COMMUNICATION
THROUGH NOISY,
RANDOM-MULTIPATH
CHANNELS
by
GEORGE LEWIS TURIN
S. B.,
Massachusetts
Institute
of Technology
(1952)
S. M.,
Massachusetts
Institute
of Technology
(1952)
SUBMITTED
IN PARTIAL
REQUIREMENTS
FULFILLMENT
FOR THE DEGREE
OF THE
OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS
INSTITUTE
June,
Signature
.
of Author
1956
=D"'-e-p-a-r~tm-e-n~t-o-f"'1-::::1:l-e---c"""'tr""'i:-?~1-'E=--n-g"""in-e-e-r--:i-n-g-,
--=-M"="a'-y--:'1-"4-=,
~i~9~56
c
Accepted
by
OF TECHNOLOGY
~~:-------=-~-r-~"--~~~~--=-~--:---=------:::---:---;::-:,---::-:---
Chairman,
1/
Th#'is
Supervisor
,.
DEPARTMENT
OF
ELECTRICAL
MASSACHUSETTS
CAMBRIDGE
ENGINEERING
INSTITUTE
39,
OF
TECHNOLOGY
MASSACHUSETTS
May 7, 1956
Mr. G. L. Turin
Lin - C377
Dear Mr. Turin:
This letter is to give you permission to print additional copies
of your thesis by the Multilith process, and to submit to the Department
of Electrical Engineering copies thus produced, in lieu of the typed
-copies normally required.
A copy of this letter is to be reproduced by the same process and
is to be placed in each copy of the thesis immediately follo\nng its title
page.
Sincerely yours,
# /'
/~
Department
N
f'Ii
'~
'V
'()
.,
w
SHC:mm
'Hr. -C~dwell
for the
Graduate Committee
'..J
ii
COMMUNICATIONTHROUGHNOISY, RANDOM-MULTIPATH CHANNELS
by
GEORGE LEWIS TURIN
Submitted to the Department of Electrical Engineering on May 14, 1956
in partial fulfillment of the requirements for the degree of Doctor of
Science
ABSTRACT
Statistical methods are applied in this paper to the problem of communication through a multipath channel which has random (or unknown)
path characteristics,
and which has additive random noise present at
the receiver end. In an introductory chapter, the transmitter of a system for use with such a channel is defined as one which encode s the ou.tput of an information source in.to a sequence of selections from a finite
set of me ssage waveforms, and transmits this sequence into the channel.
The receiver is specified as one which, on reception of the channelperturbed transmitted signal, computes a posteriori probabilities of the
possible transmitted message~waveform-sequences,
and, on the basis of
these, supplies guesses at the information source output to an information
user.
The first problem with which the paper concerns itself is that of establishing an a priori statistical model of the channel. It is shown that
this a priori model is often in.adequate, and measurement techniques for
obtaining further (a posteriori) information about the channel are discussed.
Next, the problem of determining the operational form of the receiver's
probability computer is investigated.
Si.nce this form depends on the amount
of information about the channel which is available to the receiver, several
results are obtained, one for each of several assumpti.ons concerning the
state of the receiver's knowledge of the channel. Probabilities of error corre sponding to two of the se probability-computer
forms are evaluated for the
special case in which there are only two equi.-energy, equiprobable message
waveforms and only one path; and for which the receiver makes its guess,
for each message waveform in a sequence, by choosing the waveform which
is a poste riori most probable.
-The problem of generation of an optimal set of message waveforms is
then considered, in particular, for the spe,cial case described above. In
this case, the optimization condition consists in the adjustment of the crosscorrelation coefficient of the two message waveforms.
A commentary on possible future extensions of the present work concludes
the paper.
Thesis Supervisor: Dr. W. B. Davenport, Jr.
Title: Assistant Division Leader, Lincoln
Laboratory, M. I. T.
iii
ACKNOW LEDGMENTS
It would be impossible
Dr.
W. B. Davenport,
have found in Dr.
for me fully to express
Jr.
During
Davenport
a teacher
deep understanding,
and a sincere
has been of constant
inspiration
deeply
of this thesis,
Of particular
of those
I
a counselor
As supervisor
and encouragement.
stimulating
indebted
of
of this thesis,
For
connected
the course
Prof.
J.B.
with the application
of this research,
all this;
he
I am
closely-related
discussions
Wiesner,
were
Mr.
and suggestions
My thanks
laborious
were
greatly
D.B.
techniques.
R. Price,
who has been
I should
be happy to
to him as they were
and Drs.
in this work.
and
to me.
M. Balser
Their
and
corrllnents
appreciated.
connected
Bergstrom,
Dr.
Selfridge,
interest
are also due to Miss
computations
of several
I had many enlightening
as helpful
W. L. Root have shown a generous
of his time and advice.
of the forms
problems.
O.G.
who has been a guiding
of matched-filter
with my good friend,
similar,
think that these
R. M. Fano,
value were his predictions
discussions
investigating
to Prof.
and who has given me freely
heuristic
results
During
Mrs.
of our association,
of las ting patience,
friend.
to
grateful.
I am also greatly
spirit
the four years
my indebtedness
C. L. Kemball
with Figures
Miss D.M.
Biggar,
for performing
4-1;
4-2,
and Miss
the
and 5-1; and to
M.E.
Woodberry
for
typing this report.
Finally,
supporting
I should
like to thank M. 1. T. I s Lincoln
this research
under
joint contract
Laboratory
with the Army,
for
Navy,
and
Air Force.
May,
1956
G. L.
T.
iv
CONTENTS
Abstract.
ii
Acknowledgments
iii '
List
of Complex
List
of Importa::"lt Symbols
CHAPTER
CHAPTER
I:
II:
Functions
Model
of System
B.
System
Design
C.
Notation.
Complex
2.,
Fourier
3.
Probability
CHAPTER
CHAPTER
V:
VI:
5
of physical
and probability
density
9
10
Knowledge
10
Q
2.
A model
for the multipath
noi se .
10
medium
..
12
Kno,,vledge.
17
A posteriori
distribution
medium ..
of characteristic
of
THE PROBABILITY
A.
(T.) Known:
(<5.)
B.
(-r.)
Kno'Nn:
(<5.)
C.
(T.)
Unknown.
1
1
1
s
17
Min:imum-mean-square-error
impulse
response
of medium...
PROBABILITY
..
.
A Posteriori
7
9
for the additive
1
waveforms.
transforms'4
A model
1
1
.
1.
2,
IV:
Problems
representation
THE CB.ANNEL
A Priori
to be Considered.
7
1.
1.
CHAPTER
1
A.
B.
III:
vii
INTRODUCTION.
A.
CHAPTER
vi
estimation
of
24
.a
COMPUTER.
32
Kno'wn
35
0
UnKnown.
OF ERROR:
38
40
SPECIAL
CASE
•
A,
T
Known:
<5 Known.
46
B.
T
Known:
<5 Unknown.
55
THE MESSAGE
WAVEFORMS:
SPECIAL
CASE
.
57
A.
T
Known:
0 Known
57
B.
T
Known:
<5 Unknown.
62
CONCLUDING
REMARKS
63
v
CONTENTS (Cont~
Appendix I:
THE COMPLEX CORRELATION FUNCTION.
.
.
69
Appendix II:
A POSTERIORI DISTRIBUTION OF MULTIPATH CHARAC TERISTICS . • •.
. . . . . . . . • • . . .
•
71
.
.
.
.
Appendix III: MINIMUM-MEAN-SQUARE-ERROR
MEASUREMENT OF
AN UNKNOWN IMPULSE RESPONSE . . .
75
Appendix IV: DERIVATION OF LIKELIHOODS
81
Appendix V:
.
.
•
.
.
CHARACTERISTIC FUNCTION OF A QUADRATIC FORM
OF GAUSSIAN VARIABLES . • • • • • • . • • .
Appendix VI: EVALUATION OF MATRICES OF EQUATION (4.08).
Appendix VII: DERIVATION OF EXPRESSIONS FOR P e.
.
•
83
86
91
Appendix VIII: DERIVATION OF EXPRESSION FOR pI
94
Refe rence s.
97
e
.
•
.
.
•
.
.
.. .
.
.
.
.
.
.
.
vi
LIST OF COMPLEX FUNCTIONS
Function
Definition
received
Page Introduced
signal
output of multipath
medium
output of multipath
medium
hypothe sis that;
noise
m
:: g(T)
e
-J"2TTf
0
T
th
m
14
on
34
(t) was sent
waveform
sounding
~(T)
17
11
17
signal
message
waveform
32
auto-correlation
function
of ;(t)
auto-correlation
function
of ~
cros s -correlation
m
18
(t)
81
function
of s(t)
18
function
of s(t)
34
and ;(t)
~
m
(T)
=
g
m
(T) e
-j21Tf
0
T
cross -correlation
and;
m
(t)
vii
LIST OF IMPORTANT SYMBOLS
Definition
Symbol
Page Introduced
a.
strength of ith path
13
D
decision variable
46
E
energy of sounding signal
th
energy of m
message waveform
17
1
E
f
m
carrier
0
35
frequency of bandpass waveform
7
gmi
L
gm (Ti)
35
number of paths
14
M
number of message waveforms
N
white noise power -density
11
probability of error
45,55
0
P,
e
P'e
..
a prIorI
pro ba b'l'
1 Ity
P
0f
3
m th me s sage wave f orm
32
m
Pre
]
"probability ofll
9
pr [
J
"probability
9
density ofll
s.
strength of random component of ith path
13
T
duration of message waveform
33
T'
duration of output of multipath medium
71
W
transmission
11
W
bandwidth of noise
11
n.
strength of fixed component of ith path
13
f3mi
20-~E
"I'1
n./o-.
1
1
O.
phase-shift
E.
phase-shift
1
N
1
1
1
8..
1
A
1
m
IN
bandwidth
38
0
49
of fixed component of the ith path
of random component of ith path
pha se - shift of ith path
likelihood corresponding
13
13
complex cros s -correlation
message waveforms
Am
13
coefficient of two
to m
th
message
waveform
48
34
0-.
(1/{Z)
times r.m.s .. strength of random component of ith path
14
T.
modulation delay of ith path
12
1
1
real part ofll
It..
II
N
"imaginary
part of"
7
7
CHAPTER I:
In recent
through
years
the problem
randomly-disturbed
mental
INTRODUCTION
of the design of systems
channels
has been approached
point of view than was previously
the essential
feature
of the problem
and that,
hence,
statistical
vanguard
of this approach
for communication
employed.
a more
It was recognized
is the statistical
methods
from
nature
must be used in system
fundathat
of the disturbance,
design.
The
include d Wiene r ( 1), Shannon (2, 3), and Woodward
and Davies (4, 5), to name but a very few.
Attention
dis turbed
has heretofore
channels,
that is,
one added to the transmitted
cation
through
other hand,
to consider,
through
channels
received
The more
little
point of view,
investigations
to that of Price
Model of System
The generic
1-1.
a transmitter,
See especially
additively
from
dis turbance
problem
has,
on the
of. this paper
of communication
a random-multipath
transmitter
to receiver
characteristics.
by way of
The channel
added at the receiver
to this type of channel
The present
in some aspects.
is
of communi-
distur.bed
the problem
with the noise
relating
on additively-
It is the purpose
channel:
7, 8) and by Root and Pitcher(9).
related
in Figure
may travel
to be noisy,
difficult
attention.
which have randomly-distributed
statistical
by Price(6,
+
signal.
relatively
will also be considered
A.
in which the only random
which are not purely
in which a signal
exclusively
channels
from a statistical
many paths,
closely
almost
one type of non-additively-disturbed
channel,
Other
been focused
work,
end.
have been made
in fact,
is
+
to be Considered.
model of the communication
Like Shannon's
a channel,
Chapter
system
model (10), it consists
a receiver,
III, Sections
to be considered
of aninfor:mation
and an information
A and B.
user.
is shown
source,
We shall
take
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•.-1
lit
- 3 the information
of symbols
English
source
to be one which produces
drawn from a finite alphabet;
text.
as its output a sequence
it may be, for example,
We define the function of the transmitter
encoding of the information
source
to be, first,
output into a sequence
which are drawn from a finite set of finite-duration
forms;
and,
into the channel.
the transmission
The set of message
set of pulsed sine -waves of different
telegraphy.
In the channel
or signal,
is distorted,
medium,
of this message
waveforms
first
and then by the addition of random
The received
signal, then, generally
unequivocal
indication
The receiver
through
on the basis
This guess,
or perhaps
a
sequence,
a random-multipath
input.
the receiver
message-waveform
situation
of some operation
a set of guesses,
sequence
noise at the receiver
must make do with an imperfect
sequence
-waveform
wave-
as in frequency-shift
does not provide
of the transmitted
transmitted
message
message-waveform
by transmission
wave-
may be, for example,
frequencies,
the transmitted
the
of message
forms,
second,
a printed
with an
sequence.
by guessing
at the
on the received
is presented
signal.
to the inforrration
user.
A closer
its division
encoder
.mation
analysis
of the encoding function
into two parts.
of Figure
source
output,
of say Q letters,
say M letters.
I-I,
consists
Possible
of alphabet
detection
or error
size;
purposes
translation
of symbols
of new symbols
in which written
to
by the
of the infor-
drawn from an alphabet
drawn from an alphabet
of this translation
of redundancy(ll),
symbols.
leads
which is performed
in the one-to-one
reduction
-correction
code,
of these,
which is a sequence
into a sequence
reduction
is the teletype
The first
of the transmitter
An example
text is translated
of
may be, for instance:
insertion
of the first
of errorpurpose
into a two -symbol
-4 ("mark"
and "space")
optimum
code of Huffman (12).
by Hamming(13).
alphabet.
frequencies,
An example
Generally~
ledge of the information
etc.),
The second purpose
the encoder
source
Up until now we have considered
may
ink-marks,
and as such are hardly
The second part of the encoding
of a set of M distinct
the abstract
operation,
chosen
symbols
to combat
transmitter
selection
is concerned
which require sa
waveforms
From
with the generation
consists
in the replacement
by redundancy
of the natural
of a kind more
a random
of the encoding
of a suitable
the function
the entire
suitable
of the transmitter
For
but of different
contains
redundancy
in the
a trivial
effectively.
of the channel,
point of view,
the
will do; they must be
phases~
phasefunction
of the
"codebook"
This is done by the message-waveform
the redundancy
In this sense,
through
This is by no means
the second part
abstract
to take the place-of
effect of the channel
s tatis tical knowledge
That is~
for transmission
if the channel
Essentially,
use by the encoder.
.source
alphabets.
a set of sine waves of the same frequency,
device.
sequence
1-1.
are so far merely
set of waveforms
the perturbing
would be an absurd
shifting
physical
know-
digram
function.~ then,
of the M-alphabet.
for not any arbitrary
example,
sists
eligible
a statistical
frequencies,
only abstract
of both the M- and Q..;alphabets
generation
require
in Figure
the symbols
channel.
by the
of the third is given in a paper
(e. g. ~ letter
and this is indicated
is illustrated
for
generator,
as shown in Figure
encoding
operation
con-
of the information-source
for use with the given channel.
is to "match"
the information
to the channel.
The function of the receiver
The first
of these is postulated
probabilities
of the various
may also be divided into several
to be the computation
possible
message
1-1.
parts.
of the a posteriori
-waveform
sequences.
As is
- 5 noted by Woodward and Davies (4), all of the information
signal
is implicit
reduces
there
in these probabilities,
the available
data (i. e.,
is no loss of information
computation,
knowledge
possible
the probability
of the source
probability
requires
computer
a receiver
input--for
to guess
example,
printed
knowledge.
is hence a sequence
sequence
as available
text.
a statistical
of the .various
and a copy of
at the output of the
to the information
Thus,
message-waveform
user;
the receiver
sequence
This is done by the decision
of symbols
from
the abstract
sequences.
the output of the decision
(or sequences)
exact inverse
B.
the
the user
is called
on the basis
circuit,
of its
whose output
M-alphabet,
This guessing
upon
or perhaps
operation
of course
a loss of information.
Finally,
mation
through
waveforms.
useful
English
a set of a few highly-probable
involves
form;
output which is in the same form as the transmitter
at the transmitted
~ posteriori
probabilities
distribution
is not in itself
to carry
and of the channel,
of message
merely
to an alternate
must have available
the a priori
~ posteriori
signal)
In order
sequences)
Itcodebooklt
The complete
the received
computer
message-waveform
the transmitter's
so that this computation
involved.
(i. e.,
in the received
of Q-alphabet
of the encoder.
circuit
symbols
The decoder
is translated
back into a
by the decoder,
output is supplied
which is the
to the infor-
user.
System
Design Problems.
The above description
automatically
brings
of the functions
to mind certain
of the transmitter
questions,
and receiver
which may be phrased
follows:
1) How does one design
the encoder
(and hence the decoder)?
as
- 6 -
- 7 of information
probability
about the channel
of error
will be derived
special
case.
results
obtained,
C.
Notation.
1.
Complex
Finally,
sentation
of a complex
.
.d a 1 carrIer:
.
CISOI
=
x(t) e
case,
"''''
~(t) = x(t)
to a commentary
on the
investigations.
representa-
(14)
to us is that relating
waveforms.
