).
transforms
hare
#
suppr
all ess
components
odd transforms
even
by
by -1 by # to the
may be superimposed
8209)
8,09),
the
multiplied
filters
have
9 showthe
6 to
shifts
transform
Lowpass
and
h,,(6)
sinm@
-g = v = % (lines
band
Bo(8)
of
of h,(6) andg,(6)
superimposed.
wie
transiorms
wal(O,v)
of
the
and
of
lines
—sal(7}v).
following
functions
g,(9) cos em(v,+#)9
6
to
Hence,
9
one
with
have
the
Shape
of
obtains
the transforms
the
help
of
(43):
= [g,,09) + ¢,,(8)] cos 2n(vj+8)e
(38)
= [h,,(@) + h,(6)] sin 2m(v,+8)6
(39)
to
h, (6) sin 2n(v,+4)6
as
shown
and
5
in
are
=
lines
shifted
ted
transforms
the
four possible
sine
or
rence
Sunvot
of
an
yields
The
filter
causes
carrier
lanes
upper
the
tically
no
necessary
spectrum
lO
and
used
energy
for
of
the
actually
following
be
according
fourLer
12).
in
modula-
This
a filter.
objectioFig.o5a
signals.
of asignal
0 = v = vo.
the
diffe-
signal.
sideband
by
investigation
rectangular
to
with
vrans—
The
sideband
digital
band
4
shif-
transforms
thew
single
spectrum
lines
the
are particularly
of
the
-1
(line
sideband
which
of
left;
(45).
a lower
one
power
outside
in
or
odd
yields
of
transmission
and
+1
or
signal
method
of
by
ll
transform
transforms
right
even
shown
sideband
frequency
the
The
of
as
distortions
the
for
11.
tothe
multiplied
suppression
tionis
nable
and
v,+#
products
generally
the
shows
are
cosine
The
forms
10
by
that
joaiag:
with
It
the
prac-
is
not
power
© 2 WS
Vr
3. CARRIER TRANSMISSION
AA
Fig.65b
shown.
as
having
Band-filters
the
dashed
lines,
show
thessum of
the
hatched
phase
shift
causes
linearly
not
vary
distortions.
d
separate
by bandpass
are
the
are partly
does
are
introduced
demodulated
areas
indicate
nL
TT
filters,
shows
and
The
in
and their
This
the
signals
additional
distor-
power
to
are
Fig.65c
frequency.
receiver,
N72 WAGN,
k.
spectra
of
The
hatched
improperly
attenu-
pe
a Me tO Hy VebIG
jiNN 7 RS
‘
ST
IRN
NEN ee
Fig.65
attenuated
with
Fig.65i
oscillations
DNeINS
J
the
(Fig.65f-h).
signals
where
At
by
shown
as
oscillations
The
frequency areas
signal
tions
sidebands.
upper
v,t+4vo-
sidebands,
lower
the
suppress
the
modula-
and
functions,
transmission
si-
such
amplitude
vct2v,
v9,
frequencies
with
carriers
of
tion
of
by means
bands
adjacent
into
gnals
three
of
shift
the
d shows
to
(left)
ee
Power
spectra
for
ee
the
modulation
and
demo-
dulation of three signals by single sideband modulation.
Bandwidth of the signals is 2v,; lowest frequency of the
Signals is 0.
Fig.66 (right) Power spectra for the modulation and demodulation of three signals by transposed sideband modulation. Bandwidth of the signals is 2Vo; lowest
frequency
of the signals is 2%.
3.14
METHODS
OF
ate
and d
phase
There
may
cated
ters
the
cy
bands
be
channels
given
by BENNET
vy = 4y,
is
It-is
only
wider
be
to
filters
tod
adjacent
frequency
with
Bandpass
filters
by
cause
Figs.66b
carriers
the
of
are
having
of
where
the
by
the
are
show
2,2
'Vestigial
to
demodulated
) = 4yo eas
sideband
Harmuth, Transmission
=
O <
Vv < ev,
which
the
single
is
in
power
The
and
functions
energy
by
shown
The
in
bandpass
of
2vy-
The
sum
signals
the
distort.
frequency
the
in-
vce +
sidebands.
Fig.66e.
no
signals
modulation
Ve
filters
separated
k.
such
2Vo5
shows
fre-
At
the
filters.
shown
areas
the demodulated
spectraof
non-distorted
are
signals
in fig.66a.
modulation
of Information
Av
2v,> equals
that
transmission
there
are
The
Fig.66i
O<v<2yv,
bandwidth
amplitude
vy. -
bandpass
introduced
signals
the
since
the
signals
=
band
three
sidebands
(Fig.66f-h).
10
bands
frequencies
hatched
Deuce
shift
upper
Distortions
of
the
distorted,
have
2v,
band
band
the
in
three
areas,
is
used
is
distortions.
suppress
the lower
not
tele-
signals
The
empty
band,
to
will
Its principle
important
the
lines
receiver,
tra
that
dashed
the
quency
show
the
not
is
This
method
this
empty
to
first
are
method
frequency
of the
frequency
the
than
Sideband
It
necessary
of
[8].
compared
small
fil-
frequenThe
existing
second
the
the
signals
through
lo-
the edges
modulation!.
Fig.66.
in
One
is
sideband
from
account
to
energy
energy.
digital
The
small.
locate
away
if
[7].
width
The
nor
may
sideband
energy
signal.
the
of
2v,
AY.
far
modulation
their
zero
one
division
reference
distortions.
single
of the signal
DAVEY
sideband
(Fig.66a).
neither
Or
A detailed
and
all
practically
the
useful,
time
with
discussed
where
most
Signal
distortions
most
of their
vestigial
[6].
in transposed
be
=
in
by
phony
that
filters
particularly
transmitted
cause
the
distortion.
contain
used
is
so
sideband
which
thus
keep
bands,
little
is
method
to
signals,
single
method
and
ways
frequency
cause
of
two
the
in
445
shifted,
are
shape
SSM
again
The
goes
in
to
the
and
fol-
NYQUIST
ots
distorted
back
spec-—
power
located
3. CARRIER TRANSMISSION
AAG
O
and
sev,
2
= Vv esi
binary
character
sin
30mt/T
has
T and
sin
pand
(15-1)
is
equal
of
the
wider
gained
means
quite
14
2
to
empty
than
a
+1-1
is
17
by
=v
= £1
=
band
the
=v
Fig.66.
this
of
v,(t)
= (1741).
14;
O
inan
The
signal
functiom
The
is
D,
E
Detection
V(+
sin
voltages
of
cos
3111
(B),
(BE),
30T7T@
eve
and
is
thus
of
that
sin
4078
4478
of
a
digital
-
the
nucl
be
and
>
re-—-
F by
Fie. N4G.) feet
function
sin
cos
can
width
outside
the band
show
the
Duration
The
Figs.6/7B
the
limit
1.
are
43076
from
in
=
Showman
G
Signal
follows
+1-1
to
signals
and
signal.
duration
frequency
Ay
14Av/4
ag
Fig
CG).
is
coefficients
other
Figs.66C,
it
of
concentrated
lower
according
devecvor
what
is
bandwidth
= ey,
this
interval
oscillations.
energy
2v,
unimportant
vy 218.
transmitted
transmitted
oeiu7e
oscillations
has
the
in
from
of
15
44nt/T
that
be
50nt 7) = Westin
The
Fig.68,
can
that
v,(t)
modulation:
sideband
transposed
Vit
> 4v,-
a signal
shows
Fig.67A
by
v
bands
unused
the
in
located
are
oscillations
ded-over
wrong
sig-
signals.
346);
detectors
(C),cos
(F) and
traces:
A:
output
32n8
sin
=
150)
(Courtesy P.SCHMID,D.N
of
Allen-Bradley
Co.).
for
(D).
34n8
ms
ISC
05
0 Tes
1
flHzl=8000
2
16000
3
24000
ae
32000
40000
Fig.68
Frequency
power
spectra
of
following
pulses
according
to Fig.1
HO;
$V2wal(0,6)
(a).
sin 21
2n@ {bpJcos
(b)cos
2Vewal\U,U
(a),
Sim
(c), sin 4m@ (d),
cos 476 (e), sin
(tf), cos 678
(g),
sin 876 (h), cos
(i).
8 =
MnaBhictotcval
0/71, y
wie)
Wakenaymi4
Il
fT,
-—
ake@ner Vl =
£6 24;
the
and
2 9
om
6na
879
fis
3.15
nal
CORRECTION
produces
OF
TIME
very
DIFFERENCES
little
output
447
voltage
at
the
sampling
time
3.15 Correction of Time Differences in Synchronous Demodulation
Consider a frequency band limited
It
shall
with
a
be
synchronously
local
carrier
difference
a
WieecOs gO)
[1 |:
demodulated
V2cos
with
signal
(2 )8+a)
reference
to
F(@)/2cos 0,6.
by
multiplication
which
the
has
the
received
phase
carrier
F(@ V2 cos Np6V2 cos (1,840 )=F(9 )[ cosa+ cos (20,8+a)]
Let
local
the
signal
carrier
chronously
be
2cos
by the
frequency-shifted
CO, O+a, ) and
local
by
an
(40)
auxiliary
then be demodulated
carrier
syn-
2 cos [(M)-0, )6+a, ]:
{LPC@V2 cos 2,0 JV2 cos (0,940, )}2cos[(2,-0,)94+a,]
=
)+-008(20,9+a)}
)8-a,g J+c08(2N,8+a,9
F(6 ){cos.a+cos[2(0)
-0,4
Het
Clicag
Ce OL
Ag
Equations
multiplied
be
tne
by
by
cine
and
contain the desired
high
filters.
of
frequency
There
cosa.
from
local
the
Let
us
The
sina
The
quency
ised
methods
€.¢.,
cosine
a
sine
carrier
signal
by
yields:
sin (20)8@+a)]
(42)
assume F(@) may be written as sum F(9)=14+MF‘(@)
isasignal that contains practically no ener-
a certain
right
frequency
hand
+ MFt(0) sina
second
lowpass
eimea
and
vanishes.
a
M is
term
The
Toop
thus
then
can
term
to
modulation
the
be
sina
shift
(43)
zero
suppressed
by
remains.
It
the
local
in such
in-
form:
sin (20)9+a)
f2sin(N,8+a)
equals
the
assumes
+ (1 + MF*(@)]
filter.
and
and
(42)
of
side
third
teedback
V2 cos (Q,9+a)
sina
of
can
F'(6)
gy below
dex.
F(0)
which
received
F(@ V2 cos Np 6V2 sin (Np6+a )=F(9 )[ sina+
where
terms
derive,
the
of
signal
a number
may
Multiplication
Oscillation
are
One
V2 sin (Q,@+a)
We cos (0,040).
tots
-— Ase
cosa
Temoval
oscillation
Ah
=
(40) and (417)
suppressed
for
10*
(41)
a fre-
may
a way,
or aninteger
be
carrier
that
multiple
3. CARRIER TRANSMISSION
148
of
phase
difference
Consider
carriers
be
the
are used.
periodic
unstable
signal
atime
carriers
anddo
difference,
F"(@)wal(j,¢)
multiplication
functions
The
of
with
wal(j,9)
not
demodulated
o.=
and
vanish
of
the
demo-
[2].
VITERBI
by
given
is
carriers
ex-
zero
of synchronous
treatment
the
for
holding
values
where
be
to
assumed
a is
Fig.59
corrects
that
receiver,
a
if
Walsh
(15)
shall
local
carrier
wal(j,6-8))
outside
the
are
interval
signal has the following
form:
Ft(9 )wal(j,®)wal(j,0-6,)
The
product
that
of
to that
(40).
of
This
=
the
cosacosB
sine
tion
and
simple
from
right
to
shift
theorems
(44)
a modulo
functions
(45)
this
shift
then
be
are essentially
to
left.
the
addition
theorem,
applied.
the
same
(7)
are
multiplica-
right
and
shift
Walsh
functions
theo-
have
ve-
and
not
(1.39),
= wal(j,@)wal(j,6,),
contains
the
2 addition
Certain
with
in
in
functions:
may
since
left
not
similar
etc.
(7)
theorems
from
needs
but
(Q,8+a)
performed
by
theorems
is
Vecos
one
cosine
decomposed
functions,
read
be
sina sing
and shift
if
and
problem
with
alone,
binary
wal(j,9@0,)
but
+
be
cosine
theorems
rems if read
ry
sine
is known,
The
cannot
(7)
multiplication
Multiplication
for
of
must
wal(j,9)
V2cosQ,8
theorems
theorems
V2 cos(Q,8+a)
and
and
wal(j,6-0,).
multiplication
shift
cos(a-B)
and
multiplying
multiplication
the
(44)
wal(j,6)
of wal(j,9)
be
47,
=
a
for
unstable.
then
are
-1
=
loop,
The
by
The
-4 = 6 = %.
and
correction
demodulated
wal(j,6-9)).
...
way.
sinusoidal
of
adulation
+4n,
detailed
A very
shown.
the
inthis
feedback
in the
cept
are
Let
of
diagram
block
a
shows
-1.
cosa
values
The
...
+37,
#2n,
= 0,
fora
stable
loop
feedback
or
+1
equals
cosa
and
m
special
may
periodically
be
ordinary
or
subtraction
cases
of
derived
continued
subtraction
the
readily.
functions
Sign
sign.
shift
theorem
Fig.2
sal(1,6)
shows
and
of
that
Walsh
the
cad C1565)
3.15
are
CORRECTION
OF
transformed
TIME
into
DIFFERENCES
each
other
by a
in unnormalized notation; the shift
and
cal(2,0),
a power
of
+% for
2;
equal
it =
by
in
values
(46)
Table
3 may
eo
to
9.
be
line
ee
(46)
eal
(x-sm)
and
for
following
of
(46)
and
2), 0—-2 *-*) =i
i=
see
inte-
any
for
that
the
2’.
...,
9,
sign
One
Lncesimage
4 =
1,
05
This
is
de-
marked
= ¢ for
i =
6,
by
3 is
6, for
for
i1=1
with
readily
see
thet
Lor
law
8,
are
reversed.
of
may
OF
52.
values
"image"
2'.
(48)
It
sine
ee
t=
of
5.52. sb with
images
may
be
are
the
and
cosine
relations
theorem
relation
of
sin
x
functions.
hold forthe
cal ce", 6)
= -cal(2*—j,6)
sal
functions
in-
(49)
-cal(e"+j,e)
sere
(6)
shift
tothe
(48):
Bal(o'=5,040,)
cee
special
corresponds
=
Equations
(48)
ee
gal (2%+5,649))
ib Sh
(46)
cen ew,
holding
These
the
2 =
i
functions.
The
holds:
= sal(2"-j,9)
Equations
stead
i be
gait 2 +4, 0.)
t
By a
cos
=
i =
the
=
Let
follows:
eal(2*-j,6=0,)
Walsh
may
called
i
87 to
er+j50+85,)
& S
for
i = 1 with
line
to
as
0,
i = 2“.
One
for
thus
reference
sal(2,@)
etc.
(47)
of
for
t=O,
written
+37
sal(i,9)
@, =-&
feference
Cal
=
J schows
aestar
++ or
+§ for
formula
Cn
case
of
i":
of
Calla, 6+0,)
termined
equals
general
general
more
the
Consider
Gerpmvalue
following
shift
andcal(4,6),
ENS) Sail ie Te
CES
CAIN”
pebble
the
sal(3,@)
149
to
(50)
cal(i,6+6,)
= -cal(i,9-6,)
sal(i,9+08,)
= -sal(i,6-99),
(50)
eee
yield:
(51)
special shift
and
cal(i,@)
the
9, 1
functions
9 and
Walsh
of
Table 9. Some values
periodic
of the
theorem
TRANSMISSION
CARRIER
%.
4150
Palliciy0))
ae
27
28
29
001011
001100
001101
001110
001111
010000
-1/4
-1/8
+1/4
-1/16
-1/4
4176
+1/4
41/32
-1/4
-1/8
|010001|
|010010|
|010011|
|010100|
|010101|
1010710)
|010111|
1011000]
|011001|
|011010|
17
18
19
20
21
22.
23
24
25
26
| 000007
| 000010
| 000011
| 000100
| 000107
| 000110
| 000111
| 001000
@04 004
001010
4
2
4
4
5
6
7
8
Oe
ae
|011011|
|011100
|011101
011110
011111
100000
1/2
-1/4
=A fo
| -1/16
fe
—1/4
+1/4
or
Cal(i,d+0,)
8
=
of
It
much
of
the
sal(i,o46, ) =
cond
by writing
—seita.7)
(52)
The
absolute
the
the
k
the
digit
Consider
yields
as
6)
example
The two lowest
|8,|= 2-7-7
One
An
-% for
for
can
large
obtain
6,
equals
inspection
all
digit
6,
6,
odd
is
of
Ta-
values
a O and the
equals
=e
6,
-#,
of
se-
if the
zero.
6)
digits
0;
9.
is
is
representation
binary
and
number.
a 1.
holds:
are
8)
is
binary
Generally
of
Table
binary
6,
lowest
digits
k + 2 is
9.
that
obtain
of
digit
value
binary
lowest
in Table
this
a 1.
binary
from
1.
shows
9.
to
i as
binary
-4,if the
lowest
k lowest
Table
cumbersome
lowest
is
in
i by an extension
9 readily
6,
shown
be
faster
ble
is
6, are
would
values
i.
~cal@i,e),
—2|05|«
Values
if
=
derived
of
are
9)
+|6,|if
numbers
binary
= 1/146.
|6)|
zero.
equals
the
i.
in
digits
The
the
same
equals
equals
the
i =
(k =
fourth
2%
if
—{0,|) 12
digit
20
way
and
k + 2
i =
28
2) are Zero 3
binary
digit
3.15
CORRECTION
(k+2
= 4)
the
is
fourth
rules
for
0 for
digit
i =
is
DIFFERENCES
20
and
1 and
determination
A circuit
tween
special
Let
us
is
a
6,
of
6)
4
equals
equals
6,
and
41/163
+1/16.
8,
at
received
output
was
for
i = 28
A proof
of the
given
of
produces
the
by
PICH-
The
may
once
be-
based
is
local
is
received.
passes
is
locked
circuit
=
with
a
onto
fixed
sal(i,9-6,).
by sal(i,6-0,)
is
+
= 0 34% + 6, of sal(i,6-0,).
The output voltage of
+6,
integrator
3+
6,,...
circuit
om
the
MP
by the
TP.
plitudes.
I is
This
The
sampler
circuit
following
due
to the
impervelse:
sampled
fact
-$k
AT,
the
and
times
# + 8),
over
many
average
is obtained
#90
the
interval
3 +
9,
is fed to an averaging
averages
that
+ 6,
at
orthogonality
and the
product
the
the
in
cal(i,9)isoptai-
cal(i,6-0,+6,)
is multiplied
low-
produced
carrier
circuit
delay
on
(Pie.6eo.
through
a sequency
the
during
be
functions
carrier
A further
integrated
difference
cal(i,9-6,)
The
carrier
signal
carrier
Walsh
passed
FG.
RV
a time
[14+MF(0)]cal(i,6)
RV.
carrier.
received
of
carrier
circuiG
of
local
has
generator
delay
ned
delay
signal
that
A local
the
and
theorem
the
function
the
correction
carrier
signal
filter.
Verlaple
the
shift
assume
Ft(9)
pass
for
received
the
The
TIME
[4].
LER
the
OF
imtvegratvor
sampled
at
the
I averages
am-—
output
over
= #k + 0):
cal(1,6)
1+MF*t(8)
out
Fig.69
and
Correction
local
carrier
of
a time
cal(i,@)
difference
and
between
cal(i,6+6y);
received
i = power
of
FG function generavor, RV variable delay
miltiplier,
PM
AT amplitude
circuit, D fixed delay by 6,, I integrator,
an “tie
equal Oyecxcepu
8, is put
sampler, LP averager.
feedback loop.
eo i
TRANSMISSION
CARRIER
3.
102
——
cally;6)
eall(,2) $$ —
ol)
<<a
Sal (6!) Ree
Call)@) ae
Soli)
—$$—
COAG)
ea
|
ee
ee
|Sea
ee
a
Se
ee
ts
cal (3,6)
sal (3,8)
cal (2,8) ————
Hat
sal (2,8)
cal (1,6)
Fe7,s1(8))
sal (1,8)
woe
wal (0,6)
sal (8,8)
Let
sal(7,8)
cal (6,8)
sal (6,8)
cal (5,6)
for periodic
Correlation functions
Fig.70
<i
cal(7,8)
sal (5,8)
cal(4,8)
functions.
Walsh
MF" (8 )]cal(i,6 )sal(i,0-8y)>
us
assume
that
the
average
of
(53)
the
second
term,
<MF*(9 )cal(i,6)sal(i,6-6,)>,
increases
the
more
slowly
with
(54)
increasing
averaging time than
average
Ccal(i,@ )salla,6-0y)>
of
the
output
the
first
term.
voltage
of
local
that
and
The
the
carrier
(55) and (54)
(54) vanish
(55)
term
(55)
averager.
cal(i,@-8))
vanish.
are
dominates
It
may
and
thus
The values of
obtained
be
then
used
in
to
sal(i,é@-6,),
6,
for
fromthe following
the
shift
which
so
(55)
integral:
12
Coal(i,9 )sal(i,é-6, »> = fear(i,e )sal(i,6-9,
)d® = R(6) (56)
1/2
Fig.70
shows
main
diagonal.
nal
and
some
functions
F,;,\(9,) is shown
Ricil®y)
gust
above.
F,, ,(6,)
just
The
and
E,si,si (CBee in the
below the main
interval
0 =
diago-
6,241
is
3.15
CORRECTION
OF
TIME
DIFFERENCES
VBS:
es
eS
SS
NESS
RES
eB
oe
TT
ORM
OA
te
=
——_
To
i
ee
a ee es afr
NS
BO
AA
ee
OS
cal (7,6)
——__
sll (79)
OS
A
rrr
Os
—
eS
. Lad
—
Ane
—
——
ee
a
oat
Ne
ee
ee
——
wee
KANG
Pf
Af
—————
——
——
wan
XN,
wo
SA
———<———
Ee -_
—
Se
ee
-e_
ass Ss |Se
sal (46)
shown;
cal(3,6)
the
sal (3,6)
cal(2,8)
functions
The
dashed
tions
of
Fois(9y)
or
gero
Walsh
shall
functions
Consider
the
dicallyto the
by 6,
0y
left
sal(i,@).
sal(1,6)
and
right.
+4/2i,
Things
=
...
are
into
equals
PO
result
This
of
+2/2i,
pala ,0);
the
complicated
loop
+4/2i,..
continued
again
ab-
that
feedback
...-
more
func-
The
of Fig.2
yields
periodically
+1/2i.
+3/21,
A shift
wal(0,8)
wal(0,8)
deeply
phe
...,
= 0,
functions
Samar
ames SR
from Fig.71
6y
for
cal (1,6)
into their struc-
too
(46) and (52).
stable
sal (20)
correlation
however,
+20,
of
——
enlarged in Fig.71.
lead
see,
= 41/21,
Walsh
= 0, +2/2i,
function
from
made
in Fig.69 may be
and Unstable for
al20)
the
2, 4, 8,
multiple
also be obtained
aes AYES)
ee
continued
of
would
Onemay
integer
PVA
give some insight
discussion
vanishes fori=1,
an
be
cal(4,6)
anaes
cal(1,8)
to
———
ee
ee
Fy; gi 8 y) is shown
detailed
mathematics.
stract
may
lines
amore
SS
sal(2,8)
have
outside this interval.
ture;
Eee
sl (5,8)
el 0)
es
AL SES IS
sal(6)
call(5,8)
A A A A
we
av,
——
OO
etl ee Baal A Apa
ee)
perio-
i=2*,
periodic
if
Lars
not a power of 2. Fj,,(8,) vanishes for certain values 8y
= 64, but sal(i,é-8)) is in general not identical with
sal(i,@).
the
most
Hence, the functions cal(2*,6) and sal(2*",8)
This result
synchronization.
for
suitable
are
has
TRANSMISSION
CARRIER
3.
154
eo
et
Fas)
es
functions
already
discussion
tiplex
been
systemof
according
of
used inthe
to
Fig.51.
Fig.69
AEG-Telefunken
There
are
simplify
the
a
A Walsh
has
for
such
number
of
discussed
differences.
For
been
the
to
for
the
LUKE
signal
and
MAILE
system.
generalize,
the
mul-
tracking-filter
by
a multiplex
ways
telephone
function
developed
method
instance,
of
Walsh
some
of
F cis; (8,)
F
Fig.71 Crosscorrelation
functions.
improve
correction
of
or
time
[cal(r,6)+MF*(6)]x
cal(i,9)
may be transmitted instead of
[1+MFt(9)]cal(2k,6)
if
r®i
blocks
in
Fig.69
Aine
be
in
fed
than
equals
may
be
the
of
2.
combined
Fiet5o
into
into
apower
The
into
three
one.
The
a phase
shifter
or
PS
function
or
delay
AT
feedback
“end Ccal(i,6)sal(i,@-e))»>
oscillator
I,
and
voltage
in Fig.69
generator
circuit
RV.
TP
may
rather
3.21
TIME
BASE
MODULATION
We)
3.2 Time Base, Time Position and Code Modulation
3.21 Time Base Modulation (TBM)
Any
carrier
can
writte
as time
n
will
be
amplitude
function
expec
that three
t
canbe
defined,
sequency
k,
more
this
carrier
time
base
T and
of
Modulation
=
it
can
be
modulation
One
methods
cont
the
normalized
ains
the
T is
if
Vé(k,t/T+t,/T).
individual
since
the
V.
amplitude
modulated
V8(k,6+0,)
delay
called
t,
besides
the
base
modu-
g(6@).
There
a time
lation.
The
are
basic ideaistoreplace
several
signal
and
ways
to
do
this.
M a modulation
6 byafunction
Let
F(@)
index.
One
be
the
may
modulating
use
the
defini-
GLO $
$(k,0)
= é@(k,g(e)]
7)
e(6)= f(1+MF(e)]ae =f (44+MF(t/T)
Jat
This
is
the
Sinusoidal
proach
and
are
time
$(k,@)
for
taken
carriers.
However,
strongly
connected
are
combined
as
in
frequency
the
modulation
advantages
to
the
fact
of
this
that
case
of
ap-
frequency
product,
= sin ké,
sinusoidal
general
approach
(58)
(oo)
functions.
makes
the
The
comma
following
between
k
definition
and
of
§ in
g(9)
the
more
advantageous:
2(@) = e[1+MF(@)] = gals
IMF(@)|<
The
1
modulated
Fig.72
parameter
a
how
shows,
changed
are
(60)
if
the
is
clearly
function
sine
time
now
T is
base
and
the
a
changed
time
base
T.
function
Walsh
37/4
into
and
T/e.
The
Larger
modulation
values
of
index
F(8@)
time base
se
the
increa
to
frequency
M
may
reduce
for
modulation,
be
the
M < 0.
positive
time
base
Thisisin
e
an increased
wher
or
for
negative.
M
close
>
O
and
analogy
of the
voltage
TRANSMISSION
CARRIER
3.
456
mod.
-V
;
;
carrier
{
out
sin 218
1/2
G=t/T—
0
-1/2
signal
sin 218"
T=3T
-1/2
0
@=t/T——
sin on Q”
“
1/2
Tie tT
-f2
5
0 e"=t/T—1/2
2
sal(3,8)
-1/2
0
a
err
pera
lg
bet
te
tilee al
ee at
cae
d mmm
ttt rrr iii
ALE
' im
:
ot HAAS
pe
8
28
38
48
—_—_—e
Fig.72
(left)
Time
se VHaOiG Leyla ¢
Fig.73 (right)
base
Block
modulation
diagram
for
of
a
time
sine
base
and
a Walsh
modulation
of
Walsh carriers. AT amplitude sampler,
I integrator,
SV
voltage comparator,
SP storage,
Z counter,
FG function
generator.
modulating
of
the
shows
modulation
half
may
increase
as
wide
that
the
index
M.
as
the
occupied
by
sal(4,0")
is
twice
as
functions
sin
276
or
energy
distribution
values
of
M
and
A possible
functions
(b).
is
shown
sampled
the
decrease
the
frequency
at
sampled
An
time
The
as
for
in
is
sin
A detailed
of
is
time
base
line
of the
the
stored
I produces
the
pulse
a
the
ramp
or
long
analysis
for
of
various
lacking.
amplitude
in
2n6"
the
modulation
Let
is
sequency
by
sequency
still
Fig.73.
first
functions
occupied
with
shown
or
that
function
F(6)
time
frequency
short
sal(3,98).
as
6 = O by
voltage
integrator
base
shortest
the
circuit
by the
increases
The
large
signals
shown
required bandwidth
longest.
bandwidth
shape
or
carrier.
Fig.72
the
signal
of
Signal
Walsh
have
diagram.
sampler
holding
voltage.
the
It
AT
circuit
is
(a);
SP
A voltage
3.22
TIME
POSITION
comparator
SV
in
resets
SP
and
equal.
A
duration
stored_in
The
compares
the
this
AB
ramp
integrator
sawtooth
of
MODULATION
voltage
(c)
sawteeth
is
volt
with age
the one held
I when
both
results.
voltages
The
proportional
become
amplitude
to
the
and
voltage
SP.
pulses (d) from the comparator SV which reset
I are also fed into the counter Z. A pulse
in(e)
tegrator
is
generated
meceived
(e)
by
from
clears
SP
Z if
points
is reset,
happens
the
this
and
stores
at
the
at
the
sawteeth
(d)
generated
interval
terval
®@ =
O =
8
that
pulse
(e)
is generated.
©
0, @,
is
as
the
and
long
voltage
have
as
the
these
The
before.
The
distance
into
Fig.73
Z
resetting
pulses
the
time
asinthe
a
of
Hence,
8
SV in
time
functions
at its output;
sam-—
amplitude
pulses
ted
carrier
functions
the
Counter
as at time 0.
generates
for
Walsh
40.
pulse
of
between
This
comparator
twice
Feeding
Walsh
30,
been
The
amplitude.
twice as large
has
sample
the distance
generator
tated
in Fig.74.
sampled
4@
6 3 ©.
8
pulses
amplitude
the
twice
=
is
new
Note
times
by
a
of
on
time
are
number
number
AT.
depends
when the
Signal
certain
SV;
Signal
via sampler
pling
a
in-
function
base modula-
shows
the
modu-
sal(4,0).
3.22 Time Position Modulation (TPM)
The
variable
by
a
The
parameter
case
the
function
of
time
modulating
lowing
6 of
g(9)
9
the
in the
position
signal
definitions
and
ea
ne MF(@)
modulation
9 for
largervaluesof
holds
if
M is
a
time
base
replaced
modulation.
function
h(8)
Let
again
F(§)
M a modulation
index.
in
the
denote
The
fol-
(o7)
be
positive.
of
by
was
introduced:
index
will
V4(k,6+6,)
modulation.
are
= $[k,9+h(@)]
6(k,0+6,+MF(@)]
case
is replaced
3(k,0+0,)
The
carrier
shifted
F(8)
if
This
positive
be
M.may
towards
Mis
or
larger
negative.
values
of
opposite
the ve;
negati
to phase modulacorresponds
shifts
three
forthe
2n6
sin
=
#(1,9)
carrier
a sinusoidal
shows
Fig.74
of the signal.
amplitude
by alarger
retarded
or
advanced
be
may
carrier
the
of
phase
the
where
tion,
TRANSMISSION
CARRIER
3.
153
carrier
MF(@) = 0, -# and -#. Below is shown the Walsh
(3,0) = sal(3,8) forthe same three shifts. Note that the
section
is
the
at
added
the
of
end
other
+%
limits
the
beyond
projects
which
shift
a
to
due
-#
or
EUG
of afunction
AOI
-V
sin278
mod.
carrier
signal
sin 2m (8-1/4)=cos2n8
sin 2n(8-1/2)=-sin 2x8
4i?
0
2
8=t/T——
sal (3,8-1/8) sal(3,8-1/4) sal(3,8-1/4) sal (3,8-1/8)
yy
sal
out
(3,8 -1/4)
==cal
(3,
1(3,8-1/4)
al(3,8)
Q
=
Cc
d
sal (3,8 -1/2)=—sal (3,8)
of
ll?
e
|
l
T
bacreT
=
7
Fig.74
ed
it
(left)
Time
=
position
Walsh function.
Fig.75
(right) Block
and
sr
1 AO G—— 6 GAO
modulation
time
diagram
of
for
a
b—— AbxO
sine
the
and
time
a
posi-
tion modulation of Walsh carriers.
AT amplitude sampler,
I integrator,
SP storage,SV voltage comparator,
TG trigger generator, GAgate, U divider, FG function generator.
Fig.75
time
position
sampler
put
shows
AT
signal
voltages
a block
are
and
modulation
of Walsh
samples
at
diagram
the
held
periodically
times
for
0, 0,
a
a
pulse
carriers.
the
20,
certain
diagram
The
amplitude
...
and
the
time
(>)
in
for
amplitude
of
the
in-
resulting
3
holding
3.23
CODE
circuit
MODULATION
SP.
A voltage
the
in
integrator
comparator
ramp
SP.
An
voltage
This
Positive
pulse
duration
is
Trigger
cleares
pulses
through
TG
a pulse
value
SP
of
and
voltage
(d)
the
as
voltage
resets
(c).
soon
as
stored
integrator
I.
are obtat
the output
ain
of SP, whose
ed
proportional
A divider
ger
pulses
through
duces
ger
gate
to
the
(e)
may
GA1
as
gate
(g)
output
of
produces
(e),
pulses
that
GA2
amplitude
pass
of
the
fromthe
sampled
trigger
pulse
as
long
gepre-
is
(b)
(f)
are
small
function
pulses
sampling
circuit
Demodulation
generator
é¢.¢.,
through
of
FG
(h)
from
AT
at
for
time
carriers
have been
principles
strongly
used
on
for
the
The
protrig-
the
trigger
20,
...
a
time
of
the
6,
and
turn
...
position
They
are based
modulating
transmission
(e)
divi-
that
20,
time
devised.
position
The
(g)
0,
The
pulses
(g).
pulses
times
the
which
to
is
base
pass
0, @,
pulses
the
the
trig-
They
FG,
GA2
times
the
fromthe
sal(4,6).
gate
the
that
circuits
Walsh
depend
period.
generator
with
produces
but
larger
function,
if the period
U2
same
amuch
after
the
the
(g)
function
added
compared
modulated
pulses
functions,
immediately
Walsh
trigger
have
tothe
Walsh
the
modulated
der
U1
periodic
pulses
on
the
a ramp
ti).
SeiGMey
on
SV generates
(b)
voltage.
is
I produces
reaches
pulses
merator
ES}
link
circuits,
envisaged.
3.23 Code Modulation (CM)
Modulation
Vé(k,6+8,)
is
The
evident
normalized
128
that
signals
be
may
a
are
used
for
These
signals
enals
by means
of
of
pulse
of
with
code
pulse
code
the
system,
a
k = 0, 1,
k
modulation.
of
a carrier
following
which
is substlifunctions
transmission
modulation
a generalization
of
of
the
from
constructed
system
such
of
with
wal(k,6)
form
pulses
consideration
system
k
for
functions
the
€-e.,
(kh, ep,
sequency
modulation
code
a particular
if
foro
block
as
the
called
k distinguishes
reason:
Guvca
of
is
=
1,
0,
of
-.-+
2,
7
binary
«-.,
Nees
telephony
This
normalized
modulation.
suggests
sia
sequency
160
3. CARRIER TRANSMISSION
since
i
trast
to
integer
assume
can
continuous
permit
tion,
which
tions
sal(u,9)
and
cal(u,@)
lues
of yp with
the
exception
There
tion
and
since
by
is
no
time
i and
a comma
essential
base
for
modula-
changes.
However,
the func-
defined
least
as
Walsh
other
Hence,
between
for
connected
code mo-
code
sinusoidal
product
va-
real
all
theory.
in
difference
and
for
sal(0,6).
of
at
modulation
8 are
as
position
are
continuous,
be
may
dulation
time
and
base
time
amplitude,
con-
in
is
This
only.
values
functions,
of the
change
a discontinuous
means
cal(i,§)
and
sal(i,@)
as
such
functions
of
modulation
Code
and
functions,
not
functions.
modula-
separated
It
holds:
sini® = sin#t
(62)
A modulation
of
4/7
versa.
and
vice
i
There
are many
code
connect
The
and
as
values
produce
all
of
based
on
for
Walsh
transform
to
section
a
may
1.25
use
as
Amore
was
for
$(i,8)
common
the
of
normalized
crosscorrelation
functions
of
the
functions
through
a switch
be
a modulation
modulators
and demodulators
functions
with all possible ones.
demodulator
Fourier
one
may
interpreted
integer
onemay
proper
demodulator
received
Using
i only,
the
be
possible
modulation.
sequency
may
of
the
ingenious
fast
done
and
line.
by
WalshGREEN
collaborators.
3.3 Nonsinusoidal Electromagnetic Waves
3.31 Radiation of Walsh Waves by a Hertzian Dipole
The
solution
dipole
may
scalar
potential
lar
system
A(r,t)
be
of
of
Maxwell's
written
equations
by a vector
»(r,t)
functions
for
potential
the
without
reference
to any
such
sine
as
or
Hertzian
A(r,t)
and
a
particu-
cosine:
p(t-r/c)
Ane
o(rse) = gee (eBUieale) , rwCe-a/0))
(63)
co
3-31
RADIATION
r
is
point
the
and
OF
vector
r the
is
the
WAVES
from
vector,
which
and is proportional
riable
charge
and
q(t)
t-r/e
assumed
p and
of p at the
Electric
computed
the
dipole
observation
moment:
the
direction
of
the
otu)
the
va
i(t)
e,
isthe
A
and
the
s
that
s.
on
a change
and
current
is
so
retarded
The
delay
in
arguments
between
E(r,t)
of
a change
obser-
andH(r,t)
the following
space
may be
formulas:
E(r,t) = 2A e) _ grea (r,t)
Mite
a=)
HU ) 1s
the
The
zone
the
that
small
constan
of empty
t
forces
» by means
16
of A and » at the
dielectric
and magnetic
from
and
usual,
(65)
length.
p indicat
the time
e
dipole
point.
vation
as
depend
do not
has
WOPLes
dipole
the
of
is
i(t)
of
to
is the
p(t) = p(t) = its
dipole
It
dipole
p(t)
dipole
dipole.
161
the
distance.
p(t) = a(t)s,
S
WALSH
ey
FOLAUr,tb)
magnetic
following
are
permeability
solutions
obtained
from
of
empty
space.
for E and H holdinginthe
(6%)
to
oe
wave
ale
E(r,t) = qleeex(rxp(t-r/c)]= mais
potty HCRt/e)
x(rx8) (67)
dt
H(r,t) = goderB(t-r/o)ur = groper SG
ie
ee + 377 Ohm,
The
wave
with
compared
derived
E(r,t)
11
a region,
definition,
usual
the
as
restricted
more
where
ris
definition
r
that
assumes
wavelength,
is defined
zone
near
ficiently"
H(r,t)
defined
must
be
sinusoidal
a
"suf-
will
be
large
cur-
it.) .
