JPRS ID: 8499 PROCESSING SPACE - TIME SIGNALS (IN INFORMATION TRANSMISSION CHANNELS)

• Document Type: CREST
• Collection: JPRS
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00850R000100060010-3
• Release Decision: RIF
• Original Classification: U
• Document Page Count: 233
• Document Creation Date: November 1, 2016
• Sequence Number: 10
• Content Type: REPORTS

APPROVED FOR RELEASE: 2007/02/09: CIA-R~P82-00850R000'1000600'10-3
I ' ~ ~S~

6 JUNE i979 CFOU~~ . ~ i OF 3
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICINL USE ONI.Y
JPRS L/8499
6 June ].9 7 9
r
PROCESSING SPACE-TIME SIGNALS
(~N INFORMATION TRANSMISSION CHANNELS)
.
U. S. JOINT PUBLICATIONS RESEARCH SERVICE
FOR OFFICIAL USE ONLY ~
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
NOTE
.TPR5 publicaCions cnnCain information primarily from foreign
newspaperg, periodicals and books, buC also from newg agency
tranamissiona and broadcasta. MaCerials From foreign-language
_ sources ~r~ rranslatpd; ehoae from ~nglish-language sources
are transcribed or reprinCed, with the original phrasing erd
other characterisCics retained.
Headlinea, editoriAl reports, and maCerial encloaed in brackeCa
are supplied by JPRS. Processing indicaCors such as [TexC~
or [Excerpt]�in ehe firat line of each item, or following the
lasti line of a brief, indicate how ehe original in�ormaCion was ~
processed. Where no processing indicator is given, Che infor-
mation was summarized or exCracCed.
Unfamiliar names rendered phonetically or CransliCerated are
, enclosed in parentheses. Words or names preceded by a ques-
tion mark and enclosed in parentheses were not clear in the
original but have been supplied as appropriaCe in context.
Other unaCCributed parenthetical notes wiChin Che body of an
iCem originate with Che source. Tjmes within iCems gre as
given by source.
The contents of Chis publicaCion in no way reprasent the poli-
cies, views or attitudes of Che U.S. GovernmenC.
~
COPYRIGHT LAWS AND REGULATIONS GOVERNING OWNERSHIP OF
MATERIALS REPRODUCED HEREIN REQUIRE THAT DISSEMINATION
OF THIS PUBLICATION BE RESTRICTED FOR OFFICIAL USE ONLY.
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY
JPRS L/8499
6 June 19 79
PROCESSING SPP.CE-TIME SIGNALS
~ (IN INFORMATION TRANSI~ISSION CHANNELS)
' Mogcow OBRABOTKA PRO5TRANSTVENNO-VREMENNYKH 5~GNALOV (V KANALAKEi
PEREDACHI INFORMATSII) ii'~ Rtlssi.an 1976 pp ].-208
[IIook by D.D. K].ovskiy and V.A. Soy�er, Izdatel'stvo "5vyaz
208 pages)
CQNTENTS PAGE
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . 1
_ Basic Deaignation~ . . . . . . . . . . . . . . . . . . . . . . . . 2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
_ Chapter 1. Model of a Space-Time Channel . . . . . . . . . . . . . 6
1.1. Structure of Systems for Data Tranamisaion by
Space Channele . . . . . . . . . . . . . . . . . . . . . 6
1.2. System Characteristics of a 5pace-Time Channel
and Continuous Models of It . . . . . . . . . . . . . . . 8
1.3. Different Mechanisms of Random Propagatinn of
Waves in Real Space-Time Channels . . . . . . . . . . . 11
1.4. A One-Dimensional Probabilistic Model of a
Channel with Sequential Parallel Propagation 18
2.5. Statistical Models of Space-Time Channels
Based on Correlation Properties . . . . . . . . . . . . 29
1.6. Model of Spatially Distributed Additive Noise. 31
1.7. Linear Model of Signal and Noise Fields
_ Obtained by the Method of State Variables. 35
Chapter 2. Measurement of the Space-Time Characteristics of
a Stochastic Channel . . . . . . . . . . . . . . . . . 39
2.1. Formulation of the Problem of Measurement of the
Space-Time Characteristics of a Stochastic Channel 39
2.2. Expansion of Space-Time Characteristica of a Channel
into Series and Discrete Models of a Channel 44
-a- ~ [I -USS~t-FFOUO]
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
.
FOR 0~'~ICTAL USE ONLY ~ ~~L',c I
I
~
CONTENTS (Continued) pg~e '
~ ~
2.3. Second-Order Statist~ce of the CoordinaCea of Fac- , '
ization of Channel Characterietics . . . . . . . . . . . 56 .
2.4. MeasuremenC of Channel Characteristica Using Test '
S~gnals (Gaussian field) . . . . ~ . � � . � � � ~ � ~ � 59 1; ~
2.5. Linear Measurement of the Coordinatea of Expansion ~ '
of Channel Characterietics Uaing Teat Signals. 66
2.6. Incomplete A Priori Information and Measurement I
of the Mean Statistical Parametiers of a Channel. 74
~ 2.7. Measurement of the Space-Time Characterietics ;
of a Channel Using Information Stgnals 81 j
2.8. Measurement of the Characteriatica of $ Stochastic
Channel from the Standpoint of Che Theory of ;
Linear FiltraCion . . . . . . . . . . . . . . . . . . 87 ~
2.9. Adaptive Compensators for a Space-Time Channel 99
Chapter 3. Procesaing Space-Time Signals Cantaining ~
Discrete Meseagea . . . . . . . . . . . . . . . . . . 108 ,
3.1. Statement of Che Problem of Optimal Reception of ~
Messagea in a Stochastic Channel . . . . . . . . . . . 108 ~
3.2. Optimal Processing of Space--Time Signals in a
Deterministic Channel. The Coordinated Space-
Time Filter . . . . . . . . . . . . . . . . . . . . . . 110 ~
3.3. Receiving Messages under Conditions of an '
xdeally Classified Sample by which the Channel ~
i-
is Studied . . . . . . . . . . . . . . . . . . . . . . 116
3.4. Reception of Messages in Conditions of an
Unclgssified Sample by which the Channel is , ;
Studied and the Use of A Priori Data 125
3.5. Suboptimal Processing of Signals in the Absence ~
of A Rriori Data . . . . . . . . . . . . . . . . . . . 136 '
3.6. Some Ways to Realize Algorithms for Space-Time
Signal Processing . . . . . . . . . . . . . . . . . . 148
ChapCer 4. Analysis of Algorithms for Spa~e-Time Signal
Proceasing . . . . . . . . . . . . . . . . . . 151
4.1 Qua'lity Characteristics of AaCa Transmission Systems
and Their Determination . . . . . . . . . . . . . . . 157
4.2 The Probability of Error under Conditions of an Ideal
Classification . . . . . . . . . . . . . . . . . . . . 160
4.3 Characteristics of Devices for Processing Space-Time
Signals in a Generalized Gaussian Channel (Smooth
Fadeouts) . . . . . . . . . . . . . . . . . . . . . . 168
4.4 Characteristics of ~~tection of Space-Time Signals
(Generalized Ganssian Statistics) . . . . . . . . . . 179
-b-
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~OR OFFICIAL USE ONLY
CONTENTS (ConCinued) p$ga
4.5 The Probability of Error in Discrimination of Orthogonal 188
Signals (Generalized Gauesian SCatietice) . . . ~ ~ � -
4.6 Noise Suppression of a Binary System of Oppoaite Signals
(Getieralizad Gausei~n Statistica) . . . . . . . . . . 196
4.7 Characteristica of Devicea for Procesaing Signa].s in
Channela with Non-Ga~~~saian StatiaCice under Conditiona
of a Non-Clasaified Sample Uaed to Study the Channel . 202
C011C~.U8j.011 ~ � � ~ � ~ ~ � ~ ~ � ~ ~ ~ � � � � ~ ~ � � ~ � � ~ ZO~
Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.09
Appendix 2 . . . . . . . : . . . . . . . . . . . . . . . . . . 211
Bibliography . . ~ � � � � � � � � � � � � � � � � � � � � � � 213
. ' ~ - c -
'
, FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY ' ~
~ rt~
PUBLICATION DATA
Engliah tiCle : PROCESSING SPACE-TIME SIGNALS
~ (IN INFORMATION TRANSMISSION CHANNELS)
ti
~ Russian title : OBRABOTKA PROSTRANSTVENNO-VREMENNYKH
SIGNALOV (V KANALAKH PEREDACHI INFORMATSII)
Aurhor (s) ' : D. D. Klovskiy and V. A. Soyfer
Editor (s) :
Publishing House : IzdaCel'stvo "Svyaz ~
i
~
Place of Publication. . : Moscow ~
DaCe of Publication : 1976 ;
'
Signed Co press :
Copies . .