This is essentially
of representing
The part
to the reprea generali-
A cos (2nft+<t» by A ej2nft,
a narrow-band-pass
modulating
waveform
waveform,
of
where
is represented
x(t),
and a
.2nf t
J
0
(1. 01.)
or power-density
the actual
V for this
waveforms.
concern
low-pass
of
+
fo is a suitably-defined
energy-
in Chapter
by Woodward.
practice
forms
The question
which is described
More generally,
as the product
case.
use the complex
of narrow-band-pass
The
probability-computer
for the most part,
which is of immediate
A is complex.
for future
of physical
waveforms
zation of the familiar
~(t)
VI will be devoted
and to suggestions
we shall,
special
at the receiver.
will then be discussed
Chapter
representation
tion of physical
to two of these
IV for a simple
generation
In this paper
this notation
corresponding
in Chapter
message-waveform
which is available
physical
carrier
frequency-
spectrum
waveform
-for example,
of the waveform.
is represented
the centroid
of the
As in the cosinusoidal
by the real
part
of ~(t):
cos 2nf t - ~x(t) sin 2nf t
o
0
... Following Woodward, we shall
waveforms
and English letters
use Greek letters for complex band-pass
for low-pass
modulating waveforms.
(1.02)
-8where
we have used a circumflex
to denote
"imaginary
part
(,,) to denote
of".
way of writing
x(t) represents
part
(1. 02) is,
Equation
of" and a tilde
in fact,
time (15); (1. 01) is merely
which has been in use for some
and compact
"real
a representation
a more
convenient
it.
both an amplitude
and phase
modulation.
Its magnitude,
/,,2
2
Ix(t) I = V x (t) + x (t) =
N
is the amplitude,
4 x(t) =
the phase
£(t)
where
waveforms
Il/J(T)
of lJ1(t)
denotes
A
"-
I
function of two complex transient
waveforms,
.2 ni t
.
y{t)eJ
0,
is defined as the complex function
e
-
S
j2nfT
0
"complex
is twice the cross
£(t) and 1(t).
I=
=
S *(t) ~(t-T)dt =
the asterisk
part
(1. 0 3b)
of the carrier.
=
S
and its angle,
"(t)
x
The cross -correlation
.2 ni t ,x(t)eJ
0
and ~(t)
=
(1.03a)
-1 ~(t)
deviation,
lJ1(T)
real
or envelope,
tan
S
Similarly;
x *(t) y(t-T)
dt
See Appendix
1.
*
x (t) y(t-T) dt
conj;:..tgate".
-correlation
function
the magnitude
(1. 04)
It is easily
shown
of the actual
* that
the
physical
of lJl(t),
I
may be shown to be twice the envelope
$
(N)
(1.05)
of the physical
c:::,oss-correlationfunction.$
- 9 We shall find it necessary
waveforms.
In complex
that S(t) contains
< fc
Sk
+;-.
notation
no frequency
> '-; ).
(Ie
5 IWll2
this takes
components
Then S(t) is completely
== S(~) taken at intervals
w
to use the sampling
dt =
of
.Jr seconds.
theorem
the following
outside
for band-limited
form.
of the band f
specified
(14) Suppose
c
by complex
-
W
y<
f
samples
Furthermore,
I l~kl2
(1.06)
k
2.
Fourier
transforms.
In this paper
domain
variable
transform
we shall use Fourier
is a cyclic,
rather
transforms
than a radian,
in which the frequencyfrequency.
That is,
the
pair
x(tl =
5
00
j2Tlft
X(fle
elf
(1. 07a)
-00
00
. X(f)
=
5
x(t)e
-
'211ft
J
dt
(1. 07b)
-00
will be used,
where x(t) is the time-domain
function,
and X(f) its frequency-
domain mate.
3.
Probability
and probability
We shall use the notation
"probability
subscripts)
density
of x",
density.
Pr[x]
and shall
to denote particular
for "probability
reserve
probability
of x" and pr[x] for
the letters
(density)
P and p (usually
distributions.
with
- 10 -
CHAPTER
We now turn our attention
II: THE CHANNEL
to answering
chapter,
uIn what form is statistical
receiver
and transmitter?"
knowledge
We may partially
form
of probability
distributions
very
satisfactory,
for one is immediately
characteristics,
Knowledge
posteriori.
The former
channel
labelled
A. A Priori
the
of knowledge
dispose
of the multipath
medium,
flat power-density
for example,
but this is not
"What are the pertinent
distributions?"
This chapter
ignorance
~ priori
model of the channel;
of the channel,
". ~ A posteriori
in the order
and thus be
knowledge
of channel
and ~
is based
characteristics.
on
We
mentioned.
noise.
of the question
end of the channel by assuming
the transmis
"In the
Knowledge.
Let us first
ceiving
reflect
on measurements
1. A model for the additive
receiver
led to ask,
to the
que stion.
"misknowledge
types
available
this by stating,
type may be based, on a physical
i. e.,
tV/O
answer
characteristics",
probability
of the last
of the channel
may be divided into two types:
it may merely
~ priori
soundings,
shall discuss
the latter
of the channel
on the other hand,
better
of channel
and what are their
is devoted to answering
the fifth question
that the noise
and is statistically
spectrum,
sian band.
of a model for the additive
If vie are
considering
is statistically
stationary,
at leas~ over a range
Gaussian,
of frequencies
a radio
and receiver,
and to the shot noise
noise
in the receiving
at the
independe'nt
and has a
which covers
communication
such a model might carre spond to the thermal
antenna
noise
system,
in the retubes.
*'
This will be discussed later in more detail; suffice it to say now that the
design of the receiver
and transmitter
depends not on what the channel actually
is , "but on what.the receiver
and transmitter
think it is.
.
i'
- 11 We may, without loss of generality,
spectrum
assume that the noise power-density
is constant and equal to No (watts/c. p. s.) over a bandwidth W ,
N
which at least covers the transmission
bandwidth W, and is zero elsewhere.+
Then from the sampling theorem of Section I. C.I,
completely
seconds.
specified by complex samples,
a noise waveform,
-Vk, taken at intervals
V{t), is
of I/W
N
The real and imaginary parts of each sample are Gaussianly
buted, and m~y be easily shown, using a result
and to have a common variance
of Rice (15) , to be independent
of WNN . Furthermore,
o
lation fun.ction of the noise has zeros every
Gaus sian number s are independent,
since the auto-corre-
1/WN seconds,
and uncorrelated
the sample s are independent of one another.
Then, a sample of noise T seconds long is specified by approximately+
independent complex samples,
independent.
whose real and imaginary
The joint probability-density
=
distribution
t?
TW N
parts are Gaussian and
of the se sample s is thus
N
TW
~l
12
!Yk
Using equation (1. 06), we may write for the "probability-density"
v/aveform, v(t),
distri-
J
(2.01)
of a noise
of duration T:
(20 02)
*'
Strictly speaking, the noise cannot be truly random (L e., unpredictable from
its past) under these conditions (cf. reference 1, Sec. 2.4). Practically speaking,
however, vie may ignore the inherent predictability,
for it would be difficult, if
not i!l1poSsible, to utilize.
+ + We say "approximately", for a waveform cannot be simultaneously of finite
bandwidth and finite time duration.
For T » 1/W N' the approximation is very good.
.. 12 Equation
noise.
2.
(2. 02) is sufficient,
We assume
for our purposes,
that N
A model for the multipath
sub-path
medium.
the multipath
paths ", which we shall
ik, indicates
medium
group together
is defined by a strength"
S(t), is transmitted,
in terms
in a certain
bik,
that we are considering
the k
th
of elementary
"way to form
and a delay,
the output of the sub-path
t ,
ik
"paths"
+
is bik S(t-\k)'
by amounts
of the transmission
differ
A
0
if a signal,
(The subscript
th
of the i
path.)
sub-path
v/hose delays
than the reciprocal
"sub-
such that',
is in turn defined as a group of sub-paths
much less
the additive
is known.
o
We shall describe
to. characterize
from
A path
one ano~her
bandwidth,
W.
+'h
The contribution
its sub-path
of the i ~... path to the multipath-medium
contributions:
e
where
we have written,
x(t) is a low-pass
less
than
all k.
output is the sum of
j2r:f (t-t.k)
o
1
as in equation
function
1
W ' we may,
(2. 03)
(10 Ol~,
S(t)
=
x(t) ej21Tfoto Now,
vib:.ich does not vary appreciably
to a very
good approximation,
since
over intervals
set x{t-t )
ik
=
x(t-T )
i
much
for
Then ~i (t) becomes
(2004a)
We shall
quantity
call
T.
which,
1
the modulation
practically
We shall assume
that,
delay of the ith path.
speaking,
generally,
It is a vaguely-defined
may be set equal to anyone
the terms
of the summation
of the tik's.
of equation
+This implie s the as sumption that the multipath medium is linear and its
physical properties
do not vary appreciably
across the transmission
band.
- 13 (2. 04a) are of two types:
those terms
and those for which bikand
over these
two types
=
".(t)
-'1
\k
1
are randomly
separately,
[ a.e' -J'o'1
X(t-T.)
~
+
of the i
th
terms
.. 1
s. and
E.
1
(see Figure
2-1): the resultant
and random
components
1
are
call the fixed component
The second
randomly
term
time-varying
we shall
quantities.
vectorially
of the fixed
is a vector
of length
Correspondingly,
(2. 04b) may be rewritten
(2. 04a) in the form:
(2.04b)
1
component;
1
Summing
I
in equa.tion (2. 04b) we shall
1
e..
quantitie s.
ra~dom
may be represented
a.1.and phase
equation
for a. and O. are fixed quantities.
path,
call the random
These
term
time -varying
-J'E1'] e J'2iTfot
s.e
•
and t
are fixed quantities,
ik
ik
we may rewrite
Ued
The fir st bracketed
for which b
equation
as
FIGURE
2-1
YJ i (t)
(2.04c)
We shall
call a. the strength,
We have thus reduced
acteristics:
e.,1
a.,
1
we may therefore
of.the three
and
T .•
1
describe
pr [ai'
e i'
of definition
1
These
th
the i
of
in speaking
T.,
of
T.
1
we may consider
111
to be fixed.
are
phase-shift,
generally
path in terms
higher-order
random
This will prove
functions
that,
and
distributions
for the purposes
joint
char-
of time,
of probability-density
of this
distribution,
distributions.
e.1 separately,
variations
in e.
and
ith path.
6fthe
th
of the i
path to that of three
to know only the first-order
; we shall not require
'*'We are justified
T.
+
carrier
It will be seen later
it will be sufficient
T i]
e.1 the
our description
characteristics.
paper,
sidering
and
1
for,
(i.
e.,
to be a convenient
because
of the manner'
in the t.k's)
technique.
while con-
- 14 In establishing
write
a., T.]
pr [a.,
111
"fixed"
[TJ1:
= pr
i, the random
pr [a., a./T]
113:
over the interval
i,
'iT»
(-'iT,
*
0
of amplitude
~ow,
is Rayleigh
distributed
(io eo,
random
with mean
distributed
si'
square
evenly
Rice (16) gives an expre ssion for the joint
0'£ the sum of a fixed vector
and' phase
with Rayleigh-distributed
we first
in such a w?-y that the strength,
is completely
E
distribution,
We then as surne that for a
combine
path-component
and its phase-shift,
distribution
for this first-order
sub-paths
T
of the ith random
2CT~;
the expression
amplitude
and completely-random
and a vector
phase;
using
th 1S we may wr1't e: *+
O
a.
1
--:--z
21TCT •
1
pr
[a.111
, a.1 T.] =
(2005)
o
The dependence
a ., CT., 0..
1 1 1
of pr [a.,
1
We shall leave
until a later
chapter
The multipath
described
medium
I
on
T.,
1
the question
if any,
will be through
of an a priori
will generally
say L of them.
above,
=
a./T.]
1 1
the parameters
distribution
for T., pr [T.] ,
1
1
0
L
~(t)
elsewhere
contain
The total
many paths
such as the one
output of the medium
will then be
L
~i(t)
i= 1
=
l
(2006)
i= 1
+This assumption
implie s that the quadrature
components
of the random vector,
Si, are independent,
Gaussian variables,
with zero means and equal variances.
These conditions obtain in many physical situations
(see page 16 for examples)
in
which the number of sub -paths is great enough for the central limit theorem to
apply.
++The marginal
distribution
VIII)isRayleigh
of the path strength,
a. {see equation(A8-8)of
1
for(a./<T.)=O, and is essentially
Gaussian of mean
111
CT~ for (n.ICT. )-.00 (Cf. reference
16). For curve s of the marginal
1
1 1
a function of n.1 CT., see reference
17.
1
1
Appendix
a.. and variance
dis.tribution
of
a.1 as
- 15 and the medium
I
first-order
(T.).
will be completely
distribution
of the three
We shall assume
1
described,
sets
for our purposes,
of characteristics:
that all paths are
conditionally
(ai),
by the joint
(Si)'
independent,
and
so we may
write
L
=
IT
(2.07)
pr [a.,III S./T.]
i= 1
In order
to complete
distribution
a later
the a priori
of the Ti's,
description
pr [{Ti)J; again,
of the medium,
\ve postpone
we need the joint
consideration
of this until
chapter.
One may perhaps
multipath
medium
path medium
unit height
obtain a greater
by referring
is depicted
insight
to Figure
on the assumption
and a \vidth approximately
The output of the multipath
medium
2-2,
into the above de scription
in which the output of a three-
that a pulse
is transmitted
equ.al to the reciprocal
then consists
tr ans mi tte d
pulse
T
FIGURE 2-2
of the
of three
which has
of the bandwidth.
pulses,
also
of width
- 16 approximately
phases
equal to
(relative
whose hei"ghts,
to the carrier
and S., respectively
exaggerated.
the output pulses
phase
(i=L 2, 3).
1
greatly
4-,
modulation
delays,
of the transmitted
The carrier
We shall postpone
until the next section
period
discussion
and carrier
pulse)
are a
has of course
ToJ
1
oJ
1
been
of the discreteness
(see the discussion
following
of
equation
(2.20».
We may establish
cription
the following' correspondences
of the multi path medium
path-components
may be attributed
stationary
refracting
scattering
from turbulent
layers.
of randomly-moving
diffraction
may,
regions
(either
from
correspondences,
from
of the validity
(2.05) in the ionospheric
through
our multipath
as radio
and tropospheric
cases
groups
model
transmission
fluid media.
of the probability-density
to
a randomly-moving
below or above the MUF) or via tropospheric
transmission
or from
may be attributed
(18), reflections
to such situations
The fixed
fixed objects
components
of a medium
the above des-
phenomena.
from
s (19), or diffraction
be applied
or to sonic or super-sonic
verifications
to reflections
Using these
in many cases,
physical
The random
reflector
screen(20).
the ionosphere
and actual
between
scattering;
Experimental
distribution
are reported
via
of equation
in the literature(21,
22, 23)
So much for the model of the multi path medium.
for although
we have established
and phase-shifts,
(0-.),
1
in an actual
physical
situation,
only quasi-stationary.
distribution
cription
are generally
of the
T. J
1
unknown a priori.
-:.-_-
vary
s will be unknown a priori.
of the multipath
-.:......_-
medium
is generally
of parameters
The parameters
slowly with time,
It is also likely
that the actual
Thus,
incomplete,
is quite evident,
for the path strengths
on the sets
,
1
and(2.07)
distribution
we see that this is dependent
and (0.), and these
in fact,
a conditional
Its failing
thus making
(n.),
1
may
(2.05)
probability-density
the a priori
and we are
phys,ical
reduced
de sto
- 17 one of the two alternatives:
"educated guessesll
distributions
either we may accept our ignorance,
and make
at the unknowns, i. eo, assign rather arbitrary
probability
to them; or we may try to measure the characteristics,
+
and (T.),
directly.
1
We shall consider the first alternative
(a.), (8.),
1
1
in a later chapter,
the latter in the next section.
B.
loA
A
Posteriori
posteriori
Knowledge.
distribution
of characteristic
Let us consider a measurement
s of medium.
system in which a sounding signal,
j2
x(t) e 'ITfot, of bandwidth W, duration T, and energy E, is transmitted.
received signal, t(t)
=
=
S(t)
The
z(t) ej21Tfot, is the sum of the multipath medium output,
~(t), and a noise waveform," (t):
t(t)
=
~(t)
+
(2.08)
1l(t)
We shall assume that T is small enough so that the multi path medium may be
considered fixed (i. e.,
(a.), (8.), (T.) constant) for the duration of the transmission.++
III
Because the measurement
is made in the pre sence of noise,
knowledge will not be exact; all we can expect is an a posteriori
characteristics
that the receiver
has an exact replica of S(t).
We denote this distribution
of the
by
5] .
Before deriving the expre ssion for this ~ posteriori
down expressions
distribution
of the medium, to be obtained by some operation on ~(t), assuming
pr [(a.), (e.), (T.)/t,
111
our ~ posteriori
for the auto-correlation
distribution,
let us write
function of S(t) and the cross-correlation
+The possibility of measuring just the parameters,
(aiL ((Ii), and (oi), of (2.07) may
be eliminated, for these may, in general, vary with time and thus ,:cannotbe measured
once and for all. If we are going to the trouble of repetitive measu~ements, however,
we might just as well attempt to measure the characteristics,
(ai)' (8i)' and (Ti), themselves.
++In the ionospheric case, for example,
of fractions of seconds or less.
this limits us practically
to T's of the order
- 18 function of ;(t) and ~(t). In complex notation (cf. equation (1.04»,
corre 1a t lon
o
°
1S
+
S
cj>(T) =
~* (t) ~(t-T)dt
The cross -correlation
S
1JJ (T) =
the auto-
~
* (t)
= e -j21TfoT
S
x *(t) X(t-T) dt
S
z (t) X(t-T) dt
(Z. 09)
is
-
;(t-T) dt = e j 21Tf
0T
*
(Z. 10)
The second part of (Z. 10) may also be written as
1JJ (T) =
g(T) e
-JOZ1TfoT
(Z. 11)
where g(T) is equal to the integral
containing z and x, which is a complex low-
pass function.
As we have noted in Chapter I, the real part of a complex correlation
function is twice the correlation
magnitude,
(Z.ll),
function of the physical waveforms,
twice the envelope of the physical
correlation
function.
and the
Thus, in
the real part
(Z.lZ)
represents
function of ~(t) and ;(t); and' 1JJ (T) I
A.
twice the correlation
the envelope of this correlation
'"
= Ig(T) I,
twice
function.