The
be
The
below.
given
Seay
4
ie 3x10" m/s
¢ = y=
+
A
large.
ficiently"
sent
zoneis
Stone
—
small.
for
=
The
the
following
near
p+ids
= PUt-r/c)xr
Anrr3
a region,
as
zone
(63)
"suf-
E and
H can
to
(66)
[5]:
sales ==r/e oor-s+ ASE
_ ut ter/c)
Harmuth, Transmission of Information
for
formulas
from
r is
where
Anrxr3
(69)
(70)
E and H of (67) and (68) are much larger than E and
(69) and (70). The opposite requirement defines the
zone. The following conditions are obtained:
that
H of
near
c? cHON
rss
ay
Cc
ra
Consider
integral
sin
rs
a
The
i(t)
will
with
time
in
This
is
due
in
the
wave
In
to
between
foresee
an
warning.
quotient,
zone
is
The
to
of asphere
ting's
be
power
far
vector
over
zone.
sinusoidal
func-
of
the
or
zone,
E and
to
-
however,
in
of
engineering
this
i(t)
effect
must
the
inthe
r
have
from
a
be
i(t).
The
the
or
between
terms.
One
aircraft
small
near
I
Zo
(ae
6rc2
at
may
colli-
differen-
zone
to
zone
through
the
obtai
by integrating
ned
surface
a?
dis-
wave
transmitter.
wave
is
to
wad
according
of
point
H
E will
to
thus afunction
observation
E andH
differentia-
i(t).
proportional
H is
cur-
of
in
of
the
surface
Poin-
sphere:
GP
rtEC
)xH(r tao = 220, p(t-r/c)
PCrytt)
H
near
functions
transition
the
and
as
integrated
near
and
radius
E
a sinusoidal
and H proportional
that
flowing
with
for
variation
feature
zone
the
from
-enfix
= x/en
that
well
if
other
the
zone
c/2nf
as
receiver
if
wave
The
Icos 2nft.
=
differential
a sinusoidal
E and
Note
the
rs
zone
dipole
and
sion
for
the
(67) to (70)
application
tial
and
case:
fi(t)dt
of
i(t)
current
nit
peculiar
the
transmitter
(F2a3
sinusoidal
dependence
tance
for H(r,t)
wave
(68).
proportional
time
zone
produce
to
and
wave
from
the
remain
vary
(67)
C71)
this
Thisis
not so for
both
to
in
apparent
to
E(r,t)
= .?/(2an)?,
rent
tions
for
conditions
c?/(2nr)*
is
zone
saz sin
identical
It
wave
sinusoidal
equals
2nft.
become
ted.
requirement
the
by
defined
be
now
may
zone
wave
The
TRANSMISSION
CARRIER
%.
4162
ce
(73)
(ss)
3.31
RADIATION
OF WALSH
Introduction
of
Beem ce),
yields
the
ation
power
WAVES
the
165
rms-current,
G! = t-r/c,
radiation
C74)
resistance
R, fromthe
average
radi-
P:
2
= <P(2,t)> = @(r,t')> = BS (GE,
2
Rs=
As
an
There
may
are
two
»
consider
cases
that
the
radiation
have
to
be
of
Walsh
waves.
distinguished.
One
T fi cal(k,t'/T)at'
She
(76)
currents
zt)
=r
into
the
tain
point
‘i Sale,
boy het? 5 aC tenes
-1/2
Hertzian
dipole.
inthe
e,')
wave
into
at)
=
thenvary
proportional
tou(G7)
currents
tear
ct /?),
E and H will
zone
according
Walsh-shaped
Dees
CG)
6nc2
example,
feed
eat.
(Se)
Git!
Zos*
P/Itms=
(75)
and
the
to
(6&)..One
at acer-
sal(k,t/T)
may
also,
or
feed
dipole:
Leéaltk, t/t)
C92
H will then vary proportionallyto sal(k,t/T) or cal(k,t/2)
at
a
and
certain
H will
of
in
case
the
Walsh
Fig.76
current
dipole
power
iil
will
yield
to
I,,,
into
that
R,
is
according
and
which
the
same
are
to
currents
about
in
the
is
taken
10
shows
(70).
the
according
This
ideal
R,
to
for
a
fig.76
the
average
One
may
see
and
se-
shape
functions
current
that
for sine
same
Walsh
receiver
holding
for sinusoidal
cases.
E
account.
peak
resistance
assumed
all
the
voltages.
from
into
of
functions
of the Walsh
Table
radiation
It
anintegration
Walsh-shaped
integral
(76).
shown.
are
zone
deviations
functions
shows
also
near
zone;
comparison,
the values
are
and
the
far
requires
according
For
the
voltage
cond
in
vary proportionally
to the differentiated
functions
input
point
Walsh
that
I,
rms-
Hertzian
are
fed.
currents
radiated
I,
functions
Irms
of
3. CARRIER TRANSMISSION
164
———
a
foal08)
5
eee
Fffeaitna
1 {sal(2,8)
feal(i,@)
;
oe Hl)
ee
Falls)
ARAAAS
PPA
PASS
LHS
LLG HA
DrHrSE
functions.
= JP cal(ii ,x)ax
-1/2
fsat(iye) = f' sal(i,x)ax
U2
F Seal(3,8)
SOO
Integral functions of
Walsh
the
4 feal(2,6)
ON
SO
Fig.76
8 =
Sead
Juailo.80
= Seailb8)
Sia 8)
Seal(689)
aw
foal (7,8)
PITTI
fos 78)
ELLIS
fsal (8,8)
ee
eee
-1/2
0
1/2
—
Table 10.
Peak current I, rms-current
Irms and radiation
resistance Rs for a Hertzian dipole. Z, = 377 Ohm, c velocity of light, s length of the dipole,
T period of the
radiated functions (Fig.76), P average radiated power.
f'sal(1,8)
5=0.5
3-0 289]
1
Sin 2nd
==0.225
| —==0.159
foat(1,e), fsa1(2,8)/
z=0.25
mys70 144
sin 2n9
f<-0.112 | —-0.079|
foal(2,0)..fsal(4,e) | g=0.125
By370-072
sin 816
Y2-0.056
e=0.040|
4x3=12
47 2=39.5
16x 3=48
16n?=158
64x 3=192
64n? =631
Joal(4,0)..Jsal(8,8) | 7p=0.063 |zapq=0.034 |256x3=768
sin 16n@
enor)
tke
=O: 020 | 256r? =2520
3.31
RADIATION
equal
are
sequency.
exactly
1%
One
equal
eons)
sine
OF WALSH
WAVES
may
for
further
Walsh
and
Rs
functions.
depend
While
on
the
the
determined
is
in
0 ——
now the
Consider
current
a
Ical(3,t/T)
time
sider
general
the
Isal(k,t/T);
shall
be
end
Rs
not
case
a
of
true
choice
of
k.
Fig.77
Radiation
of
a Hertzian
sequency
The
functions.
Walsh
case
that
dipole.
and
case
same
One
the
is
for sinusoidal currents, it
antenna for currents having
a
a Walsh
wave
dipole.
below
of
Walsh-shaped
Fig.77
shows
a current
At. The differential
the
used.
in
2
Hertzian
switching
dipole
by the
by
into
Develrms
frequency
Hertzian
integrated
of
shape
di
fed
that
functions
(frequency) wideband antenna
isatrue (sequency) wideband
the
see
see
a yea ye cal(oaiig)
<.egar of ae.9R)
Itms
bandwidth
165
a Walsh
L(t
the
with
finite
also shown.
current
Tealyt
following
i(t)
is
idealized
— is
approximation as for
obtains
current
the
Conl
or
in Fig.77
averages
(eS)
and ee Ct):
(=
2
&) Se -4PR,
(79)
=k
Grey «t= BY Befee = P=
(0
a
ee
ct ee
P= 20 mag Sree = “LAE Bro?
81
.54)
tz
peas oe
2
ne
Dr a beanie2
6
(82)
3.
166
yields
tion
or
Isin emkt/T
power and radia-
resistance:
p= nti? #908 = tres? SO,
2
2.2
7
om
Ry =
ZgS?
2=
Sree
2Ae
(83) and (84)
on its
frequency
f
(82)
for
Walsh
At.
large
trarily
sing
the
time
Table
At.
by Rs sal /Bssin
1 Gzps.
about
equal
of the
for
a
fora
than
frequency
switching
time
for
Table
11.
ratio
Resa /Rs,si, for
Power
antenna
and
and
swit-
arbi-
by
and
radiation
resistance
ps.
100
=
makes
-
a sequency
tech-
and
ra-
functions.
P,,,/P.i;,
aHertzian
the
one order of magni-
functions
Walsh
are
A reduction
power
radiated
de-
denoted
f = 1 GHz
At
sinusoidal
ratio
decrea-
(81) and (83)
of
to 10 ps - whichis about
present
for
be
quotient of (82) and (84)
and
resistance
higher
may
R,
quotient
power
time
switching
diation
tude
and the
limit at the
nical
sequency
the
shows
Radiated
P and
and
made
»
depend on sequency
current
(81)
relations
the
depend
current
sine
forthe
while
alone,
foragiven
by Psal/Psin
noted
(84)
Theoretically,
11
(83)
5222
2g8?
Sate
a
relations
ching
of = k/T
2
The
~ =
TRANSMISSION
Icos2mkt/T
for radiated
current
sinusoidal
the following values
The
CARRIER
and
radiation
dipole.
Rs
4
Resin
resistance
f=1GHz,
»=1Gzps.
|
772 FAt(1-~t/6 )
1 805
TO.
100.0
Let
pared
the
with
sideragate
presses
switching
the
time
average
that
At
permits
the
any pulses that arrive
ber
of
all
having
the
Bequencies
1 =
independent
pulses
at other
transmitters
same
OTS
in Fig.77
oscillation
AG
time
“he
base
may
but
receiver,
be
very small com-
period
- to
Tt =
times.
radiate
Con-
but
sup-
A large
num-
Walsh
different
the
1/9.
pass,
waves,
normalized
pulses
—
not
3.32
PROPAGATION
arriving
gate.
the
at
The
OF WALSH
the
correct
timing
of
nous
is recognized,
and
negative
gate
must
must
not
distinguish
can
distinguish
less
a timing
dio
communication.
to point-to-point
ago.
tical
applications
in mobile
the
possibility
member
that
any
of
such
verification
perimental
passed
when
useful
waves
sinusoidal
for
ra-
mobile
example
of
pos-
-
in
conlinks
some
speculate
wa-
electromagnetic
disclaiming
before
one
well
may
the
of
some
while
by HERTZ,
large
45 years
had
commu-
practical
scale
exmade
tube
electronic
repre-
their
and
by MAXWELL
70
prac-—
any
on
theoretical
the een
betw
waves
for
un-
functions
carriers
application
development
the
carriers
However,
elapsed
electromag
oftio
dic
netic
n
the
function
Walsh
non-sinusoidal
by
can-
carriers
sine
a cosine
new
syn-
of
function
communication
to
early
to
of
years
20
in
carriers
for
Sinusoidal
of
communication.
ves
transmit-
as on microwave
transmission
much
is
like
recognized
is
Hence,
first
mobile
introduction
years
as
the
is
for
trast
It
just
transmitter
a cal
and
a sine
provided.
This
carriers
radio
the
by the
in synchro-
desired
receiver
intheory
sible
since
The
and
a sal
is
least
at
used
correct
correct
carriers
sine
between
be
be
be
the
thanthe
signal
can
Suppressed
Walsh
for
recei
ver
between
any more
Sequency
same
like
of
The
frequency.
proper
be
carriers.
just
demodulation
chronous
will
howeveby
the patternof
r,
the positive
pulses,
the
167
carrier
demodulation
of sine
ter
-
time
the
phase
of the local
WAVES
Maal @eyE
OIA
3.32 Propagation, Antennas, Doppler Effect
One
ves
of
is the
For
the
invariance
plitude
modulated
practically
carrier
V2 sin 2mé.
dulated
carrier
gral:
a
may
during
carrier
sine
by a signal
constant
shifts.
to time
of their orthogonality
consider
explanation
wa-
sinusoidal
of ages
advant
important
most
F,(9).
any
Synchronous
The
period
of
signal
n
demodulation
be represented
by
the
an-
V2sin2nmn9
B,C8) es
of the
cycles
of
the
following
mo-
inte-
8'41/2
= Fp(O')bnm
dé
(2 sin 2nme
(84) Pye sin 2nne
=F,
(85)
=
dé
Vesinenme
f F,(@)V2 sin enne
9-1/2
TRANSMISSION
CARRIER
3.
168
g'-1/2
case
In the
received.
by
is replaced
F,(9)V2 sinennée
Hence,
is
shifts
time
various
with
carriers
modulated
communicationa sum of many
radio
of mobile
SF, (8) V2 sin 27n(6-6, )
n=)
and
(85)
O'l1/2,
I
i
YF
6-1/2
assumes
the
following
form:
Ceve sin enmn(6-8,)]V2 Sin 2nmé
dé
=
(86)
n=l
= F,(9') cos année
The time
talk.
The
shifts
8,
introduce
orthogonality
the
same
the
orthogonality
frequency
preserved.
The
is
to
{V2 cos k(6-9,)}
are
attenuation but not
sine
destroyed
functions
subsets
of
reasons
for
and
functions:
and
by
of
cosine
the
this
for
arethe
any
time
or
values
shift
cos ke sin ké,
cos k(6+6,)
=
cosk@coska,
-
sinké sinké,
sal(k,9@0,)
similar
shift
of
theorems
+
very
but
{/2sink(6-6,)}
sink@cos ke,
have
shifts
is
=
functions
of
frequency
sink(9+0,)
Walsh
cross-
functions
different
functions
orthogonal
underlying
cosine
of
6,
@,.
The
of sine
(87)
theorems:
= sal(k,@)sal(k,6,)
(88)
cal(k, 908, ) = cal(k,@)cal(k,
0, )
The
essential
replaced by modulo
Sional
can
ot2
ee
and
its
wave
difference
2 addition.
is
that
ordinary
Consider now the
addition
is
one-dimen-
equation,
sou
ox?”
general
(89)
solution
3.32
PROPAGATION
Wixi
OF WALSH
eC G—x/o)
The
occur
in
because
the
1169
+¢h(t4x/c).
orthogonality
preserved
WAVES
of
Walsh
ordinary
arguments
(90)
functions
and
system
{sal(k, 6+6,),cal(k,9+9,)}
for
pendent
functions
separation
of
remain
sal
orthogonal
The
Walsh
present
may
Walsh
second
column
fast
the
"left
The
column
is
column
the
ends"
a twist
third
obtained
2x360°
in
The
Fig.78.
the
hand,
are
of
of
column
the
360°
is
column
by
twisting
the
of
sense
Circularly
polarized
Walsh
re-
5
are
circularly
obtained
from
functions
inthe
to
the
a
and
sense
obtained
90°
to
first
The
the
of
by turPichu.
functions
right
e = os
Sep
78
than
of
SCrew.
Fig.
inde-
other
considered
waves.
the
screw.
first
be
of
functions
of the second
fourth
the
Fig.2
of
hand
independent
systems
on
column
ends"
the
difficult
The
functions,
signs
However,
linearly
more
first
"right
the
but
of
not
shifted.
of
their
a right
of
time
x/c.
is linearly
functions.
cal
polarized
waves
them
by holding
The
if
linearly
giving
or
t -
Separation
possible,
functions
againinthe
polarized
ning
is
orthogonal
differentiated
shown
cases.
generally
addition
and subtraction
t + x/c
except
singular
is
waves.
hand
3.
A170
polarized
right
unit
per
(turns
as
with
sequency
functions
the
of
frequencies
The
are
Hertzian
sinusoidal
or
holds
half
better
waves.
much
dipoles
more
sin onf(t-x/c)
will
standing
be fed
wave
if
+
all
known that
quarter
wavelength
sine
of
it.
based onthe
IL.
Let
A reflected
for
proper
look
for
Hertzian
the
wave
sumof both
are
of
one will
length
and the
losses
waves
Hence,
sin Onft(t+x/c)
wave
sin 2nfx
waves
yields
neglected:
=
2 sin 2nft cos 2nfx/c
(91)
dipole
may
be
Zian
dipoles,
but
feed
them
because
all
sin onf(t-x/c)
(87)
The
of
of
(91)
wave.
It
is
Hert-
is required
of
follows
due
to
to
the
wave
from
(91)
the
shift
functions.
for
Walsh
functions
is obtained
(88):
wave is produced.
along
=
a
metallic
(92)
However,
conductor
are
Walsh
waves
described
by
or
sal(gT,t/T+x/cT) rather than by the
sal(gT,t/Tex/cT)
or
sal(oT,t/T@x/cT).
It is
expressions
how
amplifier
of many
sal(oT,t/T)sal(pT,x/cT)
Againa standing
sal(pT,t/T-x/cT)
consist
transformation
+ sal(T,t/T@x/cT)
=
propagating
power
the
cosine
equation
from
sal(pT,t/Tex/cT)
to
transformation
and
following
instead
one
of
a standing
this
sine
considered
only
into
that
theorems
known
2,
= AL
This
and
equals
five
theory
functions
into
se-
The
4.
antenna
dipole
be produced
sin Onf(t-x/c)
G/f =)
a
1 and
of
radiate
for Walsh
have
1,
2.
1 and
basis
efficiently.
Consider
(t+x/e)
a
It is well
radiators
dipole.
1,
O,
again
and
line
O,
line
in
four
lines
in
dipoleisthe
wavelength
frequency
functions
the
waves
Walsh
second
the
in
appear
waves
These
1 and frequencies
sequency
for
frequencies
polarized
right
functions
are the usual
2.
and
of
case
The
O.
same
quency
the
0, 1, 1
normalized
the
the
2;
time)
special
the
here
all
having
Fig.78
normalized
the
waves,
TRANSMISSION
of
line
first
functions of the
The
CARRIER
to
make
a Walsh
wave
propagate
according to the
3.32
PROPAGATION
argument
OF
t/T@x/cT,
WALSH
than
based onthe
standing
tive.
a power
that
feeds
antenna,
poles
dred
a
either
it
that
is
metallic
wave
are individually
are
implemented
are
not
are
used.
naturally
as
a long
a
two-dimensional
small
The
nas
radiation
has
been
Its
sinusoidal
of
beamwidth
D the
diameter
of
cAt/D
occur
the
the
in
average
faned@inwie./7/7
and
width
decreases
to zero
while
the
may
thus
be
diameter
D
the
of
all
to
The
effect
pulses
increased.
on
the
last,
This
canbe
in
the
and
The
two
ratios
\/D
and
where
} is
now
switching
time
de-
be
waves,
the
of
ratio
The
many
bean-—
time
A narrower
}\/D
or
a
At
beam
smaller
Actually
byacircular
Hertzian
disc
dipoles
function.
space
empty
waves.
Upon
andthe
isnot
from
radiate
to
used
short
light.
switching
beamwidth.
replaced
by
A/D
re-
wavelength
fixed
covered
Walsh
effect
relatively
by
anten-
a parabolic
constant.
a
wave
ratio
decreasing
a fixed
would be widened
be
area.
replaced
be
be
the
velocity
for
almost
onthe
The
to
ina square
Walsh
the
is
remain
is
same
need
} is
Walsh
At
the
for
antenna
an
of
with
may
which
the
on
where
v/p,
suffice
earth.
trimental
the
D
as
antennas
not
may
e.g.,
reflector.
reflector
radiating
such
depends
18
obtained
diameter
probe
\/D
may
parabolic
Let
¢
do
simple
Consider,
case
wavelength
ratio
some
waves,
the
tech-
antennas.
resonance
antenna
a thousand
circuit
antenna.
calculated.
case
while
arranged
canbe
patternof
flector.
-ateune
active
to
di-
A hun-
dipoles
one-dimensional
Hertzian
integrated
to
lead
Hertzian
but
alongaline
arranged
Hence,
fed
current to the
Practical,
by transistors,
lead
switch
amplifiers.
perfectly
waves
attrac-
wavesisa
many
fed by such
Walsh
waves
use
more
antennas
appear
a negative
if
sine
Walsh
to
unrealistic
individually
many
or
appear
more
niques
for
attractive
switches
and
Hence,
principle do not
a positive
dipoles
circuitry is much
conductor.
amplifier
more
Hertzian
q71
but the required
complicated
Since
WAVES
would
a
have
space
no
de-
hitting the atmosphere
width
important
sectionof
of the beam
would
since it occurs
the
transmission
to
obtain
D of
the
the
same
narrow
reflector
by clouds.
and
that
A sinusoidal
Doppler
shifted
or
cae
where
relative
frequency
a given
absorbed
but
waves
once
diameter
completely
appear
more
quite
one
must
is
verification
available
yet.
wave
effect
Esin
2nf(t-x/c)
into the wave
frequency
GU VaroSeSe Slee
aero
v is the
be
Walsh
electromagnetic
The
with
thus
applications,
no experimental
transformed by the
(t'-x'/c).
beamwidth
would
harmful.
high
avery
have
electromagnetic
Hence,
promising
in certain
caution
to
have
would
waves
Sinusoidal
thisisvery
and
probe
space
a
to
earth
the
from
radiating
when
path
occurs
beam
the
of
widening
transmission
the
of
beginning
the
at
the
hand,
other
the
On
path.
TRANSMISSION
CARRIER
3.
N72
has
the
is
Esin enf'x
value
(93)
velocity
of
transmitter
and
re-
ceiver.
A Walsh
wave
Ex,t)) =B
is
transformed
vistic
t
eal (ol,t/2
into
the
nea gs
(94)
transformation
xy
On
oo
ame
vi/o2
the
following
E(x',6')
In
xfer)
equations
of relati-
mechanics
ee
=
by
—
= Esal|
order
define
the
to
ve
er
viva
(953
form:
pt,
bring
bt
axe
——_____
in
= v'/c*
="
Ve
(96)
transformed
into
(96)
the
sequency
»'
form
and
of
(94)
time
one
base
must
T'
as
follows:
o!
ee
eae ee WS,
- v/o
(97)
(98)
3.33
It
INTERFEROMETRY
175:
follows:
Bs
ce
eee
Equations
quency
f
E/N!
4 eV /ens )
((97)
and (93)
are
additional
(On
changed
change
show
equally
of
the
that
sequency
by the
time
(99)
» and
Doppler
base
effect.
T according
gene
anrate
invariant s
of the Doppler
effect
fre-
or
The
to
of
(98)
Lorentz
transformation:
T'p'
= Te
(400)
A sine
wave
with
relative
with
frequency
velocity
One
may
O.
the
Thisis
w=
have
the
The
Doppler
any
pend
on
wave,
the
be
the
on
same
of
located
would
other
wave,
a Walsh
a
with
for
wave
weak
such
for
a
of
the
=
6/1,
sallo,@).
waves
from
intelligent
These
signals
attempts
were based
would
be
raises
more
likely
and
sine
the
to
waves.
question
be
used.
transmitting
a
different
frequency
this
frequency
would
planet
could
of
wave
of
@1
signals
the
equal
functions.
reduction
= Ro yields
with
hand,
relative
differ
planet
and
one
sto o!'
not
on
with
Walsh
a
from
would
Walsh
regardless
Furthermore,
sine
so
that.
planets.
are
space
position
the
wave
received
in
not
Fig.2
sal(S,0
that
waves
direction
from
other
by a transmitter
by atransmitter
attempts
to detect
effect
these
transmitter
waves
been
from
f radiated
cannot
be distinguished
Walsh
assumption
whether
v
generally
see
B/Tiot
transmitted
frequency
radiated
resulting
There
on
f'
ireedily
pequency
but
with
velocity
in
its
always
direction
has
twice
amplitude,
from
de-
A Walsh
identified
of
the
sine
also
orbit.
be
A
as
propagation.
average
animportant
power
of
advantage
signals.
3.33 Interferometry, Shape Recognition
shows
Fig.79c
measurement.
Two
the
of
principle
receivers
the
at
parallel
pagation.
time
a
difference
and
AT
points
which
saway transmitter
from afar
wave
the rays
along
interferometric
A and
travel
angel
B receive
practically
the proof
nt
A measureme
b.
=
AC/c
yields
the
angle
TRANSMISSION
CARRIER
3.
7
sin 2n(tT)(t/T)
{Tal
oT
[Seeohi
Amin * &/f
a
sal(pT,t/T)
piel
AT
Mnax* 7
ss
pe
Brin /P
Max
* 1 *(AB/c) cosck
Otnin = (AB/c)AR
c
Fig.79 (left) Interferometric measurement
of -angles. a)
resolution and resolution range
of
sine waves; b) resolution and resolution range of Walsh waves; c) geometric
relations fortwo receivers A and B positionedonthe
same
meridian.
Fig.80 (right) Reflection of sine and Walsh waves by two
point-like
8
=
sin!
AT,
on
targets.
cOT/AB.
depends
the
-
1/p
for
denoted
the
and
angle
Walsh
pletely
must
¢
lie
wave,
in
for
+
since
a wave
the
undelayed
\/
faim
eone
The
wave.
Case
of
Walsh
b.
functions
Hence,
and
The
AT,;,
or
the
-
is
factor
is
that
is
resolution,
smallest
measu-
equal for sine
the resolution
+7T/2,
AT,,;,
proportional
is approximately
largest
range is com-
permissible
value
if
period of the
T is
delayed
by amultiple
Hence,
ATma, equals
same
difference
the proportionality
time
and
for
functions
However,
-T/2
as
time
crossings.
and
cATmin/AB,
different.
between
well
functions;
functions.
measurable
zero
sine
Fig.79a
measurable
AB
as
their
1/f
Walsh
by
smallest
sine
of
to
smallest
rable
for
gradient
proportional
to
The
functions,
the
of
T.
ATmax
T is
Since
equals
of
equal
T
AT
to
equals
ATyin/é >
4.35
INTERFEROMETRY
Certain
Walsh
T =
and
i/o
tion
is
are
functions
ATmax
shown
ipan2’
waluervof
i =
according
the
and
southern
10°° s. The distance
observation
AB
following
Cee
cA
An
11
angle
surface
one
Mars
for
of
distance
toa beacon
of
would
be
over,
42x10°
Fig.80
and
B2
An
the
1000
shows
equals
from
are
of
about
4 km
Earth.
For
of
0.05"
the
best
about
on
the
compa-
is
that
appears
about
can
be
not
be required.
to
the
10°
of
The
block
amount
2x10°
would
data
assumed
be
capacity
storage
ratio
by
Averaging
a total
require
case
would
bits
4.0% pulses.
of
multiples
a
another
to be com-
signal-to-noise
the
the
consisting
have
pulses
of
of
previously
Additional
improve
of
functions
of
capacity
Such
vincinity
Walsh
two
attractive
knowledge
A considerable
multiples
a
to
accurate
comparison.
storage
of
than
required.
storage
many
that
be
Gor)
a distance
and
measurement
probe
is
sequence
over
bits
to
AT,j,
telescopes.
that
means
to
€.g.,
fai=252°
= 10°
angle
smaller
space
needed
averaging
1 SB i
computed
‘ate
resolution
The
close
frequently
a
A minimum
for
is
transmitter.
Of aperiodic
required
be
assume
to
moon
tracking.
equipment
10°
i =
without
may
= 2.5x10%e
Mars
angle
would
or
processing
pared.
of
probe
AB
ATz.,
difference
points
and
76°.
resolvable
astronomical
guidance
A large
The value of i is obtained
the
magnitude
is
value
when
method
space
=
150.
10 000 km and the usable
corresponds
of
smallest
with
This
0.05"
surface
the
order
done
of
of
rison,
20
values
page
time
func-
usable
range
values
latitude
=e cosa/ip
the
9 on
bé >two
8
= 0.05".
period
relation:
/AU
mon
Table
resolution
AB is about
180°-
angle
= 3x10°8or AB
the
to
and
shortest
Other
measurable
GA
a
idT,;,
/€. Sucha Walsh
representative
(Joss Let
northern
have
Fig.79b.
the
smallest
following
PoOUeerie.
then
43 in
increases
increasing
The
sal(i,6)
equals
for
264
iI)
195
of
some
capacity.
radar
close
R and
together.
two
targets
point-like
Lines
a
and
b
show
B1
sine
by two
targetsorbyasingle,
pulsed
sine
indicating
two
tal
of
energy
Let
us
d and
frequency
carrier
andthe
1 us
distorted,
of them
two
only
and
targetsisin
the
f shows their sum.
or
the
waves
two
ning
andendofapulse.
tell
how
not
be
veral
Walsh
waves
may
be
the
reflection
would
a
to
solve
the
and
initial
optics
functions
as
for
waves
Walsh
would
sine
wave
be
and
wrong
to
cosine
sequency
antennas
tions
would
had
been
the
but
wave,
radar
the
dominated
systems
treat
Walsh
and
sine
and
multiplexing
have
treated
cosine
a
shifts
particular
been
done
and
cosine
no
theory
orthogonal
waves.
is
as a superposition
the
known
waves.
found,
se-
approach
sine
and the results
been
as
waves
of
investigate
not
There
of
apply
time
the
by
could
signal.
sum
proper
has
still
difference
the
for
This
begin-
distance
must
The
communications.
to
would
various
equation
been
reflected
reflected
one
dish.
wave
waves
never
wave
that
Lines
B2, and line
waves
what
show
shape
complete
optics to these
ters,
sine
as
or
of
waves.
and
Walsh
conditions.
has
much
Walsh
B1
restricted
and
shape
shaped
to-
the
of
0.1%
between
longer
are
equal
the
Wave
It
of
of
from
f in Fig.80
on
be
of
the
the energy
Hence,
although the absolute
differently
boundary
yet.
from
d to
is
insignificant.
reflection
there
are,
duration
pulse
of
cycles
1000
are
1 GHz.
A periodic
targets
lines
the
order
is
targetsisno
inferred
Since
if
difference
one
many
There
reflected
The
from
of their distances
b.
is
the
and
pulse
the
consider
e show
and
a
and end compared with lines
beginning
at
deviations
c shows
line
of
wave
The
target.
reflecting
more
reflected
whether
same
the
look
would
wave
sine
periodic
A
radar.
the
by
received
is
which
waves
sine
two
these
of
sum
the
shows
c
Line
B2.
and
B1
from
reflected
waves
TRANSMISSION
CARRIER
3.
176
if
superposition
results
Sequency
for
the
of
Walsh
Walsh
of
filwave
func-—
sine and co-
functions.
Lacking
a wave
theory,
asafirst
approximation.
lic
The
dish.
distances
one
may
Fig.81
r,
and
use
shows
r,
geometrical
a cut
+ d are
of
optics
a parabo-
equal. Hence,
a
3.35
INTERFEROMETRY
479
'‘c—_—
1+ cosa
d=r, cosol,-r, cosa,
b=,+d
Fig.81
Reflection
of waves
according to geometrical optics
by a parabolic
mirror
(a) and two perpendicular
mir-
rors
Walsh
wave
equally
time
radiated
whether
shift.
Vice
dish tothe
it
is
focal
not
Another
no
this
lar
point
D will
be delayed
B or
C and
will
reflected
add
without
by a parabolic
not be distorted
even though
in
Fig.81b.
showninthat
and
shift
distortion-free
b
are
between
without
result
also
holds
follows
figure
that
equally long.
Walsh
change
reflector
It
and
of
for
waves
It
Walsh
can
the
two
prowill
from
wave
be
the
there
reflected
the
shape.
the
Hence,
is
from
will
shown
a three-dimensional
va-
be
that
rectangu-
retlecvor.
nite
general,
dimension
the
flection
Fig.2,
a Walsh
wave
will
longer
be
will
yield
reflected
geometric
12
a
a
points
of the reflector,
In
of
D will
shown
relations
time
rious
of
mirror
paths
reflected
focal
versa,
a signal
point
example
geometric
be
at
sinusoidal.
rectangular
pagation
fromthe
reflected
(b>).
size
of
and
@ step,
no
wave
shape
like
of
the
froma sphere as shown
Harmuth, Transmission of Information
reflected
the
one
by
a Walsh
wave.
fi-
shape
about
the
the
re-
Consider
sal(1,9)
in Fig.82a.
of
The
information
target.
of
a target
at
A correct
6 = O.in
treat-
172
3.
CARRIER
TRANSMISSION
Fig.82 Shapes
of step waves reflected by perfect
scatterers of various shapes. a) sphere; b) rod of length L and
diameter d«L;
c) cylinder;
d) circular disc
of
diameter OR; e) radar reflector
(4 perpendicular mirrors).
3.33
INTERFEROMETRY
ment
would
A
first
suming
the
again
that
cident
of
time
the
sphere
functions
with
construct
the
to
shape
the
time
t =
At
half
of
of
step
of
one
amplitude,
re-
Walsh
all
Since
may
from
functions
Walsh
#ct
than
reflected.
superpositions
be
reflected
of
the
of
increase
no
negative
or
positive
as-—
illuminated
t.
of
values
considered
be
may
functions
be
will
there
larger
for
this
the
be
will
the
on
points
and
reflect,
power
flected
all
2R/c
=
t
the
on
in-
Initially
smaller
distance
a
point
surface
After
as-
will be reflected by all
power
more
by
by the
under
the
only.
S
having
much
S and
from each
Fig.82a.
on
equation.
obtained
computed
of
points
plane
sphere
the
on
step
right
the
the
wave
be
that is illuminated
elapsed, the wave
6) has
plane
the
by
to
close
sin
points
on
the
however,
reflected
shown
of
wav
is radiated
e
sphere,
reflected
Sphere
from
the
The
is
is
may,
aspherical
wave.
sumption
Ba
requ
aire
solution
approximation
surface
wave
179
Fig.82a.
The
computation
of the shape
of the reflected
is
as
amplitude
due
an
annular
area with
follows.
distance
ct
from
is
proportional
its
27mR
cos
fraction
The
sin®
of
incidence.
as
function
to
of this
The
2
angle
m/2-f
the
wave
reflection
from
plane
S
6B RdB,
in
but
Fig.82a
only
the
reflects
back into the direction
voltage
the
of
area
area
to
step
u displayed
thus
8B is
on
an
given
by
;
oscilloscope
>
is
| cos B' sin 6' dB' = KnR’ (1 - sin’B)
u(8) = 2nR’K
(102)
f
Ousacusestt,
where
K
Since
of
time
et
= RM
from
12
is
u
and
-
Fig.82a
displayed
not
dimension
ects
corrthe
amplification,
attenuation,
for
that
isafactor
of
B one
sink)
into
(102):
on
an
may
reflectivity,
oscilloscope
substitute
as
and
allows
etc.
a function
3. CARRIER TRANSMISSION
480
ee Seen
in Fig.82a.
plotted
curve
is the
u(ct)
2
= kako,
“Ulet)
KnR*[1-(1-S2) J sot stony)
ucCet):=
by a
Fig.82b shows the shape of a step wave reflected
rod of length L and diameter d << L forvarious anglesa of
incidence. Fig.82c shows the reflection by a cylinder, if
rious
angles
best
question
for
This
ambiguities,
The
cess
than
t =
T and
sion
filter
6 = % or
a target
sal(8,6),
However,
steps,
the
than
but
shape
has
the
are
from
the
the
The
point
can
from
ternative
foramore
that
sine
discussion has been
meter
be
Fig.2.
it
causes
than
gcT,
larger
than
filter
period
the
and
is
pro-
6 =1
with
or
dimen-
would
and
while
sections
Walsh
of
thus
that
offer
of
might
Fig.77
introduce
thus
tar-
waves
too.
inherently
very
a promising
al-
and
use-
obtained.
The
although
the
waves
appear much
the
of the
functions
and radar
be
complicate
about
view.
probes
to Walsh
T.
sal(7,9) has 14
information
constant
point
and
period
study of resolution
restricted
time
filter
shortest
waves behave
and
detailed
Walsh waves
to
done by sinusoidal
Walsh
ratios
simple
complicated
are
waves
these waves
switching
not
more
and
signal-to-noise
However,
this
targets
provide
tracking of space
hereis,
differentiated
to
the
only,
that
theoretical
be
different
ful
has
2 steps
steps
Interferometric
analysis
in
dimension
for
equally
also
target,
Hence,
get
would
However,
distances
shortest
is
and
itisthe
functions.
better
process.
harder
occur
a radar
cl.
sal(8,8)
of
its
forma
of
mirrors.
t = $1 and
with
is
will
sal(1,8)
sal(1,8)
and
and
<n
sal(8,6)
hasalarger
targets
sal(7,8)
that
functions
Consider
of
ambiguities
as
Walsh
to
function
process
which
easy
function
larger
The
Fig.82e
period
if
diaméter
perpendicular
three
of
orif there are several
ecT.
and
recognition.
is
a shortest
of
arises
shape
function
has
incidence,
of
consisting
reflector
The
a
shows
Fig.82d
axis.
incidence is perpendicular to the
reflection by a circular disc of
the
the
superior.
additional
the
para-
discussion.
4. Statistical Variables
4.1 Single Variables
4.11 Definitions
Consider
a
series
expansion
of
a
signal
F(06):
F(@) = 2 a(j)f(5,0)
There
are
with
three
the
Signal
help
basic
of
design.
operations
that can be distinguished
this
expansion:
A filtered
tiplying a(j)
time shifting
filtering,
signal
shifting
and
F;(@) is obtained
by mul-
with an attenuation
f(j,9) by 9(3):
function
K(j)
and
by
co
Fy(8) = >-K(j)a(s)£05,0-0(5)1
j=0
Shifting
forthe
F(@)
variable
is
j;
done
the
by
substituting
inverse
function
a function
j[k(j)]
k(j)
=
gj must
by means
of
ampli-
wal(k,@);
it
equals
exist:
Fe(8) = >’ a(j)flk(5),0]
k(j)
equals
tude
modulation
k+j
or k-j
band
The
k@j
for
of
sequency
a
must,
functions
in the one case
tions
in the
Signal
cients
c(j);
other
designis
a(j)
are
again the
carrier
shifting
modulation
{f(j,9)}
shifting
Walsh
frequency
amplitude
system
for
of
by
meansof
a sinusoidal
single
carrier
of course,
be the systemof
and
side-
sin enké.
that
of the sine-cosine
Walsh
func-—
case.
the
most
replaced
inverse
general
process.
new
coefficients
by
function
a{c(j)]
=a(j)
The
coeffi-
c[La(j)]
must
exist.
=
new
a
by
replaced
is
{f(j,6)}
system
the
Furthermore,
VARIABLES
STATISTICAL
4,
se
system {g(j,0)}:
Fa(@) = 30(3)6(38)
jz
The
a(j)
+
coding
have
been
of
Examples
set
coefficient
one
that
Note
coefficients
of
Filtering
2 and
4.
mission
and
The
shifting
extensionof
requires
A short
the
discussion
understanding
of
An uptodate
start
with
to
date
as
sults
the
are
a
and
equally
game
"if
methods
is
all
was
the
cards
is
"if
of
are
the
equally
often
‘such
of
condition
an
be
[1,2].
the
measurements
heads or tails occur when
not
is
the
mathe-
all
be
re-
results
applied
to
probability
of
equals
The
1/52,
if
condition
explained by the
once".
This
defi-
for communications,
"if
all
results
are
theory of probability
Consider
results
are
ex-
explained.
axiomatic
sets
yielding
suffice
if
52
up
abstrac-—
favorable
deck.
probable"
not
the
founded
of
the
represented
can
of measurements
of
is
probable"
theory
results,
in
avoid
to
most.
of
The
of
KOLMOGOROFF
number
a deck
less
of
an
chap-
have
century
difficulty.
equally
does
to
eighteenth
once
to
degree
may
from
two
would
order
The
the
signal
statistics.
last
definition
meaning
based onthe
ples
of
of
A somewhat
in
possible
card
nition of probability
since
by
This
card
inthe
satisfactory
defined
trans-
facilitate
discussion
here,
without
each
will
o-algebra.
prove
area
mathematical
abstraction.
represented
results
statement
of
used
number
the
applications
quotient
a certain
card
these
probable.
of
drawing
each
of
should
maticians
into
methods
mathematical
required
a
discussed in chapters
of
concept
Probability
been
theory of information
mathematical
approach
cessive
tion
the
have
the
functions
their
ters.
may
into
transformed
be
a(j)
(26
to
in Figs .€26)
given
c(j)-.
by orthogonal
design
coding.
called
is
alone
c(j)
transformation
C=
alarge
C,, C2,...
observation
flipping coins,
how
number
Examoften
or the counting
of
4.11
DEFINITIONS
NeVUCT
a
Seine theswordsvormaetext:.
statistical
C=
just
¢
dropped
variable
was
given.
a
onto
a random
as
the
yields
two-dimensional
where
two
variable.