~ COPYRIGIiT : Izdatel'stvo "Svyaz 1976
_d_
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~'OR OFFICIAL USE ONLY
UDC 519.24
PROCESSING SPACE-TIME SIGNALS (IN INFO1tMATION TRANSMISSION CHANNELS)
Moscow OBRABOTKA Fr~OSTRANSTVENNO-VREM~NNYKH SIGNALOV ( V KANALAKH
PEREDACHI INFORMATSII)in Ruasian 1976 pp 1-208
[Translation of the book "Obrabotka ProatransCvenno-Vremennykh Signalov
(V Kanalakh Peredachi Informatsii)" by D. D. Klovekiy anu V. A. Soyfer,
Izdatel'stvo Svyaz', Moscow, 1976, 208 pages]
[Text] Shor~ Description
'I'hia book sets forth the general principles of consCructing devices for
apace-time processing of aignals in digital information.tranamisaion
channels. The model used permits description of a broad class of real
physical wave channela, including channels in the optical range. The
conatruction of processing devicea is based on measuring channel char-
acteristics. The algorith~?s for processing space signals are orienCed
to *.he aquipment of holographic (in the broad sense of the concept) ~
systems.
This book is intended for a broad range of apecialists working on the
development and design of data processing systems and also for college
_ students in the correaponding specializations. ~
Foreword
Equipment based on holographic techniques gives the engineer new means
for constructing devices for space-time sigx?al processing.
Significant conCributions ~o solving the problems of optimal space-
time processing have been made by P. A. Bakut,~A. A.�Kuriksha, R.~
Kennedy, G. Van Tris, S. Ye. Fal'kovich, and certain other Soviet and
~ ' foreign suthors. However, publications dealing with this subject are
dispersed in many different periodicals and there are no books which
set these problems forth in a systematic manner. The present book
f ills this gap . �
~ 1
FOR OFFICIAL USE ONLY
f
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~'OR OFFICIAL US~ ONLY
This work inveatigatea the general principles of ~ptimal and subopCimal
signal processing in time-epace channels during ~he txanamisaion of
c'~iscrete meesages.
The book has four chaprers~ The first chapCer ia devoCed Co the search
for an acceptable statiatical model to describe the aignal and noise
field at the output of real space-tim~ communicatione channels.
The second chapter reviewa tihe algori~hma for eaticaating the parametars
thaC define the model of a stochastic channet. Primary atCention ia de- ;
voted to opCimal and suboptimal esCimation of the coordinates of �ac-
torization of the channel char~reteristics on the basis aelected. This ~
esCimation determines the most noiseproof procedure for processing the '
signal being analyzed. The special features of ineasuring the charac- ,
teriatics of a apace-time channel uaing Wiener or Kalmanov filrration
and the principlea of constructing adaptive compensatora to realize
optimal filtration in channela with scattering are reviewed.
The Chird chapter of the book is devoCed Co a synthesis of the algo-
rithms of optimal and auboptimal processing of apace-time signala con-
taining discreCe messages, while the fourth chapter analyzes their noise
auppression.
The first, second, and fourth chapters were written by the authors to- ,
gether. V. A. Soyfer wrote the third chapter and the appendices. D. D.
Klovskiy performed the general editing.
The authors express their gratitude to doGtor of technical sciences
N. P. Khvorostenko for reviewing the book and offering a series of re-
marks that helped to improve it.
Tn1e request that all comments be sent to Izdatel'stvo Svyaz' at 101000, ~
Moskva-Tsentr, Chistoprudnyy Bul'var, 2.
Basic Designations _
A, B, C, D - parameters of a function of a generalized
Gaussian distribution of a modulus
BX(t, t' r, r') - correlation function of a random field x(t. r)
B(T, p) - correlation;function af a stationary homogeneous
X field x(t, r)
DX, Dy, DZ - geometric dimensions of the spatial domain of
field analysis .
E1 - energy of the signal at position 1
F - width of the signal spectrum
2
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY
FKOp [or ~cor~ - inCerval of correlation by frequency
G(w, wg) - energy apecCrum of a channel charactierietic
+
g(t, r) - pulse aurge charac~eriatic of a apace-eime
filCer
h(f, t, r) - transfer function o� a channel
h(e, x) - pulae surge characteristic of a channe'1
h2 - mean atatiatical signal/noise raCio
K(w, wg) - trane�er function of a eoordinated space-time
f ilter _
M - number of orthogonal signals
M(w, wg) - function that determinea a regularizing func-
tional ~
, mXk, myk - mean valuea of coordinates of i8ctorization of a
channel characteriatic
NT, NF, NR - number of coordinaCes of characteristic factori~
zation by i~a~-,^~ndent variables of time, fre-
quency, and apace
N~ - spectral denaity of white noiae field output
N(t, r) - noise f ield
p - probability of erroneoua solution
q 2 - statistical parameter of a channel .
RX(t, t', r, r) - normed correlation function of a field
_ r=(x, y, z) - spatial variable of a fieldl -
- s(t) - signal at input of a channel
sl(t) - signal of position 1 at input of a channel
- T~ [or Ts] - length of an element of a signal in tranamission
1 The quantity r is always a vector quantity with the exception
. of the particular cases specified in the text.
,
3
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
.
,FOR OFFICIAL USE ONLY ,
T - interval of field analysis in time
U~(t, r) - field in the reception domain corresponding
tp poaition 1 of the ~ranamitted aigna~
wN(x1,~.., xN) - multid3mensional dansity of diatribution of a
set of random quantities ~
x(C, r), y(t, r) - quadrature components of a pulse surge
characteristic of a channel ;
z(t, r) - observed field .
a - parameter of regularization
R2 - sCatistical parameter of a channel ~
~f~~~[or ~f~x] - width of the energy specCrum of signal fade- I
outa in time '
e(t, r) - �caasurement error
e2 - mean quadratic value of ineasurement error
n(t, v) - pulae aurge characteristic of a channel
in angle-place coordinatea
r~ -,angle-place variable
xk - eigen values of an integral equation
A - space-time domain of field analysis
uT - parameter that characterizes rate of fade-
outs
vT, vT, vR - degree of selectivity of a channel accord- '
ing to the ~ariables of time, frequency, i
and space ~
[or ~max~ - channel memory . ,
p tco ~�r cor~ - interval of correlation by space
P
- Q2:=, v2y - dispersions of quadrature components of a
channel characteristic
TKOP [or Tcor~ - interval of correlation of channel param-
eters in time ~
~k(t, r) - eigen funation of an integral equation ~
i
4 ~
FOR OFFICIAL USE ONLY , , .