We may write the ~ posteriori
distribution
of the characteristics
of the medium
in the form
(Z.13)
+All
integrals
extend over regions
of non-zero
integrand.
--19 We shall
Bayes!
first
derive
an expression
for the second
factor
on the right.
By
equality,
pr
[(a.),
1
In this
(8.)/(T.),
1
expression,
1
pr [(a.),
(ai), (vi)'
a normalizing
of the a.
factor;
and 8.
fS
1
fS,
1
The remaining
function
of
So
1
l'J
1
s]
(2. 14)
We shall
-~--
of (2.07),
it ensures
1
is an a prioridistribution.
1
and (oi)'
1
[sf (Ti):~J
pr
1
that it is of the form
the parameters,
1
(8.)/(T.)]
1
assume
p r [( a .), (8.) / (T.)1 pr [s / (a .), (8.), (T.),
s] =
S,
and leave
the question
open for the moment.
that the integral
of (2.14),
of the values
pr [s/(Ti),
taken
s]
of
is just
over all values
is unity.
factor,
pr [s/(a ),
i
It is just the probability
(8 ),
i
(Ti),
that,
s] ,
from
vIe shall
call the likelihood
(2.08),
(2. 15)
given
~(t) in the form
the probability
of (2.06),
with (ai),
of ,1(t) , and assuming
(Si)'
and (Ti) known.
Using
(2.02)
for
that
(2. 16)
we easily
obtain
an expression
for the likelihood
function,
which,
when used with
(2. 07) and (2. 14),
L
TT
pr
[(a.),
1
(S.)/(T.),
1
1
S,
s]=
i= 1
o
elsewhere
(2.17)
~See Appendix
II.
- 20 In this equation
I
2
,
= ~i2 {
a.
1
,2
CT.
1
2
+
= [~~
1
+(1f
Ig(T.) 1
N
0
1
--z
v.
J
---z
=
1
+
a.
1
cos O.
1
the a posteriori
1
I
~
0
conditional
-
conditional
(a!),
(2.18c)
~(T. )
.1.
1
sets,
0
111
distribution
distribution.
(u!), and (o!),
old param eter s, and sampled
of (a.) and (8.) is of the same form
1
(2.05)
and (2.07».
depend only on the noise
value s of the cros s -correlation
and the replica
factor
R e [g(Ti) e -joi] of (2. 18a) will be recognized
sampled
1
(cL equations
signal
correlation,
of the sounding
at modulation
signal
stored
function
in the receiver.
from
delay T., in carrier
and g(T.) of (2. 18c) are twice the correlation
(2.12)
1 g(T.) 1 is twice
1
the envelope
the
of the incoming
Thus,
the
to be twice the cross-
o.;~
phase
similarly,
g(T.)
1
at T., in carrier
1
The new
power-density,
!
pectively.
(2.18a)
rr-
1
(J'.
parameter
R e [g(T ) e -jlii]
i
0
(j'.
-z
as the a priori
N
,..,
sin O.
1
Thus,
(T.
1
g(T.)
1
-1
1
2
(2. 18b)
a:
tan
}
2 a.
+
-1
1
0'
Z
1
phases
1
of the cross-correlation,
0 and ~, resG
sampled
at delay
T .•
1
Let us now return
to the question
of the a priori
parameters,
-
Because
meters
however,
choosing
of the equivalence
of the forms
may indeed be the ~ posteriori
there
has been no previous
the parameters
(a.),
1
(u.),
and (0.).
1
1
of (2. 07) and (2. 17), we see that the se paraparameters
measurement,
in such a way as to register
of a previous
we must
measurement.
If,
do the be st we can by
our ~ priori
ignorance.
Now,
.We here again invoke the property
of riarrow-band(cotrelatioil)
functions, tnat we may
speal~ of modulation delay and carrier
phase separately;
that iS,we may speak of the
correlation
at various phases for a given delay.
- 21 roughly,
we may think of the O".'s as measures
particular
is,
(T.
the more
uncertain
is our a priori
1
phase-shift
latter
of the a priori
ones;
00
indicates
(2.18)
(ai)'
(vi)'
complete
of the strength
a
and
and
(oJ)
depend
In this
a priori ignorance.
--=---
that the a posteriori
of the measurement,
in our knowledge
have u!«cr ..
1
From
1
2
2er. E
»
1
parameters
are
independent
then only on the measured
cross-
we hope,
will be to make a substantial
of the multi path medium.
(2. 18b),
this means
For this to be true,
in-
we must
that
1
(2. 19)
o
That is,
for a useful
(0".1 large),
ratio,
the larger
function.
The result
crease
=
iT.
1
we note from
correlation
knowledge
---
------
of that path;
case,
N
of this ignorance
1
~
measurement,
so that any a posteriori
.::0...
,must
either
_
our ~ priori
knowledge
is helpful,
knowledge
must
be small
or the signal-to-noise
be large.
o
Suppose
assumption
that (2.19)
is satisfied,
i. e.,
the measurement
is a useful
one.
Then
(2. 16) becomes
(2. 20)
The left-hand
side of this in.equality
fun.ction of the sounding
the auto-correlation
ment
greater
signal.
function
than the order
is the envelope
Now, this
of a signal
of
1
w.
signal
has bandwidth
of bandwidth
Thus,
(2.20)
of the normalized
W is small
will be satisfied
auto-correlation
W, and we know that
for values
of argu-
if
(2.21)
We conclude,
distribution
then,
that if the measurement
(2. 17) is valid
is u.seful at all (Leo,
only if the multipath
medium
is made
(2.19)
holds),
up of paths
the
whose
- 22 modulation
delays
differ
by amounts
too much of a restriction
ready
seen that,
delays
differ
less
as one path by vectorially
consideration
under
at least
to a good approximation.
of ~
This is not
for we have al(sub-paths)
can be considered
strengths.
(2.20) from
write
of the T.rS,
which can be derived
whose
~ priori
We can thus reduce
the number
in such a way that (2. 21) is automatically
Let us first
1
however,
a group of paths
than the order
of paths
We may also interpret
of (2.17),
purposes,
adding their
1
of W'
than the order
on the generality
for all practical
by amounts
greater
satisfied,
the point of view of resolvability
down the expression
for the a posteriori
in a manner
similar
of paths.
-
probability
distribution
to the derivation
+
of (2.17).
(2. 22)
C is a normalizing
configurations
constant
which ensures
of the T.' S, is unity.
To'S
is based
1
stored
require
replica
of the sounding
only the envelope
measurement),
of the sounding
the auto-correlation
signal,
(2.06),
If (2. 20) is satisfied,
In. particular,
of this cross -correlation
the cross-correlation
with different
(cf. equations
signal.
function
added to itself
delays,
(2.08),
ratio,
E/N
function
function.
for the case
(j'.~OO,
1
and (2.10)).
then the contribution
to the
II.
with the
all i, we
(2. 18a)).
(which is required
Ig(T)
of the
signal
Now,
for a good
I
I~~T
FIGURE
*See Appendix
1
several
once for each path
(2.09),
of the received
f
of the To'S.
distribution
(cf. equation
o
over all
distribution
-
upon which the a posteriori
high signal-to-noise
will be essentially
of (2.22),
is an a priori
1
is again the cross-correlation
for a reasonably
times
pr [(To)]
1
We see that the only operation
that the integral
2-3
- 23 cross -correlation
due to anyone
contribution
due to another
will consist
of L distinct
and hence
path will be essentially
path,
so that the envelope
pulses,
as in Figure
(2. 21), doe s not hold for a pair
i. e.,
pulses
in Figure
mente
But the two paths may also be considered
(at least
2-3 would merge,
to a good approximation),
equivalent
as sub-paths
so that our method
pair
of
by the Ineasuring
as distinct
equip-
of a composite
of grouping
the case for which we have no a priori
not even of the number
can find an effective
Ig(T) I.
the corre sponding
if (2.20),
those
sub-paths
paths
path
is
which can
by the receiver.
Now consider
medium,
On the other hand,
be unresolvable
to saying that we need only consider
be resolved
of the cross-correlation
2-3.
of paths,
nil at the peak of the
of paths
path number
which exist.
by counting
We may then use (2. 17) and (2.22)
of the characteristics
of these
effective
knowledge
of the multipath
By the above reasoning,
the nUIllber of resolved
to obtain an a posteriori
paths,
for condition
(2.21)
pulses
we
in
distribution
will automati-
cally be satisfied.
This reasoning
of course
that we can group sub -paths
accurate
1
to W.
to- group
in
of sub-paths,
however,
in a reasonable
we shall restrict
spectrum
holds,
way a priori,
assume
the paths will interfere
and destructively
at others,
medium.
constructively
as in Figure
nor to resolve
of width very
any discrete
of a continuum
are more
pulses
of paths
the power-transthan one path,
at some frequencies
2-4; that is,
close
- path case.
we consider
If there
2 -3 is an
we should be able neither
to the discrete
to (2.20) and (2.21),
of the Illultipath
pulses
that the possibility
ourselves
We have assumed
so that Figure
is a continuum
As a final point related
(2.21)
manner,
of discrete
We shall henceforth
does not exist;
mission
in a reasonable
too far.
is composed
sub-paths
Ig(T) I.
down if extended
i. e.,
representation,
If there
breaks
the medium
and
in the band
will be frequency
- 24 selective.
On the other
of the component
hand,
sub -paths
uency selective;
the interference
of a path is not freq-
it is for this reason
able to group sub-paths
and represent
path by the frequency-independent
(j
that we are
the resultant
parameters,
.. f
o ..
i'
1
FIGURE
We have noticed
posteriori
that the only operation
distribution
of the received
of the multipath
"matched,,(25)
to the sounding
same as the sounding
the convolution
=
of the sounding
can be performed
signal;
that is,
but reversed
function.
signal.
with a linear
filter
one whose impulse
in time.
In complex
which is
response
This is easily
filter
Now, as
is the
seen by writing
in terms
of its input
..L.
notation,
this is-r
J
Z1 '" S * (T) IJ. (t-T) dT
(3is the output;
let
=
;(-t)
S,
and ~(t) =
may be made physically
that is,
replica
to obtain an a
is the cross-correlation
giving the output of a linear
re sponse
where
~t)
signal,
integral,
and its unit-impulse
(3(t)
characteristics
signal with the stored
is well known, (5) this operation
which is required
2-4
setting
the in.put; and tJ., the impulse-response
i
lJ;(t), (2.23) becomes
realizable
tJ.(t)= ;(T-t).
be correspondingly
(2. 23)
delayed.
formally
identical
(;( -t) is not) by allowing
The times
at which the filter
For large ~
function.
, the envelope
If we
with (2.10).
tJ.(t)
a time delay of T seconds,
output is sampled
must
of the matched-filter
out-
o
put will look like Figure
2.
2 - 3 (four -path case),
Minimum-mean-square-error
Instead
of obtaining
of the medium,
*For
estimation
the complete
where
T
is now the time variable.
of impulse
~ posteriori
response
distribution
we might only be intere sted in obtaining
an explanation
of the factor
of 1/2,
cf. Appendix
of medium.
of the characteristics
some definite
1.
estimate
of
- 25 the characteristics.
a.'s,
111
8,.'s,
and
T.'S
the a posteriori
another
One type of estimation
which maximize
most probable
type of estimation
would be to find those
(2.17)
and (2.22);
characteristics.
that is,
values
of the
we may choose
We shall describe
in this
section
of which we shall call minimum-mean-square-error
estimation.
Let us think of the medium
let us try to e stlmate
ceived
signal,
its impulse
for the duration
that the medium,
the sounding
system
h (T),
e
6 (t)
a sounding
filter;
and
operation
on the re-
+ ,
of duration
x(t)
signal,
and hence its impulse
may be depicted
response,
as in Figure
signal as tm output of a linear
when a unit impulse,
response,
linear
stays
T.
fixed
of the transmission.
The measurement
X(T),
time -varying
by some linear
re sponse
when we have transmitted
We again assume
indicated
as a randomly
6(t), is applied
of the linear
estimating
X(T)
h
X(f)
H (f)
m
m
filter
to its input.
filter
2-5,
with impulse
We require
which makes
(T)
where
vie have
re sponse
the impulse
a minimum-mean.-
h (T)
y(t)
e
g(t)
H (f)
e
n(t)
FIGURE
square -error
estimate
estimate
of the impulse
is made in the presence
2-5
response
of additive
Suppose that we know that the impulse
inside
of some interval,
say 0 ~ T{ 6.
of the medium,
noise,
response
h
m
(T);
this
n(t).
of the medium
Then the expression
lies
essentia;lly
for the mean-square
.We abandon here the complex notation,
for the results in this section
applied more generally than to just narrow-band-pass
situation.
may be
-2.6 .. error
of the output of the estimating
filter
may be defined as
(2. 24)
where
g(t) is the estimating
over the ensembles
medium.
that is,
of possible
We minimize
noises
this error
and possible
by varying
=
the Fourier
=
(f)
opt
is with re spect to the estimating-filter
transform
the impulse
e
(2.25)
filter
is
N(f)
denotes
"complex
voltage-density
knowledge
spe ctrum;f}fm
(f) f 2, the average
power-density
-
However,
some delay in obtaining
X(f) is the
power -transmission
spectrum."
is
Thus,
the
2
1 H (f) 1 ; if we have
m
we set this equal to a constant.
may not be physically
of a realizable
not used in the derivation.
In this expre ssion,
II.
which one must have a priori
of the medium,
of (2.26)
response
conjugate
and N(f), the noise
of the medium
The filter
of h (f» which satisfies
of the estimating
(2.26)
A
of the medium;
no. ~ priori
function
impulse -re sponse.
1
the asterisk
only statistic
accept
impulse-response;
(2.25)
sounding-signal
function
of the
0
the variation
H
e
where
responses
the estimating-filter
It is shown in Appendix III that the transfer
(i. e.,
impulse
average
we set
OE
where
filte r outP'l:lt, and EN, M denote s a statistical
filter
realizable,
the condition
must be zero for negative
this is not a great
our estimate
because
of h
m
(T),
problem;
and H
eopt
that
arguments
was
we can usually
(f) can usually
+In the interest
of generality,
we shall temporarily
neglect our white -noise
tion; we shall assume,
however, that N(f) is known to the receiver.
be
assump-
- 27 made
realizable,
delay.
at least
to a ve ry good approximation,
This delay will usually
For the no-noise
case,
not be more
than the order
N(f) =. 0, the solution
i. e.,
by introducing
sufficient
of T seconds.
reduces
to the inverse
filter
He
=
(f)
opt
This is,
1
X(f)
of course,
output is H
Fourier
m
to be expected,
(f) X(f),
transform
and the filter
of h
m
(T).
for the voltage-density
of (2.27)
Thus,
restores
in the no-noise
spectrum
at the channel
this to just H (f), which is
m
case,
h
m
(T) is estimated
with-
out error.
For N(f)
be expressed
filter
1= 0,
comparison
(see Figure
with transfer
H (f)
c
where
r
2-6) as the cascade
shows that the optimum
of the inverse
filter
filter
of (2.27)
may
and a
function
=
(2. 28)
N(f)
=
IHm (f) 12
(Nr (f) is thus the noise
through
the average
power -density
medium.)
smears
X(f) is small.
nm(T).
spectrum,
referred
The output of the inverse
h m (T), but it also contains
uencie s where
so,
and (2.27)
vie have set
N (f)
tains
of (2.26)
a large
The second
The optimization
amount
filter
procedure
of noise,
attenuate
back to the transrnitter
filter
in Figure
especially
s the noise,
2-6 con-
at those
freq-
but in doing.
may be thought of as one which
- 28 make s the be st compromise
eliminating
noise
undistorted.
filter,
between
and keeping
is a trivial
phase
e
(T),
without
opt
the transmitted
is small
that the energy
2-6
of T, then the second
power-density
spectrum.
if the average
to N(f)),
term
in the denominator
If this
is small,
received-signal
then (2.26)
for
power-
becomes
which will occur,
knowledge
signal;
for example,
of the medium,
for taking
if the noise
then the filter
the conjugate
is
of (2.29)
of a frequency
is
spectrum
the time function.
So far we have considered
eopt
FIGURE
(2. 29)
with frequency,
to the sounding
implie s reve rsing
H
to
equivalently,
compared
white and we have no a priori
=
(f)
opt
*
r
H (f)
e
H
e
1 X (f)
6. Nr(f)
If N (f) is constant
matched(25)
----------J
which
of magnitude
r
=
(f)
1_-
1--1-
C
H_(_£)
I
~
the
helping
of the noise,
to N (f) (or,
spectrum
which
I
I
anyway.
is roughly
H
one,
would distort
is of the order
all f, compared
density
m
the effect
6.
of (2.26)
h
i
for any
a linear
exception)
has random
If
(except
output,
attenuate
-,
r-- ---- ----
(T)
c
phase
de sired
m
H (f) is a zero-phase
as one would expect,
other
h
X(f) to be arbitrary.
(f) , solve for the X(f) which minimize
in x(t) be fixed and that X(f) lie within
Now let us,
s
E,
subject
while keeping
to the constraint
a given band.
That is,
.
let
us set
00
S
-00
IXlf)
/2 df
=
K
(2. 30)
- 29 and
X(f)
where
=
0
for f not in F 1
F 1 is the permitted
o (E m +
=
~K)
band of frequencie
s, and solve the equation
0
(2. 32)
A is some constant.
where
(2.31)
(26)
E
m is the . minimum
mean-square
error
for an arbi-
tra:ry X(f).