Let
the
€ is
in
example
the
examples
a ball
¢ and
ball
values,
mn denote
comes
and
that
C=
to
is
the
rest.
€(C,n)
is a
a k-dimensional
generally
by k values.
defined
is
it
if
variable
surface.
points
Uy .2.Tis@caliled
variable.
a further
anirregular
of
measurement
variable,
or
C =0-C)))
one-dimensional
Consider
coordinates
Each
183
Let S denote the set of all possible results C of a
measurement. S; and Sx denote subsets of S. The sum §j+S,
is
defined
and/or
all
S,.
rence
isthe
an
by the
sible
word
with
Sctewlca
the
sail
lengths
set
with
set
with
be
stance,
the
set
by the
C = u+iv
The
diffe-
to
be
Sets
space
operations
of
of
to-
25
and
12;
8;
but
‘12.
real
maybe
be
times.
Borel
sets
The
The
sum
product
ithe
sub-
8) +8,
is
8;'5,
16
difference
Sj -Sj5,
by
intervals.
For
in-
2 is
de-
1 and
numbers
intervals
a 3 u
= b and
by k intervalsinan
Eukli-
resulting
sets.
Borel
Let
sets
setsis
are
these
infinite
ora denumerable
class of
and
subtraction
Addition,
sets yield further
R,.
be
the
A set of complex
two
defined
in
S;
S;, be
¢€ between
a finite
performed
The
Si: Let
length
10.
numbers
by
the
pos-
The
defined
1 = ¢ = 2.
defined
ina text
shortest
e.g.,
let
144 and.15.
9 and
easily
is
5..,°15.
R, of k dimensions.
following
A set
1
words
The
longest,
11
and
lengths
lengthof
the
410,
11
of these
multiplication
the
S,.
belonging
letters.
13,
3,
most
of
the
of
10,
11,12,
interval
may
Svsd.
class
§|S, is the set of
aswellas
leneths
lengths
may
number
§j
to
Sj;
where
9,
Lenpiis
Sets
dian
to
or intersection
length’,
lengths
eeu
c
belonging
to
number
the
of
with
mice
fined
elements
of all elements
example
has
lbewset.
subset
is
set
all
S,.
Consider
the
of
product
measured
eo.
set
belonging
S;-S,
to
the
The
elements
not
is
as
the
called
always
used
in
analysis.
function
assigns
a
number
to
each
element
C€ of
4, STATISTICAL VARIABLES
184
of S. Let us define
additional features
the probability
p(S)
= O and
p(S)=1.
€ belongstothe
that
belong
to
the
result
of
measuring
set
with
p(9sC=12)
9, 10,
11
probability
of
variable
Consider
a
lengthof
9, 10,
=1
de-
€ of ameasurement
Let
and
words
12.
and
C=
let
p(S;)
¢ having
to
called
is
S. p(S)
set
the
with
must
¢€ be the
S;
denote
= p(¢éS;)
one
of the
values
ine
ion
a distribut
def
of
S of
S having
no
value
of
¢€ smaller
olatsial oer
DUS) = p(k "s'¢ = x)
The
is
probability
defined
as
=
¢C.
subset
Aeweletoue
said
is
p(S)
p(S)
measurements.
all
the
lengths
isthe
iulavetiok Vs @se
Sof
set
11 or 12.
random
the
function
= p(CE€S)
p(S)
since each result
certainty,
- notes
the
aset
S is a subset
that
shows
S cS
set S, while
to the
longs
€ be-
element
the
that
shows
C€¢€S
notation
The
a set S.
of
(1)
a two-dimensional
variable
C=
(C€(C¢,n)
follows:
PCS) ip Chi S CAs 2
pekoS msi
An
example
is
the
12
letters
is
found
probability
in
a
(2)
that
sentence
a word
with
with
10,
100,
101,
the
lower
11
..,
or
125
eters:
Dec
=. p10 = Coste.
A function
LD.
lag
W(x)
can
00
aon e125)
be
defined,
if
limit
k
ee ones
W(x)
= p(-o<
W(x)
is
called
€ = x)
distribution
(3)
function.
4.11
DEFINITIONS
These
165
axioms may be expressed by the
following
formulas:
PMS Ee £0
(4)
DUS
+24...)
Os
Wx)
= p(s, ) + DG
=
O means
element;
yields
that
the
subsets
several
S;
the
results
postulate
Came
series
that
che
€, the
any
alee
and
the
of
S, have
result
no
of
measurements.
second
the
combinationof
bee stapistical
The
results
the
common
a measure-
S,.
to
as
well
as
Sj;
to
belong
not
Consider
us
5, Si. = 0
-2 4
putting it differently,
must
ment
£0T
s 4
Were) = (0, " Wes)
S,|S,
we...
n,
random
first
etc.
Let
variables
veriable,
This
third
dom variables, any combined variable (fi, -+:+ 0;) is also
For example
the
length
¢ of
the
length
ding
obtained
first
by
word
word
been
and
words
and
the
each
of
the
random
to
Grofithes
tion
or
yn.
joint
The
disbribubion-
joins
relative
[ and
to
of
[.
is
43
of
the
with
has
length
the
length
shall
of
be
no
variable.
(C,1)
of
(C without
to
relative
identical
marginal
the
second
the
random
is
is
the
example
distribution
It
of
one-dimensional
distribution
Similarly,
the
with
distribution
((,n).
identical
of
there
two
yiel-
variable
Length
where
combined
axiom
calledamarginal
y is
distribution
the
was
to
yielding
series
Another
(2),
two-dimensional
combination
variables
etc.
combination
and one
pairs:
length
following
sentence
second
combined
sentence,
sentence,
a
A
following
According
between
the
sentences.
example
variables
two
a text,
the
second
a
series of measurements
first
word of the
Consider
regard
7 of
the
sentence.
difference
one
in
combining
giveninthe
of
the
consider
with
distribu-
distribution
of
Ns
Two
variables
p(S) =p(reS)
[ and
yj have
and p(Q) = p(neQ).
two
probability
The probability
functions
function
represents the probability that a measurement
result ( of the set S and the result y of the
p(CES,n€Q)
yields the
be
can
functions
new
Two
Q.
set
VARIABLES
STATISTICAL
4,
186
defined:
(5)
p(neQg [Ces) = peGee asso
p(tes|neQ) = ee
p(teS)
> 0,
The
plneQ)
function
pability
of
S.
the
conditional
"1 the
an
the
set
defined
distribution
example,
length
us
of
Q,
by
if
pro-
C belongs to the
p(néQ|tES)
is
called
to
the
condi-
length
of
words,
7 relative
the
sentences
the
pairs
length
¢ > 1,
n-
frequency
of
¢ >
lies
on
1,
inasufficiently
of
1,
probability
of
Let, onthe
the
word
will
a
of
This
dence.
sentence
example
bination
of
of
between
length
can
only
conditional
L;
and
fromthe
between
The
L2,
in
oc-
pro-
if
¢ is
unconditional
L;
and
L>.
the
length
the
sentence
k inatext.
The
condi-
a
certain
length
will
usually
of
a certain
word
k has
be
of
length
usually
a
sequel
no
sentence
to
L;
the
=
une
n = ly),
bearing
on
the
k.
leads
to animportant
statistical
Let the following
p(CeEs,neQ)
this
word
The
differ
a value
probability
length
with
long
= n = L2.
n denote
1,
length
value
L;
¢,
of
the
all
the pair
probability
since
and
words
hand,
of
¢€ >
L,,
sentences
sentence.
S Le,
if
words
possible
This set contains
and
long
tional
conditional
of
L;
a very
by
2°)
these
all
now the pairs for which the length
n having
k and
chosen.
usually
other
Among
since
n having
than
n.
between
occurrence
depends
bability
1 are
the
containing
€,
Consider
of the sentences
larger
¢ represent
of
pairs
a text
let
consider
with
of
y of
distribution
those
cur
is called the conditional
Guero.
For
let
p(neQ|CES)
obtaining
set
DLOnw
The
> O
variables:
special
statistical
product hold forthe
= p(CEeS)p( EQ)
case
sets
of com-
indepen-
S and
Q:
(6)
4.11
One
DEFINITIONS
obtains
TS/.
from
(5)
and
(6):
pMeQ|CeS)
= p(neQ),
p(ceS) > 0
p(teS|neQ)
= p(CeS),
plneQ) >
The
conditional
distribution
pendent
of
tically
independent
y and
Let
us assume
(7)
and
in this
( and
»y are
rather
inde-
called
statis-—
pc €S)
independent.
than
(6)
Hence,
(6).
is true. Substithe equations
are
equations,
two
of the
one
each
sufficient
case
and the probabilities
yields
(5)
precisely
more
or
(7),
necessary
€ is
statistically
that
into
(7)
of
tution
versa.
variables
called
are
p(n€Q)
and
vice
of
(7)
conditions
for statistical
inde-
pendence.
Let
us
substitute
probability
the
function
distribution
p according
to
function
W for
the
(3):
W, (x) = p(¢sx)
(8)
W2Cy) = p(n3y)
W(x,y) = p(Csx, n3y)
Equation
W(x,y)
This
(6)
assumes
equation
is
independence
and
Q are
necessary
of
be
shown
that
this
that
S and
Q may
let
function
the
n(C)
and
Borel-
g(x)
for
restriction
Borel
words
which
be
real K. Hence, the values
elements of a Borel set.
is
statisti-
the
sets
However,
by the
n = n(¢)=¢".
be
for
all
real
g(x)
of
(. The
elements
= K,
the
random
The
7(C)
all
the
nun-
variable
B-measurable', real,
for
S
unnecessarily
measured
of
in S, if the
holds
if
sets.
¢ consider
defined
defined
for
n,
according
to (1).
be
of
function
or B-measurable
x,
¢ and
called afunction
uniquely
"A function
of
Instead
is
sufficient
intervals
length
general
a
Let
ments
and
variables
and
¢ of letters.
finite
the
by
Again
¢.
form:
(9)
defined
can
narrow
per
following
= W,(x)W2(y)
cal
it
the
y ofaset
subset
isaBorel
variable
function
S is
S of all
ele-
for
all
be
the
set
9 must
VARIABLES
STATISTICAL
4,
188
variable having adistribution funcn
a random
is the
tion defined by the random variable ¢.
Let Q denote a set containing n, and S a set contain(C)
ning
¢.
to S, and p,(Q)
of n belonging
bility
n belonging
to
Q.
probathete
deno
p,(S)
to S. Let
¢ belongs
if
only then,
and
Q then,
to
belong
shall
n
The random variable
probability
the
of
It holds:
p,(S) = p2(Q@)
(10)
Substitution
of
the
distribution
function
according
to
(4) yields
Wy)
= p,(n#y)
where
= p, (Sy),
Sy is the
set
of
(11)
all
¢ for
which
holds
7(¢) € y.
4.12 Density Function, Function of a Random Variable,
Mathematical Expectation
The
distribution
function
W(x)
has
been
defined
in (3)
by
Wi)
“= "pCGsi).
Assume,
x.
that
The
the
derivative
derivative
Sity
Tunction:
w(x)
= W'(x)
¢ is
called
A random
is
called
W'(x)
exists
for
distribution
a
continuous
variable
transformed
into
random
¢ with
anew
random
condition
n = y corresponds
W2(y)
variable
distribution
function
of
variable
n
is
to
in
w, (2),
this
function
den-
n
=
case.
W, (x)
aC+b.
The
obtained
as follows.
¢ = (y-b)/a
to ¢ 2 (y-b)/a for a < 0. The distribution
is obtained from (11):
This
points
or
(42)
tribution
W2(y)
all
density
for
a>
function
disThe
oO and
W2(y)
a> 0
=
formula
is
‘GES,
is
correct
for
a <
O,
only
af
W\ (2)
as
con,
4.12
DENSITY
tinuous
shall
FUNCTION
at
be
x =
189
(y-b)/a.
determined
so
At
discontinuous
that
the
points,
functionis
W;(x)
continuous
to
from
if
ErLeht .
wae
The
W,(x)
density
is
w,(y)
WiCy)
Wy (x)
= Wy)
Consider
There
function
are
=
no
The
s¢
must
is
at
obtained
oy
=
it
n and
yields
bisa 1) Bye. ye
W2(y)
for
equals
zero
sy .=70s
W,(y)
for
n is
yw Ss ©
(15)
pee)
at x = -y/2i,
if W,(x)
obtained:
is
The density
differentiable
function
for
all
ye
fe
ic
yields
s
y Wl2i-1)
holds:
Cy A)
Yntes<
be
W(x)
Let
the
function.
a certain
for
,y>0
ae Wika)
itw, Cy!)
C2i-1
=
"1
:
i
Yo
p.
1,
=
sty
7.<0
Cy M2)
St,
=
va-
ae
We et
w2 Cy)
x:
1) =8¢7'
of
7 Sy
|
W,(-y!'),
be continuous
e e Cy)! &
Hence
(3),
Coe
values
function
Sy
G
of
Thee:
©
Wey
obtained
values
s 4y
O
w,(y)
is
all
Tunckhion
relation
= {Wee)
W,(x)
RIESE
tae
negative
distribution
a(x)
for
| fw, SBD
a
surgher
POteyi—nOstnes
~y'@i
w,(y)
differentiable
function
distribution
The
Lebesgue
following
and
g(¢)
integral
shall
of
¢
exist:
E{e(¢)]
= | e(x)dw(x)
E[g(c)]
is
riable
W(x)
the
g(¢).
is.
(18)
mathematical
Equation
differentiable
(18)
for
the random
expectationof
becomes
all
x,
a Riemann
Wea)
va-
integral
if
and
if
= we),
4,
4190
a finite
at most
has
g(x)
discontinuities:
of
number
VARIABLES
STATISTICAL
(19)
Ble(¢)] = fe(x)w(x)ax
to
équal
=
x
0,
1,
...,
1 is
given
Dy whe
example
an
¢ being
of
probability
The
distributions.
discrete
for
as
distribution
Bernoulli
the
shows
Fig.83
equation
p(C=x) = X(1-a% (5) , Of 487
For
any
ae5t
S not
containing
one
of
the
points
x
=
0,
1,
Holds:
p(ces)
The
set
(20)
= 0
distribution
function
W(x)
follows
from
(3):
W(x) = a¥ (1-a)t¥
(5)
(21)
x
y=0
The
mathematical
expectation
is
represented
by
the
l
ELg(¢)] = >ige(x)a*%(1-a)* () .
sum
(22)
x=0
g(x)
has
to be
Ua
o
Fig.843
tion
of
defined
points
x = 0, 1,..,1
only.
Come C me
Po
>
(left)
2)
———
10
oo
Probability
a Bernoulli
functi
and distribut
on
ion
distributed
Fig.84
(right)
Density
of
product
of
the
at the
two
variable;
q = a
functi
and distribu
on
tion
Gauss
distributed
funcWee alsiES
function
variables.
4.14
MOMENTS
Fig.84
defined
191
shows
by the
an
example
modified
of
a
Hankel
continuous
function
distribution
K(x):
Wox) = p(cax) = 2 fKay
It
will
for
be
the
later
product
W(x)
tion.
shown
of
(23)
that
two
this
variables
with
and
differentiable
is
distribis
ution
obtained
Gaussian
distribu-
yield
the density
s
func-
tion
w(x)
= 7K (x):
K(x)
1s
(24)
approaches
+m for x = O, since
perpendicular
The
probability
hOrECOnTINUOUS
p(¢=x)
The
to
the
of
x-axis
in
the
this
tangent
of W(x)
point.
¢ having acertain
value equals
zero
Gr_SLELpDULLONS:
= 0
mathematical
expectation
E[g(¢)]
follows
from
(19):
Ble(¢)] = 2 f e(x)Ko(x)ax
(25)
4.13 Moments and Characteristic Function
Tey
o(C
matical
ini 18)
expectation
bea power
E(ck)
is
of
Cc, @(¢c)
called the
=
ase The
moment
mathe-
of order
B(¢K) = f xkaw(x)
One
Ee
obtains
\
kia
eer
for
k
k:
(26)
the
Bernoulli
distribution
(22):
71
2) I-x Cy
ee
(27)
X=
for continuous
moments
The
HCC
k
he eae
wea
distributions
from (19):
follow
Kc
(28)
| gxte wioe)dx
—oo
The
E(¢)
moment
=m
of
first
orderis
also
called
mean
value
m:
Geg)
192
The
4, STATISTICAL VARIABLES
moments
(30)
EL(¢-c)'] = f(x-c) aw(x)
the
are
LS) WESC
the point
about
moments
called
are
atone
moments
central
The
notationu,
the
and
c=m
points
the
about
moments
c.
aMlaeinig
uy = ELCC-m)] = fxm)" aw(x)
(31)
Expanding the factor (x-m)“ one obtains from (26):
Uy = 1
(32)
teehee
M2 = E(¢?) - m?
The
second
E[(¢-c)’]
order
moment
=U,
1tS
The
about
aw
+O
minimum,
+
for’
equations
4 2(m-c) f(x-m)aW
(m-c)*=
¢
containing
the
probability
of
relations
(mc)? faw
ae
Lebesgue
integrals
distributions.
¢ assuming
are
+
= Mm,
explicitely
for discrete
eng
c,
(33)
ten
general
a point
= EL(¢-m+m-c)*]
(xm)?
bas
+ ems
ZmE(¢?)
-
= EC¢*)
Us
obtained
the
value
instead
x.
of
are
writ-
Let
p,
denote
The
following
(20),(21),(22)
C27.)
p(C=x)
= p,
(34)
Wx) = Dipy, Ble(c)] = Dietedpe, B(CK) = 5° xckp,
xX =—co
Equations
m
=
(29)
ye XPy 5
xX =-Co
and
LL
=
(31)
xX =—0COo
yield:
OY (x-m)* py
(35)
4.13
MOMENTS
The
ston
moments
Of)
the
493
a
are
distribution
defined
frequently
distribution.
is
by asimple
moment
of
first
order,
point
m
order
also
is t
momen
ation
and
is
All
o
is
symmetrical
true
particularly
by measurements
distribution
variance
The
location
mean
moment
The
of
standard
odd
order
about
the mean
from
devi-
symmetry.
deviation.
vanishif
the distribution
m.
The
Hence,
yu,
characterizes
coefficient
mathematical
exp(iv¢)
is
called
variable
€;
v
e(v)
Let
is
of
w(v)
= fexp(ivx)w(x)ax
is
a one-to-one
function
identical
characteristic
done
function
of
function
the
random
(isd
p(y) is thenthe
Fourier
trans-
with
W(x)
is
a
correspondence
between
a distri-
and its characteristic
function
functions
characteristic
functions;
analogy
communications.
Harmuth, Transmission of Information
yield
rather
functions
this
to
the
is
two
9(v).
identical
Calculations
versa.
and vice
functions
complete
in
(39)
distribution
distribution
transform
13
special
W(x):
There
There
the
= fexp(ivx)aw(x)
W(x) be differentiable.
the
of
real:
of
be
skewness.
characteristic
form
Two
the
coefficient
expectation
= ELexp(iv¢)]
bution
is
(37)
called
The
of
second
The
square
Vuue= 40/07
is
of
has its
order
m.
mean
or
if
than
concentration
the
the
around
called
the
(33).
to
according
rather
function.
momentof second
characterizes
called
moments
deviation
holds
for the discus-
notation
the
used.
suited
characterizes
the
variable
statistical
the
m,
since
H,,
order,
second
of
the
about
minimum
This
obtained
analytical
distribution,
the
well
than
may
with
sometimes
easier.
the
Fourier
use
of
4,
494
STATISTICAL
VARIABLES
4.2 Combination of Variables
4.21 Addition of Independent Variables
Consider two random
¢ and
variables
7 having the diffe-
rentiable distribution functions W,(x) and W2(y). The mathematical expectations of the functions g,(¢) and g,(n)
integrals:
following
by the
defined
are
Ele,(¢)] = f e,(«)w,(x)ax
(40)
Ele2(n)] = f e,¢y)w,(y)ay
(41)
Let
from
¢ and
n
be
statistically
independent.
It
follows
(9)
GMO) = w(x,y) = wy(x)w,(y)
Mave:
(42)
iBpuAeamal (ona
(con)
= 610¢)
yields
the
ELa(e,n)]
+ 826m)
mathematical
=
(43)
expectation
E[g(¢,n)]:
f f e(x,y)w(x,yaxdy
iT]
83
(44)
Ce,(x) + 8,Cy)]w,(x)w,(y)dxdy
co
co
_JeCedw Ce ddelw, Cray + fe.(yw, Cw ayf w(x)ax
Elgi(¢)]
The
expectation
of
the
+ Elg,(n)]
sum
of
the
and g,(n) equals the sumof the
€2(n). This result still holds
tistical
The
independence
made
random
variables
a. Ccy
expectations
if the
here
is
not
of @, Cc) end
assumption of stasatisfied.
function
hCCs)
= 6, (CB, 0m)
yields
the
expectation
(45)
E[h(¢,n)]:
4.21
ADDITION
OF
VARIABLES
d95
aif.h(x,y)w(x,y)dxdy
—=Eth(¢,n)]
(46)
= T fe, (xe, (yw, (x)w, Cyaxay
= ie g,(x)w, (x)dx fe, (yw,
(y)ay
= Efg,(¢)]EL[g.(n)]
The
expectation
g,(¢)
and
of
g2(n)
the
product
equals
the
of
the
random
variables
product
of the expectations
of
g:1(¢) and g,(n).
The
results
derived
more
here
than
tions.
for
two
The
to be
about
two
Punetions
and g2(n)
products
continuous
expectations
»,(v),
and
variables
calculated
Let
sums
and
according
»2(v)
and
variables
to
for
characteristic
independent
distributions
denote
the
to
have
characteristic
= Elexp(ivn)exp(ive)]
function
random
characteristic
is
apply
distribu-
(34).
o(v)
= ELexp(ivn)]ELexp(ive)]
It
also
non-continuous
discrete
to
variables
of.¢, q and C+n. Substitution
of e7(c )=expCive)
= exp(ivn) into (46) yields:
o(v) = Efexp[iv(¢+n)]}
The
of random
functions
known
of
variables
from
of
Fourier
(39)
is the
following
w(x)
= arf exp(-ive el vay
(47)
= 9;(v)e,(v)
the
sum
equals
the
=
of
the
statistically
product
of
the
variables.
analysis
that
the
inverse
of
integral:
foo)
Denote
the distribution
W,(x)
and
and
W,(y)
w,(y).
The
and
w(z).
function
done
13*
in
density
integrals
This
(47)
function
the
for simple
integrated
tion
in
of
C+n,
¢ and
functions
(38),
(39)
n by W(z),
by w(z),
and
w(x)
(48)
may
and yield the density
functions
retransformation
into
a general
(48)
the
form
distribution
and
yields:
of
the
be
func-
characteristic
function
can
also
be
4,
196
VARIABLES
STATISTICAL
(49)
(2c)
Wz) = PPW,(z-y)aW,(y) = f W)(z-x)aW,
—0o
differentiable
W(z)
=
functions:
TW, (o-y)w a (yay
=f W,(2-x)w, (x)ax
(50)
fw, (2-y)w, (yay = fw, (2-x)w, (ax
w(z)
—oco
—oo
Denote
the
Manco
integrals
Riemann
following
the
(49)
from
obtains
One
for
—oo
means,
variances
and
moments
of
third
order
distribution functions W(z), W,(x) andW,(y)bym,
“405 OS saLiat uj) and wl3). Equation (44) yields
6,(¢)
=
m=m,
¢ and
g,(n)
=
of
m),
for
n:
+m,
(51)
Equations
(32),
oa; + o5 = BCC?)
(46)
and
(47)
— mi + B(y7)
yield:
— amg
(52)
E(¢?) - E*(¢) + E(n?) - E*(n)
EL(¢+n)°] - 2E(¢n) - E*(¢) - E*(n)
EL(¢+n)7]
The
following
easy:
a)
relation
is
= o?
obtained
in
a
similar
way:
(23)
U4
+
Consider
Gaussian
- E*(c+n)
as
(53)
an
example
two
variables
¢ and
n having
a
distribution:
w,(x) = Tana, exp[ -(x-m, )?/2? ]
(54)
w Cy) = Taner exp[-(y-m, )*/20? ]
Wi(x)
= $[1
+ erf()I,
erf(u)
is
tabulated
erf(u)
eae
eee
= aDuhJ 6)
ds! 4
erf(-u)
The
=
the
-erf(u),
characteristic
Wo(y)
error
erf@)
+ eri (t54)]
function:
“eee
Te
= #[1
y2
553
= 1
function
»,(v)
of
w,(x)
follows
4.21
ADDITION
from
(39):
OF
VARIABLES
197
m,(v) = TaVTSy J explivx - (xem, )’ /20? ax
The
substitution
y = (io?+m)/f2c0,
yields
py cvd= Tayna, oP (ivi, -bof v?) f exp-(x/V2a,
-y)’ Jax.
—oco
Using
(55)
Cv)
(us ae
:
v?). (56)
dz = exp(ivm,-#0?
Je
~to;2 v? )xre
= exp(ivm,
and
the
substitution
GAV2s, <y)
=o 25
vielas
—oo
The characteristic
function
o(v) = »,(vdp,(v)
= expliv(m,+m,)-$(0?
+03 )v?]
Comparison
have
of
(57)
a Gaussian
relationship
istic
function
between
W(x)
equal
Gaussian
a variable
shows
since
function
w,(x)
and
and
Gaussian
the
sum
€+7
must
one-to-one
function
and characterw(x)
of
and
distribution
(54)
if m,
= o?+o2.
rather
ey)
thereisa
W,(x)
o? by o?
variables
with
that
distribution
Density
by m=m,+m,
pendent
(56)
distribution,
function.
placed
and
follow
from (47):
s
C+n
of
»p(v)
than
distribution
is
Summing
re-
1 inde-
two,
again
having
the
yields
mean
\
nM
=
Mj
i=]
and
the
go? =
variance
5 o?.
(58)
1
It
can
further
be
variables
approaches
values
1 if
of
This
tribution.
tics.
It
variance
means
and
andy 52))
holds
of
the
the
is
shown
a
do
central
the
very
distribution
of
the
Gaussian
variables
under
variances
that
the
sum
1 independent
distribution
not
have
limit
general
are
of
for
a Gaussian
theorem
of
assumptions.
equal
variables
to
large
the
according
dis-
statisMean
and
sumsof
the
to
(51)
198
4,
STATISTICAL
VARIABLES
4.22 Joint Distributions of Independent Variables
two
of
sum
the
problem
¢ hasaGaussian
mean
of
¢-|n|
of
the
in
with
distribution
and
m=O
chapter
mean
also
n has
variance
Gaussian
¢ and
and
variance
distribution
function
and
w,(y)
-0< x <@
(59)
<o9
O. Sy
yO
shall
for
A variable
w,(x)
wily) = 0
holG
following
6.
are
|n|
= Taras exp (-y*/207)
x-y
the
The distribution
o2.
w, (x) = popes exp [-(x-1)?/207]
w2(y)
m=‘1
The density functions
is wanted.
variables
consider
variables
be encountered
variable
second
a
o2,
with
random
will
that
of
distribution
the
of
example
complicated
a more
As
all
ee
ey
The
density
yield
z.
values
Hence,
or
the
following
relation
must
-y:
* (60)
function
of the variable
¢-|n|
w(z)
are
and
distribution
function
given by the following
W(z)
equations:
co
w(z)
= <1. f exp [-(zt+y-1)?/20?]
0
= ee
W(z)
exp [-(2-1)?/4071[1
= Jw(e'az = a {1 #(1
Let
us
+
erf(z)]
further
if the condition
as
+ 41
exp (-y 7/20? dy
~
ert (44)
erf(u)je™
-
du,
(61)
-cOO<
us=
%Z <©
—
ertf?(z)]
calcul
theate
density
function
¢€ 2 O must be satisfied.
w,(x)
of
is
C- In|
defined
follows:
w, (x)
I
ToynUs exp (-(x-1)?/20?)
=O
ied
O
wy (x)
(62)
3
* = 8f1 + ert(4//20)]
© = qayng Jexpl-(x-1)*/20Jax
4
¢ cannot
co
be
smaller
than
zero
for
non-negative
values
of
4.22
JOINT
C|nls
DISTRIBUTIONS
thence,
(61)
199
holds
for
z=
O, but
one
has to multi-
#Oe®
wa <
Diva Dy suc.
The
zero
smallest
but
-z
due
y = -z-=|z|
One
to
x 2 O,
for
obtains
W Za)
permissible
gz
as
may
of
difference
2
10
the
between
argument
exceedingly
tion
W(z).
Pees),
ces
to
a
isthe
Viele
different
function.
sign
This
makes
the
of w(z)
since
+ Berf(1/f20)
probability
willeibe
¢ -|nl
calculatedzonly.
from
(61)
of
func-
-co to
holds
O. Ilt-suffi-
for z = O only:
=
ert (=2") jaz
- 2erf (1/20)
- ert’ (1/26
(64)
Zvdeqwert(1/V2o
integration
(64) is very cumbersome.
It was accomplishby
KASACK
by
fis
w iO)
lows
the
parameter
= WCO,s),
and
Consider
an
the
1/20,
next
the
over
distribution
independent
w,(x)
functions
joint
then
distribution
u
=
dW(0,s)/ds,
integration
statistically
density
8 =
integration.
differentiation
= V2(u-s)
of
error
thanezero
(64),
-
two
the
integration
integrate
4
The
(61)
and (63)
of
consequence,
smaller’
mow
(65)
ert (==1)
-
0
= BACs fexpl-(z-1)?/407J[1
W(O)
w
© 1S
(60):
difficult
to compute the distribution
AS
soetne
ibis-requires
ed
from
= soar fexpl-(aty-1)¥/20? Jexp(~y?/20? )ay
The
it
WV
seen
(61):
--- exp(-(2-1)?/40?][1
ef
of
be
O
IIA
instead
value
and
follows
substitutes
-(z-1)s.
the
It
fol-
substitution
s.
of
continuous
w,(y).
One
The
from
the
product
variables
density
¢n of
having
function
(42):
w(x,y) = wy (x)w2(y)
The
in
probability
the
area
w(x,y)dxdy
of apoint
element
dxdy
with
coordinates
¢ and
7 lying
equals
= w,(x)w,(y)dxdy.
(65)
xy
equals
two
ways
to
due
relation
(67)
= 2.
two
cases
may
X
to
y = =,
any
assume
the
x 2 0;
dy = &,
The
x=
probability
x
a
given
is
and
dy is
SiS
dy = -2,
of
x+dx
Cn
lying
and
if
by the
x < 0
+o
and
all
values
not
converge
terval
of
wiz)
cor
value
of
and
O,
-c
on
(68)
into
%,
between
n lies
(69)
z
and
between
z+dz
y =
if
¢ lies
- and
(70)
product
the
x
second
at
width
¢
x
=
2e
- 0
from
-o
-o
and +o.
to
is
x
O,
since
may
from
assume
However,
the integrals
O duetothe
around
y+dy=
products:
reasonableto integrate
the first
between
for
and
7s":
following
iy oe oS
O to
A
transformed
x 2 0
seems
certain
a
O
distinguish
may
one
Given
between
-=,
w, (x)ax w(Z)2
It
O.
value
0;
between
unique,
x <
value
y =
differential
it
make
O and
x 2
y has
provided
The
order
In
guous.
the
ambi-
becomes
dy
differential
the
of
transformation
The
of
the
in
obtained
be
may
z
of
value
xy = (-x)(-y)
Z,
product
2 if
@:
A certain
the
value
acertain
have
¢n will
product
The
VARIABLES
STATISTICAL
4,
200
=
factors
0 is
investigated
left
in
<. A certain
out
each
andthe
(71)
may
be
replaced
for
even
in-
limit
case:
w(z)az = -[w,(x)w,(2)daxaz
+ fw, (x)w,(2)taxaz
a
Tags
rahe
Equation
may
(71)
functions
w(z)dz = 2fw, (x)w,(2)laxdz.
==
by
(72)
-—co
Let
¢ and
m, =m,=O
n have
a
and variances
Gaussian
distribution
o? and of. Equation
with
means
(72) yields:
4.22
JOINT
DISTRIBUTIONS
204
2
os
Z\2
4
w(z) =_ ono,e)
Jexp(-x?/2a4 2 Jexpl -(2)
/207, |5dx
The
substitution
s = x?/o?
(73)
is made:
wz) = Big,G,_|ert-# (2/50) o3 +8) }4as
(74)
1
The
integral
°
u2
a
eee
jexpl-# (Ss ~ s)]zds
0
is
tabulated.
a modified
for
é¢--
Bei)
Hankel
ii
«
a
Hankel
aia)
is
function.
woke, (i)
eup= reals
function
Equations
and K,(u)
(74)
and (75)
follows
w(-z)
is
The
from
(74)
that
w(z)
is
even:
CHD
thus
defined
distribution
for
all
real”z.
function
W2) = aegr [Kogan
(78)
ra
cannot
W(z)
yield
(76)
= w(z)
weg)
is
0:
OD) = ee
eG
(75)
be
and
reduced
w(z)
if
to
one
tabulated
functions.
Fig.84
shows
substitutes
Som
(79)
and
ere
= 0,0, w(x),
The
Rayleigh
involving
and
fading
distribution
distribution
or
narrow
function
defined
)
Je =n0
W, (x)
=
1 -
= W(x).
distribution
= SE exp(-x?/6
Wee
WpCx
are
en
exp(-x?/6?)
is
as
important
band
noise.
a
variable
of
(80)
for
problems
Density
function
¢ with
Rayleigh
follows:
(81)
ZrO
6
ze NO
rar
aa@)
The
mean
equals
co
(82)
= 2Vn8,,
= f x w,(x)dx
=m
E(¢)
VARIABLES
STATISTICAL
4,
202
0
and
the
second
order
moment
equals
(83)
B(¢?) = f x?w,(x)ax = #.
0
The
variance
a. follows
from
(29),
(32)
and
(36):
o? = E(¢?) - E*(¢) = 8,(1 - én)
Let
leigh
a variable
n be
distribution
independent
with
density
wo(y) = + exp(-y?/85 )
The
density
culated.
the
function
y
w(z)
density
function
about
x
of
=
the
O.
of
¢
function
IV
and
a
product
€n shall
Rayleigh
(81)
be
used,
distribution
and
(85)
be
one
equation
207 = 8
eee
and
is
identical
207 = 6f into
with
(73)
(73)
and
if
calsince
is
not
obtains:
w(z)= ster | exp (-x’ /s, )= exp(-2? /x?6? )odx
This
Ray-
(85)
than (72) must
Using
have
w,(y):
oO
of the
Equation
(71) rather
Symmetrical
(84)
(86)
one
substitutes
multiplies
by
ee
Ge Osmoose *
The
Ot
density
function
of
¢€n follows
from
(86)
withthe
help
(€76,):
w(z) = ser K Car
=
The
distribution
=
0
(87)
function
Z
W(z)
= aye
3
De
———)du
may
be
x
2u/6,6, yields:
=
reduced
22/65,
wa) =
to
(88)
tabulated
functions.
22/5,
The
substitution
3K g(x)ax = -bin f |Cic)H!! (ax)a(ax)
0
(89)
4,22
JOINT
The
DISTRIBUTIONS
203
integral
Syl5,(y) + iNg(y)ldy = fyH'} (yay = yo!) (y)
is known.
and
J,(y)
second
and Nj(y)
order
are Bessel
(Neumann
(90)
functions
functions).
of
Equation
first
(89)
be-
comes:
W(z) = $nxH!"! (ix)
2z/6,5,
SE) - gen" (ie)] (91)
-1[54
€
Let
¢
approach
Ee (ae)
one
zero.
=.=—P/tre
Using
Date
the
equation
cecen”|
(92)
obtains
lime Hie)
= -2/n
oo)
and
Wz) = 1 - Se HIE).
The
term
in
the
brackets
is non-negative for real
values
of
Z.
Let
us
investigate
4-f6r
Z sco.
;
sae 58,
The
ea, (ix)
nishes
1.
=
Oe
equals
help
(92)
one
—
of
i
ue
to ee
O for
obtains
ae
Ot. Baza
positive
z=O
for
and
z=0:
te
approximation
\peree eu
large
thus
for
of
values
large
the
shows
x.
values
The
of
second
z and
of
functions
(94)
in
term
one
(87)
and
(94)
sum
¢+n
of
va-
W(e)
obtains
for
=
6, =
Gls
The
pendent
WCz)
(94)
ay]
Fig.85
=
the
whether
ett | eZ
=
for
holds
The
With
asymptotic
(94)
distribution
Rayleigh
= apf (1 lower
limit
function
distributed
exp[-(z-x)?/8%
of
the
of
the
variables
follows
]}xexp(-x?/63 )dx
integral
equals
zero,
two
inde-
from
(50):
z20
(95)
since
the
4, STATISTICAL VARIABLES
204
density
function
by w(x)
= O
x < O.
for
the
since
z,
is
limit
upper
The
replaced
be
to
) has
exp(-x?/8}
= x
w(x)
to
distribution function W(z-x) = 4 — exp[{-(z-x)* /87 J has
be replaced by W(z-x) = O for z-x < O. Substitution
62 +64
1/2
62
oP ae ae
yields
with
4,
of
(96)
~ 2 6, [62 +62 )iz
the
help
of
the
integral
fye? dy = -te 7
2
after
_—
lengthy
a=
W(z)
transformations:
52
a 84 exp (-25 ){ exp(- arth) +
oie
expe
care - \r pzlert(e +
Wg)
e=sO) ror
A simpler
WC)
eae
Oe
eet
is
obtained
formula
eos
Consider
functions
w, (x)
by dx
for
and
and
6,
of
is
=
the
variables
w,(y).
dy
220
= 1:
¢
(98)
quotient
and
The density
used
6,
erf(z/V2)
distribution
continuous
ert (22
mee
67
hee
\E ze 2*/e
=
the
independent
tiplied
mi
(97)
again.
n/c
n having
function
The
of
two
density
(68)
mul-
relation
Yaz
x
must
may
hold
be
if
n/¢
obtained
is
in
to
two
have acertain
value
ways,
in
just
as
z.
the
That
case
value
of
the
IROGUC UH iGrs
aa
x
-xX ag
Let
in
us
order
(99)
consider
to
make
sume
all values
value
¥ 222k,
The
of
Zz,
XS
the
the
between
provided
OF
differential
ys
is
cases
x 2
O and
differential
O and
+00 0r
y has
the
Sex
ee <0
transformed
x <
O
separately,
unambiguous.