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~'OR OFFICIAL USE ONLY
�p - statietical parameter of a channel
~k - func~ionals computed by an optimal field
processing device
: w , - cyclical frequency
wg - frequency of a spatial spectrum
The deaignaCions for the special functions correspond to those ~
adopted in (29] .
Symbola:
x -~verage value of random quantity x
s(t) - signal s(t) con~ugate according to HilberC
~ u(t, r) - estimation of field u(t, r)
h* - quantity complexly con~ugate with h
S(w, wg) - spectrum of signal (field) s(t, r) �
CNn - number of combinations from N by n.
Introduction
The problems of optimal proceasing of space-time signals in data trans-
mission channels are attracting ever-growing attention, and this is not
accidental. But what do~s optimal space-time processing offer in com- .
- parison with techniques of spatial signal procesaing already known?
Above all it points out one of a number of inethoda of spatial process-
ing that provides the best quality characteristics of information
transmission. In the second place, if we know the algorithm of optimal
processing we can always suggest a large number of suboptimal algo-
rithms whose characteristics are cloae to potentially achievable ones.
In the third pla~e, the system developer ~aill be able to compare any
processing algorithm that is proposed against the best. Specifically,
the techniques of spatial scattericig have become widespread in chan-
nels in the short-wave and ultrashort-wave ranges. The theory of
space-time processing gives aound criteria for choosing the number of
scattered antennas for such channels and the ahape of their diagrams
(space patterns) in each particular case. In the stage of system
development and design such data are extremely valuable. -
For channels in the optical range the theory of processina space-time
" signals is the only and an ob3ectively necessary development of the
theory of processing time function-signals. The processing techniques
, suggested by this theory pose new problPms for holographic engineering
5
. FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
r.
' ~OR OFFICIAL USE ONLY
~ and npen up new opportunitiiea for coherent optical data Cransmission
systema (tranamiasion in a turbulentaCmosphere, transmisaion beyond
the l~.mita of direct vieibility, and othere).
Most communicationa channels are classified as wave channels .?nd to '
one degree or another Che apatial distribution of the transmitting and
receiving structures and the route of signal propagation mueC be eaken
into account.
- UnCil recenCly Che synChesis of receiving-transmitting antennas and
pure time procesaing devices in transmiesion~and reception was done in-
dependently (separately) according to various specific requiremente
' (quality criteria). Most of the reaults in the theory of optimal
methods of transmitting discrete messages have come on the assumption
that the antennas are fixed in transmisaion and reception and the ays-
tem is optimized only with respect to time processing of the signal.
However, the limitations inherent in a system and its potential capa-
bilities can only be identified if we make maximum use of information
on the properties of Che medium of propagation and exisCing noise in
the channel and search for optimal solutions for th~ design of the
receiving-transmitting complex, not assuming a priori a separation of
the operations of time and space processing of the signal and noC fix-
ing Che type of spatial signal processing.
It may be expected thaC optimal space-time signal processing compared
to purely time-optimal processing will be more effecCive where the
quality of data transmission is more strongly influenced by external
noise than internal equipmenC noise. But the influence of external
noise on the quality of communications is becoming decisive as a re- ,
sult of advances in developing low-noise receiving-transmitting equip-
ment for space and ground channels.
Chapter 1. Model af a Space-Time Channel
1.1 Structure of Systems for I~ata Transmission by Space Ghannels
In any data transmission system it is possible to identify, in addition -
= _ to the source and recipient of inessages, the following basic blocks:
~ transmiCter, channel, and receiver [51, 104]. We will consider the
source of the messages and the transmitter, which includes the coding
device, modulator, and transmitting antenna,to be given and then we
will consider the last two blocks: the channel (medium of propaga-
tion) and receiver. We will assume here,. however, that it is possible
to control the operation of the transmitter by selecting an appropri-
ate assemblage of signals used to transmit information. '
Let us consider the concept of a continuous channel in more detail,
because in this work it differs slightly from the traditional concept. i
In consideration of the problems of optimal reception of inessages in
~ ~ 6
,FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
Fnlt OFFICIAL USE ONLY
the mo~t diveree wgve range~~"channel" or8inartly meane the enCir~
tranemiseion part of the system, from the input of Che tranemitteti
antenna to the output of the receiving antenne [30~ 51~ 104~ ~eee
2~~Sgure 1~1~ below), In thie caee ell ~he varigtione of channel
~igure 1.1. Model~ of time and epece-time channels: a) Cime;
b) epace by input and outipuC; c) epace by output
a
) ~1~ ~ jre~,~~rri,
jMe~tf'.~
~ i:n~ 6 ~
~ MtnrovNtn i~(f/ ea b� eo~ea ll0aeueo~ i nM ~e
aodrqt~nd i o~me~ia ~e vR + onmtNNa ~Puta~n?n r~exu~
1 ~
/fOMCA~8~ ~
~~~~.~J
=~~~~r
tjf fl ~ R~~ ~~fe,'
_ b~ dn??o~~aq Wtl nea b� t nprrr.~�a~ u~ ' nea o
' aaA~tAa~ ~tadb ara~ Oo + ohmtnNa Il~utMMrn ~y~d
OMfifAMG ' 11t11y~t (
~toMOA(8~i
~
R~4~J
� i ~
~ ---~r~~i - ~~r~
C) ~arr~ ~ � A~ Ilarw.c,~ rhl p~ �
� i a~mu~:e c ~ ' ~~ne?t.it Ontu""r o[
1
~ IraMQ~~YJL.._ I
. -
Key: (1) Source of Messages (lblocka directly underneath identical in
meaning); .
- (2) Transmitter; ,
(3) TransmiCting Antenna;
(4) Medium of Pro~agation;
(S) Receiving Antenna;
(6) Receiver;
(7) Recipient of Messages;
(8) Cyannel,
models can be claseified as space-concentrated models or time models.
They connect the time function-signals at input s(t) gnd the output
z(t) ~ u(r) + n(t) of the channel [u(C) is the usable signal at the
output, and n(t) is additive noise] by means of aome operator, usually
linear [40, 44J. In data transmission systems signals s(t) and z(t)
very often should be considered vector processes of some particular di-
mensionality. An example is communications systems with parallel data
input to the channel and aeparate reception.
Use of the model in Figure l.la m~kes it poasible to formulate the
problem of searching for optimal (from the standpoint of system effec-
tiveness) methods of converting a mesaage to signal s(t) in transmis-
sion and signal z(t) into the message on reception.
7
'FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
4
` X~
FOR OFFICIAL USE ONLY
Ar the present time, meChode of optimai and ~uboptiimal pro~esein~ nf '
epace-ti.me field-eignale in v~rious channele are becoming wtdeepread
~18, 46~ 52~ 82). In the opCical range this kind of treatment and
pro~eeeing of proceeeee in ep~sce and in time se 'the only poseible
one� The Cechniquee of opCimal and euboptimal eignel proceeeing in ,
nther weve band~, for examplg ehort-wave~ ultrashort-wava~ and hydro-
acouetic channels~ are a~,eo epace teahniquea~ It ie poseible to
conetruct a epatially distiributed mod~]. o~ q channel that connects .
�ieid e(t, r~) at the output of the tranemi~ter antenna where ~
r�(x ~ y, z) is the radiue-vector of a field point in tranemi~eion~
a~d thelfie~d x~t, r) � u(t~ r) + ntt, r,) at the input of the receiv-
ing anCenna whare r~(x~ y, z) is the radius vector of a field point _
in reception, u(t, r) is the aignal fieid at the channei output, aed
n(t~ r) is the noise fieid ~see Figure i.lb arove).