The solution
to (2.32) is shown in Appendix
III to be
fin
X(f)
=
F2
e -j13(f)
(2.33)
o
In this equatlon,
fin
13(f)is an arbitrary
phase
function,
F
Z
F 2 is the set of all freq-
uencie s in F 1 for which
_1_ ~ IN(f)
()::
and. F
~
Z
1H
m
l:!.
(f) 12
is the set of all frequencie
s not in F 2'
sa~isfy the energy
,constaint,
in tui tive notion s .
The fir s t factor.
some frequency,
f1, is very
uency to determine
the noise
is very
energy
Hm(f1).
(2.30).
small,
small,
then it is a waste
at f1; the energy
If we place
The intjrpretation
[ N(f)
D.] 4,
of (2.33)
signal
hand,
energy
confirms
is needed
the second factor
or the average
power
of the limited
available
may be used to more
advantage
(2.33) in (2.26),
A., is adjusted
to
onel s
indicate s that if the noi se powe r at
then little
On the other
power at f1 is very large,
The constant,
we obtain the optimum
indicates
transmission
energy
at that freqthat if
of the medium
to put much,
if any,
elsev/here.
estimating-filter
transfer-
- 30 function
corre sponding
to the optimum
X(f):
1
Z
N(f) 6 ]
/Hm(f)
He
(f)
/2
e+j(3(f)
=~
(2. 34)
opt
F
fin
We have assumed
filter
of (2.34),
noise,
here
that N(f) is non-zero
as we should expect,
and small,
or zero,
The mean-square
at all frequencies.
has large
gain at frequencies
error
Z
corresponding
The optimum
gain at frequencies
with large
to (2.33)
with -small
noise.
and (2.34)
is,
froIn Appendix
III:
(2.35)
The first
term
in this expression
smear
components
second
is the smear
of the estimate
mission
actually
band,
F l'
of H
Fl.
response
usually
instead
interested
within the passband
and
of H
(f).
eopt
lack of an estimate
the complete
That is,
we are
medium
problem,
in this case also,
of the transmission
in estimating
H
of for all f as we have done,
This impose s no additional
outside
from
of the noise
The
of
(f) .
of an equivalent
of (2. 34) is optimum
arising
to the error
op
use for communication.
impulse
which arise
contribution
H (f) in the stopband
met
We are
is the contribution
band.
m
(f) only within the trana.-
for F 1 is the band we shall
intere sted in estimating
which has zero
however;
since
The error
the instantaneous
transmission
outside
of
it is easy to see that the re sult
it is independent
of (2.35)
also
of value s of H
obtains
m
(f)
in this case,
- 31 except that the second integral is now taken over only those frequencie s in
FZ which are within the transmission
If the noise is white, i.e.,
band (i. e.,
the intersection
2).
N(f) is constant, at least over the transmission
band, we see that the optimum estimator
of (2034) is proportional to the complex
conjugate of the sounding-signal spectrum of (2.33)0
mator is matched to the sounding signal.
a minimum-mean-square-error
of F 1 and F
That is, the optimum esti-
Thus, we use the same device in making
estimation as in measuring the ~ posteriori
bability distribution of channel characteristics
(cL preceding section).
pro-
It seems
reasonable to assume that, this being the case, the spectrum of (2. 33) will also
a!
give the least equivocal ~ posteriori
distribution,
that is, the largest
ratio ~
1
cr.
1
in (2. 18).
- 32 CHAPTER III:
We shall in this chapter
the probability
transmits
probabilities,
(m
=
1,2,
P
...
receiver.
derive
computer.
a sequence
1
m
THE PROBABILITY
Let us first
of message
waveforms
M); these waveforms
Y
probability
is asked,
computer
obtain ~ posteriori
The analysis
probabiliti(:)0
becomes
restrictions;
J
the effect
(a.),
and (T.)
given in the last chapter,
1
with
= x
m
in terms
(t)eJ
.2rl t
0
which is the sum
of its knowledge
possible
medium.
The
of the channel
and
on s(t) in such a way as to
transmitted
sequences.
if we make two
of these is to allow us to describe
completely
1
£ m (t)
z(t)ej2nfot,
straightforward
medium
(8.),
The transmitter
~ (t), 'of the multipath
of the various
particularly
=
of
are known to the
Pm" to operate
multipath
1
s(t)
form
independently,
waveforms,
on the basis
probabilities
the problem.
chosen
a signal,
(t), and tIe output,
waveform
for the operational
and probabilities
receives
of a noise waveform,
of the ~ priori
restate
, from a set of M message
,
The receiver
simplifying
expressions
COMPUTER.
of the first-order
the
joint distributions
with no higher-order
of
distributions.
required.
First,
we shall
require,
as we did of the sounding
that the message-waveform
durations
that the multipath
remains
of a message
waveform.
this requirement
of seconds
Second,
we shall
medium
limits
be small
essentially
In the case
of the last
chapter
enough so that we may consider
fixed during
of an ionospheric
the waveforms
signal
to durations
the transmission
medium,
of the order
for example,
of fractions
or less.
we shall restrict
require
the receiver
that the receiver
+ 1. e., joint dis tributions
many different times.
consider
of the values
to per-waveform
each waveform
of the variables
operation.
That is,
in the received
(a.),
1
(8.),
1
and (T.)
1
at
- 33 sequence
as an event which is independent
independence
waveforms
clear
does not in fact exist,
of the transmitted
that the perturbed
of each other waveform.
for although we have assumed
sequence
waveforms
are independent
of the received
follows from the fact that the characteristics
been assumed
sequence,
message
waveforms
The assumption
tions:
history
of per-waveform
waveforms
time of each member
of the sequence;
transmission
operation
implies
have the same
of the sequence
and that either
of successive
message
waveforms
to waveform
of a
of successive
or that any overlapping
the spread
of a message
equality,
of delays
two other
(say,
of a sequence
of path delays,
probability
have
assumpT)~so that the
enough time is allowed between
the spread
Using Bayes'
medium
does not .depend on the past
at the output of the multipath
the duration
This
duration
do not overlap
(i. e.,
are not.
perturbations
waveforms
neglected
it is
are not independent ....
that all message
starting
sequence
of the multipath
the multipath-caused
that the
of one another,
to change only very slowly fr~m waveform
and hence,
This
of the medium
so that the
medium
is small
the
because
of
enough to be
is small
compared
to
waveform).
we may write
for the per-waveform
a posteriori
of the m th waveform:
./
~ The overlooking of the inter-waveform
dependences which is entailed in
per-waveform
operation results in a loss of information.
One may, by the
following argument,
gain an insight into the way in which consideration
of
these dependences could lead to additional information
about the transmitted message.
It is evident that we may use the message waveforms as
channel-sounding
signals as well as information-bearing
'Signals; that is,
on the hypothesis,
say, that Sm(t) was sent, we may compute an a posteriori distribution
of channel characteristics
just as discussed in-Chapter II.
Now, since these characteristics
are assumed to vary very slowly, we
should, for example, consider a sequence of message waveforms which all
give similar distributions
of characteristics
more probable than a sequence
for which successively-obtained
distributions
are vastly dissimilar.
Per-waveform
operation does not utilize such information.
- 34 -
Pr [S
P mpr[s/sm1
m
Now,
prEs]
the ~ priori
malizing
reduces
(3. 01)
=-----
/ S]
factor
probabilities,
independent
to an evaluation
probability
Pm'
of m,
are known,
so the problem
waveform,
V
of computing
i\.m =
of the "likelihoods",
that the noise
and pr [s] is just
prrs/s
L
m
a nor-
Pr [Sm/S]
}.A
m
is just
the
(t), is
(3.02)
1(lll) (t) is
where
assumption
that the mtiltipath
of a message
We shall
For
m
waveform,
devote
the rest
reference,
the received
~
given by (2. 06),
S
*
(T)
=
(T)
== Sz*(t)
S
medj.u;m stays
this probability
of this chapter
we write
signal
(t)
on the hypothesis
Sm (t-T)
essentially
to the evaluation
dt
=
m
(T) e
x
lll
(t).
Using
the
as:
of (3.03).
-correlation
function
of
waveform:
-j2nf
g
cross
=
fixed for the duration
may be written
down the complex
th
and the m
message
x(t)
T
0
(3.04)
where
g
m
X
m
(t-T) dt
(3.05)
- 35 -
Let us first
a priori,
assume
or that their
contained,
either
that the modulation
a posteriori
with high probability,
Then the (T.)-integrations
of pr[(a.),
(8.)/(T.)]
1
1
or have been evaluated
use the unprimed
Then,
the resolvability
m th message
a.'s
waveform,
and 8.'s
1
indicates
that they are
1
to W'
compared
We also assume
(equations
(2.05)
that the
and (2.07)),
-~--
parameters
(equations
(2.17) and (2.18)).
We shall
for convenience.
distribution
of (2.02),
and further
that
+
all i
holds for all m, where
(T.), are known
which are small
are known a priori
1
the noise -waveform
condition
(2.22),
are unnecessary.
by measurement
a priori
assuming
in intervals
of (3.03)
1
parameters
distribution,
delays,
f
k
(3.06)
ep m (T) is the complex
auto-correlation
function
+~
that the 2L-fold
integration
it can be shown
in (3.03) reduces
of the
on the
to
IIi
L
'
Am TT=
1
=. C
----.,.- .....
-exp
. l' 1
.2(J"~E
1 + No m
In this equation,
waveform,
g
.
ml
=
g
m
(J"i I g
N
0
C is a constant,
. I2
ml
+
2
2a. Re ~g , . e _J'O~
1 - 2a. E
1
ml
1 m
(3.07)
[
2No
+~
No is the (white) noise
1
+
2(f~E
~
]
m
0
Ern is the energy
power-density,
of the m
th
message
and we have written
(T.).
1
See the discussion
following
meaning of this condition.
+c> See Appendix IV.
equation
(2.20) for an explanation
of the
- 36 We thus see that the only operations
computer
signal
on the received
signal
with the Minessage
at delays
T.,
1
in carrier
of the envelopes
the correlation
probability
computer
each message
waveform,
no random
knowledge
filters.
filters,
device
Thus,
one matched
to sample
+
the
to
the outputs
+$
(3.07) for the case
contains
exact ~ posteriori
filters.
correlations
and 3) the sampling
by matched
a set of M matched
of these
of this
As we have noted before,
T .•
1
and also a sampling
Let us now consider
that the medium
at delays
of these
(3.04));
1
may be performed
contains
and the output envelopes
2) the sampling
0. (cf. equation
of the correlations
operation
by the probability
consis"t in 1) the cross-correlation
waveforms;
phases
performed
where
either
path components,
of the medium.
Then
we know a priori
or the receiver
<T.
1
=
has
0, all i, and
(3.08)
We notice
that the samples
have disappeared,
~f>J
ra
l~Hl1
of the cross
and that only the samples
e -jog1, remain,
This,
is of use only whpn there
phase -shifts
of the envelope
of course,
is phase
-correlation
of the cross
makes
uncertainty,
sense,
-correlation
for envelope
and herewe
function
itself,
sampling
know the path
exactly,
Now .. the multipath-rrlediurc
output in the case
<T.
1
()= I
n
m (t)
11m
L
a.x
(t-T.)
1
e
=
0, all i, is
j(2rrfot-oi)
(3.09)
i= 1
~
See page
24,
••
An early interpretation
of a correlation
matched filters was made by Fano(27).
receiver
in terms
of a set of
- 37 We therefore
see,
using
(3.04)
of (3. 08) is jus t the real
part
and (3.05),
that the first
of the complex
correlation
term
of the exponent
(3.10)
That is,
we may think of the likelihood
on the basis
of the cross
known signals,
~ (m) (t)
the medium.
(m
1, 2, ...
for from
may be considered
perturbed
=
between
,MJ,
This is just the correlation
as may be expected;
medium
-correlation
computer
just by additive
In the other
the received
signal
receiver
and the set of
at the output
of Woodward
of
and Davies (5),
point of view the (non-random)
of the transmitter,
and the channel
then is
noise.
extreme,
path-components,
as one which operates
which may appear
the receiver's
part
of (3.08)
when we know ~ priori
and we make no channel
that the medium
measurements,
has no fixed
we have a.
1
=
0, all i.
Then
2
<r~lg
1
1
i=1
1
+
2<r~
1
N
exp [
m
.1
]
ml
(3. 11)
-2N-~-[-I-+-2-lTN....,.t-:-m-~
o
We notice
here
that only samples
This is intuitively
function
itself,
justified,
ignorant
to prefer
much different
for if we were
function
required
env010pe appear.
to sample
the correlation
we should have to do this at a given set of carrier
we are completely
no reason
of the correlation
one set of carrier
notation,
The behavior
of the path phase -shifts
has also been obtained
of (3.07)
to the fixed-multipath
phases
for large
computer
noise
of (3.08).
over
in this case;
another.
by Price.
is of interest:
Essentially,
phases.
i. e.,
But
we have
(3. 11), with a
(8a)
as N ~oo,
o
this implies
it converges
that,
in
- 38 the limit,
the information
exclusively
reason
by the fixed path-components.
that the capacity
fluctuations
than the capacity
Price
has explicitly.
(Tj)
of that part
as well as by noise
noise
B.
which is transferred
Known:
vanish
sense,
for it stands
more
rapidly
although
Let us,
in fact,
receiver's
by noise
only.
In fact,
(6)
case.
that the parameters
if they are
a posteriori
we may know the strength
it is unlikely
assume
(oi) are known.
parameters.
(a.) and (<T.),
parameters,
Then,
are a priori
1
distributed
Am
averaging
evenly
of (3.07)
over
over
of the
1
parameters,
(0.).
.
completely
-~--
This_ may'
But on an a priori
that we know the phase -shift
that the o.'s
(1. e.,
viewpoint
independent.
by path
with increasing
1
paths,
to:
(oi) Unknown.
very well be the case
various
is conveyed
which is disturbed
which is distrubed
shown this for a' special
We have thus far assumed
basis,
the channel
This makes
of the channel
should
of the part
through
random
the interval
1
from
(-1T, IT)),
the
and
the 0.' s, we obtain ~ for the
1
likelihoods
2
<Ti / ~
o
l'r gmi
exp [
where
~mi
=
1
I
+ ~ .)
1
O[N
ml
I
a. g
0
(l
.
ml
I
+ ~ .)
ml
J
(3. 12)
m
(3.13)
N
o
I o is the zeroth-order
Equation
*
o
(1
2 2E
]
ai m
-
we have written
2
2<T.E
function
2N
/2
(3.12)
modified
depends
of the received
See Append'ix IV.
signal
Bessel
function
of the first
just on the envelope
and the m
th
stored
kind.
of the cross-correlation
message
waveform;
as in
- 39 the case
of equation
knowledge
(3. 11), this is due to the receiver's
of the multipath-medium
may now be stored
tion function
with arbitrary
will of course
One might be tempted
to any receiver
phase.
to argue
transforms
g
that,
f.1. is independent
m
(3.12),
Writing
equation
evenly
distributed
The mes sage waveforms
phase
shifts
conversely,
waveforms
a random
. of equation
ml
of i;
(3.07) over L independent
of equation
since
lack of
of the correla-
not affect its envelope.
This is not true,. for although
f.1. is random,
m
average:
phase,
in which the message
message-waveform
equation
phase -shifts.
complete
that is,
random
we must in this case
over the interval
are stored
phase
(3.12)
applies
with arbitrary
shift of the m th stored
(3.07) into g
instead
variables,
average
(3.07) as the exponential
equation
.e -jlJ.m, where
ml
of having
to average
as in the derivation
over just one random
of a sum,
and assuming
(-1T, 1T), we may easily
evaluate
variable.
that IJ. is
this
$
(3. 14)
In equation
samples,
(3. 14), the argument
which have first
relationships)
before
to the case where
In this case,
been summed
envelope
detection.
o
factor
1
the e -joi of equation
and independent
we have just considered.
See Appendix IV.
contains
coherently
Equation
the 0.' s are not known exactly,
f.1.
is again random,
+
of the I
(i. e.,
(3.14),
but their
(3.07) is transformed
of i;
the L correlation
this is clearly
in the proper
in fact,
phase
also applies
differences
are.
into e -joi e -jf.1., where
equivalent
to the case
- 40 C.
Unknown.
(Tj)
Let us finally
exact knowledge
now, or that,
the possibility
of the modulation
Then,
from
or will choose
obtain
a new expression
the T.' S.
Let us first
1
up until
1
simplification,
pr UTi)]'
to ignore,
does not have
(T.), as we have assumed
the receiver's
with some distribution,
have,
that the receiver
delays,
in the hope of equipment
this knowledge.
variables
consider
point of view,
In this case,
any knowledge
for the likeli;hoods
we choose
the
not to use
T.'S
1
are
we generally
of the o.'s,
will not
so that we may
1
by averaging
random
equation
(3. 12) over
set
2
F
1
. (x)
m1
Then,
= 1+~mi exp
using
A:
=
cr ..5
(3. 15)
10
2N (1+~ .)
o
m1
(3.15) in (3. 12), and averaging
over
the
T.' S,
1
we obtain
immediately:
L
pr [(Ti
L times
where
t
2 - 2a.2E ]
1 m
~i
-x
No
the integrations
ij
TT
F
=Jlg= (Ti)IJdT 1...
(3.16)
dT L
i= 1
extend
over
all possible
values
of the T.' S.
1
Strictly
speaking,
pr[(Till
the restriction
imposed
equation
is based.
bility
(3.16)
condition,
cannot
be an arbitrary
by the resolvability
condition,
Let us at this point,
and assume
that the paths
distribution
(3.06),
of
upon which
however, neglect
are independent,
because
the resolva-
i. e.,
that
L
pr[(T~)J
=
IT
pr
hl; and
further
that
i= 1
1
B-A
A
<
-
T.
<
1-
B
all i
o
elsewhere
(3.17)
- 41 If the number
probability
of paths is small,
of the cases
will be small.
and/or
Equations
. (x}
m:
F
(x)
of subsequent
between
derivations.