-o
and
following
into
x may
as-
O foracertain
value:
4.22
JOINT
DISTRIBUTIONS
OY R= KOZ ee Kee,
The
VA
205
ys =agee xo
probability
of n/¢ lying
between
x
and
= x(z+dz),
x+dx
and
is given
w,(2x)x
dz
w,(x)dx
w,(2x)(-x)dz
probability
for
arbitrary
values
KX =
-c
+0;
For
between
following
we
The
(100)
between
z and 2+02,
n lies
by the
wi(x)dx
w(z)dz
if
KO,
y =
12°C
lies
xz
and
yrdy
z
and
z+dz
from
products:
0)
rea,
w(z)dz
of
n/¢
lying
between
of
x is
obtained
by integrating
= -fw, (x)w,(2x)x
ae
dxdz
+ \ Ww, (x)w,
(2x )x dxdz
5
tO
X =
0
symmetric
functions
one
°°
may
write
(101)
instead:
w(z)dz = 2fw,(x)w,(2x)x dxdz
(102)
0
&
6,/54)w(26,/52)—=—
4
3
: 26,/6,
Fig.85
of
the
Fig.86
of
the
(left)
and distribution
distributed
of
two
with
(54)
and with
means
Gauss
distributed
¢ and
functions
m,
34
function
variables.
distribution
andtion
func
let
2
Z 0;,/
function
density
tion
1
Rayleigh
example
Asafirst
0
two
Density
(right)
quotient
241
Density
of
product
-+ -3
—=
= m2
Gaussian
n have
w,(x)
= O.
One
and
function
variables.
w,(y)
obtains:
distribu-
= w(x)
of
2
u = axt(S + an yields:
1
2
co
substitution
The
w(z)
=
exp(-2? x 2/20? )x dx
fexp(-x2/20?
0
=
w(z)
0, 0
Al
ia
VARIABLES
STATISTICAL
4.
206
oF +Z207
je
-u
du
(103)
0
stud eri
>
The
an
eh
O2
O02
distribution
known
with
iy
defined
by
this
as
Cauchy
distribution
or
one
degree
of
inverse
tangens
Oa;=
Fig.86
As
freedom.
the
a further
n/c
Student
function
is
distribution
distribution
function
CS ie Sh ade mraz
a
functions
example
of two
mn. Equations
as
The
density
is
function:
=
shows
quotient
w(z)
2474
(81),
of
(103)
consider
Rayleigh
(85)
and
Sy
and
the
= sar | exp (-x? /8; zx
(104).
distribution
distributed
(101)
Sa
RA?
of
variables
the
¢ and
yield:
exp(-z “x 45s )x dx
The substitution u = x? ( + z) yields
|
w(z)
=ee Ti i
+22 OF) eee
fue™~du
2
§
z6,
= ers
zB,
/8
78, 84
:
(105)
0
The
distribution
the
help
of
the
| ea
X2 +
function
W(z) is obtained
from (105)
with
integral
= te
2 ee
26/6,
W(z)
=e)
Fig.87
! CaP
shows
the
yz x
=
-
functions
Z6,
5
w(z)
7
and
.
W(z)
(106)
of
(105)
and
(106).
peveral
and
a
joint
Gauss
functions
are
distributions
variable
as
n will
follows:
of
be
a
Rayleigh
calculated.
variable
The
¢
density
4.22
Ce
JOINT
amwa (x=
mee
ent
26x
207
exp(-x?/6?),
=
0
Ws y)
=
fCon) o exp(-y?/202),
the
computation
Rayleigh
density
truncated
of
0.
image
about
wiz).
The
The
density
the
= coe
distribution
variable
holds
for
function
One
J exp(-x?/202)zx
co
-Lle
60
of
the
consider
the
densi-
for
y < 0.
variableto equal
ordinate.
0
:
substitution
(107)
oO
functionof
the quotient
= ae
w(z)
the
variable/Gauss
distribution
=2z>
IV
AVP. &
functionof
the Gauss
The
x
w, (x)
For
ty
DISTRIBUTIONS
¢€/n
all
for
zero
quoti-
computed
for this
positive
negative
values
values
x/y
is
its
obtains:
exp(-z?x?/s7)x
dx
vi IES
4
y = x(s57 + sr) yields:
2oZz/6
coz Zz 2 an
3/2
Lea)
(108 )
——
(6/V20)wi2¥/20/6)
yey
1
Step
Ue
2206/6 —~
——
W(zV20/6)
rn
1
left)
Big.8
Loa
Fig.88
1
2
2/65) —=
(right)
variable.
of
two
Density
Rayleigh
-4
i
function
Density
Stet
quotient
a
and
Rayleigh
and
3° 2
=
alt
1
i
|
function
variables.
function
variable/Gauss
dil
eh
2/2.0/6-——
distribution
distributed
distribution
distributed
«4
of
the
distributed
STATISTICAL
4,
208
w(z)
=F
Z
w(O)
must
W(z)
is
202Zz2/62 +4
defined
a set
(109)
w(z).
of
symmetry
the
to
# due.
equal
rect
formula:
by the
defined
is
function
density
complete
The
VARIABLES
Hence,
by:
W(z)
5Sra bi
ae
ae
(110)
I
nl
Fig.88
shows
The
Gauss
density
in
and
62 22
6
2 Vee)
ores
the
product
of
density
y <
function
variable
shown
of
the
O.
The
density
in Fig.89
and
of
the
the
and
a
Gauss
function
its
density
image
function
Rayleigh
variable
computed
about
the
variable
let
equal
with
the
and
of
this
ordinate
negative
= wer
co
| exp(-z?/2x?g?
)x exp(-x?/6?
substitutions
$
ma
—
v
= Yeo 5.2, eq
f
¥2e20/6z
exp(-¢X)
a
=
—"
=
zero
tax
yield
for
truncated
yield
values
variable:
=
ob-
rte)
v3 )
€
wiz)
is
(111)
function for positive
is)
The
quotient
way:
variable
functionof
distribution
random
(110).
.-3/2
computation
a Gauss
the
density
and
)
PO Ghe me a
$1
For
(109)
fx exp(-x?/8? )exp(-z?x?/207)x dx
0
6 Z
W(z)
of
distribution
a corresponding
= —
I
w(z)
functions
variable/Rayleigh
tained
w(z)
the
of
the
the
4,22
JOINT
DISTRIBUTIONS
209
_|l2o6
welt)
—>
W(z\2/05)——
TIP
epee
SO
1
se
3-290 lineaMR DT
EARP
EO
IN
ER
eee
Fig.89 (left) Density function and distribution function
Gauss distributed variable/Rayleigh disof the quotient
tributed variable.
and distrib
Fig.90 (right) Density functi
ution function
on
of the product
of a Gauss distributed and a Rayleigh distributed variable.
Using
the
tabulated
S exp(-3¥
0
one
=
=
w(z)
The
Yoos
density
values
Z
) a =
of
z
eV 22/60
holding
fromthe
for
positive
and
negative
requirement
of symmetry
about
Ze oo
(113)
O:
=
| /0P
popes)
¥206
distribution
Z
Sag dhGeeta
Harmuth, Transmission
==
function
-\ ex/6bo
= ge-V2
121 760
14
/a
ae)
function
follows
4
W(z)
\qrre
obtains
w(z)
The
integral
of Information
is
defined
ate Se
z2 <0
by:
de ge V 22/50
ates
(114)
the
sum
2?
(114).
and
PEELE
(4115)
-x?/§? Jdz
exp(=-x*/s
2)qe
q?=
example:
last
the
ert
Voop
“2.
pz )[1 + CXP\ "Ba
exp
variable
Gauss
a
of
as
:
oe
oVo\no
p?=
(113)
)°/207 2 x
(x-z
Jexpl _~(x-z)?
== sr
T1064
Wiz)
of
given
is
variable
a Rayleigh
and
of
function
density
The
functions
the
shows
Fig.90
VARIABLES
STATISTICAL
4.
ra,
62 ~2¢?
4.3 Statistical Dependence
4.31 Covariance and Correlation
It
were
has
been
assumed
statistically
of
section
to
drop
4.1
the
riables
tion
far
be
g(¢,n)
n.
The
the
Some
generalized
of
function
mathematical
is defined
by the
random
of
in
statistical
a distribution
€ and
that
independent.
must
condition
Consider
so
the
variables
definitions
order
to
be
able
independence.
W(x,y)
of
the two va-
expectation
of
a func-—
integral
co CO
(116)
i i g(x,y )dW(x,y)
a
E[g(¢,n)]
-00 -00
Let W(x,y)
and
let
number
be differentiable
g(x,y)
of
be
continuous
points.
Equation
except,
(116)
a Riemann
integral:
Ele(¢,nJ]
= { J e(x,y)w(x,y)dxdy
co
may
2
y, eee
and
for all x
at most,
then
be
at
f
a finite
replaced
CO
by
(117)
—00 -co
Let
g(€,n)
e(¢,n)
= hn
E(cK y! ) is
E(¢kny®)
be the product of integer
in
of
¢ and
(
called
analogy
n:
(118)
a
and E(¢°n!)
one-dimensional
fines
powers
moment
are
marginal
to
of
identical
(29):
order
with
distribution
k + 1.
the
of
The
moments
moments of the
¢ and
m. One de-
4.31
COVARIANCE
BGC
any
AND
CORRELATION
2A
BCC on) t= at,
The
point
the
mean
of the
with
the
(119)
coordinates
¢ =m,,
two-dimensional
about
the mean
are
called
n =m,
distribution.
central
moments
and
is
called
The
moments
are
denoted
(ee Nv:
Mat
co
= EL(¢-m,)\(m-m,)']
CO
= f f (xm,
—co
Expansion
and
of
factors
with
the
(me
help
S02
oes Myr
eomiC
ei
0.2
Ganeee= m2 = 65,
Boers
a?
the
y yields
and
ome are
DOLee
cute
called
the
tse, oe
mixed
moment
or
from
the
multiplication
zero
for
statistically
iy
=
E(C)E(n)
-
The
mathematical
m,m,
the
thus
M5,
be
be
theorem
distributions
interest
of
(46)
independent
¢ and
and
here;
it
It
follows
7.
(121)
is
thatitis
variables:
O
1122)
Reet
nC aC aalg, Cs)
too.
zero.
One
that is nonnegative
Hence,
nonnegative.
unequal
ol
wah 20)
= 0%,
the marginal
sis
of afunction
nonnegative
must
= BCcn) - mm,
of x
(119):
le ctor
4,,
of
powers
and
= ECG na
covariance
=
into
(118)
expectation
integral
be
(124)
Gm)
(116),
(Pre
i),
E{{c,(¢-m,)+¢,(n-m,)]7}
is
of
variances
mOMeCNb
(y-m,)! aw(x,y)(120)
—oo
the
right
Let
at least
may
rewrite
one
the
uno
and
hand
moment
right
must
side
of
en
ou
hand
side
£16125)%
Hap} +2
4
Cy Cp +Ho2%2 = Tage 6He0% tH
2
2
Ds
ah
+ CUpoblo2-Hi1 C2 J
(124)
A
? Dot Coxe, teriS))° + (Hadar CF J
Uo9
The
are
and uo,
in
terms
arbitrary
nonnegative
brackets
values
of
c,
in
and
for the
(124)
will
if the
c,,
same
be
reason as (123).
nonnegative
following
for
condition
holds:
Hyp Hoz ~.Hn
14*
2 0
(125)
p is
coefficient
A correlation
wing equation:
VARIABLES
STATISTICAL
4,
BE
follo-
the
by
defined
2
a
\M20 Moz
The
relations
Oe
(125).
from
follow
# p = +1
-1
or
p2 <1
statistically independent variables ¢ andy follows
0 = O from (122) and (126). The inverse relation does not
hold generally; statistical independence cannot be infered
For
‘eel
jy &
Ox
Assume
relationship
a linear
7:
‘¢ and
between
€ = Gon
+ By = a(n - m2) + B
One
obtains:
oo
= EL(C-m,)?]
=a’U,
= Ela? (n-m,)?+20(p-m, )(n-m ,)+(p-m, )? J
+ 2a(B-m,
uy, +(8-m,)?
uy, = aU,
HCE, 2
wi = EL(¢-m,)(m-m,)] = Ela(n-m,)?+(-m,)(n-m,)]
ll
=
Abba,
+
(B-m
SaOe
Oe.
QW.
1 for
and
that
1 are
as
a
the
and
=
At
least
the
Wy,
=
Myo,
be
= m,
one
one of the
will
OF
in
Woy
this
a
in
if
equal
of
m,
the
and
y,,
=
hand,
variables
inthe
must
¢
the
point
vanish
relation
distribution
The
This
m,. Hence,
of,
the
Consi-
zero.
concentrated
uy,
concentrated
definition
(126)
for
case.
equations
Mo.
FO}
is
y = m,.
the
hold
zero
points
other
for
and
moments
sign
equal
Mog Sy) Hy, S2=
the
of
andu,,
covariance
O follows
applied
least
The
On
may be inverted.
distributions
distribution
x
equality
(123)
result
both
marginal
y = m,.
inthe points
9 cannot
The
that
consequence.
=U.
(127)
concentratedinthe
two-dimensional
x =m,
pa
B=m,.
first the case
means
AL 4 >
+ (B-m, Vy
equals
der
=
2
Vs
o?
, JU,
OF
Un,
(125).
one
(124)
are
The
of the
Uy, Co+h,,
must
hold
unequal
right
hand
if
zero.
at
Let
side
of
conditions
¢, = 0,
Mor # O
(128)
4.31
COVARIANCE
is
satisfied
of
(123)
since
the
to
only
expectation
a
Shi githpae
Hence,
the
linear
from
p?
Lollows
cording
One
to
may
o # O and
an
One
this
1.
# O
(130)
for
u,,
#0
and
between
u,,
only
a measure
One
says
uncorrelated
p? # O and
Gauss
obtains
for
#0,
due
statistical
from
for
the
n
on
the
not
p =
p*
always
other
=,
ac-
the correlation
independence
are
correlated
of
for
O.
density
6x |O and
camd
gee
function
oa,,
4
independence
=ew)
2
o,
en Dor) Tama,
p = 0,
ESE
Ey Ne
because
of
G\ga2
of
a two-
two
instance,
(13%)
2
XPS
Dor?
could
and
(9)
of
one-dimensional
C132)
follows
variables
the
(x )waly)-This
pf(x,y)
NE Z|
2
22
# 0:
l
density functions
two-dimensiona
of
that
linear
variables
the
2
= Rovgas
0 = O. For
and
distribution:
p =
4
of
two
consider
210,0, \1—p2
to aproduct
€
relation,
from this discussion
p is
example
case
(130)
Fromalinear
general
a’ (W, (x)W, (y)]/axday
all
Uo2
identical
=
w(x,y)
The
=
infer
dimensional
Ce
xe
relation
in
variables.
As
everywhere.
(127).
coefficient
two
are
vanishes
func-
ena
follows
hand,
side
of anonnegative
function
jong
+ MH,
These equations
to the relation
Pichtenend)
(129):
and
(128)
C= poh nm 2) + m1,
— Ta
the
(129)
be zero if the
from
Let
hold:
= 0
mathematical
can
eng
(124).
It must
zero.
+ c,(n-m,)
follows
It
CORRELATION
according
equal
e,(¢-m,)
tion
AND
which
density
stand
relation
the
result
in
holds
»for
out infactor
functions
instead
for
of 2pxy
in
multiplied
be
could
expression
whole
the
and
(131)
VARIABLES
STATISTICAL
4.
Bas
by
1 + pga(x,y)4.32 Cross- and Autocorrelation Function
=
individual
present
sequence.
the
indices
time
at
t,
course
of
€;,
sequence
---.
a riverorthe
be
may
result
¢;.
result
the
Let
a measurement
The
C2,
not
the
sequence.
a
€,,n;3;
€2,n2; etc.
sequence.
did
instance,
after
indicate
yield
yield
shall
a time
For
measured
necessarily
the
re-
measurements.
assume
along
They
the
not
t,
the
of
ordered
was
>
distinguish
results
how
be
to
used
C2
to
only
far
us
t.
C2,
so
some
surement
€;,
were
«++
Nor
1.
n =
and
----
€ =
variables
the
of
indices
The
Let
amea-
time
at
not have
does
water
temperature
levels
at
certain
places
It makes
no
difference
—] Of ReresulusrG
for
the
computation
of the mean
siemC..
R
<c>= 5 DSi»
whether
terms
mean
the
of
(133)
index
the
square
sum
i indicates
a sequence ornot,
may
be
commuted.
The
same
since
the
for
the
holds
deviation
>o -<o’> = 8 HG Ste e
Given
two
variables
¢ and
n,
(134)
one
may
construct
the
ex-
pression
MIRE
It
is
plied
IS: tC Kin; - <n).
important
with
nj
Equations
R
=o,
ments
ifthis
out
of
for
and
the
(133)
and
limit
a total
value
with
not
(134)
exists.
of
of
nj.
(135)
Let
¢,
is
multi-
nia.
or
are
that
(135)
identical
¢;
equal
R measurements.
with
x
It
in
(35)
for
r measure-—
holds:
4.32
CORRELATION
pyeasehime
FUNCTIONS
alp
r/R.
1,R-— co
let-the
pair
surements
Oi
€ = x,
and
ets
let
g7
n =
the
y be
limit
obtained
gq times
in
R
mea-
exist
Re.
q,R=co
One
may
Pree
then
oo
ee
x,y=—00
het
che
ee
tinuous
I
Let
interval
One
integrals
to
€;,
C5,
assumes
may
run
of
(35)
all
(136)
...
be
values
rewrite
the
-#0
replaced
instead
of
(133)
to
+#@
¢;
of
by
a
time
at the times
f(8;)
this
-4#0 = 6 = $@,
from
simplify
form
the values
is written
sequence.
in the
written
my,
£(0)
infinite.
(all
in the
ta) Dy,
sequence
f(@; ) which
j = 0,1,2,.-.
or
(135)
Oy
Gime
function
cated
write
for
sequence
where
0; 5
a conbe
lo-
© may be finite
and
(144)
but
the
as
follows
limits
are not
formulas):
<£(8)>= af £(o )ae
(137)
o? =(£(8)-<¢(8)>]*>
@" f[r(e)-m,]’de
Replacing
further
tion
one
g(@)
the
= @7 [f7(8)ae - mj
sequence
n,,,,
«--
by
a time func-
obtains:
(138)
m, =<g(8)> = @ fe(e)ae
2 = ([e(8)-<e(@)
1°»
o' flee )-m,]* ae
= 9" fe7(@ ae
-
7 = (Lele )-elo IL £(8 )-<£(8 1
a i)
i]
= @” fC r(e)-m, ][g(@)-m,]de = @”' [r(e)g(e)de - mm,
e'
p
fr(e)g(e)ae
- mm,
~ {fe" fr2(e)ae - m?][e" fe*(e)ae - egy
in the
integral
The
lation
in-
8, = O if ® approaches
for
Ky (@,)
function
crosscorre-—
called
p is
of
numerator
VARIABLES
STATISTICAL
4,
206
iPala@al amare
The
@/2
@ f f(6)g(e+8,)a8
lim
Ktg (@y) =
O-e
(139)
=-On
autocorrelation
function
Kegf¢ (dy)
follows
for
f(@)
= g(9):
Kye (Oy) =
@/2
@' f £(8)f(e+0,)d6
lim
ees
The
terms
short-time
autocorrelation
what
may
assume
be
minate
zero.
yields
|p|
a measure
this
are
O/O
is
thus
avoided
that
Kfg (0)
=
1.
of
the
and
least
Hence,
the
the
autocorrelation
limits
two
functions
the
functions
and
for
of
short-time
p in
the
the
means
f(9)
are
sal(i,@)
9).
let
m,
K tg (0)
and
us
Let
us
and
n,
= K (0)
function
is
p measures
g(@) only, but
for
the
func-
Examples of cross-
shown in Figs.70
figures
are
show
indeter-
(138).
functions.
correlation
amount
To
functions,
p = O and
integrationinthose
cal(i,®)
or
finite.
crosscorrelation
by an arbitrary
and
® is
constant;
one
correlationof
The
of
not
= O yields
Kye (8@y) yield
tions shifted
+oo , Since
at
if
correlation
g(¢@)
correlationfor
Kg (Oy)
the
and
assume
equals
with
£(@)
form
further
crosscorrelation
function are used
done
that
(140)
-@/2
are
periodic
not
and
71.
-o
and
functions.
5. Application of Orthogonal Functions to
Statistical Problems
5.1 Series Expansion of Stochastic Functions
5.11 Thermal Noise
Sepsitcergs
which
be
do
not
Se1oo1
have
functions
56510),
be
orthogonal.
Each
to
of
into aseries
expanded
{f(j,¢)}
stem
suime
in the
complete
the
-4@
interval
°) \=al4n2so005
function
shall
orthonormal
sy-
= 6 = #0:
Bae e= ne, (jor 3,0)
(1)
j=0
@/2
aa(j) =f e,(e)f(
5,6 )de
-0/2
The
coefficients
a,(j)
have
certain
values
a
fixed
t functions
g,(6)
value
of j= gj, and variable
values
of
}.
yield
). Let
q,
of them
t coefficients
enV
Die
Oss
hes
reactions
sin
d,/',-
invervels
O- to.AA,
function.
Assume
values
AA
sity
by
function
calls
the
of
a,(j)
density
AA.to
that
it
be
g,(8@)
to
the
is
is
shall
2AA,
etc.
can
be
are
called
be
The
density
for
identical
result
value
The
all
and
set
of white
the
a
step
for
small
This
of
to
One
j, if
values
a,(k)
den-
j.
of
be
j.
sta-
of
time
func-
noise
with
refe-
orthogonal
called
the
functionis
system {f(j,9)}.
Gaussian distributed,
ee.
over
is
reference
for
in-
2AA,
function.
a,(j)
j # k.
a sample
be inthe
A <
plotted
each
with
coefficients
for
AA-<
approximated
distributed
functions
the
«-..
different
independent
tions
rinverval
a continuous
can
let
rence
a,(jg)
elaine
d/l,
equally
Furthermore
tistically
a,(j
Oy
for
derivative
of
the
error
if its
function.
density
The
set
For the practical
assume
aa(j)
The
index
a generator for the functions
run from zero to infinity as in
a finite
vided
of the coefficients
measurement
f(j,6).
consider
j cannot
number
in the
g,(@)
function
g,(9),
the
finite
number
values
m of
non-overlapping
into
time
the
in
of
the
m functions
f(j,8)
and
let
there
multipliers
The
in
m coefficients
the
by the
time
first
Let
us
value
plot
of
resulting
nuous
AA
) in
functions
to
= Wal1
Ag)
distribution
pends
not
only
q,/t
on
of
may
and
and
A
from
simulta-
integrators.
can
be measured
the
endof
the
the
first
measure-
coefficients
mt
(r-1)AA
be
=
shown
A <
in
sufficiently
to j causes
rAA.
by
The
conti-
Fig.91,
large.
the
a
if
The
follow-
A = Aj:
«ss
coefficients
set
<
noise.
yielding
approximated
as
ut
is thermal
measurements
a certain
Wale sag)
the
the
of
reference
for
©
\ runs
available
g,(@)
interval
small
hold
at
yields
wg(j,A)
with
by
Ce
aw
the
sufficiently
WolO,An)
denoted
, by g,(9).
ion
of
these
Repetit
functions
functions
distribution
ing relation
The
of
fraction
step
density
is
equal
the
©.
le
set
a,(j
m
The
coefficients
are represented
intervals
sila
the
be
voltages
duration
of
hans.
Assume
di-
is
j = O.....m-1,
These
output
t time
forall
m
a,(j),
integrator
BCG)
be
interval.
interval
ments
is
@.
vals
only;
of inter
is possible
Let
only
Time
interval
t.
j can
duration
interval
time
neously
1 to
(1);
O....m-1.
intervals
first
function
t
dis-
Gaussian
and
ference to j, statistically independent
j = jytributed for a certain
re-
with
distributed
equally
arej)
a,(
the
if
noise’,
mal
or thersian
noise
Gaus
white
g,(9)iscalled
of functions
PROBLEMS
STATISTICAL
5.
218
of functions
(2)
a,(j)
a,(@),
generally
but
also
deon
‘Use
of these terms is not
uniform in the
literature.
Thermal noise is frequently called Johnson noise [12] or
resistor noise. The noise generated by thermal agitation
of electrons in an ohmic resistor
is thermal noise, if
the electrons are descri
by Boltzmann
bed
statistic rather
than Fermi statistic.
5.11
THERMAL
Fig.91
Density
----m—-1;
m
NOISE
A
219
functiohs
wq(j,A4)
of
normalized
the
denotes
thermal
output
noise.
voltages
j = 0
of
the
integrators.
the system {f(j,9)}. However, it is independent
of the
system {f(j,@)} under very general assumptions for thermal
noise.
For
a
complete
{h(j,@)}
terval
proof
= 6 = $0.
be bounded.
series
The
statement
Let
the
functions
h(j,9)
co
@/2
Ay
-@/2
DiC; (x) shall
converges
g,(9)
is
converge
=
(1)
replace
the
De Org
and
be
expanded
shall
into
a
| h(g, O0£Ck,e ae" C5)
(3)
uniformly.
expanded
into aseries
of the system
{h(j,@)}:
0/2
Gd:
Oy.
by CJ)
j=0
Using
us
absolutely.
The series
co
ates)
let
The functions
f(j,6@) andh(j,9)
aes cri
£(ie, 8),) yey (ic) a
sum
then
this
orthonormal
system {f(j,8)}
by another system
that is also complete and orthonormal inthe in-
-40
nig
of
=
J g,(e
-@/2
(3)
one
obtains:
)h(j,8
ae
(4)
5.
220
@/2
eo
-@/2
k=0
(5)
or)
0/2
k=0
-@/2
eee
k=0
if all a,(k)
absolutely,
converges
sum
last
fore)
pee
=
>) ¢; (x) J g(@)f(k,6)d0
The
PROBLEMS
Ce, ee
yy eye)
| 2 6G
=m
yd)
STATISTICAL
boun-
are
ded.
The
sumof
statistically
independent,
buted variables
is aGaussian
the
b,(j)
have
statistically
Fhe
a Gaussian
distributed
distribution,
independent.
The
mean
Gaussian
distri-
variable.
Hence,
if
of
the
the
a,(k)
a,(k)
are
and
of
by J) eLsaZer0.
The
density
function
thermal
noise
Wa(k,A)
= TOPADGS exp(-A’
Integration
Wa(k)
as
wi (kA)
over
the
a,(k)
reads
for
follows:
2
A yields
°i) Wa(k,A)dA
=
of
/20)
(6)
1/n,
1
= =
—oo
and
the
sum
Aa
of m terms
wq(k)
yields
1:
eens
k=0
The
o2 =
variance
of in
is defined
4
(7)
A=1
density
function
wo [k,c, (k)A]
of
= yarn
U
2
ee
.
=|
Ola = out “ 2
2
2
cj (k)a,(k)
density
CPeGa
function
snd (en
equals:
exp[-c4 (k) A’ /202,]
2
eae
s
The
by:
L
ree ae.
.
Lim
ore
The
(6)
.
ae
|
ag
of
the
variable
(8)
L
> 24 Cs) = 0% Cc dey,
Az=1
b,(j)
follows
from
5.11 THERMAL NOISE
304
:
4
2
Wp(g,4)
=r Tarim,
exp(-A" /20; )
C9)
Ob
Pi
aks Qyhatd) = Na = def(e)
+ta
G
The
Al
last
the
pend
on! kk:
it
2
step
that
if
x
“
in
(10)
Parseval's
the
arbitrary
with
mean
square
from
(5):
of
O/2
-0/2
-@/2
is
toOllows
the
trom
Satisfied
h(j,8)
functions
assumption
o2 does
or,
de-
putting
may
be
represen-
of
a
vanishing
sense
the
not
one
{f(j,9)},
obtains
oo
SLD) o (x)t(x,e)}ae = SeiCk) (11)
(10)
k=0
and
k=0
(11),
that
the
condition
same
CE
and
values
that
e¢ approaches
of m.
The
variables
variance.
The
density
unchanged,
panded
is
in
initial
thus
eee
satisfied
large
(1.11)
accuracy
J nj(j,e)ae = 4 =
on, Men
and
by n
deviatio
the system
@/2
fr
use
of the
a,(k)
theorem
if
Gly
pt
makes
distribution
differently,
ted
ule,
if
in
a
the
samples
series
of
zero
b,(j)
for
and
functions
g,(9)
of
of
thermal
the
system
usually
defined
sufficiently
a,(j) then have
Fig.91
remain
noise
are
ex-
instead
of
literature
by
{h(j,9)}
{£(j,9)}.
Thermal
noise
a Fourier
is
series
rather
series
(1).
One
pulses
that
vanish
the
system
section,
defined
{h(j,9)}
is
outside
no
a Fourier
that
than
by
the
the
general
substitute
in (1) the
{f(j,9)}.
there
by
may
in
canbe
the
interval
According
to
difference
series
or
expanded
orthogonal
sine
+40
the
results
thermal
a
series
of
of
this
noise
is
functions
series
ina Fourier
cosine
5 6 = $0 for
whether
by
and
as
shown
by (3).
It
were
Walsh
has
found
been
to
functions,
decomposed
by
stated
have
just
in
section
sequency
asthey
sine-cosine
2.21
formants,
have
that
if
frequency
functions.
audio
signals
decomposed
by
formants,
if
Furthermore,
audio
decompose
if
and
could
ear
the
number of compo-
infinite
an
into
signals
these
noise,
thermal
of
distribution
the
had
signals
audio
If
same.
the
was
flow
information
the
sae
TiLGere,
filtered by frequency
signals
from
tinguished
hardly be dis-
could
filters
sequency
by
filtered
signals
PROBLEMS
STATISTICAL
5.
2ce
nents according to (1) or (4) one should expect such results. The experimental results
show
that audio signals
sufficiently
are
ear
decomposes
make
the
The
similar
them
results
of
results
into
section
also
functions
and
produce
decompose
light into
spectra,
Walsh
Hence,
functions
must
why
device
has
known
the
components
to
represented
by
light
spectrum.
for
an
no
that
Thereis
functions
diffraction
have
explaining
noise
into
Walsh
that
fre-
grating
extremely
pre-
Devices
oraprisnm,
filters
decomposing
practical
at
and produce
like the frequency
a device
dence,
become
sequency
an
just
to
decompose
sinusoidal
suchas
time-invariant
munications.
a
many
suchaslight.
could
and
applicable.
apply
radiation,
known that
are
noise
this
sent no device
quency
thermal
sufficiently
must
electromagnetic
to
of com-
light
fast
time
suggestion
into
depen-
for
such
a
yet.
5.12 Statistical Independence of the Components of an
Orthogonal Expansion
It
has
been
coefficients
for
h # k.
assumedinthe
proceeding
a,(h) and a,(k) are statistically
It
remains
to
be
shown
also holds for the coefficients
These
are
coefficients
statistically
client
p or the
convergence
section,
of
have
that
b,(j)
a Gaussian
independent,
covariance
oF
the
in
series
and
vanish.
a =CbaCgyba(L)) + Lim J
b,(1)
lim
tear
L
fo}
one
Using
Nee
wehea
a dC 2, 0; Ga, hy >, e| Geye,
Cry}
k=0
the
and
j#1.
they
coeffi-
absolute
obtains:
bs Ca dee ey
co
when
cotrelation
A =I
=
independent
independence
distribution
if the
(5)
this
that the
(13)
5.21
LEAST
MEAN
SMR
5
=
Gr
hz0 k=O
Denote
CoO
A=)
h=0
i
Che) iGr
a
2
CoO
double
ej Ch)
and
k=0
_
ths
L 2
ial
Cs) ee
sum
value
of
BAA)
4i
= Yay (ada, Ce) for
1;
any pair
i. follows:
Se
(14)
Ch)e, (k)
(5)
CO
lS
The
Peo
80
largest
finite
Ane
< Deke pe) & pCi)
HS,
DEVIATION
picnic,
(ka, (na Co)
ks
ye
by ¢« the
and
o5
L
EE
ert
h,k
SQUARE
converges
absolutely,
ce, (kK) converge
sum
absolutely:
since
the
sums
2 de, (a)e, (k) = K
Equations-(15)
and
(16)
(16)
yield:
o¢;-= eK
ێ
(aie)
approaches
zero forlarge values
the
of
covariance
oF
of
t by
definition
and
vanishes.
5.2 Additive Disturbances
5.21 Least Mean Square Deviation of a Signal from Sample Functions
Let
a time
functions
BAe
function
of the
F,(0)
be
composed
system
orthogonal
m-1
of
the
first
(18)
-79 = 6 = 30
y= >, ay Ci FC,9)
m
{f(j,9)}:
j=0
is
called
character
a finite
number
of
Fy(9)
ay(j)
are
not
such
arbitrary
of
an
alphabet.
characters,
but
can
if
assume
the
There
is
only
coefficients
a finite number
of
5. STATISTICAL PROBLEMS
22M
two
assume
ay( gj) may
coefficients
5 andthe
equals
m
racters;
cha-
32
contains
e.g.,
alphabet,
only. The teletype
values
values.
transmission
during
and
g,(9) is added
A disturbance
Let F,(@) be transmitted.
signal
the
R(@) = F,(@) + g,(8)
a
in
expanded
be
can
F(9)
that
assume
us
Let
received.
is
(19)
series:
PO) =. > aldie(js8) = j=0
> Layld e+ ey¢SIoL
a0
eee
j=0
@/2
a(j) =
f F(e)f(j,6)d8;
-@/2
from
j runs
O to
a,(j) = 0 for jem
infinity and not
from
O to
m-1.
a,(j)
is
defined by (1).
It must
be
decided
Fy(6@),
\ = 1....%...
sed
signal
the
of F,(@)
was
only
for
with
on
the
ance
with
bability
The
us
W and
probability
on
the
characters
The
decision
No
decision
Putting
is most
known
cases,
it
that
are
F,(@)
trans-
depends
is
then
possible
about
the
set
that adisturbless
differently,
often
than
the
pro-
g,(8)
with
energy
monotonically
with
increasing
likely
transformed
F(6). The energy!
is
is received
adisturbance
decreases
be
in many
energy
energy.
receiving
may
a transformation
probability
all
large
F(9)
of
character
probably cau-
assume
if nothing
W+AW
which
most
probability.
character,
signal
into
which
g,(@).
little
of
receiver
one
disturbances
) with
yr CO!) that
energy
The
However, it is known
gy
tween
the
depends
Let
equal
a single
g,(8).
W.
F(@)
transmitted.
mitted
one
F(6).
into
at
is the
produced
additively
be-
by a character
with
the
AWy required for this
least
trans-
'The term energy isused for the definite integral of the
square of a function. Its meaning is the same as the one
generally used in electrical engineering, if the function
represents
the
voltage across
or the current through a
Pe sister.
ULE
5-271
LEAST
MEAN
formation
is
SQUARE
given
DEVIATION
by the
225
integral
@/2
@/2
-0/2
-@/2
AWy = [F(6)-F,(9)]’de = {[F?(e)-2F(e)R,
(8)+F¥(8 )Jae(21)
The
integral
of F*(@)
yields
ed signal,
of
racter
the
gral
the integral
Fy(@) with which
of
F(6)F,(8)
relation
The
of
the
is
the
signal
contribution
characters
energy
F,(@)
of
all
and
energy
of the receiv-—
Fj(@) the energy of the chasigna
is compared.
l
The inte-
correlation
F(@)
to
the
and
the
character
AWy by F*(8)
maybe
characters
the
ey COD
same for
all
furthermore,
the
is the
ignored.
is
integral
or the cor-
If,
same,
@/2
Wy =
f[ F,(e)de = W,
G2
-O/2
one
may
mined
AW
ignore
by the
= minimum
The
if
Fy (8)
for
transmitted
AW,
has
Signal
too.
The smallest
correlation
integral
1 B(9)F, (848
-0/2
character
its
minimum
detection
which
g,(8)
such
additive
thermal
intelligent
as
(21)
least
from
type
multipliers
puted
be
15
(21)
great.
or
n
to
done
pliers
(23).
or
according
energies
AWy
Let
the
called
deviation.
are
many
conditions
disturbances
and
The
for
types
are
or
of
not
socalled
integrators,
one may
effort
alphabet
an
correlation
transmitted
integrals
n
is
however,
required,
have
de-
charac-
characters.
have
to
be
n
com-
(21) or (23). These computations
should
adders,
multi-
simultaneously.
and
is
interference.
adders,
too
usually
correctly
(24)
square
There
which
pulse
and
mean
in principle the most probably
termine
ter
(23)
satisfy
the conditions
proper.
for
deter-
case:
= maximum
of
noise
a detectionis
such
is
j=.
by means
disturbances
satisfied,
Using
of
AWy
in this
BY (8 ) will
be detected
for
detection
by the criterionof
Samples
value
alone
integrators
Harmuth, Transmission of Information
Hence, n or
are
required.
n/2
(oe)
(249:
into
(20)
and
(18)
ting
m-1
ee
ee a
j=m
j=0
AWy = minimum
(24)
j=0
j=0
S a
:
el,
oe
acs) — 2, PERLE)
AWge=)
substitu-
by
obtained
be
can
methods
expensive
Less
PROBLEMS
STATISTICAL
5.
226
= minimum
Betas
-
for
3 [a(j)
for
2 S a(gjay(d)
j=0
or
m-
AW, = minimum
m-1
-
a, az (gj) = maximum
j=0
The
sums
yield
for
co
co
i ay (gj) or
Soe)
j=m
j=0
the
same
characters
Equations
the
to
value
with
AWy = minimum
ficients
j=0
for
for
equal
(24)
j < m,
decision over which
produce
the
m multipliers
This
means
the
a
teletype
Let
us
avian,
4.
and
(25)
the
One
show
noise
character
signal
practical
ignored,
since
obtains
they
from
= maximum
(24)
(25)
that
only the coef-
sample
g,(@)
affect
Fy,(08) was the most likely
F(@).
and integrators
reduction
be
energy:
of
received
required for the
each
3 aC j)ay,(j)
j=0
(20),
a,(j),
‘
may
rather
than
n or
n/2
are
implementation of (24) and (25).
from
32
or
16
to
5 in
the
case
of
alphabet.
substitute
the
sum
ay(j)
+ a,(j)
from
(20)
for
Geans
m-1
AW, =minimum
for
2 2,Lax(5)+a,C5
je
The
effect
of the
cision
is due
to the
.
.
.
disturbances
sum
2 A Pak
jz0
m=!
lay (j)- De
ay (g) =naxinum
g,(6)
ve
on
the
The
Signal
de-
probability
522
of
EXAMPLES
a
wrong
properties
OF
CIRCUITS
decision
of the
2a,
depends
solely
coefficients
on
a,(j)
the
=
statistical
erence
—0/2
Let
cal
g,(8)
be
conditions
general
used.
{f(j,9)}
system
quite
unimportant
which
functions
for
The
statisti- under
then
are
orthogonal
the
of
to
according
probability
it is
(18).-Hence,
decision
of awrong
used to compose
are
additive
are
noise.
a,(j)
transmitted signal F,(@)is com-
the
f(j,6)
disturbances
the
thermal
independent
-
The
Tunctions
these
meccdvor
of
cocetficients
of the
properties
very
a sample
the
signal,
if
noise.
thermal
5.22 Examples of Circuits
ietwus dtecuss
signal
are
detection.
obtained
sample
Same
some
Fig.92
from
functions
as
the
efficients
ay kd ) are
one
circuits
the
shows
of
This
Fig.30,
instead
the
signal
(24) and (25)
coefficients
F(§8)
the
that
by
for
a(j)
means
of
is
basically
the
the
disturbed
co-
circuit
except
of
use
how
received
i(j,9).
a(j)
that
undisturbed
coefficients
obtained.
signal in
ae
ai)
Fig.92 Extraction
a(j)
from
the
of the
coefficients
received
MAW LplLer,
signal
F(6).