Representation of a continuous time-space channel in the form of the
modei in Figure i.ib requires eignificantly more a priori information
than repreoentation of a time c:hannel in the form of the model in
Figure l.la. In this case, however, it is poaeible to pose the prob-
lem of opCimizing all devices for conversior. of ineaeages into a signal
in tranemiesion and conversion Gack into meseagea in reception, in-
cluding the conetruction of optimal eignel-field convertora in trans-
misaion and field-aignal converCore in recgption (tranemiCting and re-
ceiving antennas). ~
' In this work we consider the transmitting antenna to be given, and eo :
we will not inveatigate the model of a channel with epace-time eignals
at the input and output (see Figure l.lb) further~ but rather will con-
centrate attention on inodele of a channel (see Figure l.lc) in which
the input eignal is purely temporal (concentrated in space) but the
output eignal is a apace-time signal. For simplicity we will consider
the fields to be scalar. ~
For vector fields such as electromagnetic fielde the results obtained
by us can be applied to any of the scalar components. ~Jhere there is
a correlation among componente of the vector field a rigoroua eolution
requirea study of the total vector field (for example, by aolving the
corresponding vector differential equations of the fie~d (43, 135)).
However, in maay situations of practical intereat thig correlation can
be dieregarded.
1.2 System Characterietice of a 9pace Time Channel and Continuous
Models of It
If ~re consider the epace-time channel under analysis to be a linear
system With variable parameters, it caa be described by vaxious sys-
tem characteristics (40, 47, 132) (see Figure 1.2 below). Among _
them are the following:
h(t, i) - eurge characteristic of the channel,
that is, the reaction of the channel at moment in time t
8 '
~OR OPFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USS ONLY
at po3et in epace r to a De1Ca pulse fed to Che inpuC
, ae moment t-~. We coneider that the inteneity of
the fieid at point r can bs meaeured by placing an
elementaYyr aneenna at thie point;
H(t, f, r)~+-+h(t, r) - tranefar function of
the chann~l, related to h(t~ r) by a Fourier
eransform by variable f;
U(v~ ~,*r)~-+h(t, r) - epectrum of channel reec-
tion at frequency v to a Delta pulee related to
h(t~ r) by a Fourier transform by variabie b;
. _
, ~tr, ~l,ryhtt1 /f
- Fourier traneform of the cor-
1i (t~ H~r, i? responding fu~ct~one according
~y `~My ~ to variables rkim I 0
L rom
_ ~ 1 J'hkt ~1i r~ Ilk~ ~~i b2'"' ~l~ r) hk, ~t ~ ~
k~.>k~~k~=10 0 ~ �
;
eo m y~
� ,
--~z~ r) ~ d r. ; f , ..,f hi ~t ~ ~1 ~la ~l~ r) h~ ~a
~ o0 0
~ - ~s~ r) h~ (1 ~ b - 5~_~ ~ r) d ~i . d ~L_~ ~ l .23)
This relation followa from (1.22) if we conaider that the operations of
- multiplying the system functions H1(C, r, r) and H2(t, f, r) are equiva-
lent to the operationa of convoluting the corresponding tranafer charac-
' teristice. _ . . . , _
. _
; Nl f~ r) Flz f, r)~ ,f hi ~irT) hl ~t~ ~ - b~,~) ~ ~i~ ~t .24) ~
0
With purely parallel propagation (disregarding multiple acatCering)
_ . . _ .
N~t~ f~ r) =~Hk~~~ f~ r)� (1.25)
k~l
In the case of sequential propagation only
~ _ .
_ _ tf (t~ r) - I1 Hk~~, f, r). . (1.26)
'a1
We shculd give special conaideration to the par~icular situation of se-
quential parallel propagation, described by the co~on tarm of the relation
(1.22) . . _ . _
� N ~k .
H~r~ f~ r) n H~k~~, f~ r)� (1.2~
, k=1l~1
When substantiating the probabilistic model of fadeouts in a channel it
is usually supposed [49, 80, 104, 135, 137] that the number of scatterers
N forming the total signal at the point of reception is large. However,
for the general situation of sequential parallel propagation this as-
sumption is not adequate to find the limiting distribution (where I3-?~)
of a random quantity (1.27). The point is that probability theory doea
not yet have a limiting theorem for distributions of the sums of the
~ products of random quantitiea. Therefore, at first we will consider the
19
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY r
, ' ~~;~t
two theoretically extreme situatione: purely additive (.1.25) and purely
multiplicative (1.26). Then we will also diecuse the intermediatie addi- ~ _
_ tive-mulCiplicative situation C~~27)~
The purely addiCive situa~ion in formation of a received field. Where ~ ,
N-?~ we may usually consider the condiCione of the ceneral 13miting
thenrem to be met. This allows us to viaur th.e.section (reading) of the
transfer function of the channel H(.t, f, r) = x~t, f, r~+iy(C, f, r) as
a composite Gaussian random quantity, Tts quadrature components
X(t, f, r) and y(t, f, r) are, in the general cas~, dependenC and have
- arbitrary (unequal) mathematical expectaCions mx and m~, and non-iden-
tical dispersiona crX2 2. The channel mudel we are discuseing is
called a four-parameter o~Y generalized Gaussian model [49, 89].
The conditiona of physical feasibility of the channel impose definite
limitations on the relationa between the quadrature componenCs of Che
transfer function x(w) and y(w). They can 6e obtained sCarting from the
condition h(~) ~ 0 where ~ 0 or from the equivalent condition
_h(~) =h('s).1('s)~. (1.~8)
where 1(~) is a unitary function~I92]� i
Now we will perform a Fourier transform on the right and left parCs o~ -
the last relation. Thia makes it possible to convert to the relation
for the transfer function of a physically feasible channel:
_ _ .
- ~
~Ef (c~) = 2~ f (co') U (ai - w') ci w', ( I .`?9)
J
where U(w) _~r8 (w)+1/iw is the spectrum of the unitary function.
~ From this integral it is easy to obtain the expressions that relate
the real x(w) and imaginary y(w) parts of the transfer function of a
physically feasible channel:..__.._. - ~ -
_ - ~ ~
x ~W) - ~ f ~ ~61~~ d cu',
, c~ - u,'
' (1.30)
~
,
y (o~~ = - - ~ X ~ d c~ . � .
n ~ ,u~-w'
The integrals in (1.30) should be considered as
- . ._.__._._._..-~Q
x(o~, ! ~im I y co~~, d~~;: -
n u-.~ w - w
For a stochastic channel the convergence of the integrals cited must
be understood in the mean quadratic aense. The relations obtained
20
FOR OFFICIAL USE ONLY
~
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~
FOR OFFICIAL U9L ONLY
aliow ua to etete thet th~ real and imagl.nnry perCe o� th~ Cranefer
function of a reA1 chennel m~et be ineerlinked by a Hilbert trenrform~
For linear deeerminietiic filt~re thie rasuiti ie not new (for exampie~
e~e (92~).
Ueing the prop�rCies of Che Hilbert traneform it ie not difficult to
ehow that ehg etatiistical characteristics of the functione x(w) and
y(w) muet me~t c~rt~in r~quir~ment~. In p~rtiicular, if it i~ aesum~d
tihaC the channel. ie homogeneoue for frequency in the bYO~d oenee~
Chen x(W) and y(w) are random proceesee with identical correlation func-
tione that are noncoherenti on coinciding frequeecies.