1
all i
m
knows a priori,
Using equations
condition
for all i, so that
<T.=<T
1
(3. 18)
taken together,
all paths are statistically
all configurations
the resolvability
effect on the validity
that a.=a and
(3.17) and (3.18),
point of view,
receiver
=
assume
the total
we may expect that the contradiction
(3.06) and (3.17) will have little
F
(A, B) large,
in which (3. 17) contradicts
In such a case,
Let us further
the interval
imply that,
identical.
all paths have the same
of path delays
in the interval
(3.17) and (3.18) in (3.16),
from the receiver's
That is,
strength
as far as the
distribution,
and
(A, B) are equally probable.
it follows
that
(3. 19)
In many cases
~ posteriori
P
m
,are
probabilities
equal,
we ask whether,
brackets
we require
of equation
this reduces
say,
in equation
only the order,
(3.01).
to requiring
Alii
1\.p is greater
rather
than the values,
If all of the ~ priori
the order
of the
probabilities,
of the likelihoods;
that is,
Alii
than or less
(3. 19) is positive,
than
1~.
q
we may then equally
Since the term
in
well ask whether
or
not
S
B
Fp[lgp(T)
A
It is important
I] dT > S
B
Fq[lgq(T)
I] dT
(3. 20)
A
to note that in order
to answer
not need to know how many paths there
are.
this question,
the receiver
does
- 42 -
A receiver
which works on the basis of -the operations
of (3. 20) is much .,
more ignorant of the state of the multipath medium than ones which ~operate on
the basis of (3.07),
inferior
or (3.14),.
(3.12),
performance .
and,weshould
expect a correspondingly
On the other hand",.the operations
implemented much more easily than those of (3.07),
no sampling equipment is required
of (3. 20) consists
computer"
this figure,
of (3.20)
(3.12),
in the former.
In fact,
and (3.14),
simply of M units like that in Figure
the cross -correlation
since
the "probability
we have indicated a matched filter as the correlation
The output of this filter,
may be
3-1.
if!
In
operator.
function of the received
signal
and the m th message waveform as a function of time, is envelope detected,
giving
i Igm
characteristic
(t-) I.
F
This in turn is fed :into a nonlinear device with transfer.
m
(x).
In general,
F
m
(x) maybe
may be functions of T; that is, the receiver
paths at one end of the interval
paths at the other end.
this information,
time-varying,'
may know ~ priori
(A, B) will be stronger,
Practically
speaking,
so that we may make F
m
The output of
which completes
the
required by (3. 20).
characteristic,
F
m
(x).
It
slowly for small values .of x, and very rapidly for large values of x.
The effect of such a nonlinear operation on the cross -correlation
envelope is greatly to accentuate the peaks of this envelope.
weights heavily those parts of the cross -correlation
probably due to the presence
+
than
we may wish to ignore
(x) non-time-varying.
Figure 3-2 shows a typical nonlinear transfer
increases
(T
that, say,
on the average,
however,
the nonlinear device is passed through an integrator,
operations
for a and
of a signal,
function
That is, it
envelope which are most
while it suppresses,
relatively,
those
The possibility of a solution in the general form of Figure 3-1 was originally
suggested to the author by R. M. Fano.
..
t".I
0:»
t".I
•
4Ilt'
I't)
I
I't)
.....
"'C
-E
.,
--
u..E
0'
-m
N
'--..<t
~
~~
..4.3-
'ac
E..... t-----------.:---lL
)(
l&..
--
,
(Y)
C\J
- 44 parts
which most probably
operator
expresses
a cross-correlation
of that peak.
Sections
operations
their
originate
noise.
the fact that the receiver
envelope
The same
sort
of reasoning
are applied
rather
to sampled
rapidly
the greater
may be applied
in these
values
Essentially,
grows
peak is significant,
A and B of this chapter;
envelopes)
from
results,
than to the complete
functions.
more
sure
that
is the magnitude
to the results
however,
of the cross
the nonlinear
in
nonlinear
-correlation
functions
and
- 45 CHAPTERIV:
Let us assume
form
that the receiv:er
was transmitted
probability.
error
by choosing
(3. 07) and (3. lZ).
case,
energy(E
in which there
The methods
for the system
at the expense
For the case under
are equal,
by choosing
Thus,
we may write
SPECIAL
evaluate
we shall
probability
increasing
complexity
consideration,
because
(3.01)
waveform
of
of equations
the simples t non(M=Z) of equal
(P 1=P z=i) , and only a single
to more
of equation
and difficulty
the a priori
that the receiver
corresponding
the total probability
a posteriori
waveforms
use can be extended
the computer
wave-
the probabilities
computers
are only two message
CASE
at which message
do this for what is es sentially
we see from equation
the message
a guess
the likelihood
containing
of rapidly
makes
we shall
1=EZ=E) and equal ~ priori
path (L=l).
at least
'containing
We shall
OF ERROR:
the one with the greatest
Under this assumption,
of the receivers
trivial
PROBABILITY
of error
general
(3.07),
cases,
but only
of computation.
probabilities
may make its guess
to the greatest
likelihood.
as
(4.01)
where
the first
and the second,
term
is evaluated
on the hypothesis
seen from
considering
are equal,
so that we may write:
on the hypothesis
that
S1(t) was transm.itted,
that SZ(t) was transmitted.
the symmetry
of our special
case
It is easily
that these
two terms
(4.0Z)
- 46 That is, we shall evaluate the probability
that
A 2 > Al
on the hypothesis
that S 1(t) was transmitted.
In evaluating
~ priori
equation (4.02),
knowledge of the channel,
we shall assume
that the receiver
has only
and that this knowledge is correc t.
that the parameters,
and 0 of (3.07),
(3.12),
we shall assume
a., cr, and N , of equations
o
are in fact equal' to the corresponding
That is,
(3.07)
parameters
and
of
the channel. •
A.
T
Known: 0 Known.
We shall first evaluate P
equation (3.07).
from (3.07),
2
1g 1 1
=
",2
g1
+ ",2
g l'
for the receiver
We may assume,
A 2 >Al
where we have,
e
which has phase information,
without loss of generality,
that 0=0.
Then,
if
of course,
dropped the path index,
i.
Noting that
'1 ar 1y f or 1 g2 12 ' an d wr1hng
,.
an d' s 1m1
(4.04)
we may rewrite
$
Cf. footnote,
(4.03)
as
page 1O..
- 47 -
(4. 05)
We shall
call D the decision
Now, from equation
variable.
(3.05) we may write
(4.06)
where
we have set T=O for convenience.
waveform
(2.08),
of the transmitted
signal
is xl (t), so,
we have for the modulation
z(t)
.S
=
ax (t)e -J
l
share
a joint distribution
parts
of the first
term
parts
of gland
Gaussian
(2.06)
and
signal:
distributed.
and hence wI'
Thus,
noise.
hence,
the real
distributed.
distributed.
It follows
variable
and imaginary
(16)
The real
(15) Hence,
that the real
w2' w 3' and w4' share
the decision
Now, a and S
is a quadratic
and
the two
and imaginary
a joint Gaussian
form
of dependent
variables.
It is easily
Gaussian
(2.05);
in (4.07) are Gaussianly
of z(t) are Gaussianly
(28)
of the received
of the additive
as in equation
parts
distribution.
waveform
of n(t) ar~ also Gaussianly
g2'
equations
(4.07)
n(t) is too modulation
parts
waveform
from
the modulation
+ n(t)
where
imaginary
By hypothesis,
shown
variables
+ that
the characteristic
function
of a quadratic
form
of
is given by
" See Appendix V. For the special case for which
result has been given by Whittle. (31)
W is a zero matrix,
this
- 48 -
Fn(J'u) :: eJuD
=
1
II-2juMQ 11/2
where I is the unit matrix,
=
pr []n
1
21Tj
5
form,
M is ~e
W is the (column) matrix of the means of the
"t" ,denotes "transpose
The probability-density
(4. 08)
1 W M-.l{I (I 2jtiMQ)_-I}w]
t
- - -
- 2"
Q is the matrix of the quadratic
moment matrix of the variables,
variables,
exp [
ofi', and I...
distribution
I
denotes "determinant
of n is given by the Fourier
of".
transform
of F n:
joo
Fn(s) e -sD ds
-joo
Now, the probability of error
Pe = pr[D<O] =
5
is
o
pr[D]dD
(4.10)
-00
Substituting
(4.09) in (4. 10), changing the order of integration,
and integrating
on n, we obtain
Pe
=
1
- 2iTj
5
joo
Fn(s)
s
(4.11)
ds
-joo
The path of integration
in (4. 11) is taken to be indented to the left at all j-axis-
s ingula ti tie s .
Let us now evaluate equations (4.08) and (4.11) for our special case.
first useful to define the complex correlation
It is
coefficient of the two message
waveforms:
(4.12)
- 49 The real
values
and imaginary
of the normalized
waveforms
I AI,
parts
of this,
physical
at the origin,
at the origin.
A and NA,
are,
respectively,
cross -correlation
and at a displacement
is the value of the envelope
function
"
function
of l/L1£.
o
of the normalized
It is easily
shown,
the
of the message
The magnitude,
physical
cross -correlation
using the Schwarz
inequality,
that
IAI<l.
It may be shown'" that the moment
m ..
IJ
=
matrix
M, the elements
of which are
w.w. - w.w.
1 J
1 J
(4.13)
is
,...
(13+1)
0
0
M == 13
N
1\
A(13+ 1)
It may also be shown
*"
-A((3+1)
,..,
-A(p+l)
2
13IA/ +1
0
A(13+1)
A(13+1)
1\
N
(13+ 1)
"A(f:>+I)
N
A(f:>+I)
A(13+1)
(4.14)
0
2
13/A/ +1
that
"'( (p+ 1)
w=
0
(4. 15)
"
r(13A+l)
N
113A
where
'Y= ~.
.,. See Appendix
From
(4.05)
it is seen
VI for outline
that the matrix
of method.
of the quadratic
form
is:
- 50 -
Q
=
Substituting
derable
1
o
o
o
o
1
o
o
o
o
-1
o
o
o
o
-1
(4. 16)
equations
matrix
(4.14)
algebra
through
(4.16)
ln (4.08),
we obtain,
after
consi-
+~.~
kl s(l+kZs)
]
exp [ 1 - k s(1+k s)
3
Z
(4. 17)
1 - k s(1+k s)
3
Z
whe re we have written
kZ
=
Z((3+1)
k3
=
Z(3Z (1-
The integral
cannot
(4. 18)
IAIZ)
obtained
be evaluated
In the former
by substituting
in closed
form
except
equation
(4.17)
in the special
into (4.11)
cases
CT
=
apparently
0 and a
= o.
case ~.1/11
1 [1 - er f { (1 -x')a
Pe _
- 'Z
IN ZE ~
o
nhere
+
See Appendix
VI for outline
+.
See Appendix
VII.
of method.
(4.19)
- 51 x
. 2
So
= - ..-..
erf(x)
{if
e
-z
2
(4. 20)
dz
This, as should be expected, is the probability of error
for a simple correlation
detector operating in the face of white, Gaussian noise , (34) since in this case
the path has no random component.
= 0,'+
Fora
p
e
=
(4. 21)
where r 1 andr 2 are the roots of
k2y
.
2
1
+ Y -;:-.K
=
(4. 22)
0
3
r 1 is the positive root.
In the general case (aI=O,
erIO),
we may put the probability of error
form which is more convenient for numerical
- S
pe - k6 .
1T/2
k . 2e
s1n
e- 7
o
1+k4tan
2
2
de
evaluation.
in a
This is +:
(4.23)
e
where
k4
= VI
kS
I
+ (4k2!k3)
(kl/k3)
k6 =[lk~-1)/1Tk4]exp
k7
tit
= ~k;-1)/k4]
See Appendix VII.
kS
[-
ks/k;
J
(4.24)
j
1
,I
I
- 52 -
I
It is to be noted that the only characteristics
on which the probability
correlation
of error
coefficient.
depends
of the messa~e
are their
(Note especially
energy
the dependence
waveforms
and their
on the quadrature
N
component
'of correlation,'
dependence
on Aderives
complex
A
A, as 'well as the in-phase
component,
A; this
/'oJ
set of channel
this value,
parameters,
system
the channel
parameters
there
curves
instability
N , there
o
the relationship
of Figure
the two message
any given'
P ;
-waveforms.
These
The family
the values
values
of P
e
are presented
parameter
e
waveforms
A t as a function
op
Substituting
we may compute
4-1.
For
is a value of, Awhich minimizes
We shall evaluate
(4.23),
message
of the path.)
between
in the next chapter.
into equation
may be written
CT,
performance.
with optimally-rel,ated
unbroken-line
the phase
a,
A t' indicates
op
for optimum
obtained
from
of
of At
op
,
-
for a system
as the
of these
."
curves
as
(4.25)
Now, it is easily
shown(16)
that the mean-square
value of a,.' the path strength,
is given by
(4.26)
Hence,
the family
average
received
parameter
signal
along any ore curve,
of the average
energy
of the curves
energy
to the noise
of Figure
power-density.
this ratio
is held constant,
received
via the random
via the fixed path-component
is varied.
4-1 is the 'ratio
_ In progre.ssing
2/'12
while the ratio,
path-component
The effect
of the
=
2 2
2cr /a ,
to that received
of an increase
in the
- 53 average
strength'
decrease
of the random
in the ~trength
probability
of error.
the system
deteriorates
=
fixed (2/1'2
and random
component,
at the expense
of the fixed component,
It is interesting
0) to one with the same
components
=
(2/'(2
shows up as an increase
to note how rapidly
as the path changes
from
average
illustrated
the two limiting
(see Figure
5-1),
=
(J"
the performance
of
total strength
but with equal fixed
I).
P
by considering
cases,
in
one which is completely
The effect of path dis turbanc:e-s on the performance
strikingly
of a corresponding
e
=
0 and a.
for the large
we have from equation
is
signal-to-noise
In the former
O.
of the system
case,
letting
ratio
.
A. = A.
in
op
t = -1
(4.19)(35):
.
a. E
exp [ - ~ 2 ]
~
P
e
(4.27)
E
In the latter
P
e
2
case,
channel
ratio
by additive
noise.
has random
2
of 2/"{
opt
=
0, we have from equations
(4.21)
and (4.22)
(4.28)
of error
very much more
of operating
+ To estimate
A.
the probability
path distur.bances
is capable
=
No
As would be expected,
solely
A.
1
2 (J"lE
N-"'oo
o
severe
N:
letting
E
signal-to-noise
N
1111
~~OO
0
It is interesting
without
error
zero with increasing
slowly when the channel
as well as by additive
+
approaches
noise,
is perturbed
by
than when it is perturbed
to note,
however,
in the absence
of noise
that the system
even when the
path disturbances.
the rate
of approach
than the ones considered
of Pe to zero for large E/N
above (0 and (0), see Figure
for other
0
4-1.
values
-540.5
M
_-----------
------
02
o
2
3
2/y2
Fig.4-2
4
6
- 55 -
B.
T
Known: 0 Unknown.
The probability of error
for a receiver
of equation (3.12) is much simpler
tonic function of
Ig m I,
with the likelihood computer
to determine.
Since
it is evident that equation (4.02)
A m is
I
a mono-
reduces to
(4. 29)
where we have primed Peso
error
as to distinguish it from the probability
calculated in the last section.
assume that ~
=
0;
In evaluating equation (4.29)
this is the optimum correlation
of
we shall
coefficient for a system
which has no phase information. +
Let us first assume that the path strength,
It is easily shown +. that the probability
of error,
a, is known to the receiver.
conditional upon knowing a,
is
(4. 30)
The total probability of error
p~
=
S
is then pi (a), averaged over a:
e
00
pr[a] P~(a) da
(4.31)
o
pr[a] may be obtained from equation (2.05)
this result in (4. 31), we immediately
by integrating
it over 8.
Using
obtain +++
... See Section V. B, page 62.
++
See Appendix VIII. For the general case of this result,
equation (37) of reference 34.
+.+ See Appendix VIII.
for kfO,
see
, i
- 56 -
j
Pe,- -
I
1
fj+Z
exp
[~12 J
"(4.32,)
- 2(~+2)
Equation (4. 32) is plotted as the broken-line
f
i
along with the curves of equation (4.23).
I'
of the receiver
,I
t!
curves of Figure 4-1,
As we might expect,
which has knowledge of the mean path phase-shift,
"
great,
except in the region of high path phase ..stability
advantage is expressed
gives the increase
in power which would be required
system which has phase knowledge.
(average received-signal-energy
=
2/12).
This
in the system without
of error
The family-parameter
to noise-power-density
-
to that of the
values of Figure 4-2
ratio) are those for
system.
Equation (4.32) again illustrates
CT
(small
0, is not
in terms of power in Figure 4-'2, in which the ordinate
phase knowledge in order to reduce its probability
the latter
the advantage
0 and a
=
the change, between the limiting cases
0, of the rate at which the probability
with increasing
signal-to-noise
ratio.
For <r
=
of error
approaches 'zero
0 (and, hence,
~
=
0), the
approach is exponential:
(4. 33)
while for a
=
1=
For other values of
small values of E/N
(large (3);
1= 0), the approach is inverse,
0 (and, hence,
o
a./<r,
as in (4.28).
the approach to zero is roughly exponential for
(small ~), and roughly inverse
for large values of E/N
the point at which the change in behavior occurs depends on the
values of the path-strength
parameters,
a and
CT.
0
- 57 CHAPTER
V:
In this chapter
in Section
we shall turn our attention
B of Chapter
the channel
under
sage waveforms
criterion,
As in the last
by choosing
chapter,
only for the simple
lTIessage waveforms,
general,
T
multi-me
Known:
and only one path.
ssage -waveform,
forms
in the special
Thus,
of the systelTI.
of probability
of
im-
rs which make a decision
waveform;
and these
are but two equi-energy,
We shall postpone
multipath
chapter
case under
complex
comments
case until the concluding
merely
in specifying
the probability
of error.