U2 Lotecratboi.
f(2,8)
The
are
cients
of
the
products
by the circuit
produced
composed
15>
of
sums
three
By(O),
ay (1)
and
ay(2)
according
Pie.So. Lue
of
functions,
a
UG
m
=
4.
occur
Hence,
that
to
Characters
three
are
(25)
alle
coeffi-
represented
5. STATISTICAL PROBLEMS
|
228
nts
.
coefficie
ages
by voltThe
= 1,
2,
..-
amplifiers
terminals
a,(0O),
ay(1)
and
ay(2),
4 =
ted . The operational
by resistors
represen
A have differential inputs. The inverting input
are
the non-inverting
are denoted by (-),
Va(0) Va(t) Va(2)
by (+).
ones
+V
Va(0) Va(t) Va(2)
Fig.93
(left) Signal
characters have equal
~a(2)ag(2)]13
detection by the largest sum. All
energy. V, =V[a(0)a,(0)-a(1)a,(1) +
Vi= VL alO)a, (0)+a(1)a,(1)-a(2)a,(2)];
=V[a(0)a,(0)+a(1)a,(1)-a(2)a,(2)].
Fig.94 (right) Signal detection by the
characters
do
not
have
to
have
equal
smallest
sum.
Vz =
The
energy.
Vee Vlad (0 )+ag (1)+a2(2)-a(0)a,(0)+a(1)ag(1)+a(2)a,(2)];
vies VL a? (0 )+ap(1)+a?(2)-a(0)a,(0)-a(1 a, (1)+a(2)a,(2)];
V2 = VL al (0)+a7 (1)+a3(2)-a(0)a,(0)-a(1)a,(1)+a(2)a,(2)];
ll R/([at(O)+
Ry =
oat
as(4
(2)ise
ear. 2
eee
522
EXAMPLES
For
the
of a*(j)
to
sum
as
V,,
sos
uses
a
parators
with
the
the
voltage;
ramp
tage.
An
voltage
largest
of
does
back
non-instantaneous
the
instantaneous
voltage
equals
at
the
the
largest
the
largest
put
terminal
(-)
non-reversing
and
A,
sing
input
A;
terminal
is
voltage'V;,
indicated
alarger
diode
j=0...7,
by
the
vol-
that
the
time
and
The draw-
the
52
4 diodes
Let
than
must
V3
at
am-
non-rever-
(-).
one
denoted
be
the
The ampli-
at the
Bs
be
in-
negative
at
be
reversing
and.
The
sufficient
by representing
characteristics
of
by B,=-1.
voltage
at
Fig.95.
non-reversing
will
B,
By,
The
is
in
larger
voltages:
+1.
com-
ramp
with
Assuming
output
B,-=
The
A,
positive
driven
n
n
increasing
voltage.
at the
(+).
(+) than
an
group
to
are
number.
shown
each
voltage
be
via
unimportant.
saturation,
Both
largest
receive
type
of
determined
positive
amplifier
shall
which
saturation,
of
linearly
of
voltage
terminal
output
the
plification
fiers
of
input
The
One
smallest
circuit
is
point
applied
voltage.
ouc—
operation.
comparator
common
vol-
comparator to fire
case
are fairly
The
output
a decreasing
have to vary
opera-
andewhich
compared
of
the
value.
or
is
of
type
fluctuations
is
in
voltage
this
not
is
but
con-
smallest.
The first
voltage
that voltage
An
that
re-
with
which
largest
be
of the
line
proper
is
ati Jaya)
must
in Fig.94.
iseilearcest
the
sum
without
Fig.93,
as
determine
to fireincase
advantage
voltage
of
n voltages.
the
comparator
of
of
terminals
in Fig.94
smallest
first
to
.-.
sum
sign
additional
by an
dt) LPiclo>
V,,
The
the
interchanged
determines
voltages
ramp
be
the
determined
circuit
input
that
sum may be disre-
be
that
the
required
that
determines
shall
resistors
and
+V
are
circuit
ramp
must
5) Vos
voltage
of
mayuse
note
This
of AWy.
except
is produced
Circuits
BICCEVasmv,
AW,
value
non-inverting
voltage
stant
all AW,.
before,
amplifiers
of a5 (5)
put
the
one
and
for
snallest
Hence,
inverting
same
know
produced
versed.
ey
implementation of (24) letus
wit, the
any need
tional
CIRCUITS
is the
earded,
is
OF
by B2
indicabe
j as
very
=
the
binary
similar
of
2' voltages.
of
several
are required for comparison
r amplifiers
results.
good
for
magni-
smallest
or the
has the largest
voltages
which
can detect
circuit
of the
Variations
PROBLEMS
STATISTICAL
5.
230
tude.
to
between
the
largest
the
isthe
be
of
table
but
smallest
positive
tive
than
comparison
(n-2)
total
of
voltages
third
Vo,
V;
diodes.
Its
required.
of
are
A
(n-1)
+
Hence,
a
needed.
The
n
= yn
for
requires
amplifiers
may
sensi-
fed
differences.
hand,
that
are
measurement
amplifiers
only
V,
and
more
amplifiers
voltage
shown
-
voltages
the
Z3l=
not
largest
is much
other
onthe
are
denote
circuit
of
The
V,.
voltages
voltages
requires
differential
lg,n
and
diffe-
by the
and
V,
They
since
differential
Fig.95,
are
rather than through
= $n(n-1)
+1
A;
Az
A,,
output
number
n voltages
...
of $#n(n-1)
circuit
Fig.95,
large
the
of
+
The
This
amplifiers
directly to the
+
of
and
caption.
second
voltage.
one
is
the
negative.
the
drawback
also
-
or
the
of
figure
the
the
V,,
voltages
permutations
6 possible
in
three
the
of
saturation
negative
or
positive
driven
rences
determination
amplifiers
three
The
voltage.
largest
for
circuit
another
shows
Fig.96
only.
5.23 Matched Filters
It
are
has
been
obtained
f£(j,9)
and
equivalent
matched
rather
of
pulse
approaches
the
the
of
D(@)
product.
very
response
matched
having
the
coefficients
the
signal
the
Consider
delta
method
further
a
bank
of attenuation
and phase
filters.
§(@)
of
uses
response
amplitude
function
with
pulse
Consider
1/e
for
a narrow
inside
= 6 = $e and the amplitudeO outside.
the
a(j)
F(@)
A mathematically
different
It is customary to.use
characterize
-ge
that
technically
than the frequency
block
e.
integration
but
to
of
so far,
multiplication
filters.
shift
terval
assumed
by
the
vanishing
filters.
Let
in-
This pulse
the
values
pulse
6(0+%) be applied at time 6 =-% to the input of the filter j- The output function f(j,6), -4 = 6 5 +4, of (18)
and (20) shall be produced. f(j,6) is the pulse response
525
MATCHED
FILTERS
ee feet
a rs
See Ss STE
may
aera
Beige
0es eee
aon
é
eis
Fig.95 Detection of the largest positive voltage Vy to
V,. The largest
voltage
is
determined by the values of
B,,B, and B, shown.
R
Seow
Fig.96
Detection
of voltages.
largest
second
third
Our AeabGene
of
the
relative
values
voltage
largest
largest
voltage|
V,
voltage]
V,
ae
The time function F,(@) of (18) canbe produced by applying
the pulses a,(j)6(6+8) to m filters with pulse response
f(j,9)
denoted
The
Fy(0)
as
and
receiver
or
summing
transmitter
These
filters
are
invert
the process.
a(j)
time
the
inputs
functions
time
Let
appliedto their
The
the
6rtnouermal)
are
outputs.
interval -& = 6 = #, and the coefficients a,(j) or
in (20) are obtained
at the output
of filter j at the
@ = +4.
F(6)
filters
the
filters.
during
functions f(j,6) be represented
of pulsés D(6—ke)s; ki =O, +1,
system
by the
+42,...:
STATISTICAL
5.
Coe
PROBLEMS
(26)
a(3 £03,802 aC) 24; (k)D(@-ke )
ke+e/2
ke-e€/2
-1/2
Peo
ee any sor
exactly
by
small
ciently
the
function.
the
that
the
can
However,
square
mean
if
in (26)
sum
the
e becomes
deviation
suffi-
f(j,8)
between
arbitrarily
becomes
be
receiver
since
{D(@-ke)},
function
step
functions
Let
system
the
represented
not
generally
are
f(j,9)
a step
represents
and
eae e
functions
The
eve
> Cj
uf = £¢3;09d0"
[ £(3,8)D(0-Ke )ae =
)e=
dpc
small
for those
generated.
filter
j
produce
the
output
h(j,8),
4+ = 0 5s 4, ifthe input 5(6+%) is applied. The input function
D(6-ke) = D(6@'+#)
produces the output
h(j,8@') =
h(j,@-s-ke) if e is sufficiently small. Hence, the function a(j)f(j,9) applied to the input produces the following
output
signal:
ad) 2,d;(k)h(j,0-#-ke),
This
signal
has
the
ke )
at
Let
d,;(k)
9 =
in
(28).
e andheight
an
+#.
The
if
41/2¢
(27)
(28)
us
sum
substitute
f(j,ke)e
yields
areaof
f(j,ke )h(j,-ke).
integral,
....
value
a(j) d, 4; (c)aC§
time
k = 0, #1,
e is
the
This
sum
sufficiently
from
(26)
for
stripes
of width
may be
replaced
by
5mall:
1/2
a(j) J f(g,0)n(j,-9)ae,
-1/2
This
Cj
integral
equals
58)
=
The
coefficient
© = lim ke
ans
= lime
bansd
(29)
1 for
f£(j,-8).
(30)
a(j)
is
obtained
recei
filterver
at the time 6 = $.
on the
a6
e—
other
hand,
if the
at
The
function
the
output
a(1)f(1
output
Ois
6);
of
the
obtained
a=4eg,eas
5.24
COMPANDORS
applied
to
Bae
the
input
of
receiver
filter
aS
v2
a(1)[£(1,0Yh(j,-8)a8 = a(1)f2(1,8)£(J,8)48=0 §41 (31)
-2
-1/2
The pulse response of the receiver filter j must be f(j,-8)
if the pulse response of the transmitter filter jisf(j,é@).
Transmitter
functions
=
and
receiver
f(j,8)
filters
= f(j,-8),
are
and
for
identical
odd
for
functions
even
f(j,9)
-f(j,-@).
Matched
ene
filters
coeiticieny
vantage
say
over
whether
for
Walsh
filters,
from
like
one
shown
multipliers
or
In
matched
other
and
in
general,
filters
e.g.,
on
determine
ad-
cannot
are
superior.
arevery
accurate.
but
page
an
one
hand,
do not have
capacitors,
Fig.36
to
is frequently
circuits.
functions,
onthe
coils
need
(20).-This
correlator
structed
the
not
in
correlators
Multipliers
Matched
do
aCj)
may
to
be
be
con-
circuits
80.
5.24 Compandors for Sequency Signals
It
is
well
known
that
instantaneous
frequency
limited
signal
produces
frequency
limited
anymore.
The
compression
of sine functions
Thisisnot
Walsh
ters
so
for
sequency
functions.
F,(@)
and
Fig.97a
limited
shows
compression
signal,
reason
always
for
i=
= -wal(0,6)
an
example
+
these
S Gi
Fi(9)
and
same
the
they
are
FJ(8)
Walsh
that
harmonics.
composed
two
of
charac-—
r
eer celia,
iz]
i=5
i
pelCl.6)
shown
through
cal(i.d)
These
compressor
having
the
signals
signals contain
exactly
produces
by Fig.97b
of Fig.97c.
a
F,(6)
as the characters
functions
only
multiplied
by different
a
compressor
characteristic
Consider
is
a
not
i=l
characters
characteristic
the
is
5 CEleL
sO
iz]
Sending
this
generates
functions
as
of
that
F,(@).
F,(9) =wal(0,6)+ of Pup) sal(i,@)-
Fy(@)
a
andF,(6),
coefficients.
n
= Eerf(¢/f2o).
be the amplitude distribution func-
W,(x) =W, (-©<¢sx)
Let
(4.11):
from
follows
W, (-co<nsy)
= Veo erf ¢ = 20
=
W,(y)
function
The
compression.
before
of asignal
tion
PROBLEMS
STATISTICAL
5.
234
erf't
WoCy) = W, (-co<¢sV2e
erf 2) = W,(\2o
erf 4)
w2(y) =Ver Sy exp(erf£) w,(\2o erf£)
Consider further
tions
val
of
Fig.2.
All
0 < 6 < 1/16.
can be produced
character
=1.
with
a signal
16
functions
Among
from
the
the
have
equal
a
16
amplitude
aes
composed of the
16(41) =16
the
CucCre
holds
same
result
binary
characters
a Bernoullian
composed
of
amplitude
m Walsh
probability
for
any
of
that
is 1 = ee
interval;
time
16
Let
a
an
12;
interval.
functions
with amplitude
sampling
inter-
14(41)+2(-1)=
other
distribution.
p,l (m-2h )a]
the
characters
there
composed of Walsh
functions
func-—
15(4+1)4+1(-1)= 143
have the amplitude
ie
Walsh
in
in this
amplitude
120 = ("a5 characters
Hence,
+1
binary
functions
16
have
character
+aor-a.
amplitude
be
The
(m-Zh)a
equals:
Bgl (ms2n je) =e):
The
distribution
yo" lew Opty reds am
function
is W,(x),
[x]
W(x) = GP SY,
h=0
m
where
[x]
Wa(x)
denotes
can
be
error
function:
Wa(x)
= #(1
The
and
+ erfraR)
derivative
racteristics
cE
in
the
largest
approximated
w,(x)
Fig.98a.
for large
= W,(x);
is
n = Eerf
integer
equal
values
m by
are
of
x.
the
= ma’.
showninFig.98b.
(¢//2c)
The
E?
smalleror
shown
corresponding
Compressor
foro
density
cha-
= 0.5E,
E
functions
5-24
COMPANDORS
eas
i] ye E ertlf20)
]
Fig.97
Compression
of
sequency
multiplex signals. a) original
signal, b) compressor
characteristic, c) compressed signal.
Fig.98
(right)
Compression
sequency multiplex signals.
compressor characteristics,
of
a)
b)
density functionof
the statis-—
ecole
VarLaple,
tC,
Gc) density
functions after compression.
wo(y)
are
shown
in Fig.98c:
wo(y) = # & exp(1-02)(ert” £)’]
Wo(y)
Note
= #[1
that
formed
the
into
Figs.98a
hold
Pep
This
for
+ erf(o erf| £)]
Gaussian
an
and
equal
distribution
distribution
c also
show
a non-reversible
cene=scevore
compressor
lines
all
Fig.98b
o
=
denoted
compressor
|Claeds>neand
clips
of
for
with
r= 21.58
amplitudes
is
trans-
E.
by "13%".
the
for
They
character-
{¢| = 71.55.
absolutely
larger
TMEMNOremCetad
Lalo
discussed
be
will
clipper
This
distribution.
a Gaussian
of
case
amplitudesinthe
the
of
13%
are
which
1.5E,
than
PROBLEMS
STATISTICAL
5.
236
eeud One Oncrls
5.3 Multiplicative Disturbances
5.31 Interference Fading
Let
The
a radio
signal
samples of the
at the receiver.
be
same
transmitted
signal
Consider
as example
viatwo paths. The samples
with
two
a delay
samples
A, cos 2nv,6
(A,
ll
of
be
6,
are
written
in two
receiver
the
The
phase
A,+A,
and
The
tudes
evidently
shift
and
large
tigation
plicated.
has
the
(32)
+
A, Sin nv,
sin amv,by
receives
A,-A,
of
or
one of the two terms
the
signal
betwéen
A,
received
and
-A,.
determines the amplitude
line
of
(32).
It
varies
reason for this variationof
the
a time
same
=-cos
functions
reversal
8,
of
It
would
A simpler
transmission
useful
is,
only.
functions
however,
It
A
of the
between
cancel
application
does
by
not
is
the
amplitude
rever-—
reasonable
hold
between
or
does
theoretical
mathematically
Walsh
ampli-
m7 of anoscilla-
appears
Ageneral
obvious
differentiated
not
8,=
effect as an
emv,9.
amplitude
of
shift
for which the equivalence
values
shifted,
Fig.77
third
that
cos 2nv,(@-1)
other
of
forms:
amplitude
and
receiver
the
sum
O.
cos emv,8
sal,
time
in
mathematical
is
tion
for
A;+A,
insensitive
oscillation
use
line.
The
+ wy" cos (2tv, 8-a )
A phase sensitive
second
another
=
+ A, cos 2mv,
6, ) cos 2nv,9
between
paths.
one
a sine wave transmitted
received.
+ A, cos anv, (6-8,)
(pA‘are 2A,A, cos amv, Sy
varies
several
with
A, cos 2nv9@ and A, cos 2mv, (8-8,)
difference
may
via
interfere
to
time
hold
inves-—
very
com-
that a superpositicn
functions
of
according to
interference.
of orthogonal functions for the
through an interference-fading medium follows
5-31
INTERFERENCE
from
the
tion
2.15
narrow
follows:
for
Let
ated.
a harmonic
Using
receiver
e(8)
the
input
answer
an
before
necessary
oscillation
Rayleigh
several
limitations.
The
more
and
10
onto
of
secdiscusin
sed
carriers
many
carriers.
can
be
spread
to
worthwhile
diversity
with
for
can
of
fading
a voltage
= v(8)cos[env,9
so
over
results
known
into
method
as
for bandrequirements
itis
whether
is
4 carriers
or
2
system
use
to
concept
known
modulated
than
by the
power
transmitter
a fixed
digression
The
und
the er
influence
excessive
without
discusin
sed
sec-—
well
bandwidth
to
due
questionis,
The
width.
a
are
More
one.
possible
it
2or%4
of
is
reliability
used
makes
2.15
instead
system
transmission.
required
bandwidth
the
Signals
than
be
not
generally
narrow
of
diversity
fading.
rather
carriers
is
teletype
transmission
interference
237
bandwidth
Frequency
improving
tion
FADING
A short
transmission
given.
frequency
model,
one
v,
be
radi-
obtains
at the
e(@):
+ a(9)]
Geis)
v(@) is a slowly fluctuating envelope, which is practically constant
during an interval 6, -$9, = 9 3 9,+#6,,
and which has a Rayleigh distribution with the following
density
function:
2
= 0
6?
equals
(34)
vz0
wiv) = & exp(- 4)
We S00
expectation
the
E(¢’)
according
to
(4.83):
LADS GO
The
phase
angle
have
a constant
wa )
ll
ro)
ns
also
a(@)
density
=fsed
Cis
(35)
fluctuates
slowly.
It
shall
function:
(36)
2 +7
6
+
sion reliability requires, that
ent
in transmis
improvem
two ormore statistically independent 'copies' of the sigof the
function
density
the
Hence,
received.
are
nal
An
of
density
functions
(34).
narrow
between
beam.
Two
independent
quency
diversity uses
diversity
problem
basically
which
val
has
6,
gain
the
largest
summation).
ring
by
an
For
cally
interval
that
6,
g,(9)
of
thermal
the
copy
on
as
copies
them
of a signal
arises.
a)
The
There
copy
during
their
ratio
three
is
a time
inter-
methods
1 from
copy
(equal
before
average
ing
is
Gi 40)
v,
represented
equation
F(8)
9, -29,
The
£9
to
let
F(@)
into
G,(8).
q statisti-
be
available.
G, (8).
Hence,
A sample
the
which
fol-
Cre
during
a short
according
to
assumed
time
6,
by the follow-
(43):
to
be
09-29%
= 8 = 6,486,
constant
in
(38)
the
interval
smaller
than
= 6, +#0,.
probability
threshold
time
are
du-
1:
"= 7 COE )cos[ atv, O+a ,(8)],
anda,
sum-
power
H,(@) = Gi(8)
+ gale)
G,(@)
are
used
summation).
signal
noise is added
received
and time
carriers
copies are multiplied
copies of the
transform
is
of
power
depend
of
Fre-
b) All copies are added
(maximal
a comparison
independent
lowing
average
c) All
factors
fading
use
available.
selection).
provide
region.
repeatedly.
independent
best
methods
(optimal
mation
Let
of making
three
wave
short
sinusoidal
signal
several
waves
polarized
the
several
the
obtained
in
discriminating
antennas
circularly
copies
transmits
Having
the
polarized
right and left
fairly
antennas
by means of directional
copies
obtains
diversity
with
Angle
spaced sufficiently far apart.
antennas
several
uses
diversity
Space
copies of asignal.
independent
sta-
receptionof
the
for
known
are
A number of methods
tistically
product
a
be
shall
amplitudes
the
of
distribution
joint
PROBLEMS
STATISTICAL
5.
238
vg
v,
p(v,<v,)
of
or,
putting
it
is
smaller
than
p(y, <v, ) = Wv,)
Vg
v,
being
differently,
v,
follows
the
from
fraction
a
of
(34):
= f - exp (= wav =1- exp(-v’/s? ) (39)
0
5-51
INTERFERENCE
Let
FADING
259
q statis
independen
tic
t copies
all
be received,
y
havin
the g
same distribution.
p,(v,<v,)
that
q copies
the amplitudes
v,
of
all
isthe
are
all
probability
smaller
thanv,:
Pa(v <vy) = WoCv,) = [1 - exp(-v2/s?
)]'
(40)
The average power of the copy G,(@) inatime interval of
duration 0,, thatisan integer multiple of 1/v,, follows
from
(38):
+8, /2
8-8,/2
Let
with
;
;
Gy (8 de) = “Sy
i(8p)
P,
copy
Peere
is
“
ee
ib
8k
= Py (85)
denote
the
1.
signal-to-noise
The
average
noise
= avi
= PF
power
ratio
power
(41)
received
ratio,
Tee.
a quantity
probability
(42)
that
of
P,
fluctuates
being
due
to the
fading
belowathreshold
P,
only.
follows
The
from
(39) and (42):
eT 7 F;
=
vagiges
W(P, )
=
DAP) /2.<
WP,)
=
1 -
Let
from
the
the
smaller
if
all
Wecee
The
as
=
Pa
copy
with
the
available
than
copies
this
of
value
The
all
statistically
P,/P,
probability
copies
follows
be
selected
that
P,/P,
from
(40),
independent:
distribution
(44)
us
;
co
denote
copy
Dae lee Sh
the
average
by Ps /P, :
U Ae,
OT
Ee
calculated
was
J SL aW,(P,) = J qy(1-e""
Go
each
W(vg)
exp(-2P, /6* )
largest
for
Coe
Of
(43)
= (1 = exp =2P,/6)]
mean
Let
4, =
copies.
P,;/P,
are
Mere.
p(y, <vy ) =
exp(-v, /6") =
q
is
<
th
8
by BRENNAN:
q
4
)" eY dy = 2 rt (45)
signal-to-noise
power
ratio
signal-to-noise
average
The
= Py:
CPi > = $6’
relation
with the help of the
is obtained
copy
best
the
of
ratio
power
PROBLEMS
STATISTICAL
5.
240
by the
ratio (Peq/Pp/ CPs Fy) is shown in Fig.99
denoted by 'a'. One may readily see that the aveise ly
power ratio o-no
increases insignificant
signal-t
The
points
rage
if more
than
three
or
four
copies
used
are
optimal
for
selection.
— So
Fig.99
|9
Increase
of
signal-to-noise
8
the
power
average
ratio
by
diversity reception according to
BRENNAN.
q number
of received
copies of the signal; (Psq/Pr)/
a
BG
at.
a”
ae
ASS
(P;/P, = (average signal - tonoise power ratio of q copies)/
(average
signal-to-noise
power
ratio for 1 copy).
a) optimal
selection,
tion,
=
Replacing
q copies
optimal
yields,
b)
equal
c) maximal
selection
by equal
according
to
gain
ratio
gain
BRENNAN,
the
summa-
summation.
summation
following
of
re-
lation:
Psq/Py
=
Psq/Py
stands
ratio
The
points
of
(Ps/P, )[1
the
ratio
of
dB
for
of
the
all
by
differ
'b'.
q copies
over
Optimal
only
equal
(47)
average
(Psq/P,)/(Ps/Pr)
However,
4.5
sum
denoted
summation
2).
now
+ ¢n(q-1)]
slightly
signal-to-noise
of
is
the
shown
signal.
in
selection
if
gain
summation
optimal
selection
Fig.99
and
2 copies
yields
if
power
are
an
q =
10
by
the
equal
gain
used
(q =
improvement
copies
are
used.
For
in
a
maximal
time
ratio
interval
summation
of
the
duration6,
amplitudes
is
of
multiplied
copy
l
by
a
5-31
INTERFERENCE
weighting
of
copy
the
FADING
factor
1 and
which
copy.
pression
replacing
Psq/Py
=
CPs/P ha
Psq /Pr
now
of
the
weighted
The
ratio
points
denoted
than
dB
1
a
during
measure
(39)
(43)
and
fraction
of
threshold
vg,
average
threshold
gain
of
range
for
of
yield
such
ratio
of
ex-
signal.
in
of
by the
is
some-
difference
q shown
of
and
the
the
various
signal.
is
possible
measure.
methods
fraction
is,
however,
Equations
The
gives
first
which
second
the fractionof
time during
Pg = Py for
power
now
voltage
ratio
rewrite
which
W(P,)
of
of alink.
during
us
a
The
the
Let
ap-
q.
time
signal-to-noise
is
signal-to-noise
power ratio
reliability
a
The
values
comparing
of
Fig.99
summation
values
transmission
the
the
shown
summation.
infinite
copies
for
Pg /P; -
median
W(Py)
Maximal
means
of
which
a better
the
'c'.
for
good
utilization
rms-value
following
signal-to-noise
power ratio
q copies
equal
dB
the
rms—value
(47):
increase
of the average
provides
the
of
forthe
1.05.
the
the
derived
esr AG sey Samp is
by
less
time
and
(46)
sum
than
for
BRENNAN
to
to
(48 )
CoP
better
The
proportional
proportional
denotes
the average
what
proaches
is
inversely
noise of that
24
v,
is
P,;/P;
(43)
equals
by
the
below
is
a
which
below
a
introducing
all
3:
= # = 1 - exp(-2P,/8*)
(49)
ihe, ae@ullkeyigess
Ory
foesely ast 0.095
(50)
P/F,
DPWca =e crs/ Py) Une
4107693"
Equation
rewritten:
W(Pg)
The
(43)
may
be
(2)
= 1 - exp(-0.693P,/Py)
probability
of
P|/P,
than Pg/P,
beinglarger
becomes:
(52)
/Py)
p(P /P,>P,/P,) = 1 - W(Py) + exp(-0.693P,
p(P\/P;>P,/P, ) is
The
16
ordinate
of
shown
that
Harmuth, Transmission of Information
«in
figure
Fig.100
shows
by
the
the
curve
percentage
q = Ane
of
the
5.
STATISTICAL
PROBLEMS
than
a threshold
Pg.
eye,
P,
Here
normalization.
PR, for
median
divided by the
are
Pg
and
is larger
P,
which
during
time
q copies are received one obtains from (44) the propability that P,/P, is larger than Pg/P, forat least one
If
copy:
)]"
#1 - £1 - exp(-2Ps/6*
Dq(Pi/Pr>Po/Pr)
One
may
rewrite
this
equation
using
(50):
(53)
Da(Pi/Pr>Py/P,) +1 - [1 - exp(-0.693Pg/Pu)I]
Pq(P,/P,>P,/P, ) is shown
q =
2,
time
4 and
during
optimal
ratio
These
which
P,/P,
time
during
ger
than the
of
time
if
equal
which
Pg/Py
Fig.100
P,
of
may be computed
methods
is
Pg.
of
used
than
is
of
the
possible
8 copies
and
if
if
a
required.
the
percentage
2, 4 or
Hence,
they
of the
8 copiesis
lar-
give
the
fraction
transmission
is
possible
with
Ps /Py
withthe
have
percentage
4 or
show
diversity
larger
the
2,
a sum
summationis
by the solid lines for
transmission
with
in
which
P;/P,
numerical
used
threshold
gain
give
than
lines
during
if aratio
is
larger
dashed
curves
diversity
selection
The
ves
8.
in Fig.100
help
of
to be used
2,
4 or
8 copies
is required.
(4.95)
for
for
larger
These
and
cur-
q=2
‘while
values
of
q.
5.32 Diversity Transmission Using Many Copies
The
methods
taining
ally
provide
cannot
sity
discussed
statistically
only
yield
could
more
in practice.
several
short
only
hundred
wave
a
few
than
yield
the
apart
section
copies
Polarization
required
for
space
time
that can provide
diver-
However,
limit
ob-
usu-
diversity
Space and angle
antennas
and
for
of a signal
many copies.
space
Frequency
methods
previous
copies.
instance,
meters
region.
practical
and
For
the
copies.
two
theoretically
sideration of cost
ber
in
independent
con-
this
nun-
have to be
spaced
diversity
in
diversity
many
are
the
the
copies of the
Signal.
In
order
to
apply
the
curves
of
Fig.100
to
frequency
5.32
DIVERSITY
TRANSMISSION
243
Fig.100 Relative time A® during
whicu the normalized signal power of adiversity transmission
exceeds a threshold
ved
from
NAN).
figures
q number
Pg/Py(deri-
due
to
BREN-
of received
co-
pres;
solad lines
:foptimal’ ge—
lection;
dashed
lines:
equal
gain summation;
dashed-dotted
line: reception without diversity.
and
time
diversity,
Signals
space
are
and
mitter
than
angle
power,
smaller
by
for
quency
Instead
and
time
using
each
Let
frequency
is
Fig.100
Py/Py
the
wer
16*
may,
use
of
This
'equal'
trans-
signal
time
for
is
diversity
drawback of fre-
course,
be compensated.
reception
one
antenna
the
per
and
for
power
ordinary
diversity
will
signal-to-noise
ratio
Signal
diversity.
compensate
transmitter
increased,
power
just
q
average
radiated
frequency
q antennas
Replacing
average
angle
that
asinspace
with
and
q-times
reduced
signal
the
power
copy.
the
fixed.
for
one may
gain; this would
of
power
diversity
diversity,
in mind
Givenacertain
average
1/q
keep
only one signal is radiated
diversity.
the
space
of
must
while
a factor
and
angle
one
radiated
despite
of
Py/q.
fraction
exceeds
the
Given
of
time
qPg/Pm
an
1/q.
1/q
by
be
of the
input
signal-to-noise
the
average
in
the
median
Py
threshold
Pg the
ratio
reduces
q-fold
larger
be
onlyif the
receiver
Reductionof
which
antenna
(q = 1) by q-fold
improvement
certain
a
during
must
receiver
ratio at the
decrease
by
Using
qPg/Pm.
becomes
bring
copy
the
transmission
power
copy
per
power
to
each
and
frequency
the
average
than
the
diversity,
signal
fraction
poof
5S. STATISTICAL PROBLEMS
244
exceeds
Pz Pu
The
curve
an example:
Consider
transmission.
ordinary
for
power
signal
average
the
which
time'during
q=1 in Fig.100 yields A®@ = 95% for 10 log Pg/Fy =-11
curves
the
while
power
signal
One
may
if
the
by
at
ted
is larger
readily
least
10
'4 dB'
by
Evidently
see
that
log
q =
show
10
time,
curves
and
between
order
to
The
separation
equal
with
operation
is
deno-
just
4 dB.
summation
gain
occurs
operation
possible
points
is
selection
optimal
make
selection
on
for
more
be
will
possible
is
than
worthfor
tion
is
best
with
It has
sine
bandwidth
Hz.
fold
cuits.
by
more
onlyif
80% of
points
the
'3 dB',
least
been
discussed
bandwidth
Six
may
in
be
teletype
according
frequency
40%
number
of
to
2400
diversity
of
'6 dB'
and
'9 dB'
the
2.15
utilized
circuits
Table
diversity
satisfactory
time
and
is
gain
are
using
opera-
is
then
copies.
section
well
operation
equal
Frequency
of
Optimal
For
= 40%.
at
indi-
time.
worthwhileif
in
worthwhile.
'9 dB'
9 dB.
satisfactory
+= 6 dB
8 = 9 dB
and
6 and
is
largest
log
diversity
are just
between
10log4
10
'6 dB'
A@
A total
The
least
line
the
pulses.
at
least
summation
possible
frequency
the separation
eightfold
70% or
three
be
at
denoted
worthwhile
the
gain
must
q =8
separations
least
all
located
q=4
fourfoldor
these
at
summation
and
in Fig.100
is
possible
considerations
q = 1 and
where
equal
same
q=‘1
points
cate
is
while
is
99.3%.
horizontally
separated
2 + 3 GB.
log
average
the
improvement
an
2 are
diversity
99% or
95% to
dB
52 1of thesvine.
the
240
P, from
this
satisfactory
onlyif
Basedonthe
The
which
where
40%
tiame5
during
such
if satisfactory
while
time
diversity
twofold
the
twofold
q =
worthwhile
of
Hence,
than
1 and
q =
curves
99.3%.
of
fraction
the
increases
and
= 99%
A@
values
the
dB,
-8
=
10log 2P,/Py
for
yield
qg=2
for
4,
that
by
require
twelve
a
certain
sine
about
circuits
and
120
coHz
about
Hz
bandwidth are required for tentransmission of these twelve cir-
spacing of the ten copies by multiples
sufficient in the short wave region.
of
240
Hz
6. Signal Design for Improved Reliability
6.1 Transmission Capacity
6.11 Measures of Bandwidth
It
was
recognized
communications
of
bols
very
filter
pee
of
KUPFMULLER
and
Af,
bandwidth
[2,3]
the
states
per
time
frequency
low-
transmitted
be
may
frequency
its
on
Forinstance,
+ through an idealized
of duration
interval
pass
[1]
symbol
independent
one
shift.
phase
and
of
rate of sym-
transmission
depended
channel
of attenuation
famous theorem by NYQUIST
that
during the development
that the possible
a communication
response
early
where
17K Ts
The
cae
transmission
case by the
per
unit
number
time
rate
1/1
[4,5].
transmission
tistical
disturbances
attenuation
lebrated
formula
information
under
the
Goeent
where
per
Af
filter
power)/(average
is
It
(2)
as
tent
theory
as well
for
ason
shift
the
of
the
additive
that
response
obtained the ce-
transmission
frequency
rate
lowpass
and
contain
of
P/P,
noise
of
filter
noise,
C2)
is
power
for
the
the
the
bandwidth of the
quotient
in the
band
present
frequency
communication
e.g.,inbits
stated,
capacity
frequency
isthe
the
depended on sta-
He
thermal
this
transmitted
account
frequency
[6,7].
possible
transmission
important
well
symbols
into
in
P/Es es
C isthe
second.
took
defined
rate of information
phase
influence
is
independent
through
an idealized
ted
lowpass
and
of
symbols
SHANNON
possible
of
of
idealized
(average
boo
0 sf
purpose
bandwidth
basedoncomplete
signal
that
Af.
(1)
as
A consis-—
systems
of
6.
246
an
input
the end of
mean
performed.
Pat
does:
power
<Vy PRG >=
Using
caused
of
equation
where
V
are
that
integrates
Va»
voltage
output
zero,
is
1 inte-
of
will be denoted
at
V,.
For
equal to zero.
The
multiplied
by
1/R,,
by
just
the noise
characterize
is
like
(3)
this
result
a voltage
noise.
g,(9)
is
from
5.11
be
by thermal
instead
integrator
through
applied
A total
At.
let
L
- SEVIS
lim
(5.1)
integrator
voltage
may
and
section
of
The
mean
deviation
square
average
g,(9)
the
an
interval
integration
\-th
the
noise
an
a time
over
is
grations
thermal
R, to
resistance
voltage
this
be
voltage,
noise
so
do
To
= Af.
O =f
band
a
by
represented
noise,
the
frequency
the
in
Pyy
power
frequency.
average
the
from
eliminated
is
First, frequency
of
concept
the
need
not
does
that
capacity
requires a definition of transmission
functions
orthogonal
DESIGN
SIGNAL
and
replaced
defined
as
The
the
by
may be generalized.
acrossaresistor
notation
V,(t)
functions
the
will
f(j,9)
normalized
Let
Rg which
in
be
used
the
same
VGj.t)
voltages
7
.
4
-@/2
J
follows:
[tC,e)2Cc,0)ae = mtr f VCd,t)VCk,t)at = 1; Bee
Q2
is
T/2
(4)
mt
“172
The coefficients of (5.1) are represented by normalized
voltages using the notation Aia es
Equation (5.1)
then
assumes
the
g(a) eV
following
form:
Vat) = Dax63)2(3 48) =v" S v,CaVG,t)
65)
j=0
;
a,(j)=V'
af
@2
i
V,(3) = f ,(e)£(5,8)ae =m |
Va(t)VC5 tat
-O/2
-T/
Let
the
voltages
grator
and
integrated
tage
at
the
time
V"'V,(t)V(j,t)
#T'
from
-#T'
equals
to
-V,(j)
be
applied
+4$T'.
if
The
the
to an inteoutput
time
vol-
constant
6.11
of
MEASURES
the
The
OF
BANDWIDTH
integrator
quantity
derived
is
247
chosen
equal
VE@IIZRa , with
from
to
the
dimension
unit
of
of
power,
time
7.
may
be
the
output voltage. Let Vi (tyaf (5) be squa—
red, dividedby TR,, and then integrated from -#T' to $1':
4
T'/2
-
-
co
mn ah Wi CeR erate
;
Since
noise
the
left
sample
meaning.
oy v2(3)R
=
hand
(6)
side
is
g, (8), the right
Acertain
term
the
average
hand
ViGao/Re
side
inthe
power
must
sum
of
the
have
the same
represents
the
average power of the component j, or f(j,9),
of the noise
sample g,(6). Averaging the term Vei07 8 over
1 samples
of noise g,(6),
L
BP, = Xvi(G)RI> = Lim’ t >)veg Re,
yields the average
samples
the
or
same
makes
no
case
one
over
j.
m
of
for
power
“the
any
the
the
average
of
average
average
one
j of the noise
distribution
component
the
the
The
case
of thermal
which
replace’
Furthermore,
times
of the component
noise".
j in
difference
may
Pj
(7)
is
over
of
m
of
noise.
V,(j)
is
Hence
it
averaged.
In
} by
average
the
components
this
equals
component:
Te
C20,
URa?1 = mV;2 (j)R]> = mP,
.
-
:
(8)
-)
j=
Mae
value
of
system
{f(j,9)}.
by the
functions
have
the
and
(5.3),
yield
however,
can
from
is
quite
independent
of the orthogonal
of the noise samples g,(6)
Multiplication
of an orthogonal system {h(j,@)}, which
intervals
orthogonality
same
f(j,9)
Pj
be
expanded
into
Cy
instead
voltages
(5.4)
to
a
as
series
of
the
functions
according
to
V,(j). It follows,
(5.12):
(9)
CGha = <1, (a )Ra>
and
e
of time
‘This exchang
the ergodic hypothesis is
ensemble average
satisfied.
requires
that
6. SIGNAL DESIGN
248
functions.
cosine
and
sine
of
independent
is
that
a parameter
by
equation
that
ean
OfSAL
replacement
the
to
turn
now
us
Let
(2).
Pree
of
replacement
the
about
investigation
the
finishes
This
,9),
al
f£(0,8),f<(1
ogon
functions
2141 orth
e
the m =
Assum
a
through
£,(1,0),---,f¢(1,6),f,(1,8) may be transmitted
communication
-++ = 6 = 4.