Within the framework o� a Gaueeian probabilietic modei of a channel
thes~a properties of x(w) and y(w) lead to the Rayleigh or Rice dietiri-
bution of the modulus of the tranefer function (amplitudee of the sig-
nal received). In the case of a channel that ie Lnhomogenaous in fre-
quency the correlation functione of the proceeees x(w) and y(~) have
the following relation
~ ~ , , d to' d a~~
g, ~c~~~~ ~~1 f f ~r (~i, wl) , ~ (1.31)
~ , . ~ tn~ n, . .
~m � .
and are~ in the general cese~nonidentical. 'Che mutual correlation
functions ~ ~
~ H~ ~tn~ ~ uip~ ~
B.v(~~~, wi1=: f , dwi,
~ w1
/ ~ ~ ~N~, f01, ~
~Yt \~~I ~ ~2~ ~ , d ~l
11 _ � y~ 41! , �
are nonidentical and in the general case do not become zero on coincid-
ing frequenciea; in other worde, the procesaee x(W) and y(~) are not
_ noncoherent.
Within the framework of the Gaussian probabiliatic model of a channel
these properties of processes lead to a geaeralized Gaussian four-
parameter [49J or Hoyt (eub-Rayleigh) (135~ dietribution of the modulue
of the tranefer funct~~n.
It is also poeaible to make the opposite assertion, that the occurrence
of a four-parameter or eub-Rayleigh di$tribution of the modulue is in-
evitably linked to frequency nonhomogeneity of the channel. Because -
many actual communicatione channels are inhomogeneous for frequency,
it may be expected that the generalized Gauesian or sub-Rayleigh model
of fluctuationa will occur in many cases.
It is always possible to pass to x and y, the independent quadrature
components of transfer function H, by rotating the axea of the coordi-
nates (orthogonal transformation) [26, 49j.
' 21
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~AR OFFICIAL U9E ONLY
Raly~ng on theee reAUlee, we wi~~ here~fter s~~a~ma thA~ the real 8nd
imag~nary componente of a~eceion of the eransfe~ function are inde- _
pendent nnd heve ehe paremater~ mx, ox~ and n~, , v? reepectively~
Tteating the pulee eurge charecterietiic of th~ eh~nnei in ehe compoaiee
form of rdpree~nretion H(e, r) ~ x~~, r)+iy(ti, r)~ w~ w111
maka eimilar aeeumptian~ wi,eh reepect eo ite real and imegin~ry com-
ponanee. Thue, tha quadrature componente of the trenefer function
(or compoe3te pu~ee eurge cheracreri~tic) ara indapandent and hava a
normai d38tribution: ~
! .
(~1: : : r,;p _y._~~
~�~~:~n, [ ?e~
u', ~y1 exp ~ ~y -"'y~t f~,12~~
. ,
~ 7 a e~ er
Ie ~hie cas~ the one-dimeneionai dietribution of Lhe modulue 1'~ ya~
can be obtained in the form beloW ~89~
~rn ~ ~
~2 ~n ;
~Y) ~ n~ ~ dmj dn~~~ }
n-o
z. a
~ n exp to \ o ~/at;~~~~,~ (~.331~
{C ~
where the fo~lowing designationa have been introduced:
~ _ as~n:
4
~ ; nt - m~ ~ a' = ~ R ~ ' -r + '
, = y7 y7 ~
There are also other formg for ~+riting this dietribution ~49, 135).
Where certain conditions are met, a number of particular cases follov
from the generalized dietribution (1.33):
1. The Beckman (or three-parameter [49~) distribution follows from
(1.33) with a c2rtain p~hasing of the regular component � 0,
mX, ~F 0, and o ~ a . Let us stresa that Within the~ramevork of
the generalize~ Gauss~an model the existence of a regular component of
the aignal being received is not necesearily linked to the hypothesie
- of the existeace of a"regular" beam in the channel; the regular com-
ponent n~2 + my2 ~ 0 caa alao occur ag a reault of special character- ~
iatics of aave scattering (49, 51, 104, 125~.
2. The Rice (or generalized Rayleigh) distribution is obtained from
(~.33a Where there is chaneel sqmmetry by dispersions of quadrature
components cX2 R oy2 ~ c2,and R~ 0. ~
3. The Hoyt (or aub-Rayleigh (49j) distribution follows from (1.33)
where vx2 ~F cy2 and in the abaence of a regular component mX � my ~ 0.
22
_ FOR OFFICIAL USE ONLY
.
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~'OR AFFICIAL t18E ONLY
if o 2� 0 in thi~ caee, e1~a di~trl,bueion of amplitudee i~ deearm~,ned
~ by e~ia ona-eided normdi laa ~49~ wh~,ch correaponde eo the d~apeat fade-
- ouea wSthin eha frameaork of the four�parAmete~ model.
- 4. The Rayleigh dierribution is obeainad from (1.33) in eha abaenca of
boeh aeymmatry ~x2 = cy2 � 02 a~nd a Yegu~ar compon~nt mx � my ~ 0.
' ~e i� ~a~y eo tr~e~ eh~ eondiCiana und~r ahinh the inr~rf~r~nc~ rum (i.2S~
aiChin indapend~ne componente N~~t, f, r) gtvee riee tio ona or anotiher
fieid dietrSbue~on H~e, f, r)~
!?eeuming thae ~ha ampl3tudee and ph8aee of the alemeneary compon~nts
nk
yR~~N~f �l'~x�=~~a~ and 6~~arrtg !
are indapandent, ie ie poaaibie to arite
- _ _ _ . .
_ _ ` L
mx ~,e ~~~}~R cos HR, n~,, y___ C~ ~:R ~In(~R;
R~t , R~I
L
C~ = ~ ~ C~! k- ~y'4CASHR~=~;
L
Qy = ~~si~~ ~~~~~s~~~~rR~=~~ ~~.3a)
R~1
t.
I3;~, (.r-mR)(I~--myl ~ ~ ~~Rsir~-.~. ~
~ r.
Sit1 dn COS Hl Ilt~ Nl y.
~r
n~. ~ t� ~ ' .
~n+~ t
If the elementary componenta have identical statiatics~ then
' m,~ ~ L y cos H, mY ~ L y� sin E~;
~ mi ~
= o, ~ Ly~ cos~- L~ v;,--�L~sin'(~- (1.35)
~
B,~� z L y= sin 2 c-t _ n~~~~ .
L
Aaalyzing (1.35), it is possible to draw certain general concluaions
about the poasible model of the channel:
1. If the phases of the elementary components are dietributed evenly
in the range from -n to +~r, thea mX ~ m~, = 0, cx2 ~ vy2, B~, � 0 and
, the scattered field ie a Rayleigh vector. ,
23
~OR OFPICIAL USE ONLY `
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~t : i
::3 i
FOR OFPICIAL~ VS~ ONLY
E
2. Wieh fiuceu~~ion~ in ehe ph~ee~ of th~ elem~t~ea~y compon~ne~ wiehin
~imit~ eignif~caeeiy ~xc~eding ~r, ~he reAUir~ng fieid is aiao a Rayleigh
vector. Thie conciu~ion fullowe from the fect thae for the periodic
functiona ein A~and coa 0~ in~tead of ehe dis~ribution functic~n given
wiehin i~rge limiti~ it ie poesibla to uea ~noehe~, reduced to eh~ ~ne~r-
val of periodicitiy ~64~. ~f on~y ehe in~eia~ dietribution o� ehe phaeae
of the ele~enCary compon~nte ~e not a periodic funceion, ehen eh~ con-
v~iu~~d di~tributi~n aiehin th~ i~mi~a +~j ~?i1i b~ ~ia~~r ea ~v~n
' Where the iimiea of rhe fluetiuetions of the pheee of ehe el~meneary com-
ponenr~ are greatar. ~
3. Where the flueeuaeions 3n phaeea of the elemeetary componente are
eymmetrical relaeive tio their average v~lue,equal eo zero~and ehe dia-
pereioe of phase fiuceuatione ie no~ eoo great, then oaing to ehe parity
of the dietribution func~ions from ~1.35) ~t folloWa ehat mx ~ 0, m~, = 0,
0~2 ~ oy2 and 8~, � 0~ that 3s, the scattered fiold forme a thrae-
parAmeter vector C49, 135~.