(4. 18), (4.23),
(see equations
waveforms
consideration
cros s -correlation
for a given energy,
consists
lTIinimlzes
the set of mes-
which is of primary
receive
message
of the two message
E, and ,their normalized
(4.12».
for use with
receivers
equiprobable
o'ri the more
chapter.
0 Known.
the only characteristics
depends
we asked
engineer.
case in which there
As we have noted in the last
error
the minimization
performance
most probable
special
waveforms
the performance
we shall only consider
the ~ posteriori
CASE
we wish to determine
we shall choose
to the communications
message
in some sense,
of system
SPECIAL
to the second question
'suitable'
That is,
which optimizes,
for this is the yardstick
portance
A.
II: "What are
consideration?"
As an optimization
error,
THE MESSAGE WAVEFORMS:
specification
on which the probability
are their
coefficient,
of an optimal
(common)
we require
of
energy,
A (see equation
set of message
that value of cross -correlation
That is,
and (4.24»,
wave-
coefficient
the value of A
= X. + jr.
which
for
which the conditions
ap e
ak
ap e
-at
=
0
(5.0la)"
=
0
(5.0lb)
- 58 -
are simultaneously
a2Pe
d~A 2
()2Pe
d12
satisfied,
and for which
>0
(5.02a)
~O
(5.02b)
In addition,
if we obtain more than .one solution for (5.01) and (5.02),
we require
the one which yields the absolute minimum of P •
e
Now, it is easily seen from equations (4. 18) that P
N
component~ A,
e
depends on the quadrature
.
of the cross -correlation
coefficient only in the square.
Therefore,
we may write (5. Olb) as
N
=
2 A
;)Pe
B(kZ)i
= o
l
(5. 03)
~
We thus have immediately
N
A
=
a possible
0
(5. 04)
It is difficult to show precisely
i. e.,
solution for A:
that it satisfies
that this solution is indeed the one which we require,
(5.02b), and yields,
in conjunction with some solution of
(5. Ola) and (5. 02a), the absolute minimum of P.
.
.
e
plausible,
but not rigorous,
The probability of error
We may, however,
construct
a
argument that this is so.
is the probability,
on the hypothesis t~at
S 1(t)
was
sent, that the decision variable, •
D
=
(J"
2
(5.05)
Nlo
is less than the decision threshold,
.Cf. equation (4.03).
zero.
Now, roughly,
we should expect that
- 59 -
the larger
less
D is on the average,
than the threshold.
Thus,
the smaller
will be this probability
the problem
of minimization
to X. would seem to be related
to that of ma,ximizing
course,
we should not expect
in general
between
the two problems
- - - one may envisage
in A, while increasing
change
of D, that P
variance
e
interested
the average
of the problem
to
will be a rigid
of D,
instead
solutions,
of P with respect
e
D with respect
special
value
would also increase,
at this point not in exact
an inve stigation
that there
that D is
x..
relationship
situations
in which a
may so increase,
of decreasing.
but in trends,
Of
say,
the
But we are
and for this purpose
of D with re spect to A would seem
of maximization
to be justified.
The average
value of D may be obtained
~asily
from
the characteristic
function,
F D(s), by use of the relationship(36)
=
D
(5.06)
s=o
Applying
(5.06) to (4. 17), we obtain
(5. 07)
t
Now, remembering
other
than zero will cause
the second term;
perhaps
'i
I A 12
that
that is,
obtain a better
understanding
terms
on this basis
envelopes
in the first
D is maximum
of (5.07)
zero decreases,
"2
X + NA 2 , we see that setting
a decrease
and second terms
in (5.05):+
=
are,
on the average,
in (5.05) while leaving
term
with respect
of this result
respectively,
N
X. equal to anything
of (5.07)
to
1. for
=
~
O.
We may
by using the fact that the first
the averages
we see that settil1.g1
of the corresponding
equal to anything
the difference
of the squares
the difference
of the correlations
+-This may be shown with the help of equations
(A6 -13) of Appendix VI.
while not affecting
(A6-6),
(A6-7),
other
than ..
of the correlation
themselves
(A6-9),
(A6-12),
un-
and
- 60 affected,
again on the average.
corresponds
to the minimum
On the other hand,
lie s somewhere
from
somewhere
between
should,
in fact,
the following
0 and -1.
0 to -1,
these
For,
depending
(5. 07),
= 0, 'Y = 00,
1312
(3) for a very noisy
(4) for a noiseless
contains
average
and third
'"
op
channel
of these
(N
o
of
f3
s as
1.
maximuin
y.
and
We
reasoning,
that
the two message
envelopes;
only the correlation
by making
13 =00):
= 0,
11
the two message'
AOp~
top t
= 0
variable,
equation
is maximized
antipodal,
that is,
hand,
Ig2 ;
and
-1
difference
on the other
g
A
the decision
waveforms
J
phase
with completely-random
13...... 0):
gl and g2; their
tains
their
waveforms
'
on the
by setting
the decision
difference
orthogonal,
(5.05),
variable
A
A
=
-10
con-
is maximized
that is,
by setting
O.
The re sults
probability
predicted
of error
above may indeed
with respect
thus found is shown in Figure
2
131
physical
decrease
wilLbe
i. eo , -with stable
i. e.,
conditions,
conditions,
"A =
on the values
value of A
t = -1
(No~ 00,
chann~l
For the second and fourth
on the average
andD
as well as from
component,
finite):
only the correlations,
by making
the fir st term
increases,
(2) for a path with no fixed component,
~
~
pha s e (a = 0, "I = 0): A t = 0
op
For the first
A
that the optimum
will obtain:
(1) for a path with no random
( 0-
although
extremes,'
A.= 0
.1\.
(5.07)
term
rJ
that the solution
for any given value of -A.
of error
from
-
plausible
the second
expect from
re sults
probability
we may guess
between
progresses
It is therefore
.
The first
strength
of these
of the random
to
1,
with
be shown to be true
'X.
5-1 as a function
parameters,
path-component
=
10pt=
O.
The optimum
of the channel
it will be recalled,
to the strength
by minimizing
parameters,
is the ratio
the
value of ~
~
2 and
of the average
of the fixed path-component.
-61f-
a.
o
{3y2 -=
<...<
..=
z
0
CO
r-:::::p::===C:::===?==SEiiiiT~~::t:==~
w
(.)
It
-0.2
1----I----]jIf-----+-::~-----f-::..ItIfII!=---___+_---_+---__I
-0.4
...-.-.--.....-4---#---4---I--=----+-----+-----+---_---l
W
o
(.).
z
o
~
...J
W
~ -0.6
-+-
1--I--.~-4__#_-__#-+--------4------lL-+-
,
o
U
:E
~ -0.8
~-J.--+_I_-__,f.-_I__---_t_---_+_---__...--__I~
~~
~
~
~
"
o
-1.0
L.L---L __
f3y2:0t
....L.-....:...__
J.L.._-L._..L.-
o
2
3
2/y2
Fig.5-1
NEGATIVE
PEAK
o
Fig.5-2
~
......l...
4
--'-
--I
5
6
- 62 2
2(J"2E
131=N
The 'second,
,is
o
random
path -component
by numerical
to the noise
minimization
Physically,
the above results
N
The fact that k
op
=
t
period
of the average
energy
power -density.
of 'equation
shows a pos sible physical
of a carrier
the ratio
received
via the
,
5 -1 was obtained
Figure
(4. 23).
may be illustrated
cros s -co~relation
as in Figure
function
5-2,
which
of the two me s sage waveforms.+
~
0 indicates
that this function
from the origin,
and hence
passes
"
through
through
zero
one-quarter
an r. f. peak at the origin.
A.
The fact that X,
conditions
op
t is negative
indicates
(2) and (4) above,
at displacements
tion envelope
the correlation
of one -quarter
the mean fixed-component
just
B.
results
considered
stored
T
Thus,
the origin;
at the origin
channel
as well as
hence the correlation
that the modulation
0, of the path are
an identical
(cf. equations
when 0 is unknown,
I g 11 - I g21,
In the light
is maximized
satisfying
is zero
For
func-
restores
delay,
(3.05)
T,
zero.
delay,
T,
and
If this is not so,
the problem
to that we have
and phase -shift,
0, into the
and (3.07».
0 Unknown.
the difference,
ference
from
still apply:; the receiver
waveforms
In this case,
page 55).
function
for convenience
phase -shift,
by introducing
message
Known:
period
peak is negative.
is also zero at the origin.
We have until now assumed
the foregoing
that this r.f.
of the discussion
on.the average
to the intuition)
optimally~'
is greater
the receiver
makes
than or less
in the last
by setting
its decision
than zero
section,
k
=
of the correlation
(see Section
it is clear
function
IV.B,
(as well as
the probability
in Figure
as
that this dif-
0, so it is plausible
that this value of Awill minimize
the'envelope
according
of error.
5-2 goes to zero
at the origin .
• Of course,
both the carrier
period and the amount
are exagge rated for the sake of clarity.
of phase
modulation
in this Figure
- 63 -
CHAPTER VI: CONCLUDING REMARKS.
In answering
ing re strictions
(1)
the questions
and as sumptions:
The additive
noise is stationary,
independent
of the multipath
(2)
The paths
of the multipath
one another
white, Gaussian,
medium
medium
(see equation
ceding equation
(3)
1. B, we have made the follow-
we asked in Section
(2.07),
and statistically
(see page 10);
are
page
statisti'cally
independent
15, and the discussion
of
pre-
(3. 17), page 40);
i. e.,
The paths are resolvable,
their
modulation
delays
satisfy
con-
dition (3.~06) (see page 35);
(4)
The medium
is non-time
sage wa;eform
(5)
The system
In addition,
Chapter
performs
on a per -waveform
the performance
III, and in determining
(6)
for the duration
of a me s-
optimal
basis
of the various
relations
(see page 32).
systems
derived
among the message
in
waveforms,
that
The receiver's
knowledge
and correctly
represents
and we have restricted
ourselves
are
at least
(see page 32);
in evaluating
we have assumed
-varying,
of the multipath
the medium
medium
is ~ priori
knowledge,
(see page 46);
to consideration
of the special
case in which there
only
(7)
Two equi-energy,
equiprobable
message
waveforms,
and one path,
(see page 45).
Avenues
restrictions;
igated,
shall
for future
that is,
work are immediately
suggested
we may ask for the solutions
briefly
The assumption
here
on these
that the additive
enough in most practical
cases,
possible
noise
is Gaussian
extensions
and
we have invest ...'
and restrictions
var.ious
and extension
assumptions
to the problems
but with any or all of the above assumptions
comment
by these
removed.
of the present
We
work.
would seem to be realistic
of the analysis
to non-Gaussian
noises
- 64 would not at present seem to be worth the concomitant severe complication of
the mathematics.
Similarly,
the assumption of the independence of the noise and
the multipath medium is in most case s justified.
For,
even if some or all of the
noise arrive s at the receive'r from distant noise sources by way of the multipath
medium,
it would almost certainly traverse
that traversed
independent.
a different region of the medium than
by the signal, and these two regions would in general be statistically
It is of course this independence of the noise and the signal-utilized
region, of the multipath medium which we have in mlnd in the last part of assumption (1).
A clue to the extension of our probability-computer
re sults to the non-white
noise case may be taken from the equivalent analysis for a channel which is disturbed 'by noise only (37, 38,9:).
signal is correlated
In this case it has been shown that the r'eceived
not with the stored me ssage -waveforms directly,
but with the
stored message ~waveforms after their modification by linear filtering.
icular,
for T»
~,
the modifying filters
is equal to the reciprocal
have a (common) transfer
of the noise power-density
one would expect the same results
spectrum,
In part-
function 'which ,
N(f).
Intuitively,
to apply in our case.
We may easily extend our previous work for the case of stationary noise to
the quasi-stationary
case,
in which it is assumed that the noise is stationary
least for the duration of a message waveform.
results
still apply, but now the parameter
,
For this latter
case,
N , or more generally,
0
,
a.,
1
CT.,
,1
0.,
1
T..
1
our previous
the noise spec-
trum N(f), will vary from message waveform to message waveform,
other channel parameters,
along'with the
The more general non-stationary
is of course considerably more complicated,
at
and of dubious practical
case
interest.
The assumption of independence of paths is most probably a realistic
one, for
different paths generally pass through independent regions of the multipath medium.
One exception to this, which is of possible interest,
is the case in which some or all
- 65 paths have a common random attenuation (e. g., in the case of an ionospheric
medium, where some paths may pass through a common region of an absorbing layer; say, the "D" layer).
to analysis.
This case would probably lend itself easily
Besides this special case, however, it is doubtful whether a
generalization
to the dependent-path case would be worth the labor.
As we have noted in Chapter II (see page 23), assumption (3) may be almost automatically satisfied,
for if two paths are completely unresolvable,
they may be considered ~ priori as a single path.
To be sure, there is a no-
manIs-land in which two paths neither can be considered to be completely unresolvable nor can strictly satisfy (3.06); this is the small region in which
the difference of modulation delays is approximately
ever,
~.
In most cases,
one would expect few delay differences to fall into this category,
would feel that our re sults would apply with no great error
sharp line of demarcation between "unresolvable"
ferences,
say,
~;
we would then aIbitrarily
how-
and one
if we established a
and "resolvable"
delay dif-
consider as a single path all
paths whose modulation delays differ by less than this amount, and consider
as completely resolvable all paths whose delays differ by more than this
amount.
The only important case in which this technique would be suspect,
and in which a more general analysis which does not make use of assumption
(3) would be in order,
is the case where there is a continuum of paths; for,
in this case, large numbers of delay differences would fall into the no-man'sland category,
that is, near the line of demarcation.
The more general anal-
ysis of the probability computer has in fact been performed by Price for the
special case in which the paths have no fixed components(8a).
Assumptions (4) and (5) were made in order to avail ourselves
city of analysis which evolves from use of only the first-order
of the multipath characteristics.
The obvious generalization
of the simpli-
joint distribution
is to eliminate the
- 66 -
nece s sity for the se as sumptions by taking into account higher -order distributions.
*'
Again, Price has done this for the special casei~;'~hi~h'
no fixed path-components(8a).
which fixed path-components
Extension of his excellent work to the case in
are present would be of great interest.
Elimination of assumption
on system performance
there are
(6) would allow the determination
of errors
in the receiver's
of the effect
a priori knowledge of the
multipath medium .. It would also allow the evaluation of the performance
the system when the receiver's
of
knowledge of the medium is based on measure-
ments, and would enable a comparison
to be made between system performan-ce s
with and without the benefit of measurements.
Extension of the analyses of Chapters IV and V to more general ca,ses than
that of assumption
great difficulty.
(7) would be of great interest,
but also, unfortunately,
A simple extension of the probability-of-error
Chapter IV which would give an insight into the relative
analysis
of
of
effects on system perfor-
.inance . of the different paths of the medium would be the evaluation of P e for
the case in which there are two paths of equal.total
esting to determine how rapidly P
purely-fixed
to a purely-random
e
increases
energy:
it would be inter-
as one path changes from a
one, while the other path remains,
say, purely
fixed.
In regard
form,
to the extension of the work in Chapter V to the general M-wave-
L-path case,
it seems clear that the minimization
would be with respect
to M~M-l). [L
2
- (L-I)]
complex ~ariables
just one as in Chapter V.' These are the values,
delays,
Ti - Tk (i, k
=
1, ... , L), of the
function modulation-waveforms,
++
of probability
at the
L 2 - (L-I)
of error
instead of
.',distinct
M~M-l) complex cross-correlation
"
.
\ X*(t-T.) x (t-Tk) dt
. J p
1
m
(p;f m).
See footnote, page 32.
We say M(M-l)/2 modulation waveforms, instead of M(M-l),
waveforms come in complex-conjugate pair s.
When the
since these
- 67 fixed-component
.
probability
phase-shifts,
of error
(0.), are unknown,
would occur
when the envelopes
when all of the se variable
of all the message-waveform
at all delay difference
L l
M Li - )
L(L-l)
which is not based
variables
.
modulatlon
2 ..
function
delays,
+)
to be zero in the resolvable
(S
IXm (t) 12
dt) enter
of error
.correlation
probably
-path case.
functions
envelopes.
vanish
have to be taken into account
should be considered
M energy
at the
energies
.
are
variables
of the probability
We sayl~(L-l)/2
value's,
functions are (Hermitian)
for
of probability
M2+M
2
or,
more
autopre-
one would like to have all of the cross-
for all values
constraints
of their
problem
as an integrated
this,
but there
are
and these
would
problem.
up into nearly-independent
of analysis,
and that,
whole.
of integration
between
arguments;
which prevent
in the minimization
connection
of error
Minimization
waveforms,
and probability-computer
should be some intimate
J
if the waveform
of the M message
I we split our analysis
channel-measurement
the values,
of the complete
that should be taken in the direction
+
Finally,
an additional
but noted that this was for. convenience
problem
probabil-
an additional
specification
Ideally,
physical-realizability
In Chapter
problem:
1
envelopes
general
(3) is employed,
the T.'S are unknown (Section Ill. C).
in this case would involve
their
and a more
of intere st is the evaluation
and cross-correlation
cisely,
go. to zero
into the problem.
Another problem
.
resolvable,
on assumption
to be given originally,
the case where
that is,
Ti - Tk (Ti >Tk), of the M complex auto-correlatlon
~
(t-T ) dt. These are assumed
k
J\ x*m (t-T.)X
1
m
, modulation-waveforms,
.
not considered
zero,
cross-correlations
are added to the minimization
.'