Fourier
Consideras
= 1, £,.(41,0)
0's 8) b=
These
Sine
terval
ched
case
the
of the
functions
=
V2cos anid,
£f,(1,6)
= ¥2 sin2nid,
Vowe ky 0 Sebea
and
by the
cosine
elements
(10)
are
orthonormal
inthe
in-
substitution
= 6/€
as
in
section
1.21:
(11)
,04) = \2cos
f,(i,@')
The
2ni(6/5)
-$§
duration
increased
from‘
= V2 sin2n(i/e)e
3058
of
to
unit
of time
be
= Ve eos aniseJG = FayeG7
= V2sin2ni(0/é)
4 =e0' ss,
index
6@'
Yo= (0/870)
C1
must
special
interval
-4 = 6 s $ andundefined
outside. Let them be stret-
PCOVe
per
orthogonality
the
series:
£(0,0)
=to5
during
channel
the
€.
shall
6
orthogonality
The
numberof
remain
transmittedinthe
i runs
from
1 to
= £,(i/€,8)
functions
constant.
interval
k,
interval
where
been
transmitted
&(214+1)
€-times
k is
has
as
functions
large.
The
defined
by the equa-
tion
(21+1)&
Let
Sine
kee
§ approach
elements
tions
runs
= 2k+1,
with
from
vy,
difference
Av,
is
AV sho
given
=,
the
periodic
frequencies
= 1/§
The
let @-4+ 1722 )-
(42)
infinity.
The time limited
become
the
OC
to
v,-v,,
v,
i/€
= k/€
denoted
=
sine
vy =
since
as
the
fT.
and
The
i runs
sine
and
cosine
co-
func-
frequency
from
frequency
1 to
v
k.
bandwidth
by
Lim
E—oo
Uva
ee
a
k/§ = #(214+1).
C13)
6.11
MEASURES
secltd
is
unit
ted
oT
the
T.
k of
The
orthogonal
bandwidth
orthogonal
during
of
ihe)
numberof
time
number
the
elements
and their
infinity.
instead
Af
The
fréquency
of
On
sure
sine
the
of
transmitted
and
cosine
the
concept
tical
section
of
domain
the
occupied
interval
occupied
inthe
finite
of
by
cosine
a
that
be
m/T
ele-
interpreted
is
of
as
a mea-
that
reference
a
the
canbe transmit-
functions
without
m/T
measure
can
to
be
sine
generalization
Af
and
m/T
every
goes
of
beyond the grea-
cumbersome
function
for
theore-
occupies
time-frequency-domain.
=
function
that
tz
Fig.101b
only.
= f = f,.
by truncating
t
there
since
use
bandwidth.
that
a
= t
f,
only
Itisoften
only
f or
may
a frequency-limited
interval
number
< 6 s ee
may
and
the
The
an
in-
hatched
shows the section of the time-frequency-—
by
t,;
is
time,
the
-#€
one
of
transmit-—
if
sine
per
T.
orthogonal
Hence,
the
section in Fig.101a
Af
of
functions
m/T
of
investigations
finite
time
of
m/T.
to
unit
between
of
measure
elements
orthogonality,
number
frequency
generality
cosine
the
hand,
functions.
The difference
ter
transmitted
a
(14)
unit
of
is
According
cosine
number
per
of
bandwidth
other
the
or
Af
interval
per
and
or
orthogonality
whichis
transmitted
number
ted.
of
functions
Av
sine
interval
approaches
ments
249
S.4nm
aia
m/T
OF BANDWIDTH
The
them
are
no
differs
function
hatched
and
zero
in
section
that is non-zero
areas
arbitrarily
time
from
shows the
can be made
at
some
value
frequency-limited
functions.
It
of
shown in section
been
has
ting
the
replace
a
system
Walsh
= t,
which
and
-
are
zero
Consider
orthogonal
outside;
unnecessarily
It
by
it temp-
a time-
distinguish
better
to
introduce
a system
of
functions
is
in the
j = 0,
class
thereisa
This makes
frequency-domain
functions.
'time-function-domain'.
{f(j,9)},
st
of
time
But this would
sequency—domain.
the
functions.
time and sequency-limited
to
that
1.33
1,
finite
interval
--edryeeeda2ee0-
t,
Let
signals
be
composed of functions
j;
to
j2.
from
along
the
abscissa
and
along
the
ordinate
of
The
system.
coordinate
a cartesian
j/(t2-+,)
=
j/T
or
j
indices
the
time is plotted
the
Fig.101c,
to
According
j running
index
the
with
DESIGN
SIGNAL
6.
250
signals considered occupy the hatched section of this timeThese signals are exactly time and "func-—
function-domain.
limited.
tion"
Let
us
investigate
bandwidth
Ap
and
the
m/T.
The
system
CECOLG))
tei. Gus te Ci 70)
shall
be
orthogonal
crossings inthe
rations
tions
apply
quency
Au =
eh
uw has
2i
for
to
the
are
be
(Ue-Wa 2) =
ne
oo
S
Sri.
of
(13)
ee
Comparison
normalized
number
of
the
is
again, but the
the
replaced
and
functions
that
can
be
duration
1.
(14)
with
(16)
bandwidth
of
the
of
c) The
frequency
duration
normalized
normalized
fre-
(16)
Av
and
is
system
transmitted
canbe transmitted
terval
Equa-
by Au:
in
(17)
shows:
a measure
{V2cos
is
a
systems
which
the
isnot
applicable.
ber
of
of
concept
functions
time
measure
system
of
(FEC yy
in a normalized
time
in-
1.
bandwidth
Af
=
Av/T
isa
special
case
the sequency bandwidth Ap = Au/T, but m/2T is aneven
general measure of bandwidth since it applies to all
plete
of
enve,
a normalized
b) The normalized sequency bandwidth Au
the number of functions of the more general
that
conside-
elements.
k/— = $(214+1)
frequency
interval
f5(u,8)}
number of zero
The same
cosine
for
(15)
(17)
V2 sin 2nv6}
of
aa uuman
interval.
and
sequency
functions
equal
obtained
ee
00
i+
shall
substituted
Av
Se2
of
between
Ack sbaa=e
sine
Furthermore,
a) The
the
and
v.
hk
Sy
«iia =
orthogonality
as
(13) and (14)
sequency
connection
orthogonal
functions
of
sequency
in
m/2T
equals
"one
transmitted
per
its
including
present
half
unit
the
time
of
more
com-
those
to
definition
average
1".
num-—
6.12
TRANSMISSION
CAPACITY
Corl
Fig.101
Time-frequency-domain and time-function-domain.
a) section
of the time-frequency-domain occupied by atime
limited signal; b) section
of the time-frequency-—domain
occupied byafrequency limited signal; c) section of the
time-function-domain occupied by atim
and function
e
limi-
eed
cignel..
f,—f;
= Af;
t,-t,
= 7; j,-(j,-1)
=n.
6.12 Transmission Capacity of Communication Channels
Consider signals F,(8) that are composed
of the system
Sieruncyi oie 21200756 .7.(1,0),f.(1,9)}°
“orthogonal: in the
interval -4 = 6 = #.
co
Fy(6)
= a,(O)f(0,8)+ >) Lacy (2)F, 4,9 +asy(A) PGi, 82)
Elke
Let
Fy(@)
Then
and
6(0O)
ficients
for
be
assume
£.(i,8)
time
i=]
Ms hot .o
transmitted
for
the
time
f,(i,@)
through
a communication
being
channel.
that the functions
are only attenuated
and
f(0,6),
delayed
by the
during transmission.
Using the attenuation
K(O),
the
(18)
K,(i)
signal
at
and K.<(i) of section
the
coef-
1.32,
one obtains
receiver:
F,. (8) = by(0)£T0,0-9(0)] + > (dey (i)£eLi,0-8(0)]4
(19)
Dsy (ifs [1,9-9(0)]}
by (OyeK(0)a,(0),
The
receiver
characters
deviation
Signals
as
be 744
shall
F,(8 ) was
criterion
Fyy(8 ) must
similar
as
aK c(i acy (1),
determine
transmitted.
shall
be used
be
The
Loe,
one
of the
forthe
to the received
possible
which
Ch,
possible
least-mean-square-
decision.
signals
integral
Sample
which
at the receiver
produced
to decide
is then necessary
lest:
which
Way Wi) she
Byee(O).
I(¥,x)
is
are
It
smal-
Bor 1/2
DESIGN
SIGNAL
6.
ee
2
8o=1/2
tn
cs
rca
Let
us assume
exactly
equal
the
to
inteeral
difference
AI
possible
(19)
to
and
GORE
is
(21)
zero
by
The
at
afinite
to
the
integral
least
cannot
be arbitrarily
due
could
be made
signals:
then zero.
from
determine
(20)
# yib8)
= x
I(¥,¥)
differ
functions
received
for
) # x must
Pe
sample
the
Fyw(8) = Pye(8)
The
Pd
I(¥,x)
AI.
small
The
for
minimum
sinceitis
only
It follows
from
difference.
orthonormality
of
the
system
pra GU CW Mee Gilet we
TCV5x) = Coy(0)-bAO +> (ib. vba
(22)
iz]
+ [bey (i)-bey(i)]?} = al
Consider
in only
one
of
those
one
the
of
signals
the
following
feyc-ny
0)
conditions
minimal
values
from
(19)
|ay(O)-ay(0)|
| acy (2 )-acx
ay(O),
tween
then
To
IIA
Pj
IIA
+A
Aa(O),
and
IV
agy(i)
given
by
or
Fy_(6)
agx(i);
hold:
Aa,(i)
must
and
erece
Aas(i)
by which
-A.
CAT)
Ae, (i)
and
The
x,, Ce
PARCO)/ (CAT
the
differ
at the transmitter
(23):
= aa(O) = (ar)? /K(0)
(i)|
and
must
from
agy(i)
atbeptig—be (ipl
agy(i)
number
ene
ee
(24)
/E CT?
(ar) /K,(4)
ecg Gime, Gil ei heete)
Let
differ
(23)
coefficients
of two signals
follow
that
a,(0),
rar,
fila ycber ipl. Siht,
The
Fy_e(@)
coefficients
be
of
restricted
possible
to
values
coefficients
is
ret
4
%
2AKe(L)/(AT)41,
be-
(25)
yj
41.
Ss 2AK (4) /(O1)"?
6.12
TRANSMISSION
The
'ones'
on
CAPACITY
the
right
possibility
that
The
integers
largest
must
be taken
for
Bose (ee
small
that
ee
No
the
let
0, 2hal0O),
they
are
+#Aa(0),
> slo,
and
TyS|
be
eyes naHey
transmitted
or
permissible
the
zero.
(25)
values
it 7a
+2Aa(0)....
+3Aa(0),...
Kei);
relations
account
inequalities
are
=si.
us
the
ayo)
> 1.
iirc er ores)
satisfy
into
have
the value
ro; - The
r,
can
take
may
and
following
information
sides
r,;
toni
tOr
fication
that
r,,
even
for
is odd;
hand
the coefficients
coefficient
of the
AO)
for
i>
714 sbe- se
hold:
ISS
with
£6150), .tori
(26)
a
single
function
> lcesPor
simpli-
put
eet rane ol",
where
1 is
mitted
Mae
(27)
called
the bandlimit.
beyond
the
eoetficients,
Gl so),
genal.
leo
This
nuation
bandlimit
but
the
of at least
two
functions
must
type
Information
of
be
changed
processis
trans-—
different.
f£,(i,8)
and/or
ro, obtain.adifferent,si-
transmissionis
increases
so rapidly
canbe
impossible
if the
beyond
the bandlimit
atte-
that
the
condition
co
be Ai)
ea,
yt)
(28)
ys 81
tole]
is
satisfied
for
The
number
mitted
during
by the
product
any
pair
y and
of distinguishable
a time
yj.
can be transthat als
sign
al
of duration
interv
T is
then
given
l
T
Tre: esi:
i=]
mission
time, orthe transper unit of ted
transmit
capacity of the channel, isthe logarithm of thas
product
divided
The
information
C = mig r, +
by T:
l
2 (GN
ee Leen
(29)
6. SIGNAL DESIGN
254
(25
forty
ep=e rae
air
Geeen.ot
@%
(29):
from
follows
It
1, arean—
1.3
and £,(1,8),
£.(1,8)
£(0,0),
equally.
tenuated
aatiger
If
and
the
(30)
system
cosine
from
ee
ECON aR eCe yree eed
All. functions
(29). It follows
of
case
special
considera
Let’ us
of
functions
functions
one
used
may
are
the
substitute
periodic
Af
from
(14)
sine
and
obtains:
G=
2QAf ler=
This
formula
although
will’
Af ler
be
has
it
the
was
shown
(31)
structure
derived
in the
of
under
remainder
SHANNON's
different
of
this
formula
assumptions.
section
that
(2)
It
r in
(30) and (41) is replaced by (1 + P/P,, )? if the same
assumptions are made as in the derivation of (2).
Consider signals F,(@) composed of r functions f(j,@).
The orthogonality intervalis -4 = 6 = # or -4T st = $1.
r-)
ByGh ta l)
cas jat Gd. 00
(32)
j=0
aGiRcey
The
=
UECO
integral
Ley ye
of
ery Get
re(6)
era gas
yields
the
Pisa
average
s
powerof
the
si-
gnal:
1/2
335
= Py
F2(t/T)at
= 4 f
i Fe(0)d0
-1/2
Sara
i
“¢s)
V2.
4
=a -T/2f >j=0 a2(ge2G3,t/mat = =S' a2(3)
Instead
Fy(8),
one
cartesian
the
of
unit
the
unit
of
r
representing
may
signal
by
represent
it by a point
signal
vectors
space,
e;,
coordinate
vectors
a
equals
J =
axes.
the
according
O...r-1,
The
a
time
function
in a r-dimensional
to
point
section
in
the
2.11.
Let
direction
square
of the length
of these
integral
of the square
of
f(j,6).
6.12
TRANSMISSION
CAPACITY
ape
2
T/2
RE 2,8 de = e2" = 7),
f £7Cj,t/P)at
-1/2
= 2 = Te? (34)
=1/2
A signal
is
represented
by the
following
sum:
r-1
Fy= >, ay( ie;
(35)
j=0
F, rather
and
than
F,(8)
F, represents
space.
Its
is
a
distance
writteninvector
certain
from
the
r-1
point
in
origin
the
is
r-1
Meee sCE j>,=0 a2(j)e2]° = [1 j=0Saat j)]
A sample
of
thermal
representation,
r-dimensional
Dy:
e=r(OP,7) « G6)
noise,
g,(@) = Sia,(3)t(i,8),
(37)
=0
may
also
be
represented
by
a vector:
co
G,=
>, aalide;
(38)
j
20
According to (5.24) and (5.25) only the
or e; that occur in the signal
(38). Hence, g,(9) is divided
gi(6);
the
part
gi(9)
may be
r-1
are important in (37)
into two parts g}(@)
r-1
j=0
j=0
fo.)
gf = Ya,(ae,
Ma (gee)
Diraut
distance
and
and
ignored:
jer
The
f(j,9)
(39)
gh = Yialdre,
gi(8) = Yia,(j)t(5,8)
gi,(8)
r components
" =
of
the
point
9; from
the
origin
equals
Paez
fal
Ta,
(i))
Dj):
(40)
j=0
The
P,7
average
of
power
; the indices
r
many
and
noise
T indicate
samples
g
is
denoted
by
the number of orthogonal
6.
256
U
under
very
the coeffiofon
distributi
the
that
assumptions,
general
5.12
and
5.11
sections
shownin
been
It has
the
(41)
ateheya Ca)
lin
=
Pe
of
duration
the
and
sample
of the noise
components
orthogonality interval:
DESIGN
SIGNAL
cients aj(j) is the same forall gj, if the g,(9) are samnoise. Equation (41) may thus be rewritten
s
of thermal
ple
follows:
as
The
L
j=)
average
average
eo
The
over
over
} for
j for
a
=
Pe,
j
may
be
t =
C
>)a4Le(i)
(44)
jel
of
gi, from the origin approaches
of
of
(40)
and (44)
r.
in
The
points
signal
space
a r-dimensional
The
average
Tges
Rieeamer
of
shows
that
the
representing
arbitrarily
sphere
power
with
thermal
close
radius
t signals
distance
Di of
(TP ,,)"* for large
to
noise
the
CaP ey )>
F. follows
irom
This
equation
may
the
distribution
tistically
eins“
=
be
rewritten,
if the coefficients
for
all
(39):
a
J
have
are
surface
eae
te Sia es
cut
same
the
r yields:
ald points
located
by
(43)
Comparison
values
replaced
}:
L
iin
lim
=
fixed
fixed
2(4
2,84
Coy
substitution
din
Lim
(42)
ay
pa, (seer
dime.
l[—oco
Deitew=
rT
j andif
they
are
ay (35)
sta-
independent:
pe
L
2,8x(d) =
2
G
dim
q
= 518263)
L
A
(46)
6.12
TRANSMISSION
The
CAPACITY
substitution
1. =r
oe
Comparison
with
(36) shows that
all points Fy are located
close
to the surface
of a r-dimensional sphere
(TP)?
for large values of r.
arbitrarily
radius
A signal
Pome
yields:
T
ee
with
2o7
with
an additive
pre Scole Cm Diveuine
noise
sample
g
Superimposed
polit
pai
Fy ms gi
=
Ss faycg)
cts a,(j)le;
°
j=0
The
points
close
for
to
F+g,,
the
large
% =
surface
values
1,
of
of
2,
a
--.
Piya fascj4e, CIs)
(ee
with
located
arbitrarily
radius
VECPLPs,
r:
r-1
dimes
are
sphere
r-
-<11in
j=0
{7 STO
Weta
eee
j=0
= lim ¥OCP+P, 7)
One
may decide
disturbed
Signal
points
of
points
is
equal
this
unambiguously
which signal
signal
F, +9)
is
at
having
to
the
if
least
this
the
2E0P, 72)? -
minimum
possible
the
radius
R
The
distance
number
number consider
the volume
having
distance
of
(48)
Fy
caused
between
possible
from
number
one
signals.
To
the
any two
another
determine
V of ar-dimensional
sphere
[6,7]:
mia
Y* Ttaret)*;
The
volume
R-e
R and
n
V-
between
approaches
most
closeto its
ber
17.
of
r
r
2
Shae
Ve = fGreiHence,
a)
of
the
points
Harmuth, Transmission of Information
concentric
for
large
"
mtl2
spheres
numbers
3
= FG@r-) Rta
volume
surface.
signal
two
of
A good
is
the
r the
é
with
radius
volume
r
4
C50)
= =) 1 eV
r-dimensional
sphere
the possible
of e
estimat
obtained
V:
by dividing
the
is
num-
volume
of
with radius
ine
PEP
rate
thus
may
see
functions
that
interval
of
infinite.
average
orthogonality
ratio
P/P,;
power
may
tion
(44)
for
is
F,(6),
The
shows
beyond
the
number
must
-3T
of
of
approach
average
be
The
finite
signal
or
to
the
Equa-
that the average
noise
infi-
T;
(47)
according
signal
all
be
orthogonal
infinity.
= t = $T may
the
also
(52)
finite orinfinite.
finite
for the average
becomes:
lim sy 1e(1+P/P,;)
which
noise
nite
grows
r,
in a signal
limit of the
The
T.
duration
the
has
F,(@)
(51)
lim
plg(1+P/P,,)"? =
r=oo OL
One
of a sphere
by that
/2
B/Pe
transmission
error-free
C=
me Gers
signal
Each
V2
(TP,, Nor
rt r/2
tte
VTC P+P, 7)
radius
with
sphere
the
DESIGN
SIGNAL
6.
258
to
power
bounds
P.
if
T
The
is
power
the
P,;
same
transmission
finite
and
is
holds
true
capacity
P/P,,;
is
not
Zero.
Consider
gonality
ple
two
let ususe
that
special
interval
asystem
vanish
k =
system
of morthogonal
ted
-1
tions
an
orthogonal
interval
integer.
may
f(j,9).
These
functions
interval
#kT'-T'
functions
= rj;
The
average
are
because
have
s t = -$kT'+2T'.
{f(j,9-k+1)}
is
m,
vanish
Continue
which
orthoexam-
{f(j,8)}
s t s -#kT';T',
furthermore,
that
shape
as
noise
are
shiffunc-
outside the interval
this
way
until
in
power
P,,
is
the
in
the
vanishes outside the
The total number of orthogonal
then:
r/2T
non-zero
a
the
k, r = integers
factor
T'
is
functions
-skT'
same
the
For the first
{f(j,9-1)}
the
reached
=t s $kT'.
produced
forwhich
Consider,
functions
which
system
only
is
(52)
infinity.
and
—ekT'+T'
mk
m
the
where
by
of
outside
T/T'
casesof
approaches
any
(52)
(53)
becomes
becomes
one
durationof
of
the
P,,
the
mk/2kT'
= m/2T.
because
m functions
k time
intervals
orthogonality
The
and
interval.
6.12
One
TRANSMISSION
obtains
f
Ca=
from
apr
to
wait
the
signal
the
ends
As
(52);
1E
derivation
have
(14+P/P ak
of
this
F,(6@).
of
the
Part
example
inthe
interval
Dy artactor
of
The
number
St
= #2T
It
of
is
=a]
is
from
replaced
Signals
and
cosine
cies
in
C=
=
= tri
Some
the
620)50031),
mission
variable
signals
magnetic
also
has
of
must
It
are
in
available
at
m
=
21+1
sine
According
to
for
1 in
per
(12)
orderto
unit
functions
of
in
and
cosine
elements
one’
keep
time
the
are
has
the
to
nun-
constant.
interval
-éT
a)
=
the
factor
Af.
The
follows
reP,;
that
band
the
is
average
noise power
in
Fyyy;
(52)
= Fy,
O = f = Af
noise
received.
r/2T
samples
SHANNON's
since
andall
with
the
sine
frequen-
formulais
thus
am At le 14P/P,,))
(56)
ey
be exercised
functions
They
hold
with
the
one
to
the
transmission
corresponds
This
represented
in interpreting
the formulas
(54) and.(56).
by voltages
travelling
independent
positions
formulas
to each
apply
ling ina
wave
of
or
variable
in
guideindirection
trans-
independent
An
of
electro-
z-direction
or-
and
the
However,
z,
the
the
but has two
t only,
polarization
them.
for
currents.
space
infree
the
for
thogonal
12%
of
orthogonal
time.
the
not
(52):
€52),
wave
does
information
1.
frequency
iySe
care
the
These
lim ser Ls Cy ry
Af
one
= t = #1.
that
m/2T
band
from
(54)
= me
components
this
obtained
7)
by:
by F,,,;-
occupy
that
-4T
§ >
(14)
by mé/2eT
shows
consider
orthogonal
given
(1+P/P,
informationis
transmitted
= (2147)€
follows
placed
the
k = §(1+$41/2€)
functions
are 2S
intervals.
k time
second
substitute
=
long
to obtain
of
a
sureuwcned
kn?
formula
infinitely
elements
ber
eo
mk
oa
The
CAPACITY
the
vector,
ina
wave
variables
travel-
x
and
y
better
way to increase
this
numberisto
use
variables
are
are
capacity.
and
used
this
in
prin-
the
A possible
that
channels
of time
the
that
show
functionsis
orthogonal
a guide
as
but
(30) and (52)
transmission
telescopes
Optical
nates.
can
exis-
what
of
transmit,
and
for-
SHANNON's
limit
the
as
Equations
which
signals
Hence,
channels
channels.
number of transmittable
cipal factor determining
mit
them.
only be viewed
communication
ting
to
not
should
mula
show
modes
as
up
freedom
of
apply to each
formulas
the
of
degrees
additional
These
variables.
independent
as
t
to
addition
in
appear
may
DESIGN
SIGNAL
6.
260
space
transcoordi-
way.
6.13 Signal Delay and Signal Distortions
Several
simplifying
derivation
nation
of
of
the
these
assumptions
transmission
assumptions
have
been
capacity
will
be
made for the
(29).
The
investigated
elimi-
in
this
section.
Let the functions
f,(i,@)
by 6¢(i)
and
6(0).
functionsin
The
and
8(0)isno
more
general
fco(i,®@)
and
6>5(i)
rather
longer
definition
with
are
sample
time
the
than
by
a
common
delay
then
no
longer
orthogonal
delay
time
of the
of asignal
fs5(i,8@) be transmitted
tions K(0)£[0,8-0(0)],
§-8>5(i)]
f,(i,8) in (19) be delayed
are
(19)
the
and
then
functions
difference
between
Let
£(0,6),
6
delay time let
them
be
f-(i,6)
= O and
signal.
individually.
Ke(i)feli,6-8¢(i)]
received.
the
time
For
a
f(0,8),
The func-
and K,(i)f,[i,
crosscorrelated
and
f.5(i,@).
absolute
crosscorrelation functions yield the delays
maxima
The
of
9(0), 9(i)
and @,(i). The values of the maxima yield the attenuation
coefficients K(O), K.(i) and K,(i). Using these coefficipak - may derive
a sample function Fy,.(@) from F,(@)
L718
y's
BY Coy teak
Ove, (O)6COye
mes 2, (K(i)aex(i)tc (458) + (57)
+ Ke(ipesta
lee Cantal
The
received
signal
F,,(6)
has
the
same Shape,
but
6 must
6.13
be
DELAY
AND
replaced
hand
side.
end
ey. (8),
er
DISTORTIONS
by
Let
9-6(0),
the
261
@-@,.(i)
or
@-6<(i)
crosscorrelation
on
the
function
of
right
as Cale)
ee EY (8-0 de = £69"),
co
(58)
—oco
yield
an
absolute
maximum
for
a certain
This
value is defined
as the delay
time
of
ceiver
to
the
which
define
for
signal
propagation
average
signals
to
to
or
the
not
define
6y,
0'=
6x.
propagation
re-
itis advantageous
independent
values
of
known
at the
arrive,
time
the
Fy(@)
time
Since itis
signal
is going
a
instance,
ferent
Fy(6@).
value
if
of
y.
there
a propagation
One
are
may,
R dif-
time
6,:
R
a = 7dox
(59)
ed
The
propagation
a statistical
approximation
Signal
The
delay
but
side.
only
The
with
the
in
has
function
by
6-8,
a series
Towing
equations,
which
fen.
(i,0-6 -(1))
and
the
or
group
on
the
shape
delay
of
(57)
on the
or
if
9
right
too,
hand side.
Let
andf,[i,9-6,(i)]
{f(0,9-6, ),fc(i,6-
analogy to (2.26)
v = 9-6.,
is
first
has this shape
right
system
in
in
[2].
6-@.5(i)
Fym(8)
of the
obtains
in
of
f,[i,6-9,.(i)]
6,),£,5€i,6-8,
)}. One
information
identified
in optics
6-8c(i)
f[0,6-9(0)],
be
concepts
F,-(@)
sample
carrying
can
defined
6-9(0),
6 must
be replaced
expanded
a signal
which
signal
by
the functions
be
of
originally
received
is replaced
hand
time
variable
the fol-
f,) = f[0,9-8(0)],
£- = £.{1,0-0,(1)]
is written
for
abbreviation:
co
fy
= K(0,0)£(0,v)+ >, [K(O,ck)f£,(k,v)+K(0,sk)f,(k,v)]
(60)
k=]
Pree =k
(1,0 )f(0.v)+
;
fo
S" [K(ci,ck)f,(k,v)+K(ci,sk)f£,(k,v)]
k=]
= K(si,0)£(0,v)+
>, (K(si,ck)f,_ (k,v)+K(si,sk)f5(k,v)]
k=1
Let
Fy-(0).
these
The
series
first
be
term
substituted
of F,,-(8)
has
into
the
the
formula
following
for
form:
DESIGN
SIGNAL
6.
202
(61)
[K(0)a,(0)K(0,0) + 5 [Ke(iacy(i)K(ei,0) +
} Ky CHacy Cl RCs, 00) 200, 0—oe
t=]
correcting
tion
£(0,0-0,)
Let us further
are
not
but
only
also
£(0,0),
g.(i,6)
K(0O)a,(0O)x
Fyy(e)
and
Fye(8)
(61).
identical.
then
are
of
is obtainedinplace
that
so
measurement,
of
accuracy
the
within
crosstalk
the
compensate
that
circuits
distor-
devise
to
ble
in principle
possi
is
It
efficients.
co-
the
between
crosstalk
or
interference
is mutual
There
assume
that
£(0,8@),
fc(i,@)
attenuated
and
delayed
during
transmission
invariant
distortion.
suffer
a
linear,
time
and
fs(i,8)
f-(i,9) and f,(1,8) are transformed into
and gs(i,@) according to section 2.22.
functions
f£(0,8),
vidually.
The
g(0.,8),
Let the
fc(i,8) andfs(i,8)
be transmitted
correlation
indi-
functions
of the received
func-—
tions
g(0,8@), g,(i,9)
and gs(i,@) with sample functions
£(0,;6), £.4,6)
and .(1,0) is produced. The time farce
between
8(0),
their
6,(i)
attenuation
and
be
and
6,(i).
K.(i) =K(si,si).
6, and
with
(59)
distorted
expanded
maxima
The
coefficients
constructed
yields
the
absolute
= K(0,0),
functions
these
coefficients.
g(0,8),
of the
resulting
same
(60)
and
the
K,.(i)
Sample
functions
in
yields
Fiy(6)
defines
a propagation
inaseries
those
8 =O
values
of the maxima
K(O)
f,(i,9-6,
)}. The
as
and
the
of
(57)
may
time
9,.
and
(58)
Now
let
gs5(i,8)
be
{f(0,6-6, ),f-(i,6-6, ),
expressions
the
yield
= K(ci,ci)
Equation
g,(i,8)
system
delays
same
are
formally
conclusions
the
apply.
6.2 Error Probability of Signals
6.21 Error Probability of Simple Signals due to Thermal Noise
Consider
the
transmission
of teletype
presen
of thermal
ce
be
computed
tection.
The
for
The
noise.
several
general
of
inthe
error
shall
probability
methods
form
characters
such
of
of
transmission
characters
and
de-
represented
6.21
ERRORS
by time
DUE
TO
THERMAL
functions
NOISE
263
is:
4
F,(8) = 2 on GE GGG
eer a age
The
orthonormal
functions
6 = 4.
T is
usually
150
or
167
or
ms.
The
interval
character
coefficients
a,(j)
have
g,(@)
(63)
co
Ba(@) = 2 an(5)F(5,8),
as(5) =
aC )t(3,6),
energy
from
fe ,(oF(5,8)a8
aCj) = a,(3) + a,(3)
j=0
system.
1/2
-1/2
es
The
of
all
Using
sample
the
characters
is
the
same
in
a
least-mean-square-deviation
functions
balanced
criterion
F,(@),
4
Pee
of
into
FY Co)
BOO) = Fy(6) +g, (0)
ee
is
system;
A sample
system.
the character
noise transforms
-% s
which
forabalanced
-a,
and
+a
O foranon-off
and
inthe
of ateletype
(62)
F(6@):
signal
the
duration
thermal
additive
are
+1 and-1,
+1
are
they
the
100,
values
the
f(j,9)
ea
Yai
)tC) 8),
(64)
j=0
one
may
decide,
according
to (5.25), which value
of
y will
give
(65)
Ss a( jay Cg)
j=0
maximum
its
as
sign
same
occur
value.
the
All
coefficients
coefficients
z elec,
j20
If, for example,
(65)
the
if
must
ay(j)
= +a
or
a(O) had the
d
be larger
woul
for
-a;
y=
opposite
the
the
have
is
maximum
y = xy. The sum (65) then has the following
for
4
sum
a,(j)
a(j)
to
value:
¥
sign of ay(O),
the
with
the
character
F,(8)
6. SIGNAL DESIGN
264
for
a, (k) = ay (ik), k=15..4
than
+|a(o)|+ |a(1)| + a2) + faC32 + aC,
ven
= -ay(O),
a,(0)
coefficients
Fy(@):
- Yas )ay(3 y={
-|a(0)|+
js
|a(1)| + faC2)| + faC3)| + [a(4)],
to
have
according
must be satisfied,
conditions
The following two
to (63), in order
different
signs
V=X
for
a(j)
and
aie:
a) sig ay(j) # sig a,(Jj)
F
:
(66)
5
a
Bye layCi)
|< laiCd7| 5 equivalent
sig ay(j)
means
In
case
of
thermal
positive
is
% and
the
being
‘sign
of
]
a
Sayld > 4 or
j
ey
<1
ay(j)'.
noise
the
the
probability
of
probability
of being
a,(j)
negative
is also #. Hence, the probability of condition (a) being
satisfied equals 4, independent of the sign of a,(j).
The distribution of x = a,(j)/|ay(j)|
is needed for
the
computation
of the probability
satisfied.
constant.
The
Since
Therefore,
density
bility
since
for
that
the
tion
x
for
function
all
for
one
coefficients
the
follows
from
m=5
have
noise.
and
is a
a,(j).
(5.6)
by
the conditional
k=j.
values
the
being
|a,y(j)|
from
follows
condition
a,(j)
(5.6)
only,
obtained
the
case
of thermal
-a
(b)
same distribution
as
is
of
condition
same
The
of
proba-
j is
1/m,
distribution
Thus the density
func-
(4.5):
re ee
(67)
Cdl/ [exCd)|” = etsfe
Each
coefficient
energy.
oe
has the
or
A. Fromw,(k,x)
ren) @ foe)
x Sey
+a
wa(k,x)
w(x)
k equals
j in
w(x)
x
function
substituting
density
a,y(j) canbe
of
4
Hence,
1/2
SP
2
CD
ay(j)
the
in
average
ee
4
ye
j=0
(62)
is
signal
peer
transmitted
power
P
with
equals:
equal
6.21
ERRORS
DUE
TO
THERMAL
This
result
may
be
generalized
NOISE
(A)
and
solved
for
a’:
(68)
a? = P/m
Equation
for r = @ = 1
(42) yields
+
ah
The
:
mean
2
:
i
square
2
where
of
Eat
0
deviation
Sie PE (Pati
becomes
AP dePe se mP,
is the average
thermal
(69)
= Ay
Mim, 7 aad) = RD
(70)
power
of
m
orthogonal
noise
in an orthogonality
components
interval
of
duration
a
Using
(56)
one
pero
where
+1
is
the
band
The
o2:
of
width
smaller
7)
average
probability
or
rewrite
AL = nor,
P,,
quency
may
of
thermal
noiseina
fre-
x is larger
than
Af.
p(x>1)
+ p(x<-1)
than
p(x>1 )+p(x<-1)
power
-1
follows
that
from
(67)
by integration:
PRES iaexp(-x?/202 )dx
C72)
seal
A=
The
probability
of
(66)
Pp,
=
The
#4[1
-
is
of
probability
the
m
(a)
as well
that
the
the
the
conditions
of (66) arenot
probability
that
they
m = 5 coefficients
they
that
coefficients
equals
are not
ay(j)
satisfied
are
is
at
for
CONG
The
{f(j,9}
tlOnGmOmEse
numerical
does
not
used,
provided
CUulOMS
mont
values
satis-
satisfied
(1-p,)
least
m
; the
one
of
p,:
of error pm
probability
of functions
Tem
(b)
C738)
Dm = 1-(1-p,)" = 1 - ()"[1 + erf(\P72Pat
)I”
The
as
becomes:
erf ( \P/2P ny )]
1-p,;
any
p,, that conditions
satisfied
probability
fied
for
are
eer
Cr Co, ) = 1 — ert C\P/er.,)
m=
these
deeamd
5 and
depend
Af
on
(74)
the
functions
system
satisfy
5). die.
= ei =
poz
16.6
Hz
Curve
racter.
Ost P/E
GLO,
cosine
and
sine
of
ting
according
pulses
consis-
{f£(j,9)}
with the system
shown in Fig.30
equipment
‘ls
Fig
to
the
of
version
early
an
with
measured
The
Af.
and
m
of
values
obtained
were
'a'
points
of
these
of (74) as func-
p, =p,
shows
Fig.102
'a'
for
cha-
per
ms
150
of
standard
apply to the much used teletype
DESIGN
SIGNAL
6.
266
Ben
Fig.102 Error probability p for the reception of teletype
signals superimposed
by additive thermal noise. P/Py;
=
average Signal power/average noise
power
in a 16.67 Hz
wide band. a) balanced system, detection by crosscorrelation; b) balanced system, filtering by a120 Hz wide ideal
lowpass filter, detection by
amplitude sampling;
c) same
as
(b) but
on-off
synchronization
Let
the
:
_
ore
d=
OF
system
ay same
by the
{f(j,9)}
Sinm(md-j)
as
consist
sind
(c) but
of the
functions
'-j
ly
2,43.
(74)
Boy hae seed ear cr
applies
functions
a
is
to
this
system
concentrated
in
too.
the
o7a
The
According to section
for
the
the
functions
F(8)
filter
(74)
coefficients
is
16.6
also
(75)
and
passed
Hz
holds
wide
for
2.14
the
same
energy
frequency
-- = v = fT/m s $ with the bandwidth af=m/2T
ther
start-—stop
noise.
bmg ee Br
Equation
these
system;
disturbed
values
are
whether
F(@)
the
product
is integrated,
andthe
an
ideal
amplitudes
filtering
and
band
= 16.6 Hz.
a(j)
through
obtained
is multiplied
frequency
are sampled.
amplitude
of
or
by
whe-
lowpass
Hence,
sampling
of
6.21
ERRORS
DUE
TO
Bhe=pulses.(75).
the
average
means
dB;
of
the
Consider
power
curve
'a'
curve
is
on-off
conditions
cient
ay(j)
instead
+b
to be
of
denoted
system.
or
must
120 Hz wide
increases
by 120/16.6 + 7.2. This
in Fig.102 by 10log7.2 +
(74)
the
an
267
filter
in
shifted
assum
the values
e
ing
NOISE
A lowpass
noise
a shift
8.58
THERMAL
The
O instead
be
of
satisfied
detected
as
+a
in
The-conditions
power
P =
have
of
(66)
to
the
get
pestliy
block
the
OLecurves:
as O
thus
<
with
the
and
(76)
(> 7
-1
a,(j) may be +b
are
same,
-%b.
values
'c'
were
pulses
of
then
ceiver.
The
“a
The
+b
or
or
O.
but
+a
average
O equals
adjusting
adding
characters
the
receiving
points
block
the
pulses
lowpass
after
of ateletype
The
sampler.
not
filter
of
The
and
the
'd'
depend
teletype
channel
measured
to
the
the
disfil-
a teletype
well
not
only
with
the
re-
curve
shape
ideal,
very
points
'd'
of
and
roughly
hold
pulses
for
were
for synchronization.
strongly
receiver.
means
lowpass
but start-stop
transmission,
the noisy
through
re-
sys-
This
which
have
was
works
measured
be
noise
wide
fairly
did
receiver
'‘'c'.
thermal
magnet
agree
to
by TOtog
2 $5" dB.
by
teletype
the
'c'
system.
Fresi0e
denoted
by
fedto
same teletype
points
(bin
is
P has
fora balanced
on-off
were filtered
by a 120 Hz
as anamplitude
transmitted
and
that
holding
an
obtained
pulsesin(79),
magnet
for
'b'
measured
although
shows
equations
curve
ter and
in
eG
by +#b
(68) and (77)
equations
signals
The
0, or
follows:
of
turbed
the
a coeffi-
of
(77)
shifted
points
the
(67)
replaced
by #P in the
temto
the
may
= P/2m
placed
'c',
for
x\d
since
coefficients
it
Comparison
The
and
be
m
$mb’ and
(4b)?
ay(j)
-a. The follow-
+b instead
a, 3)
7 sig
may be +#b or -3b,
-a
or
order
») lax(4)-#b]< lax(3)], equivalent
and
Dee
+b:
ey sie fe,(i)-8b)
a,(j)-gb
bye.
coefficient
on
the
care
taken
DESIGN
SIGNAL
6.