4. With agymmeeric fluctuatione of phaees of the elementary componente
m~ 0, m� 0, cxa � vy2 and Bxy ~ 0, tha~ ie, the ecattered f1e1d is
e four-parameter vector.
Thus, vith the aesumptioae made, the genaralized Gaueeian atatistics of
a acateered field are e consequence of esymmetry, which can be expiained
in the dietribution of phaBee of elementary waves on ehe baei~ of the
phyeicai proceeees relaeed to the propagation of vaves in random media.
If there ie a regular beam at tha receiving point in addition to the
ecattered field~ it is natural that caees 1 and 4 laad to a Yeeulting
field in the form of a Rice vector, ahereae cases 2 and 4 yield a re-
sulting field in the form of a four-parameCer vector.
Experimental dar.a shoW that the generalized Gaussian diarribution and
its variou~s particular cases cover a very large class of communica-
tions channels (49, 135~. SolvYng the stochastic wave equation of the
:ield for different mechaniems uf aave propagation aleo leada to a
generalized Gauesian model and a number of its particular caaea (32, `
33~ 49 50, 116, S0, 124-126, 135]. in addition to parameters ny~~
~ my , cx~, and oy2, it ia convenient to introducp four other parametere
Wf~ich have graphic phyaical meaning:
~ m~~
9= ( i .36)
oR �f- o~ '
- the ratio of the average povera of the regular and fluctuating paYts
of the transfer function or surge charanteristic of the channel;
a~~__ Q~ar _ ~1.3i~
- the coefficient that characterizes asymmetry by dispersions of quad-
rature components;
24
1~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~
?
FOR OFFICIAL U9B ONLY
~ ~
;
;
? ~o � arc tg(?n~l?nx) (1,36)
� - ehe phaee angle of ehe ragul8r componentf
~ _
~ ~ ~R my a; ~ a~ ( t ,39)
~ - the m~,an �quera o� ~he Cranafer fun~:tion Caurge charactiariseic).
- Far ~ fu11 d~ecription of th~ channei ie is euffic3ent to coneidar the
folloWing rangae of ehange in ehe paramaenre introduceds
. _
0~9s~~i~~~'U~diS~~ O~~Pn~~nl~~t n~Y'G~.
A`rhola anries of ueaful formnl8a reigeieg to the four�paremaeer dieCri-
~~urion of the modulue ie coneained in the literature (89~. Let us ob-
eerve that ehe t~+o-paramaear m-diseY3bution of Nakagami (137~ eatie-
factorily approximatae the four-paramatar diseribution of ampiitudee [49~.
Tha dieeribution of the indapandant varia~ile of tha eurge funeeion
arc eg (y/x) for a genarel3zed (iaue9ian chg:u?ei is conesined in the
litarature (49, 64~.
The purely mulCiplicaCive eituation in formation of ehe field being re-
ceived. If we ~rrite the tranefer fuection of pertigi filter k in (1.26)
~ in ehe form ~~xR e~~R , it ie not difficult from (1.26) to obtain '
the following ' - . _ ,
H'~ ex o= y e~ ( f.A~.
For the quantitiee
_ L `
x' ~P Et.41)~ `
+~~i 4~,~
ahere L~?~, the conditions of the central limiting theorem are met, mek-
ing it poseible to coneider them Gauasian random quantities.
Nith multiple ecattering, in particular for a stochastic optical channel~
the aadulue of the transfer function Y~ eX and i~s independent variable
~ can be considered statiatically independent [67, i11J. The one-
dimeneional di.etribution of modulus Y is logarithmically normal
~ - e_ ~~n y - ~n= oX - ~ 1.42~
, (Y) y- �
Pnrameter Q2 (dispersion of the logarithm of nwdulua Y) may be related
to Nakagami~s parameter m(57, 137j
m (y`s)~ / (Y~ (j~)~]~ ( I .43).
- which, changing in the range from 0.5 to is a convenient measure of
the depth of aignal fadeouts (the depth of fadeouts increaees with a
decrease in m). The folloaing relationa are correct:
,
25
~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~i
~
.
i ~~iz
,FOR OFFICIAL U3E ONLY
t~x 4 IIt f 1}� l, 1!i ~
e~ , (1.~~1)~ i
~ ~ ~ x... ~ .
For emall digpereions of the logarithm of amplituda ~nz ~~~5) , the -
logarithmically normal dietribution (L.42) ie eatis�actor~.ly approxi.- ,
maeed by Nakegami's m-die~r~.bution where m>3 and therefore also by ~he
Rice diatributiion ~49, 57~. '
~or large dispereion values ~~z~~,~) the approximation ehown above
is uneatiefactory because under these conditions the logaritihmically
normal distribution~ unlike the m-dietribution, is charncterized by a
very elow decrease in probabilitiy deneity in the domain of large values
of the independent variable.
Let ue pass on to a consideration of the queetion of the distribution of
phaees for the purely multiplicative situation of formation of the re-
ceived fieid. As can be seen from (1.41), ttie distribution of phasea
~ in an infinite interval is governed by the Gausaian distribution.
However, for the problem of optimal signal proceseing, the distribution
law of phases in the eegment [-~r, +~r), that ie, the distribution reduced
to the interval of periodici~/jr, is most interesting. Beginning from the
result in (64], 1.t ie not difficult to show that the disCribuCion of the
random quantity ~ in the interval of periodicity (-n, ~Mr) has the form
_ . - . _ _ . _ .
b
= 2rc 1 2 ~ ~Di ctu r ~ . ( I :45)~
?..i .
where 61(u) is the characteristic function of the quantity
In this caae the quantity ~ is normally distributed. We will suppose
that its average value is equal to zero. Thia can always be done, con-
eidering the diatribution of the initial phasea relative to the average
phase incremenC. Then, from (1.45) it is not difficult to obtain
. _ az ~~~s 2
~'i(V'~ " 7n 1-f- 2~ e ~ cas r~p 2:c C 2' e~�~~ J. (I .46)~
where ~93(z, g) is Jacoby's Theta function (29~.
From a practical point of view the m~st intereating valuesto consider
are the valuea of the parameter v~�1. In this case~ it follows from
(1.46) that k~l ~~p) I12:c~
(l.4i)
in other worda, there is an even distribution of the initial phase of
the transfer function of the channel in the segment [-~rr, + n]. The
even character of the distribution of the initial phase in channels ,
26
. ~OR OFFICIAL USE ONLY
e
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~OR OFFICZAL U3E ONLY
with eequanrial wave propagaCion hae baen poinC~d outi more Chan once in
Chaoretical and experimential worke ~67, 97, 111~~
Zn order to compare tihe probabi.lietic mode],s of ehe channels For Che
putiely add3Cive and the pure~.y multipl~.cative eituaeions o� formation
of tha field received, it ie relevan~ to coneider tihe diatributiions of
the quadrature componente of the tranefer ~uncti.on in both caees. W3Ch
an independent logarithmically normally di~tributed madulu~ and a uni-
, formly dietributed independent vari8ble the ~oint distribution of
quadrature components can be written in tihe foliowing form
. 1 _ _ .`In V�~,--~ "!.~~s ~ (1.48)
'~s (Y~ y) ~ ,Zn /'1 rc o~(xs~ r ~ o%
V
Beginning from (1.48), it ie easy to observe that the quadrnCure com-
ponents x and y have the eame distribuCion lgwe with identical statis-
tical parameters, for exgmple: .