,
s are
s, Ti - Tk.
When the paths are not completely
ity computer
one would expect that minimum.
1
more
Perhaps
strictly,
the first
is that of considering
problems
together.
the two problems
parts,
the
step
the
That there
would seem to
instead of L(L-l),
since the complex
symmetric
about the origin.
auto-correlation
- 68 -
be indicated by comparison of the re sults of Section II. B. I and of Chapter III;
the se show that precisely
the same operations
used lor channel measurement
such questions as:
of correlation
and sampling are
and for probability computation.
Should the message waveforms themselves
sidered by the receiver
as channel-:-sounding signals,
.channel-characteristic
distributions
ation?+
should known channel-sounding
Or, perhaps,
waveforms be sent alternately,
first be con-
and then the hypothetical
so obtained used for probability
co~put-
signals and message
and if so, what proportion
time should be allotted to each?
One may ask
of
the transrpissio.n
Is there some form of measurement
implicit in the probability computers
already
of Chapter III, as some of Price's
work(8a)
would sugge st ?
Besides the questions relating to integration
vestigated
separately,
there is a whole group of questions relating to problems
we have not even considered.
and receiver
How, for example,
can we supply the transmitter
with identical information about the channel, as we have assumed
to be the case?
By transmission
which is available at the receiver,
disturbances
of the problems we have in-
of the transmitter-to-receiver
back to the transmitter?
affect this transmission?)
receiver -to-transmitter
sounding link?
Or, perhaps,
sounding data,
(How will channel
by establishing
(Is the channel reciprocal?
We may put the gist of this chapter in just a few words:
a separate
)
there are still many;
many que stions which must be answered before we may say that we have a thor0ugh understanding
...
of the problem of multipath communication.
Cf. footnote, page 33.
- .69..-
APPENDIX I: THE COMPLEX CORRELATION FUNCTION
The real part of equation (1.04) is
(ll-l)
in which
N
A"
!(t)= X(t) cos 2nf t - x(t) sin 2nf t
o
~
(Al-2a)
0
A
N
!(t)= x(t) sin 2nf t + x(t) cos 2nf t
o
1(t) and
"
and similar expressions hold for
r-I
,(t+~)
1
o
"(
(Al-2b)
0
.1)
= x t+ ~
f'oJ
N
~ (t). From (Al-2b)
cos 2nf t - x(t+~)
0
0
1
0
sin 2nfot
(Al-3)
Now, it has been assumed that ~(t) represents a narrow-band waveform.
This implies that x(t) remains essentially constant over many cycles
of carrier, so that we may write x(t+
.
N
1
~ (t+ ~)
4~ )
= x(t), and hence
0
=
"
~
(t)
o
(Al-4)
Similarly,
N
1
~(t+ ~)
A
=
o
~(t)
(Al-5)
- 70The cross-correlation function-of 'the physical waveforms~
,..
1\
~ (t) and
~ (t),is
(Al-6)
But from (Al-4) and (Al-5) ~e have also
(Al-7 )
from which it follows that
(AI-B)
which was to be proved.
By inserting (Al-2a) and a similar expression for, ~(t) in (Al-6),
one obtains, after some trigonometric manipulations and the use of the
narrow-band assumption to eliminate integrals of double-frequency terms:
U>('t)=A('t) cos 2nf 't + B('t) sin 2nf
Too
(Al-9)
't
where
A(1:)= ~
j
B(1:)= - ~ )
The envelope
of
Ijl(t) is
just
+ ~(t)y(t-'t)]
[~(t)y(~)
[~( t)y( t-'t)
[J:;j.
(1.05), is just twice this, or 2/A~'+B2.
-
x( t)~(
dt
t-1:)J
(AI-IOa)
dt
But the magnitude of
(AI-IOb)
~ (t),
, which was to be shown,
equation
- 71 -
! POSTERIORI
APPENDIX II:
Using
DISTRIBUTION OF MULTIPATH CHARACTERISTICS
(~5), and (~.07), (2.14)may be written as
(2.0~,
-I
i
2
a.-2a.a. cos
J.
J.J.-
(A2.-1)
2
20.
J.
where the factor
K
1
=------
(A2-2)
(2nWrlJ6) T'WN
is independent of (a.)
and'(S.).
J.
J.
TI
is the duration of n(t),
the output of
.1
the multipath medium; it is greater than T, the duration of the sounding
signal, because of the spread of the delays in the medium.
for ~(t)and writing ~(t)
=
Using (2.06)
z(t)ej2nfot, the first term of the exponent,
of (A2-l) may be written (dropping, for convenience,the
limits on the
integral) as
La.x(t...,;.)c-jSi!2
•
J.
J.
J.
dt
(A2-3)
We write, as in going from (2.10) to (2.11),
Jr z.( t)x( t-"t. )dt
= g(-r.)
J.
-
J.
,
(A2-4)
- 72 and also
. (A2-5)
(cf.
equation
Substituting
(2.09».
(A2-6) into
l
prr(a.),
(e. )/(". ),t;,~1
L
J.
J.
Using .(A2-4) and (A2-5),
J.
~.
=
(A2-l),
we may r~wri te (A2~3):'
we get
K"[TT al.J exp["2l \'.Lcokaoak L
1
~
~
{
J.
+
~
d.a. ]
~
0
~
~
(A2-7)
where
(A2-8)
(A2-9)
+
a.
J,.
2
0.
(A2-l0)
~
&
ik in (A2-9) is unity for i=k, zero for ilk.
useful
to us, we should like
distributions.
a single
This will
summation for all
it
In order for (A2-7) to be
in the form of the product of first-order
occur if the double summation can be written
values of (a.),
(A2-7) is a product of exponentials.
J.
(e.),
~
and (~.),
Thus, we require
J.
that
for then
c
ik
=
0
as
- 73 (ilk)
for all
(a.),
that
f(O) :::ep(o)
l~st
condition
~
=
(e.),
1:-.
and (~.),
~
or at least
c.k«c ..
~
this
leads to
from which (2.16) follows immediately.
pr
Noting
2E, where E is the energy of the sounding signal,
all
(~-ll)
(ilk).
~~
in (A2-7), we have, after
=
[(a.),(e.)/(~.),'"
~1T
~
~
~
K'
Using (A2-9), (A2-10), and
some algebraic
2
a~
~. ~
exp
(A2-11)
ilk
[
manipulation,
t
a.-2a..a. cos
- ~
~ ~ 2
2o.I
wherea!~ Jcr!,
and <f!are given by equations
~
.'
(e.-d.)]
~
~
(A2-12)
~
(2.18).
To find K', we integrate
.~
(A2-12) ov~r all values of (a.) and (e.)
~
~
and equate the result
to unity ••
(A2-l3)
(2 ~17) follows directly
from (A2-12) and (A2-13).
In order to derive (2.22) we note that
(A2-14)
But from (A2-2), (A2-8), and (A2-13)
(A2-15)
• For the evaluation
of (A2-13), cf.
reference
24.
- 74 where K". is independent
, r:' y.
( A2-.J:;)
i
1n
(14)
A2-
of ('1:.).
1
Equation
.
, and l'e-tt1ng C'=
l
.~
(2..22.) results
K' ,
pr[z:7,Ef.
from :i:.ns.erting
- 75 APPENDIX III:
MINIMUM-MEAN-SQUARE-ERROR MEASUREMENT OF AN UNKNOWN
IMPULSE RESPONSE
We desire the solution of equation (2.25):
(A3-1)
Expanding this, we obtain
~,M
.[
r
g(t)
Og(t)dt
=
]
EN,M
o
since dh (t) ~ O.
m
[r
hm(t)
(A3-2 )
&g(t)dt]
0
Now get), as indicated in Figure 2-5,
estimating filter, whose unit-impulse response is he(~).
is the output of the
If yet) is the
filter input, then
g(t)
=
~
lXl he
(A3-3)
(1:)Y( t--t}ch
-00
yet) is, in turn, the sum of the noise, net) and the output of the
unknown filter, whose unit-impulse response, h (~), is to be estimated.
m
That is
y(t)
=
~a>
h (1:)x(t4)ch
m
+ n(t)
(A3-40
-00
where x(t) is the channel input (sounding signal).
Combining (A3-3)
.and (A3-h):
00
g(t)
=
J~
-CD
he (~)hm (a)x(t-~-cr)dadt +
(A3-5)
. - 76 The variation
ofg(t)
is thus
00 .
(])
Og(t)
=
~ ~ hm( 0) x( t4-0)Jhe
('I;) dam- +
~
-00
and assuming that
m
=
(A3-6)
-00
Remembering that net) and h (~) are independent,
EN[n(t)]
n(t4)Jheh)m-
0, we obtain for the right-hand
side of (A3-2):.
+'00
EM[~:
~ ~ hm(t)hm(o)X(t4-0)Ohe
-00
Similarly,
(A3-7)
side of (A3-2) is
he ('I;' )hm (crl)hm (cr)x( t4'
-0'
00
+~
]
,
the left~hand
~ [):) rS I
h )dom-dt
[ll~~
he ('I;I )'PNh-'I;'
)dhe ('I;)m-' m-
)x( t4-crHhe
J
('I;)do' m-: dom-dt
J
(A3-8)
-00
In deriving
(A3-B) we have assumed statis~ically-stationary
auto-correlation
noise with
function
(A3-9)
pince we have assumed (cf.
the duration
page25) that
of hm (1:), the limits'
on the first
may be extended to (- co, + 00). without
II is
essentially
integral
sign in (A3-7)
.
The same state"me'nt may be made apprOXimately a bout the first
term is the product of the integlral
component. of the estimate
•I .•e.,
the first
than
changing the value of the integral.
.
(A3-B); for this
greater
of h ('t') and the variation
m
term in (A3-5).
term in
of the n signall'
of this
component,
- 77 and one would not expect the signal component to last appreciably longer
than h (~) itself.
m
Thus extending the limits, and equating (A3-7) and (A3-8), we get
~ ~he(-r
)droEM[ \
\00\ ~ he (-r' )hm (a' )hm (a)x( t...,;'-a'
-co
- 00
)x{ k-a)da'dr'
co
00
+ A ~ he (-r' )'PN(-r...,;' )dr'
-co
-
~ ~ hm(t)hm(a)x(t--T:-a)dadt
~co
(~-W)
We neglect the physical realizability condition,he('t)
't(
o.'"
dadt
Then Oh e ('t) is arbitrary for all~,
=
0 for
and in order for (A)-IO)
to be satisfied, the factor which multiplies dh ('t) must be zero for
e
all '1;. Setting it equal to zero, Fourier transforming the resulting
'to..
equa ~on,
an d averaging over the ensemble of all possible unknown
filters, we obtain
IX(f)'~+LlHe
op
t(f)~(f) - IHm(f)(2 X" (f) = 0
(A.3-11)
Equation (2.26) follows from this immediately.
In order to show that
this solution yields a minimum, rather than a maximum or inflectional,
error, one merely finds the second variation of
is positive for H (f)
e
= H
e opt
e,
and shows that this
(f).
To derive equation (2.33) for the optimum spectrum of x(t), we
start with equation (2.24) for the mean-square error.
.See discussion, page 26 •
•• Cf. equation (I.07b), Chapter I.
Using (A3-5) in
=0
- 78 this, and remembering that the average of the bracketed expression in
(A3-10) is identically zero, we obtain for the minimum mean-square error
E m is also expressible in terms of frequency domain functions; using
Parseval's theorem in (A3~12) and averaging:
(A3-13)
Using equation (2.26) for He .(f), (A3-I3) becomes
opt
Eel
m
7i'
r
JIm
-co
2 .[1 _ ~x.(f)1
H (f)
I
Z
]
.- df
N (f)ill+!X(f)12
r
(A3-J.4)
• We now constrain the energy in the
transmitted waveform to be constant:
iIX(f)1
00
2
= K
df
(A3~15)
>--00
In order to find the optimum X(f), we must solve the variational
problem (26)
J (E m + AK)
=
0
(A3-I6)
- 79 where A is some constant.
I{
00
....
2
.,
N (f)
H (f)
r
A -
-00
Using (A3-14) and (A3-1.5), (A3-16) becomes
m
[Nr(f);L\+
} t
I.A
F
IX(f)12
a
. 2
X(f) df
I
I
=
°
.
(A3:-17)
If we constrain IX(f)12 to be zero outside a certain band, Fl (cf.
equation (Z.31»), then 8IX(f)12 is also zero there, and (A3-17) is
satisfied for those frequencies.
For frequencies within the band, on
the other hand, we must try to set the bracketed term in the integrand
equal to,zero.
•
This l~ads to the equatio.n
I
2
).. I X(f) 14 + 2A.L\N(f)
r X(f) 1 + Nr(f) {lL\N r (f)
-
IHm(f)
12
j
=
o.
(A3-18')
If (A3-18) and (A3-15) can be simultaneously satisfied within the
band FI by a non-negative function IX(f)12, then the solution is.
complete.
If, however, there are frequencies at which IX(f)(2 would
be negative, then the correct solution is simultaneously to satisfy
(A3-18) and (A3-l5) at all frequencies for which
non-negative, and to set IX(f)
Iz
as indicated in equation (~.33).
I X(f)12
turns out
equal to zero at.all other frequencies,
That this is indeed the correct
solution may be shown by taking the second variation of (E
m
+ lK):
• One might be tempted to obtain another solution, X(f) = 0, by
writing S IX(f)12 = 21x.(f)1 81 X(f)l. This is a spurious solution,
,however, for the problem is actually phrased completely in terms of
IX(f)12 (cf. (A3-14) and (A3-IS).
The second solution would
disappear if we replaced IX(f)12 by, say, S(f).
, - 80 (A)-19)
is positive
for all variations
that the solution of (A3-I8)
further
(A3-18)
s'olution, the larger .is'tJieerror.
as closely as possible,
frequencies for which the solution is negative
setting
this
and for those
is achieved by
IX(f)l~eqUal to zero.
Finally,
(2.33)
This implies first'
is in fact a minimum,and second..that the
IX(f)12 is varied from this
Therefore, one must satisfy
of IX(f)12•
equation (2,.35) follows directly
into (A3-14)
0
on substitution
of
- 81 APPEND!X IV:
DERIVATION OF LIKELIHOODS.
Using-(2.02), (2.05), and (2,.07),
the 'integrand of the (a.)-and
1.
(e.)-integrations
of (3.03) may be written as
J.
[IT
1
a.
J.
2
2.no.
J
exp [- ~No
1.
r
_
L
i=l
L
I (t)_~(rn)
(t)/2
dt
a~+a~-2a
..acos
(e.-J.)]
).
).
). J.
).
J.
2
.
. 20'.
:
. ).
(A4-1)
where T' is the duration
x(t)
=
of ~(m)(t ), which is given by (2.06),
.1ith
x (t~. The expansion of the first term of the exponent of (A4-1)
m"
..~
follows exactly the derivation from (A2-j)
to (A2-6) of Appendix II,
with the waveform index, m, inserted at the appropriate places.
Assuming the validity of (3.06), and noting that f (0) = (D (0)
m
(A4-1) becomes, usi~g -(A2-6):
L
cD
i=l
i
a
exp [2
2noi
="
2E ,
m
'
;~d {(Em
exp
1m
.1)
- N +:-2
0
2CJi
2..
ai ~ Re
[(gmi.
ai j di) ":',Wi]
N+ 2" e
e
0
0i
ai
(A4-2 )
where
(A4-3)
}
- 82 -
is independent of (a.), (e.), and m, and we' have written g . = g (~.)
~
~
(cf. equations (3.04) and (3.05».
.
ID1.
m'~
We obtain equation (3.07) by
integrating (A4-2) over the variables (a.) and (e.)p
J.
J.
For these integrations
we need the results(24):
(A4-4)
and
exp
[~t ]
(A4-5)
4c
3
In (A4-4), cl is generally complex.
In using (A4-4) in (A4-2)'we
note that
(A4-6)
The derivation of equation (J,.12)from equation (3.07) involves
merely a straightforward application of equation
(A4-4).
In deriving equation (3.14), we first write equation.(3.07)
in the form
(14-7).
where
Pm
.
'th
is the random phase-shift of the m
.
message waveform.
Then, (3.14) follows directly upon use of (14-4).
- 83 APPENDIX V:
CHARACTERISTIC FUNCTION OF A QUADRATIC FORM :OF GAUSSIAN
VARIABLES.
We are given a quadratic form,
(A5-1)
of n variables,
WI
'W
2
W=
(A5-2 )
•
W
n
which share a joint Gaussian distribution}c9)
pr[WJ = __
I_
n
(A5-3)
1
(211)21 M 12
Q, is the matrix of the quadratic form,
W
is the matrix of the means
of the variables:
-
WI
'W2.
w=
•
•
..!.
'W
n
(A5-4)
- 84 and M is the moment matrix of the variables, the typical element
of l-vhich is
m. .
1J
11
=
(w.
tit denotes
-w. ) (w.J-w J.)
11
=
w. w. - w. w .
1 J
~ J
transpose of1l, and
n
1 •..
(A5-5)
1 "
11
determinant of" •
We require the characteristic function of the quadratic form:
Fn(ju)
juD
= e
=
r~I
juD
e
pr[W
J dW;t •••
dwn
(A5-6)
-00
n times
Noting that M, and hence M-1, is sym:netric, we may write
The last equality follows from the fact that wtM-lw is a one-element
matrix.
Using (A5-7), we find
(A5-8)
Then placing (A5-l) and (A5-3) in (A5-6), ~nd making use of (A5-8),
we obtain
- 85 Using a result given'by Cramer(30), (A5-9) becomes
•
, (A5-10)
Finally, noting that
we may immediately obtain equation (4.08) from (A5-10). I is the
unit matrix.
APPENDIX VI:
- 86 EVALUATION OF MATRICES OF EQUATION (4.08).