268
6.22 Peak Power Limited Signals
same.
tudes, are the
The
Ave-
and
probability
error
‘Theoretical
denoted
Let
The
these
curve
carrierisa
Walsh
carrier
would
average
power;
holds
for
binary
a
Only
of
time
same
one
PCM
must
if
curve
be
of
while
no
increased
signal
=
1
the
the
in
same
Fig.103
modulated
signals
and
n
access
activity
factors
activity
factorof
curve
for
now
is
be
channels
multiple
denoted
the
factors
satellite
of
the
the
m/n
the
the
would
activity
activity
3/4
that
the
m/n
or
of
by
a telephony
number
of
systems,
is
or
Walsh
pulses.
large
peaks
although most
The
1/4
least
not
very
The
peak
to
=
obtain
would
all
0.25
ground
the
is
transponder.
in
amplitudes
number
channels.
stations
of
sum
of
the
equal
to
the
A represen-
Fig.103.
binary
resulting
time.
Fig.103.
the
available
the
deli-
the
in
m being
since
shown
used
time.
stations
transmissionof
block
at
amplified
occurinthe
0.05
are
amplifier
by m/n
ground
Using
the
# 6 GB
factor,
satellite
=
in
amplifiers
by 10 log4
power
curve
very
m/n
signals,
are transmitted
is
cosine
yield
channels
the
ratio
Consider
by
if
of a sinusoidal
to
amplitude
the
resulting
tative
power
larger
a carrier.
apply
during peak traffic.
The
low
peak
denoted
transmission,
useful
busy
still
carrier,
The
Very
onto
would
4 dB
quarter
only,
average
ver
for
(73)
pulses.
signals,
power
ampli-
of P/P,, =P./P,
modulated
The
be
system are busy
for
the
the
about
pulses
useful
to
sinusoidal
block
multiplex
limit'
carrier.
have
of
p,
of
limit'.
pulses be amplitude
'Theoretical
consisting
negative
digit is plotted in Fig.103 as function
one
of
Pe.
power
or
positive
having
pulses,
block
binary
that
a signal
of
P,
power
peak
P and
power
rage
an amplifier
a peak
delivers
and
at +E
amplitudes
clips
Consider
power.
average
the
than
rather
power
peak
the
limit
generally
amplifiers
power
However,
probability.
error
the
in
factor
determining
is the
wer
po-
signal
average
the
that
so far
assumed
been
has
It
digits by sine-
signals
F(8@)
are much
have
smaller
6.22
PEAK
43%
POWER
=
\
SIGNALS
4.6%
\
x NG
\
say\,
6B
"
748
WV
va
ye
\
\
ea
\\ |
‘a\
Ss
ea
a \
:
Theoretical
limit
8
\
\\\ \ \\
—\—
10%
015
we
\
e
269
0.64%
al \\32%
Ms
LIMITED
Ofh
:
44 dB
pal bts Be
>
\
g,
Ais¥
EAST
\\
i.
A
Ss
ced
eS
Lh RS
Ws
eS
a
0,05
Nett
\
ee
Why
\\ \
|
ws
\
\
ie
Gale any 1h) Kh) GG oe, 22 Zh 25> 200 30
R/P,p [dB] —=
0
[NEP
eis
Geko i cal,
F(8)—=
Fig.103 (left) Error probability p as functionof Pe/Pyt =
= peak signal power/average noise powerina bandof width
Af=m/2T.
Solid lines:
time division, sine carrier,
actiViny factors 1,.0725
and 0.05; dashed lines: 4 sine and
4 cosine pulses, percentage of clipped amplitudes shown.
Fig-104
(right)
Propability
p[F(98)]
of the
amplitudes
of
the Sie signals Fy(6) being
in intervals
of width 0.1.
Gaussian density function with equal mean and mean square
deviation shown for comparison. ay(0), ay(i), by(i) =+1;
Fy(8) = a,(0) + V2 Di Lay(i)
cos 2nie + by(i) sin 2nie).
than
the
such
a
signal
wide.
the
Fig.104
having
Superimposed
same
results
very
peaks.
mean
an
is
the
amplitude
a Gaussian
within
density
deviation.
an
square
of section
5.24
this density function
signals
consisting
of
Fig.104
are
average
of asum
symmetrical
power
if the large but rare
of
peaks
for
the
interval
Walsh
negative
signals
of
0.1
having
According to the
approximates
function of the
of
p[F(6)]
function
mean
accurately the probability
The
probability
and
of
peaks
shows
pulses.
values
The
of
plots
F(6).
be
very
small
would be transmitted.
The
large
must be limited to increase
the
would
amplitudes
average
signal power.
indicate
32%
carrier.
Walsh
similar
cosine
curves
pulses
[2],
The
following
clipping
while
no
energy
conclusions
transmission
of
lowest
error
if
lower
The
rates,
error
rates,
The
clipper
Fig.98
for
crease
of
the
case
of
constant
D.ROTH
of
increase
equal
error
is
of
shapes
shows
the
increase
the
sine
The
right
solid
by AP_
the
the
Hochschule
yield
better
in
energy
of a pulse
while inthe
power
be
will
be
increased
used.
shown
that
characteristic
dis-
has
results
functions,
required
peak
block
in
serial
required
than
clipping
provided the er-
domain'is
power
pulses
is
are
needed
replaced
transmission.
for
in Fig.104
apply to these
pulse in frequency
shown
4057.
the
APe
less.
criti-
that ade-
Aachen
function
curves
1.
Note
average
error
used
the
function
is
transmission,
the
or
not
density
clipped.
keeps
Walsh
Fig.103.
0.5
is
is
if
to
is
amplifier
the
pulse
factor
control
of
ad-
iscloseto
will
below
rates
other
shapes.
factor
of apulse
5.24
composed
to
produces
energy
the
and
or Walsh pulses yields
and
serial
Technische
probability
An
the
gain
cussed in section
ror
pulses
to
sine
pulses.
from
distribution
transmission
and
using
signals
factor
of
of
Walsh
amplitudes
amplitudes
case
automatic
activity
clipped
of
drawn
activity
amplitude
activity
parallel
compandors
for
be
block
characteristic
13% of the
kept
an
the
of
Gaussian
unchanged inthe
if
binary
if the
percentage
a clipped
sums
transferred
is
may
yield
transferred
is
of
transmission by sine-cosine
exact
cal.
of
energy
by
Serial
Parallel
Little
bands
by a
pulses
Walsh
bands by clipping of sums
sequency
jacent
[1].
frequency
adjacent
carriers.
sine
of
The
single
for
approximately
hold
also
They
clipped.
transmission
for
or
and
13%
4.6%,
0.64%,
amplitudes
of
transmission
DC
modulation
sideband
very
percentage
the
for
hold
curves
parameters
The
noise.
thermal
tive
addi-
of
presence
the
in
pulses
sine-cosine
for
clipping
of amplitude
results
the
show
Fig.103
in
lines
dashed
The
DESIGN
SIGNAL
6.
27.0
some
have
pulses.
defined
Table
typical
'raised
by the
by
12
pulse
to be shifted
The
for
to
co-
equation
6.25
PULSE
Table
power
TYPE
DISTURBANCES
12. Increase APe
of a block pulse
ea
of peak signal power over
the peak
for equal error probability.
pulse
shape
DC block pulse, & foreQs<9G =<) 1/1; .Ovotherwise
raised cosine pulse in frequency domain; rollQrirltacvor 1 = 74
Sei
WS ORs
Sane
raised
Oa
cosine
NWFW
Psprienpular
”
=
f(t/T)
pulse
pulse,
sinnnt/T
muy.
E
mins
the
socalled
used
for
pulse
and
to
T/n
be
isthe
in time
Fig.39
-T/n < 6 < 0
ECl=nt/?).
-Os<°¢
<sT/n
cosnrnt/T
1-Cernt/T )4
roll-off
shaping
factor,
[3],
duration
transmitted
domain,
E(1ant/T),
of
during
n
of
is
the
the
lowpass
number
of
filter
channels,
a block pulse if n of them
the
time
have
T.
6.23 Pulse-Type Disturbances
of
The
error
the
particular
their
probability
of digital
transmission
ditive
thermal
turbances
which
lines.
are
Let
us assume
much
turbed
plitude
are
that
of
for
not
so
are
for
more
important
than
the
amplitude
of
caused
thermal
largest
these
If
the
rise
and
short,
block
pulses of various
will
be
shall
for
during
as
r
fall
pass
of
the
R time
in
r
intervals
of
on
pulse
undis-
through
timesof
addis-
noise
an
the
am-
pulses
length
obtained
at its output.
Let
but
these
duration
T;
pro-
r/R
is
the
during
an
interval
intervals.
of a pulse
occurrence
if
and
function
pulse
one
the
T,
be
amplitude
for
by
a disturbing
the
pulses
used
pulse-type
let
duration
bution
is
Then
amplitude
r/R
disturbances
This
than
limiter.
bability
functions
larger
be observed
pulses
there
the
orthogonal
signal.
sufficiently
equal
if
noise.
telephone
is
systemof
signals
is independent
W,(T)is
written
W:(T) isthe
R approachinfinity;
distri-
r
for
and
the
R arevery
occurrence
large.
of
a pulse.
gq out of
infinite values
tistically
of
the
joint
distribution
W(T, 7; )
=
W(T,t,)
cannot
W,(T)
is
function
by the
defined
then
WCE
product
(95).
be
2
(78)
determined
by
separate
measurement
of
W,(1,) if statistical independence does not hold.
of
RQ
rather
than
R
+
Q measurements
would
then
required.
The
one
distribution
pulse
pulses
is
W, CTW,
and
A total
be
The distribution
independent.
sta-
be
pulses
the
lengthof
and the
occurrence
the
Let
pulses.
the
of
length
the
for
q/Q for
by W,(t, ), the dis-
Q is denoted
of q and
function
tribution
let
and
duration At, of the pulses be observed
Q have a duration At, = 1,- The limit
the
Let
DESIGN
SIGNAL
6.
eve
occurs
occur,
assumed
function
in
an
computations
that
more
W(T,t,)
interval
than
applies
of
when
duration
get
very
one
pulse
T.
involved.
occurs
only
If
more
Hence,
very
it
infre-
quently.
Denote
T is
At,
Pp»
is
by p the
changed
probability
beyond
= T,-
The
conditional
under
the
condition
The
conditional
various
pulse
puted
if
of
suffices
p,
signal
a
probability
that
of
duration
pulse
of
of
error
a pulseof
CULL
probability
shapes
and
an
duration
duration
for
p,
equals
At,
#
shown
m
in
Fig.3
A positive
amplitude
= T/m
for
or
5.
negative
sampling.
causes
=
an
1,
methods
Each
with
pulses
one
m
amplitudes.
the
sampled
p
pulse
m,block
has
com-
pulses
duration
be
detected
with
duration
shall
the
of
the
probability
The
canbe
The knowledge
pulse
change
for
to disturbances.
of
amplitude
disturbing
a
calculated
susceptibility
consist
half of the
of
the
A disturbing
error
be
methods.
of
shapes and detection
character
as
may
RIF
measurements.
a comparison
Let the transmitted
T/m.
iS
detection
W(T,rt, ) is known from
various pulse
Ats
a
by
received:
P, = p(tsl,At, st)?
by
that
recognition
sign
p,
of
= # since
at
probability
least
p,
in-
6.243
PULSE
TYPE
DISTURBANCES
aio
0
mlOn=
0
02
O4
O08
06
125-0 erie
O20
46
mb, /a—>
10
Ats/(1/m)—>
Hig). 105
Clert)
wurbime
repulse
(right)
disturbing
sisting of
Presses
Shown
2.
same block
r,(k)
by
with
curve
sign
As,
1 in
of
in the
amplitudes
of
This
The
ceived
turbed
pulse
signal
has
lest
one
the
amplitude
the
of
the
if
the
2aht,
< al/m
s T/m,
have
an
At,
following
means
and
+a
-a
or
a
since
have
amplitude
the
relation
on
re-
undis-
a
signal
completely.
is
allow.
pulses
inte-
the
the
On
an
zero.
disturbing
be
A disturbing
negative
+a-2a
the
of
if
-a.
be suppressed
of
pulses
that
superimposed
would
of
block
On
disturbing
-a, the smalthe
average,
amplitude
No
error
pulses
+2a
will
is
so
holds:
(80)
72m
conditional
=
-2a
limiter
duration
the
to
disturbing
other
that
<
the
the
will
the
amplitudes
+a
amplitude
amplitude
+a
short
Ata
at
amplitude
occur
18
limited
hand,
amplitude
-2a,
aT/m
be
would
be limited
half
The
sampled.
positive
with
other
pulse
or
can
with
pulse
the
signal
is
O = Ay,
con-
Fig.105.
the
pulses
b, of
signals
interval
gral
the
of
correla-
pulses,
limiting;
determined
by crosscorrelation.
of
bya dis-
“pulsés
of the amplitudes
pulses after amplitude
m = 8 Walsh pulses.
the
caused
block
1.
correlation.
Probability
linearly
Let
Of anerror
sampling;
pulses,
Walsh
3.
tion;
Fig.106
p,
Ars/(T/a).
"duration
amplitude
Fig.3,
as
Probability
“of
2akt,
probability
= 2aT/m
Harmuth, Transmission of Information
or
p,
T/2m
depends
# At,
for
= T/m
Ce)
on
and
=T/2m
At,
4 at
O to
from
disturbing
the
of
positionintime
the
jumps
p,
pulse.
values
larger
for
increases
DESIGN
SIGNAL
6.
274
There
2inFig.105.
n
by curve
of Art, linearly to # as show
is a strong threshold effect at At = T/oan.
the
Consider
Walsh
-a/m.
of
m
functions
such
clip
may
thus
bed
signal.
Let
m
the
aie =
The
at
be apower
+a/m
ct-ek/mja,
of
-a
2.
have the amplitude
smallest
and
-a.
At
the
+a/m
a sum
of
amplitude
limiter
undisturthe ing
chang
a certain
Walsh
moment
functions
amplitude
a
character
have
the
am-
-a/m:
Oy 150
5ail
of
a,
ocurring
(82)
is
denoted
by r(k):
p(k) = 2")
The
the
(83)
amplitude
amplitude
following
b,
a,
probability
-2a
and
+2a
the
after
signal
pulse
may
amplitude
have
limiting
superimposed
one
of
at
+a:
the
= 2ka/m
r,(k)
follows
An
example
of
distribution
The
of having
from
an
amplitude
b, between
(83):
(86)
r,(k)
is
as negative
after
shown
pulses
m-]
for
m
disturbing
amplitude
crosscorrelation
of Walsh
two
(85)
Sonata)
tive
as well
on
(84)
= -2(1-k/m)a
= -a-(1-2k/m)a
The
of adisturbing
of
values
by = a-(1-2k/m)a
or
bk
m
amplitudes
An
without
if m-k
k have
Ks
probability
+a
and
a,
and
the
is
+a
amplitude
plitude
and
largest
The
of
composed
characters
of
Let each function
functions.
or
has
transmission
8 in
pulses
-bosi=
have a Bernoulli
limiting.
of a binary
signal
wal(j,9),
B,(0)~ & Say
(j)wal(d,0),
j20
Fig. 106.
a,(3) = #1,
F,(9)
composed
6.31
CODING
WITH
BINARY
ELEMENTS
eye
yields
1/2
ii F,(9 )wal(1,6)dt
-T/2
According
to
(84)
plitude
of
enc
no)
error
that
the relation
Bers
This
is
Can
ey hee hen SOR
(85),
the absolute
disturbing
Occur
pulse
cannot
its
duration
if
eatt,
< aT/m holds.
eg fe
(87)
of
am-
value
be
the
larger
AT,
is
Hence, p,
than
so
the
very
same
value
the
error
pulses
is zero for
transmitter
plitude
is
are
as
limiter
by
somewhat
probability
pulses
curve
lower
[6].
3 of
than
The
Fig.105.
for
block
thresholds.
pulses
discussed
at
error
disturbing
obtained
sine-cosine
pulses.
conditional
longer
several
results
or
block
calculationis
shown
has
Better
Walsh
for
the
for
probability
and
as
of
tedious
of
2a
smart
(88)
calculation
is
result
The
and
Ae
2 /2m.
The
Pp,
the
= earn
the
if
is
in
the
signal
amplitude
section
receiver
may
6.22,
then
composed
limited
since
be
set
of
at
the
the
am-
to
lower
levels.
6.3 Coding
6.31 Coding with Binary Elements
It
be
has
been
discussed
in section
represented
by
a
a set of coefficients
is
of
the
characters
be
used
to
of
turbances.
18*
Some
of
F,(@),
of
function
problems
here
for
a vector
R different
Fy(8)
of
Fy
is
a
designing
which
may
or
signals
cha-
the
orthogonality
a character
may
it to be
cause
at
the
receiver.
character
R characters
the
probability
certain
that asignal
advantage.
foradifferent
choice
A set
be discussed
A disturbance
ken
A
alphabet.
will
function
a,(j).
called
an alphabet.
racter
may
time
2.11
of
Some
this
of
happening
methods
for
an
alphabet
for
may
certain
g
a suitable
makin
mista-
A suitable
reduce
the
of
dis-
choice
will
types
m
by
represented
be
characters
R
the
Let
investigated.
be
DESIGN
SIGNAL
6.
276
coefficients:
functions:
time
by
representation
the
obtains
one
of
& ?,
-$ #9
interval
inthe
orthogonal
{f(Jj,¢)},
functions
system
a
Using
codes.
block
called
are
alphabets
Such
(89)
x = = 1.+---R
ay(m-1);
alpiebe amet
“ERD
m-1
20
Generally,
be
a time
in
other
It
appears
(90)
ee ies),
PAG
a
=
F, (9)
the
signal
dependent
at
the
electric
instances
a
time
reasonable
to
or
functions
when
However,
it has been
functions
CeACA.
alone
additive
decide
thermal
{f(j,9)}
mission
f(j,@)
require
and
the
and
detection
the
error
A further
by +1
additional
cient
This
means
be
or
a
changed
+4+%4.
It has
cient
a,(j)
by
+1-1,
been
is
a
a number
or
shown
even
of
in
by
O.
though
reasons
by
not
influ-
One
are
often
usually
makes
leaves
ay(O)
one
and
5.21
value.
=
not,
disturbance
ay(j)
+1
the
only
and
+1+1,
the
four
e.g.,
into
the
into
canbe
only
of
that
the
a coeffi-
permitted
as the character
only
by
restricting
which
into
why
of
genera-
characters
coefficients
section
a
case
functions
case.
achieved
notation
-1-1
that
in
do
the
values
disturbance
-1+1
changed
have
any value
are
with
pro-
5.12
their
they
that adisturbance
character
error
of
for
but
by 1 and
time
bandwidths
for trans-
special
two
written
in short
+1+1,
There
-1,
assumption
forms
may
and
to
by
coefficients
error
represent
in this
low
systems
unchanged
or changes it to the other
ay(1) =4+1,
can
may
current.
section
of
may
strength,
or
and the
frequency
different,
ay(j)
in
difficulties
simplification
is
coefficients
denoted
Different
a,(j)
with
shown
probability
One
field
voltage
alphabets
the
are
a receiver
representation
noise.
rate.
coefficients
the
unimportant
different
ence
the
are
practical
tion
the
for
of
magnetic
dependent
use
bability.
the
looking
input
coeffi-
a(j)
which
+1
or
-1.
values
+1
and
6.31
-1
CODING
are
WITH
often
BINARY
permitted
velopment
of coding
functions
f(j,@)
amplitude
sampling.
as
+1
were
block
onthe
than
the
on
previous
the
time
-1.
was
of de-
that
the
are
by
interpreted
changes
the
increases
F,(6@)
the
only
with
The
the
the
that
called
block-alphabets
turbed
as
well
elements
numbers.
as
+1
Number
in
used
has
starting
case.
the
f(j,9)
a(j)
for
and
be
and
acode
the
error
rate
and
this
compati-
coding
of
[1-5].
An
ay(j)
as well
+1
and
called
Since
characters
consider
a large
-1,
that
can
be
or
the
binary
undis-
contain
only
themtobe binary
numbers
problems.
may
then
Binary
of publications
number
excellent
for
alphabet
the
a(j) are usually
are
block-codes.
in
on
was gisummary
ven in a bookby PETERSON [6,7]. Non-binary alphabets
also been investigated using number theory [8,9].
of
dis-
consisting
of cha-
applied
to binary
treated
value
is
alpha-
positive
reasons
can
elements
disturbed
theory
The
and
it
elements
is based
a,(j)
-1, onemay
HAMMING
with
The
dis-
pulse-type
between
Alphabets
investigation
the
been
to
coefficients
of
binary
and
in the
results,
implementation
by binary
number
or
shown
pulse-type
these
only
coefficients
equal
rather
equipment.
elementsinthis
with
‘their
coefficients
of code
O, only. The coefficients
racters
been
of
resistance
the undisturbed
disturbed
It has
coefficients.
existing
thermal
of
a,(j)f(j,9)
effect
distinguish
theory
of coding
assumption
of
Despite
for
values
of the
additive
shape
of the functions
amplitude.
to
than
anvestigation
a,(j).
that
designed
and
«an
requirement
of simple
bility
1 and
other
functions
depends onthe
negative
on
the
decoded
quantization
generally
and
sources
6.23
clipping
that
coding
were
This
as
coefficients
turbances,
be
and
one
customary
to consider
the
assumed
amplitude
in’ principle,
section
turbances
as
beginning
A positive
from
require,
effects
are
At the
pulses
(5.24) and (5.25)
in
Disturbances
mOlse
bets
a(j).
ANTS.
EIMIAOW
on
for
a7)
theor
it y
was usually
andanegative
sums
the
ELEMENTS
communications
achieved.
have
depends
Computation
of
of
It
denotes
+14+14+141-1
and
tance
for
block
disturbed
The
tance
1.
without
energy
of
following
smallest
One
from
may
m'
racter
an
transmitted
the
shows
is
increased
characters
energy
kept
is
reasonable
of
and
partly
to base
the
with
is
increased,
is
also
m'
>
lation
which
effect
reduction
additive
distance.
thermal
It
+1o0r-1.
the
2”
cha-
the
energy
of
the
error
the
construction
larger
on
energy.
two
equal
must
Hence,
the
per
pro-
of
It is
the
often
alphabets on equal
average
then
energy.
contain
the Hamming
errors
is
characters
transmitted
if
for
are
cannot be decided
the
parity
error
check
rate
digit
under
noise by increasing
Consider
elements
A
m/n'
distance
one
element
caused
without
by
ad-
calcu-
dominates.
one
of
m
probability of error
noise.
with
or
if
dis-
m'/m
elements
e.g.,
limited.
Hamming
decrease
from
element.
but the
thermal
Alphabets
m
per
increased,
ditive
factor
The
comparisonof
characters
energy
from
by their
the
rather
d between two characters
energy of each
derived
Hamming
The
is
of
number
probability of error.
constructing
by the
partly
character
a
by
The
constant.
times
the
d
elements.
sequence
power
peak
a large
meanalow
distance
the
signal
that
constructed
increase
is
bability
canbe
be
the
Con-
pulses in which
occurring.
if
a
of
of
may
error
useful
example
Hamming
> m
element
of
larger
does not necessarely
2™ characters
The
the
particularly
is
the
than
differ,
pulses
distance
consisting
in-
with
characters.
number
the
larger
The
charac-—
decreases
y often
characters
instance,
pulses.
characters
the
a disturbed
decoding
Hamming distance between the two
creasing
sider,
of
character
x into the wrong
ter
of
probability
The
1.
dis-
Hamming
the
have
11110
and
11111
or
+1+1+1+1+1
characters
the
instance,
For
differ.
racters
cha-
two
in which
elements
binary
numberof
the
elements.
by binary
theory of coding
alphabet inthe
an
quality
the
judging
for
distance'
'Hamming
the
montouse
com-
itis
Hence,
difficult.
very
rateis often
error
this
DESIGN
SIGNAL
6.
2738
the
2” =
32
are
the
the
characters
an
example
influence
smallest
of the
of
Hamming
teletype
6.31
CODING
WITH
BINARY
ELEMENTS
eps)
alphabet:
“Ales = le alee alle Saha ta
eens
Fle 4-4
eee
e144
Le
ed l—1—4
etc.
The
smallest
digit
+1
Hamming
be
of
elements
an
even
(91)
added
distance
to
all
1 andacheck
digit
1. Let
-1
aparity
having
to
all
an
odd
check
number
characters
with
1:
elements
of
number
equals
characters
el
1-1
a
sfFurpo
‘lig
The
smallest
to
2.
The
nerally
large
(92)
Hamming
energy
to
per element
m/(m+1).
values
distance
of
m,
has
thus
been
must be reduced
The
factor
while
the
m/(m+1)
increased
to
5/6
or
approaches
Hamming
distance
ge-
1 for
is
still
doubled.
The
smallest
Hamming
an
alphabet
may
of
adding
are
be
sufficiently
called
made
many
systematic
decode
all characters
have
been
reversed
1-1
between
43, 4,
check
...
the
or
generally
digits.
alphabets.
characters
These
Making
d =
d,
by
correction
of
1 reversals.
21+1
one
may
disturbances.
d
=
21
permits
the
reversals
and the detection without
1 errors-correcting
by
alphabets
correctly,
if nomore than 1 elements
rection
of
distance
Hence,
andl
one
distinguishes
errors-detecting
cor-
between
alphabets.
This
distinction
is necessary only if the disturbed
coefficients
a(j)
According
are
(5.24)
limited
and
to
(5.25)
the
the
values
relation
hold in order to make anerror
possible.
The
the
zero
if
are,
however,
not
probability
disturbances
disturbances
+1
or
AWx
= AWy would
detection
AW,
that
are
due
for
-1.
without
thermal
which
this
to
correction
are
AW,
and
to
have
to
equal
noise.
is
There
probability
is
zero.
The
2™ characters
constructed
from
m
binary
elements
Walsh
the
called
is
bet
ters
form
a binary
group
group.
this
A
charac-
whose
code
systematic
a
is
code
group
systematic
alpha-
of
subgroup
a
are
characters
its
if
code
or
alphabet
group
a binary
An
[11].
feature
the same
have
functions
that
Note
2.
modulo
addition
under
a group
form
0
4 and
DESIGN
SIGNAL
6.
280
a group.
Reed-
the
are
codes
group
binary
of
class
A special
Muller codes [5,10]. Their characters contain m elements,
m being a power of 2. The numberof check elements is m-k
the
and
number
ee
of
characters
is
the
k has
2",
value
eerie te:
(93)
1=0
The
smallest
Hamming
Consider
Muller
and
an
alphabet
k=14+4=5.
constructed
distance
example
of
first
This
from
16
m
elements,
This
asa(m,k)-alphabet.
of this
by +1
first
The
16
signs
tudes
of
of
alphabet
is
-1.
Compare
characters
correspond
the
Walsh
characters
follows
with
the
The
17 through
32
are
characters
16
through
functions
ters;
the other
the
functions
belong
to
the
minus
half
of
-wal(j,6).
class
yield
the
of
which
are check
shows
the
elements
and
signs
1.
or
charac-
represenof
Fig.2.
negative
ampli-
of
the
elements
the
orthogonal
One
may
thus
the
con-
characters
as follows:
j -0...455,
represented
one
the
characters
Thus
=e
obtained
by reversing
wal(j,9®),
Signs
a
characters
functions
positive
functions.
a Reed-Muller
alphabet with m
and
=
32
(Reed-
(16,5)-alphabet
13
Walsh
struct
by $m plus
r=1
d
=
of
the
of
$m Walsh
2°
and
signs
of the elements
of the
the
the
16
as
Table
Signs
The
the
It
denoted
with
to
=
16-5=11
(16,5)-alphabet
and
2™".
o*
contains
elements.
ted
d =
=
order).
alphabet
generally
ters
is
where
half
are
of
charac-
represented
Reed-Muller
by
alphabets
alphabets.
6.32 Orthogonal, Transorthogonal and Biorthogonal Alphabets
To
save
space
let
of a (16,5)-alphabet.
us
It
considera
contains
(4,3)-alphabet
instead
2° = 8 characters.
The
6.32
ORTHOGONAL
Mable
156
The
ALPHABETS
coefficients
pape gegen bebet
dl
=
eeee
|x| 0
281
a,(4j) of
according
the
characters
to REED-MULLER.
y = 1s.
of
32,
e
Cc
aoe
ee
are
the
first
Cue
ee
Oe O01 eie-714.
1k 45
dl
2
3
A
2
6
7
8
2
first
four
-
+71 +7
«47 +7
-
-1
-71
+17
+7
NN
Os~
-1
+1
+7
-1
+1
+1
=1
=-1
The
elements
of
X.
Interchanging
matrix
1This
four
Walsh
functions
is
Fig.e:
(94)
(94)
rows
may
be
and
considered
to
columns
yields
matrix
[7-9].
X*:
matrix
of
a Hadamard
form
the
a matrix'
transposed
+1
foe
Siem
-1
-1
+74
[et d'—t +0 4
erdee |r
teat
+1 +1 -1 -1
+1
-1
+1
-1
+1
+1
+1
+1
+1
-1
+1
-1
-1 +1 +1 -1
A matrix
-1
-1
-1
+1
+1
-1
+41
+71
-1
-7
called
matrix
constant.
An
can
be
The
alphabet
ters
+1
+1
as
(94)
16
of
us omit
alphabet
the
the
unit
is
an
product
matrix
of
form
an
with
multiplied
an
alphabet;
by a
elements
orthogonal
orthogonal
its
matrix.
the
charac-
aiphabet,
as
sal7 muOmoer.
the third element
with
its
orthogonal
13
“OO: +4
40
orthogonal
if its
elements
Table
domuinemchiaraci,eir
Let
yields
if
(96)
oO
29°40
Go-O
6
alphabetis called
written
1 to
| Oe
orthogonal
by 4:
E multiplied
(2029 ©:
+1
O54 Te Oe
ae
ol ee eee
is
tvansposed
(95)
unit matrix
the s
yield
XX*
The product
An
DESIGN
SIGNAL
6.
26e
three
of
all
elements
characters
in (94).
and
four
characters
is
obtained:
“ie, | Sd) edhe
2a
ele
ie
Cet erg
-1
(97)
4,
+7
The
product
+1
Yar
+1 +1
ie)
ee
=
eq)
al
-1
etd)
of
the
matrix
a
Y
and
the
transposed
+1 -1
[= =1
Y* =
+7
+7
-1
+1
+14
-1
-1
-1
matrix
Y*
yields
LN
The
>
ne
=1/3
=1/3,
+7
1/9 <1
=<I7/5 =<173
“1/3
-1/3
-1/5
Aan We
difference
gonal
for
(97)
and
eee
ets
ee
betwee
the elements
n
the
others
is
the
unit
matrix
(96).
is
called
larger
Both
on
for
this
transorthogonal.
The
contain
four
from
characters
the
the
For
transorthogonality
is evident
(97).
(98)
+"
principal
matrix
reason
the
practical
the
(98)
than
alphabet
meaning
alphabets
and the
dia-
Hamming
(94)
of
and
distance
6.32
ORTHOGONAL
ALPHABETS
265
betwe
ary
en
two characters
(94)
requires
four
equals
elements
2.
and
However,
the
the alphabet
alphabet
(97)
only
three.
Let
the
characters
of the alphabet
(97)
be
represented
by vectors:
q.
Fo=
+@,
+e,
+e,
Bie
F,=
-e,
-e,
+e,
4.
F,= +e,
-e,
-e,
The
end
points
of
a tetrahedron,
coordinate
dron
and
these
as
shown
systemis
the
four
vectors
in Fig.28a,
placed
coordinate
at
the
system
are
if
the
center
is
the
corners
origin
of
rotated
the
of
of
the
tetrahe-
into
a
proper
position.
The
are
off
the
principal
closeto zero
for
transorthogonal
than
terms
four
Let
the orthogonal
alphabet
—-t
+1
+1
-1
+t
-1
47
-1
The
—1
-1
-1
-1
(4,3)-alphabet
also
the
one
Let
which
shown
Any
added.
by
that
the
be
of
character
replaced
a
by
system
An
If the functions
in
in
the
of
Table
f(j,8)
any
are
and
Table
except
other
of
example
a bior-
alphabet
is
elements
or
Fig.28b.
characters
by
representation
16
orthogonal
of the
by one
functionis multiplied
a character
(94)
of
a biorthogonal
Reed-Muller
a
octahedron
of
of
d from
2d.
not
is
characters
(16,5)-alphabet
distance
representation
Each
supplemented
by the
signs
of the elements:
of the
The
distance
the
Consider
functions.
ficients
nas
alphabet
the
be
the
consisting
biorthogonal.
biorthogonal.
coefficients
£(j,9).
more
(99)
has the Hamming
alphabet
thogonal
with
+1
+1
-1
-1
(99)
is called
one
(94)
obtained
by changing
ae
6.
Ze
8.
for
alphabets
YY*
characters.
characters
14% is
diagonal
of the matrix
13
and
block
the
by
time
functions
16 coef-
products
are
,
the first
pulses
46
multiplied
by -1.
function
Walsh
one
by
fifteen
represen-
then
are
characters
The
add the products.
© and
multiply
well
as
other
the
and
-1
or
by +1
just
could
one
products,
the
adding
-1
or
by +1
pulses
block
16
the
Instead of multiplying
and
functions
characters by the same Walsh
16
Fig.2, the second
of
functions
Walsh
the
by
represented
are
characters
DESIGN
SIGNAL
6.
284
Table 14
by the coefficients +1, -1 and O as
ted
where the first row lists the index j of wal(j,8) and the
in
shown
first
column
functions
-1
or
one
O.
of
The
product
One
14
the
alphabet
are
Both
of
8
in
spectrum
sing
T =
type
signals,
150
there.
The
tered
about
53.33
row
bandwidth,
to
add
no
that
Fig.9
of
by
the
Hz
Hz.
that
Table
the
above
60
One may construct
cients.
It
is
The
to
may
say,
the
rate
the
ternary
one
may
use
pulses
the
a,
the
according
first
b
and
to
5 pulses
The
six-
Its
c.
v=8inFig.24.
used
would
standard
have
would
alphabet
be
of
and
resulting
32
-1
for
tele-
energy
cen-
practically
Table
multiply
Choo-
frequencies
its
conclude
+1
from this
14
is
pulses
of
better
according
Table
signals
no
nar-
13
have
and
almost
Hz.
2'° characters
The (16,5)-alphabet
usual
yield
error
of
unnormalized
there
coefficients
products.
energy
the
One
13
alphabet
= V2sin (16n0+in).
at
should not
the
13.
Table
of
curves
F,,(8)
since
character.
ternary
pulses,
a much
and
trivial
disturbance.
7 cosine
Fi<(6)
One
each
of
spectra
is
+1,
is
the
of
Walsh
obtains
The
characters
centered
signal
60
the
the
be
which
one
shown
than
kind
be
character.
coefficients
must have the same
and
by
would
for
that
alphabet
power
would
ms,
above
zero
any
sine
Fig.24
the
products
result
14 by 16
character
energy
of
frequency
shown
power
unequal
representing
£(0,8),
xy of
the
alphabets
Table
The
teenth
of
andthe binary
of
constant
number
multiplied by the
curious
influence
Instead
Fig.9.
is
the
signals.
under
the
are
summation
only
has
Table
Same
lists
wal(j,8)
that
from
16
binary
of Table 13 uses
this
alphabet
coeffi-
2° of them.
contains
5 infor-
6.32
ORTHOGONAL
ALPHABETS
265
mevlowi4+wwhe coefficients ay(j) of the characters of a
ternary biorthogonal alphabet. y = 1G
sees, al
ORAACaee
O
“AI
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
+
+
+
+
+
+
+
+
+
+
+
|
|
I
|
I
|
I
|
1
(|
O
O'O'
C1O
OO)O
lO
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CO
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Ae
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tation digits and
character
contains
uses
to
16
2° of
each
sign
ternary
the
considered.
One
Without
considered
to
be
or
1g,(3'®
derived
may
the
a
the
11
r'®
each
bits
re-
be constructed
Table
of
The
certain
there
is
bits
5
to
concept
order
no
14
as-
of
are
reason
should
not
be
characters
of
an
(16,4)-alphabet
from
that
be reluctant
- 2°) to them.
of
-
information
however,
restriction
the
and
alphabet
The
if alphabets
of
better
information
assign
will,
this
-
characters
may
one
useful,
characters
of
3'°
of
redundancy
is
digits
coefficients.
character.
redundancy
why the
them;
check
5 bits
A total
dundancy.
from
11
6.
286
r rather
alphabet of order
an
alphabet
The
order
of
concept
of
theory
of
coding,
from
the
has
also
proven
longer
no
in
useful
of
Hamming
the
binary
to
restricted
the
of
generalization
a
For
elements.
2'° characters
than
2.
distance
general
DESIGN
SIGNAL
distance
consider two characters represented by time functions F,(6)
and F,(@) inthe interval -$ 3 6 = %. The energy required
to
transform
Wey =
Fy(@)
into
F,(@)
is
Wy:
f cE, Ce - F,(@)]’ae
(100)
-1/2
The
energy
of the
character
Fx(@)
is Wy:
V2
Wy =
f Fy(9)de
(101)
-1/2
The average
energy of all
R
characters
of an alphabet
is
W,
R
We
a Py Wy s
where
py
The
(102)
isthe probability of transmission
energy
distance’
F,(8)
is defined
dxyy
=
Wyy/W
Let
F,(@)
tions
dyy
the
by normalization
characters
of
the
F,(@)
energy
and
F,(@) be constructed
func-
£(35,0):
m-1
obtains
for
Wy,
Fy(@) = j=0Diay(3)f(j,8)
and
m-1
=
all
(105)
j=0
characters
term
eae)
(104)
Wy,:
=I
'The
and
Wyy:
fromm orthogonal
Wey = 2, Lay(j) - ay(j)]? 9 Wy I= 5’ a2 (3)
Let
y.
(103 )
Fy(8)= j> ay(j)£(5,8),
One
of
of character
have the same
'normalized
distance
if
energy
non-similarity'
the
integration
W= Wy. It follows:
has
been
interval
is
used
for
infinite
6.32
ORTHOGONAL
ALPHABETS
2o7,
m-1
rs
>, ax (ay(3)
oy = Hey 2 1 - 1 py (ayn (eae = 1 - EB(496)
m-]
-1/2
Ya 2(3)
j=0
It
holds
for
eG)
=
the
characters
Table
14:
at
-1
for
2 aed
ay dl Je=a
erletor
Meai(g)
=
j =0
- u oO
of
—
ay
2,8 x65 )ay(d)
=
fs
The
following
characters
peepee
The
Table
distances
4
are
thus
obtained
for
tor xy = 42 - 4 + 41
(107)
xy =
mee Or
yee 4, 52. — tet)
m-]
y
of
Table
15
prea Gd
pay)
j=0
j=0
=
13
yield:
=016
(108)
“3
Dey
jay.)
= 4 for
=O
=
x
ay(jday(gj)
=
32
-
y+
=;
-16
for
xy =
+16
for
x = ¥
32-y+1
1
fory=y
a 2 ior
the
14:
= O for
characters
dyy
32-y+1
y=,
O for x # ¥, 32-y+1
energy
of
y =
(109)
y # ¥, 32 - 4 + 1
The distances d,y, of the characters
of Table 14 would have
the values 16, 0 or 8, if Wyy in (103) were divided by
W/lg,m=W/4
elements
ming
in
energy
square
signal
distance
of
the
space.
by W.