- _ C~> ~ - . i . ~ ~ cxp- ~In j'a,a y~ rnx .
dy. (1,49)
. , ?n ~''2 n ox x' -t- 2 ox
These distributions are symmetrical relative to the ordinate axis. Thie
meana that the logarithmically normal diaCribution of amplitudea and a
uniform dietribution of phases preclude the poesibility of the appear-
ance of quadrature componente with non-zero mathematical expectatione.
F'or emall values of the parameter vX, the distiributions of the quadra-
ture componente are bimodal and very far from the Gaueaian law.
The addiCive-multiplicative eituation of forroation of the field being
received. We write expression (1.27) in the following form
N
N~l ~ f~ = L; HR f~ r) = x-I- ~ y=1' e~ o~ (1.50)
k=1 _
where
~k
= ~�k e~ ~R rI Hrk (r~ r)� ( I.b 1)
(u~
If no constrainta are imposed on the aet of random components Hlk and the
quantities Lk and N, it is extremely difficult to find one-dimenaional
distributions of H. It may be asaerted that, in principle, situationa
are possible that yield the mos,t diverae distributions for H. However,
it is worthwhile to undertake at least a qualitative Creatment of the
relations which will enable us to emphasize the apecisl importance of
the two limiting typea of distributiona: four-parameter and logarith-
mically normal. With this purpoae in mind, let us consider first the
,
27
,FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
- . ~ ~ ~ . . . . . ~~rt , 1
~~i
FOA OFFZCIAL USE ONLY ~ f` i
1
~
;
case whare ehe random ~ompoeite quantitiea
~ ~ ~ (~~~i~~ !
HiR ~ ~,~~k ~ UiM ~
are mutuaily independant and Che number of fgctiora forming H does noti i
depand on k and ie equ~l to I,k ~ Q. Figura 15 below shows the model
of saquential parallel propagation of wavee being coneidered~ includ- ,
ing QN linear filtere with characterietice tt~k.
~
~
; r~~ . . . _ , I
.
j
Nn ,~7 ~ N1N ~ NiM ,
~ ' !
Kt~ ~:i ~ Hu ~ Ni~+ ~
1 ~ ~
N� N,~ ; H,R I N,~ Figure 1.5 ~ Chennel with 3averel
, Independent patihe of Sequential
; ; i~ Propagation of Transmitted Signals. ~
~ NiJ I Nie j N~~ '
~ � .
3
tl/Rff , ' ~ .
, ~
As a result of the independence of the parallel patihs of the model (com- :
ponents of Hk, see Figure 1.5 abova), it is natuYal that where t~f-?~ the '
distribution tende toward a generalized Gauasian distribution regardleas
of the dietribution of the componenre. Thsrefore, let us consider the ~
case of a limited number of componeinte N. `
As the results of digital modeling ehow, for the model in Figure 1.5 '
on the condition that the parametera of the filtere are random but in-
variant in time _ - ~ _ . _ . i ~3 .
H?k(t~ i, r) = NiRti? r)~ ~
the law of distribution of the modulua H(t, f, r) in (1.50) where Nc10 ~
is determined more by the multiplicative character of the relationship
am~ng components than the addi.tive aspect.
Apparently it can be expected that as the intensity of the relations
amoag the particular components Hk:ia(1.50), Chat is, of the signals
in the parallel paths of propagation, grows strongex, the dominating
role of the multiplicative aspect of the relaeionship will increase.
As for distribution ~ with a limited number of components N in (1.50),
beginning from [67] a uniform distributioa of phase may be conaidered
typical.
When the parametere of the spatial filtera of the model change randomly
in time in ttie interval of the analyeis, a lessening of the impact of
28
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
,
FrOR OFFICZAL U3E ONLY
Che multiplicaCive aepecti pf the relationehi~~ ehould b~ axpecCed. In-
' deed~ ~.f ~he width of the energy epectrum of time fadeaues Qf~g � 1/'rCnx
3e coneidered ~3mited ~TCOr ~e the interval of tiime correlation~, ehe
funcCion H~,k (C, f, r) may be represenCed by a Kotel'n~.kov aeries witih.
uncorrel,a~ed references, whi.ch ~hould lead Co an increaee in the number
of componenta in formation of Che total tranefer function.
Theae reeulte c8n also be applied in look~.ng for the dietr~.bu~ion of
- more complex additive-multiplicaCive formation (1.25).
Thus, it may be seated that the generalized Gaueaian probabilie~ic
model of a field ie acceptable for describing a broed clese of real
channels witih both single and multipie ecaxtering, but in tihe latCer
case the ephere o� appiication of thie model is definitely narrower.
1.5. Statietical Models of Space-Time Channale Besed on Correlation
Properties
In solving Che problems o� optimal processing of fielda, ae will be
demonatrated below, correlation characteristice are decisive for de-
acribing not only Gauesian but also etochaetic fielde of arbitrary
~ ehape. ~
~ Yn this connection, we ahould conaider the classification of fields by
degree of correlation in time, by frequency, and space. To do so Che
correlation function of any syetem charac~eristic of a channel may be
investigated [47]. �
An exceptionally important property of the correlation functiona of
real space channela which makes it considerably eaeier to conatruct
optimal procesaing diagrams is the fact that they are partially or com-
pletely factorable, that is, they are represented in the form of prod-
ucts of correlation functions by separate variablea. In particular,
a review of the correlation functiona computed for a whole aeries of
_ channels [32, 33, 40, 80J shows that in many cases they are epatially
distinct, that is, the space correlation coefficient ia factorable.
As will be ahown below, factorization by the epace variable makee it
possible to greatly simplify the algorithms of optimal proceaeing and
to separate epace and time proceasing of fielde received.
In engineering practice it is often convenient to characterize a par-
ticular apace-time channel depending on the relations among the corre-
lation intervals of the field by frequency F~or, in time Tcor~ and by
apace Pcor~ gnd among such important characteristics of a communications
system as length of signals T8, widCh of the epectrum of channel signale
Fs, and spatial extent of field R~nalyzed at the receiving place.
Let us observe that the signals used to transmit information are always
finite (TS is limited). But this meana, etrictly speaking, that their
~ spectrum ie not limited. Nonetheleas, when solving applied problems we
assume that FS ia also limited.
' 29
~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
r
FOR OFFZCIAL USE ONLY ,
We daCermine duration T~nd epectrum width F of tihe e~.gnal at tihe ouC-
puti of a channel bound to an i.npu~ signal w~.th parametere Te and F~ by
ehe Eollowing relatione (46):
where ~~kc ~~nax~ ~~'~~car ~'n tihe ineerval of dignal ecatrer-
ing in time (ahannel memory) caueed by ~he imperfectnees o� the frequency ~
charac~erieCics or the lack of a eranefer characterietic from the De1ta
func~~,on (owing to multiibeam wave propagation, nonlineariCy of the phasa- '
frequency charactierietic, and the like); Ofrogx �~~TCOr ~e the ~.nterval
of signal scattering by frequency (nr width of Che energy spectrum of
time fadenute) caused by change 3n channel parameters over time and mu-
_ tual dieplacement of the areae of signal formaeion and reception.
The channel memory ~~x may eometimes exceed the duration of eignals
tiransmitted TB eubetantially~ for example in high-apeed sequenCial trana- ,
mission of inessages in ehort samples. When there are no protective time
intervals and sma11-base eignale (2FBTe ~ 2) are usedy this gives rise to
intercharacCer interference [49, 53]. ,
For moat radio communicatione channels, the interval of frequency ecat-
tering ~f~X � Fs. Where long-duration complex aignale are uaed the
correlation time Tcor � 1/~fmBX may be considerably leas than aignal
length Tg.