.~. ~ .
~"'''''
~,' .. t
equation (4.07) 'in. (4.06),
Substituting
g
.e
1
~Ixi(t)12dt
= aeJ
~ n~(t)~
+
we obtain.
.,';':' ~_
t.
i
.
(t) dt
(A6-l)
+ ~ n~(t)~(t)dt
~ ~*(t)Xz(t)dt
;,
~
Noting that (cf. equation '(4.12)
'.
~ h(t)\2dt
}~lf(t)
=
~
Xz(t)dt
2
IXz(t)1 dt = 2E
=
(A6-2)
2m..
and letting
p = 2.aEeJ
.e
(A6-3 )
and
i
= 1,2
(A6-4)
we may rewrite (A6-1) as
gl
=
p + q1
g2 = PA + q2
(A6-5)
_
.•
- 87 In order to obtain the means and second selfof equation (4.04),
of the w-variables
and cross-moments
which are required for W- and
M-matrices, it is obvious that we must have the means and second selfand cross-moments of the real and imaginary parts
real and imaginary parts are, from
=
"
gl
N
f'J
"
= ,,'"p~
N
=
(A6-5):
P + "ql
=
g2:
IV
P + ql
- NN
pX + "q2
I\N
rJA
(A6-6)
r.J
pX +. pX + q2
Since we have assUmed-(without loss of generality)
joint
of a and e of equation (2..05),
distribution
for the various momentsof
1\
P
=
These
1\
gl
g2:
of gl and g2.
p
andp,
that
8=
0 in the
we may write immediately
using results
of Rice(16):.
2.aE
(A6-7)
"N
P p
= 0
From (A6-4), we may vTrite
~i
= ~
[~(t)~i(t)+ ~(t)~i(t)]dt
i
~i
• cr.
=
~
=
1,2.
[h(t)~i(t)- ~(t)~i(t)Jdt
Fig. 2.-2, in vThichlet
0=
0, and multiply all vectors
(A6-8)
by 2E.
-A-
=
Noting that n(t)
- 88 -
---;v-
n(t)
=
0, we have immediately
!,
.'
i = 1,2
(A6-9)
Now , for all practical purposes, we may consider the bandwidth of the
noise, WN, very much larger than the transmission bandwidth, W.
Then, to a very good approximation, we may consider that
A
A
n(t) n(s)
=
N
~
n(t) n(s)
= No
~
o(t-s)
where O(x) is the tQrac delta-function.
We also note that(15)
3(t) 'it(s) = 0
(A6-11)
'.
Using (A6-2), (A6-10), and (A6-11), we obtain for the various second
self- and cross-moments of the real and imaginary parts of ql and
~
of equation (A6-8):
2EN
'0
(A6-12)
Finally, because of the assumed independence of the noise and the
multipath medium, we may write for the cross-moments of the real
- 89 and imaginary parts of p a~d ql and q2:
A
A
p q.
l.
-;::;::r
p q.
,J.
= "p "q.l. =
0
=
0
-;:;:;p q.
J.
=
i =1,2
-,::; ;:::PoJ"
p q. = p q. = 0
l.
J.
f'J tv
p q.
J.
(A6-13)
~~
= p q.l. = 0
We have thus evaluated (equations (A6-7), (A6-9), (A6-12), and
(A6-13»
the.means and the various second self- and cross-moments of
the real and,imaginary parts of p, ql' and q2,.,1!sing these, we may
obtain the means and second self- and cross-moments of the real and
imaginary parts of gl and g2:'which in turn may be used to evaluate
the
w.
I
s and the m .. f s of equation (4.13).
J.
l.J
In order to obtain the inverse of the moment matrix, which is
.
required in equation (4.08), we apply the followi"ng identity to
equation (4.14):
a"
0
b
0
a
-c b
c
=
b -c d
0
b
d
C
0
Post-multiplication
1
(ad~b2_c2)
d
o -b -c
0
d
c -b
(A6-14)
-b c
a
0
-c -b
0
a
I.
of the M-matrix by the Q-matrix changes
the signs of the elements of the last two' columns of 'the M-matrix.
Multiplication of the MQ-matrix by the scalar, 2ju,and subtraction
of the result from the unit matrix leads to:
- 90 -
1
~+1- 2juf3
1
0
=
1-2juMQ
-2juf3
1\'
{3+1-2ju{3
A
N
l(~+l)
-A(~+l)
r-J
l.{~+l)
.
-l(~+l)
0
A.
1({3+1)
,.J
-l({3+1)
"
l(~+l)
-{31 AI
2
1.
o
-1 2juf3
0
>
1
2:
-{3llf
-1-2jU{3
(A6-1.5)
In inverting the (I - 2juMQ) -matrix, as required in equation (4.08),
we again make use of (A6-14) •. The determinant of the (I - 2juMQ)
_
matrix may be evaluated through the use of the relation
a
0
b
c
o
a
-c
b
=
K
b
-c
c
bod
d
0
_J.
~(ad-b
2
Z Z
-c )
(A6-l6)
- 91 APPENDIX, VII:
DERIVA'!'ION OF EXPRESSIONS FOR Pe •
We start with equation (4.11), in which we substitute equation
p.
e
= -
where kl'~'
1
(A.7-1)
2nj
ana
kJ
are given by equations (4.18). The path of integration
is taken to be indented to'the left at the origin.
obtain Pe for the special case, a
k2: =. 2 and
kJ
=
= 0, by.noting
We may immediately
that, in this case,
0, so that
(A7-2)
"
2~s2
Equation (A7~2) is in the form of the Fourier i:nansf.om of __e __ ,
s
which is available in tables(32).
Thus, we obt~in for a
=
0.:
(~7-3)
• " The"tables referred to actually giV~ the right-hand side of (A7-3)
"
2k s
as'the sum of the transforms of -e 1 /s and l/s~ However, the path
of integration in the tables is taken t 0 be indented to the right ,at
the qrigip. NOH, the left-indented integral in",(A7-2) may be written
as'the'sum'of two,other i~tegrals having the same integrand: one along the,
j-axis 't-li th an indentqtion to the right at the origin, and one alo~ a
closed contour of infinitesimal radius which encir~les the origin. The
'..
'
'2~s2
fi~st of these is'the right-indented transform of -e
Is, and the ,
second may 'easily be shown to be equal to the right-indented transform
of lis. Thus, the.right-hand side of (A7-3) is also equal to the
2k s2
left-indented 'transform of -e 1 Is, equation (A7-2).
- 92 This, with equations (4.18), leads directly to equation (4.19).
We may evaluate Pe for the case a
=
0 by the method of residues.
For this case, kl = 0" Writing (A7-l) in terms of the roots, .r and r ,
2
l
of the quadratic, 1 - k S(1 + k2:S), we have
3
(A7-4)
where
(A7-5)
,.J
s.
Since the integrand of the integral in (A7-4)
vanishes as
.j- as s+co, we
may close the path
s
of integration by enclosing either the left
or the right half-plane, without changing the
value of the integral.
,..
s
Since the j-axis path
is taken to be indented to the left at the
origin, the former contour encloses only one
pole of the integrand (see Figure A7-1), that
FIGURE A1-1
at s= r2(r2 < 0), and the integral may be eval'uated by calculating
the residue, R, in that pole, and multiplying it by 2nj (33):
(A7-6)
- 93 R is easily calculated:
(A7-7)
(4.21) results from the substitution of (A7-7) into (A7-6).
Equation
In the general case (0 I 0, a
put in a better form for numerical
method.
I 0).1 equation (A7-1)
computation
We first shift the path of integration
may be
by the following
from the j-axis to
the left by an amount 2~l' this does not change the value of the '
intBgral because no singularities
of the integrand
the pr~cess (~ee Figur~.A7-1; the singularities
essential
s ~ 2~
singularities).
are crossed in
at r1and r~'are now
Then, making the change of variables,
(jz-l), and noting that the imaginary part of the resulting
integrand
is odd aboutz
=
0, and the real part even, we obtain
2
_ ~(z
2
p
=
e
k4-1
n
exp
[
+1) ]
z2 +k~
~co
(A7-8)
o
where k and k, are given by equations
4
further change of variable,
1
+
e
2
tan
= sec2e~ we
(4.24). Now, making the
z2 = k~ tan2e,
and noting that
obtain the desired result, equation
(4.23).
- 94 APPENDIX, VIII:
DERIVATION OF EXPRESSION FOR Pl.
e
We v1ri te, as in equation (A6-5) of Appendix VI:
gl
=
2.aEe
g2
=
q2.
je
+ ql
(A8-1)
where we have used the assumption that A
=
o.
represented vectorially as in Figure A8-1.
Now, the real and ~maginary parts of ql and
q2 can be expre~sed in terms of linear
operations on the Gaussian functions ~(t)
and ~(t).
"
~
(Cf. equation (A6-8).) Hence(28),
and f\J
qi are.also Q-aussian. Furthermore,
FIGURE A8-1
we have from (A6-9) and (A6-12):
-;:
~
q. = o. = 0 .
.~
"2
o.
"':t
'"J.
=
1\.12
q.
1.
~A '"
~ =
=
2EN
i
0
==
1,2:
(A8-2 )
0
That is, the real and imaginary parts of ~
are independent,
Gaussian, of zero mean, and common variance, 2EN ; and an identical
o
statement applies to q2.
Thus, using a result of Rice (16), we may
write
~J
2
pr
[I~IJ
exp[ -
i
=
1,2.
(A8-3)
- 95 That is, the lengths of the vectors ql and q2 in Figure A8-1 are
Rayleigh distributed.
Now, we first assume that the path strength, a,
is known to the receiver.
fixed length~
Then the vector 2aEejS in Figure A8-1 is of
Again invoking a result of Rice for the length of the
sum of a vector of fixed length and a vector whose length is Rayleigh
distributed(16), we obtain
(A8-4)
We may finally show, using (A6-~)
A
=
and the assumption that
0, that
(A8-5)
Since uncorrelated Gaussian variables are independent, we infer from
(A8-5) that ql and q2' and hence gl and g2' are independent.
Then
we may write
00
=
pr [[gll
~
1
00
/a ] dlgll
pr [Igzl
Ig11
o
Since, from (A8-1), g2. = q2' we may write pr
Using (AB-3
exp [-
,r
the
d:~J.
/a
I~ I -
integration
Using this
result
remaining Igll-integration, we have
[lg21/a]
1
=
d Igzi
(AB-6)
pr[lg21
=I~I] .
of (AB-6).becomes exactly
with equation
(AB-4) in th~
- 96 -
exp [_
P~(a)
=
2EN
a~:]
(A8-7)
o.
o
With the help of equation (A4-5) of Appendix IV, equation (A8-7)
reduces immediately to equation (4~30).
The marginal distribution, pr[a], of the path strength, obtained
by integrating equation (2.05) over
pr[aJ = a2.exp
o
[
_ a2:+~ 2 ] 10
2:0.
e,
is(16):
[a~].
(A8-8)
0
Substituting (4.30) and (A8-8) into (4.31), we have:
p~
=exp~a:l ]
r
a exp [~ ~~ (~2
+
~JJ
10
[:~
]
da
o
Again using (A4-5), and remembering that 'Y =
a
-
C1
io2.E
and ~ = -,
No
we may obtai~ equation (4.32) directly from equation (A8-9).
(A8-9)
- 97 -
REFERENCES
1.
N. Wiener, Extrapolation,
Interpolation,
arid Smoothing
Series,
Wiley and Sons, Inc., New York, 1949.
2.
C. E. Shannon and W. Weaver,
Univ. of Ill. Press,
1949.
The Mathematical
3:
C. E. Shannon,
37,
4.
P. M. Woodward
5.
P. M. Woodward, Probability
and Information
Theory,
with Applications
Radar,
McGraw-Hill
Book Co., Inc., New York, 1953.
6.
R. Price,
"Statistical
Theory Applied to Communication
Disturbances,
" Tech. Rpt. 266, Res. Lab. of Electronics
Lincoln Lab.),
M.LT.,
Sept. 3,1953.
7.
R. Price,
8.
R. Price,
(a) IINotes on Ideal Receivers
for Scatter Multipath, II Grp. Rpt. 3439, Lincoln Lab.,M.LT.,
May 12, 1955.
(b) "Error
Probabilities
for the Ideal Detection of Signals Perturbed by Scatter and Noise, " Grp. Rpt. 34-40, Lincoln Lab. ,
M.LT.,
Oct. 3,1955.
The essential
results
of these reports
are to be presented
in a projected
paper,
"Optimum Detection of Bandpass,
Gaussian Signals in White, Gaussian
Noise, with Application to Receiver
Design for Scatter-Path
Communication,
"
to be submitted to Trans.
1. R ..E. -Po G. 1. T.
9.
W. L. Root and T. S. Pitcher,
Proc.
Trans.
1.R. E.,
and 1. L. Davies,
of Stationary
Theory
Time
of Communication,
10 (1949).
Proc.
1. R. E. -P'. G. 1. T.,
1. E. E.,
Part
1954 Symposium,
Trans.
1.R.E.
Ope cit.,
III, 99, 37 (1952).
through
(Tech.
Sept.,
-P.G.1.T.,
to
Multipath
Rpt. 34,
1954, p. 163.
Vol.IT-l,
No.3,
p.33.
10.
C. E. Shannon and W. Weaver,
p. 5
11.
Ibid.,
12.
D. A. Huffman,
13.
R. W. Hamming,
'14.
P .. M. Woodward,
15.
S. O. Rice, B. S. T. J., 23, 282 (1944) and 24, 46 (1945), or Selected Papers
on
Noise and Stochastic
Processes,
N. Wax, ed., Dover Pub.,
Inc., New York,
1954, p. 133 et seq. ; Sec. 3. 7.
16.
Ibid.,
17.
D. Middleton,
18.
F.
19.
J. A. Ratcliffe,
p. 25
Sec.
Villars
Proc.
I.R.E.,
40,
B. S. T . J., 29,
Ope cit.,
1098 (1952).
147 (195 0) .
Secs.2.
9, 4.9,
6. 1.
3.10.
Quart.
Appl.
Math.,
and V. F. Weisskopf,
Nature,
~'
Phys.
162, 9 (1948).
445 (1948),
Rev.,
94,
Sec.
5.
232 (1954).
- 98 20.
H. G. Booker, J. A. Ratcliffe,
London, 242, 579 (1950).
and D. H. Shinn, Phil.
21.
R. W. E. McNicol, Proc.
22.
R. A. Silverman and M. Balser,
23.
K. Bullington, W. J. Inkster,
24.
W. Magnus and F. Oberhettinger, Special Functions of Mathematical
Chelsea Publishing Co., New York, 1949, pp. 26 and 35.
25.
J. H. Van Vleck and D. Middleton, J. Appl. Phys.,
26.
F. B. Hildebrand, Methods of Applied Mathematics,
York, 1952, Sec. 2. 6.
27.
R. M. Fano, unpublished lecture notes,
Noise," M.L T., 1951.
28.
H. Cramer,
Sec. 24. 4.
29.
Ibid.,
30.
Ibid., Equation (11.12.1).
31.
P. Whittle, Hypothesis Testing in Time Series Analysis,
AB, Uppsala, Sweden, 1951.
32.
G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications,
D. Van Nostrand Co., Inc., New York, 1948, transform pair no. 727.
33.
E. A. Guillemin, The Mathematics of Circuit Analysis,
York, Chap. VI, Art. 15.
34.
C. W. Helstrom,
35.
W. Magnus and F. Oberhettinger,
36.
H. Cramer,
37.
R. M. Fano,' "Communication in the Presence of Additive Gaussian Noise, II Communication Theory, 1952 London Symposium on "Applications of Communication
Theory, Ii W. Jackson, ed., Butterworth Scientific Pub. , London, 1953, p. 169.
38.
F. A. Muller, "Communication in the Presence of Additive Gaussian Noise,
Rpt. 244, Re s. Lab. of Electronics,
M. I. "T., May 27, 1953.
LE.E.,
Trans.,
Royal Soc. of
Part III, 96, 517 (1949).
Phys. Rev.,
96, 560 (1954).
and A. L. Durkee,
Proc.
I.R.E.,
.!2.,
43, 1306 (1955).
Physics,
940 (1946).
Prentice -Hall, Inc.,
"Communication in the Presence
Mathematical Methods of Statistics,
Princeton Univ. Press,
New
of1951,
Sec. 24.2.
Proc.
Ope cit.,
I.R.E.,
Almquist and Wiksells
Wiley and Sons, New
43, 1111 (1955).
Ope cit~"
p. 96.
equation (15.10.1).
II.
Tech.
- 99 -
BIOGRAPHICAL NOTE
George Lewis Turin was born in New York City on
January 27, 1930. After having obtained his primary and
secondary educations in the public schools of that city, he
entered the Massachusetts
Institute of Technology in 1947.
At M.1. T. he participated
in the Cooperative Course in
Electrical
Engineering with Philco Corporation,
and was
awarded the S. B. and-S. M. degrees in Electrical
Eng-
ineering in 1952. During the summer of 1952 he was an
M.1. T. Overseas Summer Fellow at Marconi's
Wireless
Telegraph Company in England.
In the fall of 1952 he became a Staff Member of Lincoln
Laboratory,
M.1. T.,
in which position he remained for the
next two year s, engaged _inthe de sign and development of new
type s of communication systems.
From the fall of 1954 to the present he has been a Research
Assistant at the Department of Electrical
on assignment to Lincoln Laboratory.
has completed his doctoral studie s.
Engineering,
M. I. T. ,
During this period he
He has also during the
past two years done consulting work for the Boston firm of
Edgerton,
Germeshausen
and Grier.
He is a member of the Institute of Radio Engineers,
Eta Kappa Nuand Tau Beta Pi; and an associate
ber of Sigma Xi.
and of
member of mem-