This
characters
evident
meaning
dyy
vector
These
the signal
between
an
than
the
is
just
differ,
the
i.e.,
number
their
of
Ham-
distance.
The
the
rather
inwhich
vectors
points in
in
of
the
two
characters
connecting
their
is
equal
signal
to
points
are represented
by the rods
Fig.28.
vector
The
term
distance
representation.
Due
has
to
6.
SIGNAL
require,
forthe
288
such
in
ted
a
gravity.
of
lisbeet
alphab
of abiorthogonal
R characters
the
Let
sequence
relation
the
that
vector
average
the
have
points
center
common
their
1 from
must
one
signal
the
that
representation,
distance
dyy
of
normalization
the
DESIGN
pegeC e e ames
is
satisfied.
(110)
It
follows:
7
1 fory=y
af Fy(@)Fy(@)d0 = | -1 for y=R-4
ae
It
+1
O fory#y,
follows
gonal
ter
from
alphabet
R-y+1
ters;
(106)
has
that
an
energy
and anenergy
yy =
the
(111)
R-y
+1
character
distance
distance
y of
4 from
2 from
all
a biorthothe
charac-—
other
charac—
No. salts
6.33 Coding for Error-Free Transmission
SHANNON's
formula
communication
sion
of
is
possible
that
bets
as
formula
may
be
for
channel
the
in
section
obtained
presence
Consider
transmission
that
a limiting
which
pacity
of the channel
in
the
proves
of
and
case.
6.12
capacity
an error-free
it
From
is
approach
the
additive
thermal
a system of Fourier
how
alpha-
transmission
which
have vanishing
a
derivation
evident
the
of
transmis-
ca-
error
rates
orthogonal
func-
noise.
expandable
tions f(j,9) in the interval -4 = 6 s #. Random numbers
aj(j) with a Gaussian distribution are taken fromatable
and the character F,(6) is constructed [1-3]:
m-1
Fo(8) =), a,(s£(5,8)
(112)
jz0
One
may
Fo(8)
assume
is
then
distinguished
beyond
all
Using
that
the numbers
a
time
from
a
variable
sample
of
aj(j)
represent
voltage.
thermal
voltages.
F,(8)
noise
cannot
if
m
be
grows
bounds.
another
constrauct
second
set
of
m
character
random
F,(8).
numbers
The
a,(j),
general
ome
may
character
6.34
TERNARY
Fy(8)
ted
can
be
random
these
the
COMBINATION
ALPHABETS
constructed
by means
numbers
characters
channel
of
a,(j).
equals
(54)
follows
of m Gaussian
distribu-
The
unnormalized
duration
of
The
transmission
capacity
of
from
m,
T andthe
average
si-
re
ratio
power
gnal-to-noise
T.
ASS)
este; (1 4-2/2,7)
(113)
Let n be the largest. integer
n
characters
ee
EN
mhese
n
miotey,
n
one
(114)
formthe
alphabet
drawbacks.
alphabet
is
point
It
is
for
is
in
of
the
a finite
probability
[4,5].
The
much
combination
and
zero
very
Now
and
There
L
but
to.)
from
to
the
practival
how
good
per
of
the
er-
character.
approaching
the
energy per bit of infor-
rate
also yield
that
zero.
however,
alphabet
SHANNON's
close
in-
satisfying
information
finite
than
way
approaching
are,
L al-
in this
what the probability
of
for
let
approach
close
interesting to see
transmission
alphabets
n
very
non-random
smaller
come
rate
are
amount
error
however,
only
limit,
found the first
mation
error
view.
not
If
arbitrarily
alphabets'
ELIAS
lities
random.
is:
an
alphabet.
each be constructed
at
yields
'random
theoretical
first
characters
alphabet
scuesprobability
These
ror
constructed:
cis cin yg —T
with
pick
this
be
characters
phabets
and
F,(8@)
smaller than 2°' and let
of
information
limit.
vanishing
SHANNON's
was,
The
socalled
error
probabi-
limit.
6.34 Ternary Combination Alphabets
m
ents
orthogonal
ay(j).
functions
A fotal
may
Ged) ifa,(j)
Writing (1+2)"
R =
of
of
can transmit
m
coeffici-
can be construcvalues +1, 0 and -1.
3™ yields the following expan-
the
assume
instead
£(7,9)
3™ characters
three
sion:
Bes (442)" =72°(5)+2
This
19
decomposition
(4)+.. 42"
divides
Harmuth, Transmission of Information
the
4... +2")
set
of
R characters
Glee
into
function
or
+1
-1.
the
form
characters
These
one
of
consisting
ay(j)
equals
biorthogonal
alpha-
coefficient
one
only
because
each,
2m
=
ENG»
are
there
more,
Further
zero.
are
characters,
func-
no
containing
a,(j)
coefficients
all
because
tion,
character
= Bake
£Gj G0.) lnSresis
functions
many
equally
containing
characters
of
subsets
DESIGN
SIGNAL
6.
220
are ane characters, each conbets. In general, there
taining h functions ay(j)f(j,8), where ay(j) equals +1 or
Ge is the
Siemoiuce
m
functions,
tion
ber
alphabets
these
alphabets
for
h #4 0,1
characters
2 Ge of
in
combinations
of
number
are
0r
Table
m.
such
of
h
the
shows
15
out
of
combina-
ternary
called
num-
alphabets.
Table 15. Number of characters internary combination alphabets. According
to KASACK [2], the numbers above the
line drawn through the table belong to 'good' alphabets.
1
2
3
4
5
6
7
8
9
_0
Equation
(115)
yields,
for h = m, the 2"(7) =2™ cha-
racters that contain all mfunctions
equal
+1
or
-1.
These
are
the
ay(j)f(j,8)
characters
with
of
the
a,(j)
binary
alphabets.
Consider
an alphabet
tions
f(j,9).
equal
to
+a,
with characters
Each character
or
-a,
and
Let
these
the
received
signal
with
coefficients
a,(j).
Let
characters
be
imposed
on the signal.
which
have
a Gaussian
m-h
containsh
functions
additive
distribution
equal
a,y(j)
to
zero.
Crosscorrelation
f(j,@)
thermal
The coefficients
h func-
coefficients
coefficients
transmitted.
the
containing
yields
noise
a(j)
with amean
are
be
of
the
super-
obtained,
either
+894
6.34
TERNARY
O or -a),
COMBINATION
denoted
j-->
by a®'(5),
= <>
+A,
ee TN
(0)
<
a
>> <a »
The
variance
o*
ALPHABETS
aol
al(5)
(2D
ana
aca ys
- Cal
= 1
(116)
e
of
these
distributions
follows from (60),
(70) and (71):
Bifis
2?
eee a
=< az
iy 2 Pate HPA
DO Ga eee Gliors
h = number
n=
ee
of
coefficients
Ge
= information
characters
P,,=
average
noise
P = haj
Py
=
power
in
an
average
ite
tween
of
of
value
character
with
thermal
Af
or
if
of
of
binary
snoisee
and
2gG.)
T;
= n/2T;
noise
in
a
frequency
power
is;
inorder
ternary
to
rather
then
h,;
facilitate
ork;
and
any
sums
(118)
produced
coefficients
one must be determined
(5.25).
m-h
Consider
are
terms
sum
cients
largest
those
the
S,.
sums
positive
are
whose
a(j)
for
O.
The
those
the
the
for
The
terms
largest
largest
transmitted
of all
terms
a,(j)
+a,
or
magnitude.
character
Fy,(@)
will
remaining
Sy will
the
The
re-
yield
‘sume
2”
sums
2G)
contain
The
and
the
while
O for
coefficients
-a,
these
of
‘accor-
are
j=0O...m-h-1.
a(j)a,(j),
largest
received
decoding
certain
cing ca, equal
non-vanishing
with
for
a(j)
coefficients
sums for which
coeificients
h
contain
of
O, forinstance
h
2" aifferent
19*
m
largest
ad)
the
the
to
feaining
m—-h
from
the
.-
“is
be-
alphabets.
j=0
ding
band
comparison
m-1
be
all
thermal
duration
eye=) > 8d )ay (3)
must
Ete
probability;
components
interval
power;
+a,
in bits,
equal
orthogonal
signal
a reference
The
n
with
Af.
average
as
per
orthogonality
power
width
ay(j)
transmitted
= average
of
used
are
117)
h
sum
when
be
coeffi-
will
be
the
ab-
6.
Zoe
DESIGN
SIGNAL
dolute value of the h coefficients a’ (9) and alge
of the m-h coefficients a!°(j), andifin
larger than that
e,
the following
Henc
two
error-free
(see
decoding
4. All coefficients
are
gal
None
than
of
the
the
ae
are
for
satisfied
Fig.107):
al = al(j)/a,
and -al!) = -al')(j)/a,
(119)
h
absolute
Condit vom
eeu
zero.
ico
al%)_ al%(5)/a,.
Tie
be
must
conditions
than
non-negative:
Gasca
2.
a'(3)is smaller
and
a*'(j)islarger
addition
coefficients
value
This
|
}tor
|=
one
condition
of
the
-at')
m-h
is
smaller
coefficients
needs
to be satisfied
only
1 Sasabish Led:
a
Sala
of
+a'*') and
Otsya'Y
eC
nes
(120)
O
The density functions w,(x) of a!) and w,(y) of la |
given by (4.59). The probability p(at!!<O) = W,(0) of
condition
plal’<0)
(119)
=
not
W, (0)
being
satisfied
= ae.
equals:
exp[-(x-1)?
/20? )Jax
(121)
#(1 - erf(1/Af2c)]
The
probability
p(-al"<0)
= #[1
pi?’ denotes
the
not
satisfied
and
al!);
Ph = 1-1
Consider
Its density
p(-a
-
(-1)
<0)
has
same
value:
erf(1/f2c)]
probability
for
the
at
least
(122)
that
one
of
the
the
-W,(0)]" =1-2"(14
condition
h
(119)
coefficients
ert(A2c)]
is
al*!)
Cas)
the distribution of a't!) —|al®l], 0 < alll< co.
function is given by (4.61) to (4.63). The
probability
that the condition (120) is not satisfied for
Gens one
of the h(m-h) differences a TK jat@l and —al-!)
-|a'
| is
al [al <o
Se eae a
WOO)
=
Y
f[ w(z)dz.
(124)
6.34
TERNARY
COMBINATION
ALPHABETS
20
a(+!)
Fig.107
gta
Density functions of
ea (Oe
| at Oe
wand
a!) -la°|
for
a
ternary
combination
alphabet.
hatched
areas
indicate
TOL.
The
er-
alt). Jq()
This
integral
was
es ee denotes
is not
satisfied
-al-Ne [gl]
ec
and
ea
(4.64).
for all h(m-h)
differences
(120)
a'*!’— ja!) and
dew WOODIEY
(123)
and (125)
ternary
binary
eee
in
;
Equations
of
evaluated
the probability,
that the condition
combination
(125)
yield
the
alphabets,
error
probability
biorthogonal
pee
alphabets
(m,m)-alphabets:
Cle ,
o
a - UG
(126)
)
:
;
4 = 21
+ ert(1N20))"
{+
Equation
(126)
yields
a binary
alphabet
with
for
m
h = m the
+ eebio
tie ti |
h(m-h)
error
coefficients
probability
ay(j)
and
of
2™ cha-
ie
Ge
Coes
mP.
m,m
Tr eae
- pl?) 24
GPU
+ srt Acer
m
ee. =m
Giese
follows
alphabets
biorthogonal
of
probability
=
eyes
/BP es ba, /2y) Le=
error
The
C7):
as
same
is the
which
racters,
DESIGN
SIGNAL
6.
204
\ifopan aly ta
(128)
leesty
oie)
Wasted
M
Ke)36
1
1 = #11 + ert(1W20)1 {+ eee}
4 fl t7eg
Gta=n by, /E = Py /oP,
Fig.108
gonal
shows
alphabet
(curve
n=16,
m=
=
5,
The
n=
are
the
of
the
as
noise
power
Pit
noise
in
orthogonality
reference.
sion,
mit
since
amounts
P.
(P/n)/P,,
signal
This
bet
per
P/P,
= EYE
P/P,,
of
(n = 5, m = 16) one
price
paid
give
requires
for
this
11
-
gain
is
of
thermal
T is used
a false
impres-
transto
information,
is used
use
ra-
in Fig.108:
cease
with
(129)
5 bits
n
probability of 10°. Accoralphabet (n = 5, m = 5) re-
dB
andthe
of 8 dB.
8 =
mea-
The average
alphabets
of
ratio
The
It is better
bit
which
curve
power
duration
a Pf)iim
11
The
component
of
(5,5)-
Fig.102.
evident.
the transmission
of characters
a ratio
alphabet
P is
information.
of information With an error
ding to Fig.108, the binary
quires
in
explanation.
would
power
gives
= P/oP, 5
Consider
comparison.
‘a”
interval
P/P,
biortho-
binary
signal-to-noise
power
of
some
(16,16)-alphabet
characters
of the various
P/n,
the average
than
for
orthogonal
Plotting
the
different
ther
one
for
of the
the
requires
Signal
of
and
curve
average
along
the abscissa
an
5)
shown
same
ning
of the average
as
probability
probability
5, m=
m=16)
5 is
choice
plotted
error
The error
(curve
n
niedle,2'
Ci je= oe oe
the
alphabets.
Ta
3 dB
an
biorthogonal
Hence,
less
increase
alpha-
the biorthogonal
signal
in the
power.
number
The
of
6.34
TERNARY
COMBINATION
N
N
\
\
GEN
ALPHABETS
29)
|
“N
A
NENG
NN
NYE
ING
\
3
ates
oe
e
9
“
cae eatae
PI Ra¢ [4B] —=
13
"1
15
8
PIR «(d8]——
Fig.108 (left) Error probability p of biorthogonal alphabets. P average signal power; P,,
average power of thermal noise in a frequency band of width Af = n/eT; n information
of
the characters in bit;
T duration
of the
characters; m numberof orthogonal functions inthe alphabet. Solid lines: biorthogonal alphabets;
dashed
lines:
binary alphabets (5,5) and Aendens
Fig.109 (right) Error probability p of ternary combination
alphabets; P, Pa}, n and m definedinthe caption of Fig.
108. h number of orthogonal functionsina character.
Dashed lines show the error probabilities of the binary alDHapeter(S,5)- and *(16,16).
orthogonal
times
what
functions
larger
required
sectionof
the
from
less
precise,
a 16/5-times
required.
Consider
further
with
n=‘16
bits
of
m=5
the
information
wider
with
an
alphabet
dB; the biorthogonal
m= 32 768)
one
requires
binary
tions
only
alphabet
required
11.7
of 5.8
about
dB.
one
band
is
error
probability
(n = 16, m = 16) requires
Thus
the
quarterof
(11.7=5.8=5.9
increases,
a 16/5-
or, some-
transmission of characters
of 10°? . The binary
of
m=16;
frequency
ratio
P/Py,
to
time-function-domain
biorthogonal
the
dB).
however,
alphabet
signal
The
from
alphabet
power of the
number
16
to
a
(n= 16,
32
of
func-
768.
-
n
than
16,
=
m
This
for
shows
shows
P/Ps+
one
than
bability
Che
shows
information
the
n
=
n=8
ones
n
=
m=5%12
9.9,
m
need
alphabet
P/P,,;
a
but
For
with
larger
a
smaller
instance,
yields
an
the
error
pro-
of 8 dB; the comparable
=
n
there
with
8.8,
the
8,
h
=
4 requires
m
=
8,
which
combination
is
alphabets
same
ara-
error
with
is
signal
3.
m
less
=
that
a curve
functions
of
yield
do more
than
power'.
the
n
=
the
that
ex-
a binary
the
8.8
requires
a
Consider
of
8 functions
than
alphabet
by
of
A character
probability
represented
m
transmit
nevertheless
alphabets
'less
h =
which
number
and
These
for
transmits
The
are
(m,m)-alphabet
bits,
ternary
alphabet
n=10,
combi-
A comparison
(m,m)-alphabets.
functions'
(8,8)-alphabet
functions.
particular
biorthogonal
probability.
'more
curve
these
that
binary
error
change
the
ternary
(126).
and
[NO'5) Gls.
than
mation
of
to
foraratio
more
the
probability
alphabet
Elen
Fig.109
lower
error
alphabet
combination
Wa
32768
the
the
of 10°5
=
AYES
binary
biorthogonal
m
of
that
the
16,
probability
according
than
=
somewhat
to
amounts
n
alphabets
of
14.8
by
power
error
alphabets
Fig.108
difference
same
the
while
ratio
a
an
at
16
=
Fig.109
nation
ratio
dB.
7 dB
=3.6
11.2
more
in Fig.108),
signal
the
of
reduction
aB, apossible
44.8
error
requires
5)
=
m
5,
(n =
alphabet
m=16
n=5,
(curve
of 107
binary
the
P/P,;
a ratio
requires
it
ample,
probability
ex-
for-en
dB
11.2
Of
For
alphabet.
a biorthogonal
of
use
the
is
justified
more
the
probability
error
required
the
smaller
The
DESIGN
SIGNAL
6.
296
infor-
bits
also
of
m=8
binary
lies
(8,8)between the
curves
curve is
n= 5, m= 5 andn = 16, m= 16 in Fig.109. This
about 3 dB to the right
of the curve n = 8.8,
h=%3 forerror probabilities between 1074
and 10-7.
m=8,
Consider
values
of
x
etits)
.
=
one
m
obtains:
-
the
error
and
n.
eae
ee?
probability
Using
MoS
the
Tg
aad of
(126)
for
large
approximations
BEI el
ye
RA
ok
6.34
TERNARY
COMBINATION
Fenn
Soames
ete
\Fat
Vr
Pp
f=
Let
n and
Thus,
OD
Tor
the
Using
in
ia
[h(m-h)
P
(21 b= ZDPat
infinity:
4 > 0
ROT
enc
(137)
fO
transmission
is
achieved
in
the
relation
=h
le; i>,
transform
condition,
the
holding
(132)
condition
for
7 < O into
a constant
value
the
following
of h:
P/ Pao > A hne
A
ratio
Gigs)
P/P,,
transmission
larger
for
The limit
function
and 109.
gonal
alphabets
>
Hence,
1).
ones
from
red,
since
proportional
Let
to
m >>
in place
of
of
ee
Error—-free
of
and
is
the
the
for
same
O then
are
or
functions
yields
finite
biortho-
the
(h
superior
bandwidth
requi-
increases
but
only
alphabets.
approach
m°;
to
the
h;
4 ine.
alphabets
required
mandn
still
the
alphabets
combination
is
for
combination
proportional
(132)
and
n+co is shown in
alphabets
as
n,
error-free
is smaller than
biorthogonal
constant
h of
m
of functions
m
for
2h
n <
condition
The
P/P
to
2°
situipecrease
condition
limit
number
large
yields
for m+oo,
combination
h not remain
Let
Duce
This
41n2
1 if-P/P,,
standpoint
the
proportional
is
lim p3)
(h = 1) and
the
the
than
infinitely
the error probability
Figs.108
limit
a <0.
n = ieee)
may
(130)
Or
error-free
OS
one
=
approach
(cy
;
Se
Lim
ies
sel7
iq
im igs =r
C77,
Pie
2
m
ALPHABETS
infinity,
0 2.a.<
satisfied
41.
The
forlargenm.
following
condition
"(135):
1
tO.
-a
transmission
ees” ae Ous¢0..<
01.
is possibleif
a
(134)
is smallerthan
‘1.
mission
nary
combination
alphabets.
must
be
replaced
by Ppt in
It follows:
(130).
(135)
Eee
ee
vee
Sa
The
condition
Py,
power
noise
average
The
by ter-
is approached
(54)
of
form
capacity inthe
trans-
the
of
limit
Shannon's
how
investigate
us
Let
DESIGN
SIGNAL
6.
206.
n < O becomes:
h < snpm
44 an Ch(m-h) 2)
(136)
m,T
The
approximation
stituted
on
the
n =
left
side
m
term inthe brackets
and
h remains
time
is
1.
character
it
n/T
are
reordered:
(137)
1 when
becomes
m
n
becomes
(1-a)/(1+a)
transmitted
since
transmitted
duration
n
C=m<
finite;
to
terms
h, is sub-
Pmt
becomes
The information
equal
the
m >>
P
In mh - $1n(1g2>)
The
<
h let,
IR
and
ol
n< pel
O Za
Lapel
is
infinite
for
h = m‘%,
error-free
per unit
the
information
during
an orthogonality
of
each
interval
of
T:
m
+ Lia
eaei
ial
1
-a
Cee
ur<es
ta
The
logarithm
transmission
h = constant
m
4
iz
to
the
base
oT Sine Dene
capacity
is
(138)
i Rome
2 must
to
be
be
Got
used
obtained
se eeeey
in
(139)
(54)
in bits
if
per
the
unit
time:
C= or 1e(1+ P/P,.) * Seep
The
right
hand
2(1+a)/(1-a)
Hence,
side
than
a ternary
transmits
half
by Shannon's
Talequale) Pee
(140)
right
asmuch
small.
hand
(140)
by a factor
2 or
sides of (138) and (149).
alphabet
information
provided
The
pT<< 1-
is larger
combination
limit,
is
of
the
Ee)
the
physical
with
h
=
constant
errorasfre
permite
ted
signal-to-noise
meaning
power
of the condition
6.35
ALPHABETS
P/P,,t <<
phabet
OF
1 is evident;
have
only the
of the average
less
be
of
cients
an
more
-1.
than
P/Fh,t
D/Pea useworti—
the
zero. Use could
only
three
if
the
values
coeffi-
+1,
O and
by
be replaced
must
alphabets
al-
An increase
order.
higher
was
ratio
combination
detailed
alphabets
of a ternary
+1, O and
probabilit
has reached
y
assume
of
A more
coefficients
values
increased
ternary
alphabets
the
299
signal-to-no
powerise
ratio
could
The
2r+1
three
onee
the error
made
-1.
ORDER
investigation
recently
published
of
ternary
by KASACK
combination
[2].
6.35 Combination Alphabets of Order 2r+1
Let
characters
tions
f(j,6),
These
coefficients
as
for
F,(@)
mial
(ra
the
the
Let
4
expanded
in a bino-
Cor): Green (or)?
of
characters
coefficients
ay( 5) are
non-zero;
These
2r+1.
equal
bits
equals:
Let
characters
The
information
a
are
denoted
coeificient
by
a,,
ay(j)
be
per
# O may
p =
+1....4Tr.
by p(p). Let p(p) be independent
of the functions f(j,@) is P,:
from
1
alphabet
character
in
(142)
a,(j)
assuming
that
transmitted
moo h
Each
of the h coefficients
They
y runs
formacombination
probability.
alphabet
means
all
1g,(2r)’"(@) =n 1g, ,
of
This
characters
these
(Pyra44)
in the
f(j,9).
with
power
than
characters
functions
Gry).
noted
be
a,(j).
rather
(1+2r)™
m
order
lity
values
of
(1+2r)"
number
h of the
of
lues.
2r+1
A total
er) ee
G) is
to
n=
func-—
multipliedby coefficients
assume
alphabets.
produced.
et
containing
of
may
orthogonal
series:
eee
h
be composedof m
-+=6 54%,
ternary
canbe
Fy (8)
the
of
assume
value
er va-
probabi-
The
a,
j. The
is
de-
average
+r
4
v/2
:
7
DESIGN
SIGNAL
6.
300
+r
(143)
> Plod@ J aff? (3,8 dt mee p(p daz
P=
a
-1/2
coe
+r
yi p(p) = 1
Pio
The average power of the characters
BG),
of
composed
h functions
Lees
(144)
The
following
a) The
lue
b)
assumptions
probability
of
a, is independent
The
difference
= ag.
Sn?
This
Eg
The
tu
average
made:
a coefficient
of
p: p(p)
|a,-a,,|
condition
teeta
are
is
1s
a,(j)
having
the
va-
= 1/er.
independent
satisfied
if
of
a, is
p. la,—ay4l
a multiple
=
of
2S {headin cassie
power
P;
of afunction
f£(j,8)
follows
from
(143) and (144):
r
+7
>) pai /er = (ai/r)
=
P
a
> p* = fret ere) a? = P/h
a=
af = 6P/h(r+1 )(2r+1 )
Let
a character
Fy (8)
with the functions
the
receiver.
these
f(j,6)
with
(+ ple
into
means
coefficients
are
|pla,,
denoted
+
The
=
(al)
variance
of
transmitted.
Crosscorrelation
yields the coefficients
additive
a(j).
or
by alel(j),
0;
a Gaussian
|p|
and
es
at
changes
distri-
= 1...r.
al-rl(j)
.
ay |Petey = a
I
ay(j)
thermal noise
They have
-Ipla,
(- pl;
SnD a ty
—(O)(
be
Superimposed
coefficients
bution
(145)
These
alt)(j):
(146)
O
these
distributions
follows
in analogy
to
Cade):
oO
2
(ai(j)/ay> = h(r+1)(2r+1)P,
,/OP
hee
)(2r-4 Nae /onP
= h(r+1)(2r+1)P,,
(147)
/6nP
6.35
ALPHABETS
OF
ORDER
2r+1
301
h = number of non-zero coefficients ay(5)3
>) =I een) VG) = information
per character in bits, if
all characte
are transmitt
rsed with equal probability;
er = numb
of non-zero
er
values which the coefficients a,(j)
assume;
may
ae
average
noise
power
in
an
of
average
of
=
of
components
interval
average
signal
thermal
noise
of
of
power;
in
thermal
duration
a
1;
= nyeot
Af
frequency
band
of
than
Af.
characters
of combination
third
must
power
width
The
orthogonal
orthogonality
P = haé(r+1)(2r+1)/6
aie a
n
order
are
not
transmitted
determine
the smallest
for
the
detection
ae
ey
sums
alphabets
of
with
energy
the
equal
AWy
signal.
higher
energy.
One
accordingto (5.24)
This
means
that
By = >,
ats) = ay G1’
j=0
must
be
computed
termined.
where
The
be
is
-
h
occurs
to
with
if
of
a,(j)
in
following
value
smallest
for
de-
} = yx,
F,(@).
obtainedif
the h smallest
added.
# 0
smallest
character
Sy is
termsinthe
the
the
S, isnot
transmitted
value
smallest
if
one
Bey Gai As are
equal
noise
(148)
the
the
smallest
[a(j)
the
mal
An error
x denotes
terms
acgo
and
the
The
the
h
terms,
for
noise-free
presence
conditions
of
which
case,
will
additive
are
ther-
satisfied
(see
Bie
1 Os
4.
None
of
the
h coefficients
= -al-el(j)/ayis farther
=
2.
|p|
None
from
one
than
one
of
These
ditions
the
m-h
of the
Satistied
The
from
the
of
only
two
the
ale)
from
other
its
means
coefficients
h
means
condition
conditions
a'®’.
are
1 is
of
the
1....r,
and -al-r)
the
This
-al?!=
la,/ayl
=
# pis farther
absolute
condition
value
of
must
be
satisfied.
essentially
error
and
mean
|p'|=
a't?)
(119) and (120) for ternary
calculation
correct
1..... r than
coefficients
if
= abel(j)/a,
to the conequal
combination
probability
is
alphabets.
much
more
6.
302
Values
Teo
r
h
using
2r+1,
h,
m,
values
large
For
functions.
m
of
out
order
of
alphabet
combination
a
of
noise
thermal
to
due
probability
error
the
denote
Ca
Let
here.
stated
be
will
results
the
Only
complicated.
DESIGN
SIGNAL
small
and
Of o°,
Seii>
Sk)
Mare
ore
red
Snes
(149)
n = 1g,(2r)"(7) M2= h 1g,(xm/n),
one
obtains
ili
the
following
SAG
gueeneiee
yl
UY
formula:
a
(150)
m,h
1
nm =
ZP
inh - perp)
Let the information
Limp
N=
CO
cea
case
nor
n, <
0,
yeah
grow beyond all bounds:
eroamnony
terminthe
n,
>
Gibrae.
O is
ac a
because
it holds:
eeeee }+(tinn-g 4]
Shr Pay
Shr? P,,
(152)
n,
not
bracket
must
be
probability
probability 1 for
for
>
second
O that
yields the error
condition
y= Geer
n,
ag
from
character
1, soe
e.
== O- S61"
Wake tng es
The
n per
:
+
(2r+1)
Lim
Dinh
The
4
(2 = yg InG@-bh - Piha
Eeeti?
n +o,
possible
is
equal
larger
to
than
n,.
O.
It
follows
Hence,
n,
O and
n,
> O yields the error
n,
yields the following
error-free
PP
4
of
a...
SS 3 in
transmission:
(153)
n from
Let us investigate
The
average
formals
noise
(150)
(149)
yields:
2See
ln(m=h )h
can be approached
for
by
O
Rewriting
P/Py, > $2 r?1n(m-h)h
Substitution
<
how
(154)
Shannon's
combination
power
yy:
Pyy
must
limit
alphabets
be
inthe
of
replaced
form
order
by
(54)
2r+1.
Bo
aa
6.35
ALPHABETS
OF
ORDER
2n+1
303
Fig.110 Density functions of
all) alt, alt)" abel, altel) ate
and afer),
The hatched areas
indicate
oa
errors.
af?)
: N
~|pl-1
~lpl
ap)
~|pr1 et a
{pi-1
'p!
Ipl+1
ys NX
-rel
Pap
=
Ent
=
One
obtains
dim Be , pa
Using
the
n
nP,;
Sn
from
(151):
e= Out or
at)
it
r-1
r
r+)
faa
tee
4h
ra
r?1n(m-h )h
CASso
relation
ree 22 m
which
(156)
follows
from (149)
n < Sle {Caer
One
of
this
tain
is
One
inequality
of
value
too
may
see,
large,
the
be
» 1, one obtains
from (155):
eal
1579
right
the
that
so
as possible
mes
as large
beco
a fixed
to
however,
should
r
= h(m)
h
and
m
complicated
logarithm
too
choose
must
for
ratio
P/FR,,;-
find
a maximum
that
the
as
terminthe
large
factor
as
brackets
The
side
hand
for
a
cer-
expression
by differentiation.
h
possible.
becomes
in
front
If
h
smaller
of
the
becomes
than
sles
creasing
Tae as
woke
equation
the
that
so
chosen
constant
suggests
This
satisfied.
is
h is
Hence,
m.
(158)
choice
the
h = n/\fin = :
(159)
it followertrom
ial, <<
3 CE
The
information
C
=
td K< a
(157):
1g ,(KP/P,, ,)
n/T
CHa
of P/P,,
difference
equal
[ 1g,(P/2,,,
is
bution.
The
from
only by the
probably
distribution
(160)
transmitted
This formula differs
values
in-
with
small
arbitrarily
become
then
would
term
This
DESIGN
SIGNAL
6.
304
) +
unit
time
becomes:
1g,kK ]
Shannon's
factor
accounted
for
physical
per
(161)
limit
(54) for large
(1n m)*
for
. This
by having
chosen
p(p) rather thana Gaussian
meaning
of
the
condition
small
an
distri-
P/P,;
>>
K
is readily understandable. r > 1 had been assumed in (149);
many different values for the coefficients ay(j) will permit an error-free
to-noise
power
transmission
ratio
is
large.
onlyif
the
average
signal-
References ordered by Sections
Introduction
‘s
MANN,
Nachr.
P.A.,
Der
Zeitablauf
Technik 20(1943),
von
183-189.
Rauschspannungen,
El.
2. STUMPERS, F.L., Theory of frequency modulation noise,
Proc. IRE 36(1948
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4. VOELCKER, H.B.,
Toward a unified theor
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y
Proc. TEEE 54(1966 ), 340-353, 735-755.
4. RADEMACHER,
H., Einige Satze von allgemeinen Orthogonalfunktionen, Math.Annalen 87(1922) ,122-148.
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actions
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A closed
set
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Amer.J.of
Mathematics
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10.HOWE, P.W., The use of Laguerre and Walsh functions in
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11.FRANCE, M.M., Walsh functions,
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Technical Report AD-621360(1965).
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1. TRICOMI, F., Vorlesungen uber Orthogonalreihen,
Berlin/New York:
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2. SANSONE,
G.,
Orthogonal functions,
New York:
MInterscience
1959.
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J., Reihenentwicklungen in
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de Gruyter 1953.
4, MILNE-THOMSON,
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and H.FESHBACH,
Methods
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3. LENSE,
J., Reihenentwicklungen in
der
mathematischen
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de Gruyver 1953.
Laguerreschen Polynomen,
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4, HIER, R., Signalanalyse
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anae
of modern
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5. WHITTAKER, E.T. andG.N.WATSON,
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lysis, chapter IX, London:
er
- integral,
Sinecry of the Fouri
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C.eTITCHMARsH,
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-Oxtord
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;
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20
Harmuth, Transmission of Information
306
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Oxford
University
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1937.
3. BRACEWELL, R., The Fourier-transform and its applications, New York: McGraw-Hill 1965.
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1. WALSH,
J.of
2.
J.L.,
Aclosed
Mathematics
RADEMACHER,
nalfunktionen,
set
of orthogonal
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55(1925), 5-24.
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Einige
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Some
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in
Control
Ubertragung
MIT
In-
Il.
Ar-
biund
3201968),
Index
171
134,268
Active antenna
activity factor
addition
modulo
2
205
aircraty collision
amplitude clipping
amplitude sampling
angle diversity
angle measurement
antenna, active
astronomical telescope
attenuation
audio
coefficient
signals
27
162
270
122
238
175
eo
175
95
oscillation
period.
4,166
average wavelength
4
axioms of probability
184
Banach's
theorem
system
14
264
Bernoulli distribution 190
- method
88
- polynomials
9
Bessel functions
204
- inequality
a
binary character
66
- shift theorem
148
biorthogonal
67,280
block codes
277
Boltzmann statistic
218
Borel measurable
- sets
Cauchy's
principal
value
Cauchy distribution
central limit theorem
channel routing
character
group
characteristic
function
function
correlation
matrix
94
density function
diffraction grating
diode quad multiplier
dipole
- moment
- vector
distortion
free
line
distribution function
distribution density
Doppler effect
dyadic correlation
- group
- rational
Eigenfunctions
electrically short
energy distance
187 ensemble
average
183 equal gain summation
ergodic hypothesis
error correction
45
- detection
206
- function
197 Euklidian space
ce
26
152,214
212
correlation coefficient
120
coset
135
cosine channel
covariance
211
crosstalk
}
4205146
- attenuation
105
— matrix
94
222 Delay
average
balanced
continuation of functions 27
188
continuous variable
Fermi
statistic
188
222
78
1'70
161
161
87
184
188
172
53
26
23
55
87
286
O47
248
247
279
279
196
183
218
circular polarization
closed systems
coaxial cable
194
formants
-,Ssequency
914
221
169
12
88
code
fourth method of SSM
frequency channel
- diversity
141
ics
238
159
-
162
185
- filters
—- limited
270
—-
modulation
Wz
1"
255
+
-
shifting
synthesizer
theory
modulation
collision warning
combination
compandor
completeness theorem
complete systems
compressor
compression of information
conditional
division
- tracking filter
function detector
probability
186 function Limited.
45
62
56
58,249
“Sp
181
76
eye
447
279
INDEX
ey
Gaussian
distribution
geometric optics
group code
- delay
- theory
Haar-Fourier
half adder
196
number
215
476
Nyquist
transform
pe
-,
46 -,
PIS -,
78
278
Hankel function 191,201,203
Hermite polynomials
18
Hertzian dipole
163
signal
110
incomplete system
Nee
integral,
Walsh function 164
integrator
He
intelligent interference 225
intersection
ASS
interval
a5
Johnson
joint
2ny
rate
82
280 On-off system
261 Open wire line
NZS operator, differential
Hall multiplier
Hamming distance
Image
theory
noise
218
distribution
185
87
55
55
54
54
eigenfunctions
linear
time variable
optical
telescope
260
optimal
selection
248
orthogonal
outphasing
Parabolic
division
method of
SSM
62
141
cylinder
functions
18
parabolic reflector
laa
parameter integration
199
parity check digit
278
Parseval's theorem
2
partial response
84
PCM
Ari owsalinese}
periodic continuation
29
phase channel
SiS)
- modulation
AIS)7
-
Legendre polynomials
linear independence
263
shift
method
ANE
9, 38 — jumps
Suryai) Plancherel
- operator
Lerentuce pranstormab
Lom
lower sideband
1/4
108
Marginal,
185
distribution
mathematical expectation
maximal ratio
summation
189
248
Maxwell's equations
mean square deviation
mean value
160
194
191
85
theorem
14
Pointing's vector
162
polarized Walsh waves
164
power loading
85
prism
eon
probability,
axioms
184,185
-, defined
184
-
function
product
eS
of
random
variables
propagation time
polarization diversity
7o9
261
293
259
mixed moment
211
- vector
mobile radio
communication
167
Quadrature modulation
114
modified Hankel
V2oe 150
function
1915201
modulation index
AISI». Aa? quotient of random
variables
204
modulo 2 addition
Zone
moments
1911
19,1271
multiple access
268 Rademacher functions
166
multiplication theorems
22 radiated power
multipliers
ee tes) radiation resistance 163,165
radio
Near
zone
AGE Mee Cree
Neumann functions
non-synchronized groups
normalized
systems
204
127
communication,
mobile
raised cosine
radar target
6 receiver
pulse
filter
GH
Sener d
180
24
520
Target analysis
180
tracking
180
transorthogonal
68
, 280
87
oo telegrapher's equation
84
y, teletype transmisstion
85
APY) TELEX
relativistic mechanics
random alphabet
- variable defined
Rayleigh
172
289
135
distribution
rectangular reflector
Reed-Muller alphabet
resolution range
resolvable angle
rise time
roll-off factor
Sampling
theorems
scalar potential
Schmid multiplier
second method of SSM
seus
sequency allocation
- bandwidth
— definition
-
formants
- limited
280
174
7S
1241
thermal
noise, definition 218
third method of SSM
141
time base
51542
Za
- division
time-frequency—domain
-
diversity
248
61,1730
249
Fag ae as time-function-domain
160 time-sequency-domain
79 time-shifts
57 topologic group
184 transposed SSM
124 two-dimensional filters
99
50 Uncertainty relation
unsynchronized groups
OAeeed upper sideband
58,249
249
249
167
26
444
105
25
128
108
aliclizs Variance
193
160
Se vector potential
- shifting
181 - representation
62
- spectra
101 voice signals
90
- tracking filter
bec vocoder
91
Signal classification
45 voltage comparison
229
SSM
145
- detection
2e5 vestigial
- delay
261
- space
62, (63.0 ao Walsh functions, integral 164
shift theorem,
sine 148,168 - multiplier
(Oy eee
-, Walsh
25,148,150 - tracking filter
154
Single sideband
1075108 - waves, polarized
169
simultaneous
wave equation
89,168
transmission
83 - guide
250
Sine channel
155 - optics
AS
skin effect
88 — zone
1671, 162
Space diversity
238 weak convergence
35
- probe
“7 wideband antenna
165
special shift theorem
149 Wiener-Chintchin theorem
17
speech analysis
a
standing wave
“AO zps defined
50
statistical independence 186
-
multiplexing
- response
- variable
Student distribution
sum of random variables
Superconductive
cable
184
206
196
88
supergroup
switched telephone
network
synchronization
125
Systematic
code
85
ak
279,280
Date Due
LATTE
EET
Demeo
38-297
217.2
Hast
Aarmuth, Henning
F, _
Transmission
of
in-
formation by orthogon
=e
a
517.2
H28t
HUNT LIBRARY
CARNEGIE-MELLON UNIVERSITY
PITTSBURGH, PENNSYLVANIA
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on