Let us introduce parametera ehat charac~erize the number of degrees of
freedom of the atochastic field received
vr I nNr = In [ I-{- Tl,r,io~1, -
vF In NF -1n (1 F; F,:opl, (1.55)
~R = In NR ~n ( t-}- R!!?~~oNl~
and call them respectively the degree of channel selectivity in time,
by frequency, and by space.
The quantity NT =[1+T/TCOr~ determines the approximate number of non-
correlated (and therefore, independent for Gaussian processes) time
readings of the aignal in interval~T; NF is the number of non-correlated
frequency components in the apectrum of the field received. The quan-
tity NR hae an analogous meaning.
It is apparent that the larger the number of vT, vF, and vR, the greater
the set of possible realizationa of the received field will be and the
more complex the model of the channel that gave rise to them will be.
We will call a channel nonselective for given parameter P if vP = 0.
But if vP > 0, then we will consider the channel selective by this
parameter. Thus, if vT = 0(there is juat one uncorrelated reading in
30
~`OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
i ,
~'OR 0~'~'ZCIAL US~ ONLX
interval T) or T ~r~~or, (1,56)
we w~.l~ ~gii the channel nonse~ective in tiima~ Such a channel ie often
called a channe]. with aloW fadeout, nr faee fadeout wh~re condition
(1~56) is not meti.
~ Coneidering whati hae been said above, it is posaible to define eight
tiypea of epace-time channels by degres of aelecCivity (considering ~us~
one space coord~.nate) and group them achematically as ehown in Figure 1.6
below. The aimplesti one o~ them ie the channel ehat ie nonselective by
frequency, time, and epace(vT~ vF ~ vR~ 0) and the mos~ complex ie
. selective for all theee parameters (vT > 0, vF > 0, and vR > 0).
!fC no vccmame npo�
~ ~ cmpcncm0~ u no Ap~ue�
Nn V; V, V~~ 0 ~
IfC no DpcMeH~t r~ A'C'no vacmeme u IfC no A~eMrnre u ,
npetmpc~cmo~y ~b rrpocmpaNtmey ~ ~ ~~otmome
~fC>0.
If the spatial variable is not conaidered (the~ signal is a function of
time), an analog of function (2.165) is
~a ~ 1 z (t~l .1G6)
N c~) = , ~
I a A! (w)/.S (u~) � S (w)
In such problems Tikhonov-type regulariaere of~ order p[9] are uau-
ally uaed, aeaigning a function of the type
� ~S! ~W~ a W~p~ ~2.16~~
which determines the set of regularizing operators and, using a certain
algorithm,the value of regularization parameter a is found. In works
[98, 99] it is shown that use of regularizers of type (2.167) makes
it possible to obtain a stable solution to the problem.
In the situation of space-time signals under consideraCion, Tikhonov-
type regularizers should be used, choosing
~?'f = u~~p ~ue . (2.163)
The other possibility, frequenCly employed in practice, is to choose
a function M(w, wg) that affords a truncation of the spectrum of the
space-time frequencies of the function under study.
0, -2:sF~~ 3 In 4:t the opposite is true.
Let us move on to a consideration of the upp~r boundary of probability
of error. To determine the upper boundary it is convenient to write an
algorithm to distinguish two signals in the form
.1 .1 r~-u~(r~ r)~1dtdr < J 1 ~Zli. ~)-^i~l,`~)J~ dldr. (4.12)
00 00
It is not difficult to show that on the asaumption that the l-position
signal s(t), 1= 1, 2 is transmitted, the probability of an erroneous
decision is determined by the prabability of fulfillment of the in-
equality
164
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~OR OFFICIAL USE ON1,Y
; ~t ~ r ~ ^
I~ tl~ r) ~2 (r~ I~J rf ~i� (r ~ r)ljdJdr;< ,11,~ I rri U~
uo
~
---i~o(1, r)~~idldr~ ~4.13)
where e~(t, r) exprea~es the errar of e~eimaeion of ehe 1-poaieion signal
caused by inaccuracy ~.n measurin~; the ch~nnE1 characterigtic:
~
~ r~l~, ~l=�Ite f ~hlr~ ~1---1i(t, s, r)~sll~)~~.
0
Entry (4.13) enables us tio draw qualiCative conclugions on the ef~ece of
inaccur~cy in meaeuring a channel characteriae~.c for noise auppression.
It ie not d3fficult to noCe thae the effect nf inaccurate measurement on
Che characteriet~.ca of ~n algoriChm finds ex~ression in Che appearance
of addiCtonal additive noise correlated with Che signal.
The linear, unbiased estiima~~s gynehes~.zed i~~ Chapter 2 are used to mea-
aure channel characteriseics. Tf~en, for Gau;3aian additive noise n(t, r)
the addiriottal noise el(t, r) is also Gaussi;in with a zero mean and cor-
relation func~ion Bel(t, r, r').
~
It is perfectly obvious that the probability of fulfillmenr of inequality
(4.I.3), which is Che probability of error, will be greater if additional
naiae el(t, r) is white noise, which staCistically does not depend on the
usable signal, and has a spectral density of output of:
lt~r (~,0) /3f; (O,U) ~
IJ�~ r ~i (4.141
This circumstance makes it possible to write the upper boundary of the
pr.obability of error in the form
? ~
/~ut~~~~i;.y ~ , _
,
~D ~ ~ r'ri n l
~ i1'o n~ I 1~"~ ll~ i) i~t It. r1 J~ dldi 1 (4.15)
o'o ,
_ tf optimal linear estimates are computed to :,tudy a channel, then (see
Chapter 2) the mathematical expectations and dispersions of estimates
coincide with the mathematical expectations and dispersions of the quan- -
tities being estimated
- ' n n
A1 {Fk} tlt~k; AI {tlk~ = m:~R~
/){.Y,~} =0ak: D{yk} =Oui�
i65
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~ i
~FOR OFFICIAL U3~ ONLY
it ie nlear from thi~ th~t ~he upper boundaz~i~~ o� probability of error
nre determi.ned by eha eame exprQSaiong as ehe lower boundariee, which
were obCained above, excepC ehat instiead of parsmeCer No, ehe parameter
(No + Do) ie involved, which ~eads tn g decre~~~ ~.n ~he signal/noige
ratio of (lo+ do/n ) times. For the good li.near estimates constructed
in Chaprer 2 in Che domain of large eignal/noise ratios, ehe following
relation is norrect
n~ ~ Nn ~a, i~~
where (E~] is the energy of tkie signal being ueed Co oeudy Che channel.
~
Assuming that signals of equal etiergy are uaed (Ei = E, ! ~ 1, 2), we
conclude ~hat ehe iaaccuracy of opt~.m~l linear estimates leads eo a de-
cr~ase of (1 a~/E ) times in the signal/noiae ratio h~. Thie facC en- -
ablea us to conclude that tihe effect of tih~ inaccuracy of ineasuring
characteristica of a aCochaseic channel on noi~e suppreseion and recep-
tion of diecrete measages can be eYiminaCed in praceice by studying the
channel wiCh a aignal whose energy ie 10 tin~es greater than the energy ,
nf Che information samples. Let us note that a completely analogous
conclusion concerning the ratio of energies Ei and E was drawn by ~46J
for a Etayleigh channel as a result of finding exact formulas for the
probabiliCy of error. The exact values of the quantity bo for different ,
correlation functions of the channel should be computed from the rela-
tiuns in Chapter 2. Their use makes iC poagible to construct ~raphs of
the probability of error for a broad class of channels with an arbitrary ~
probabilistic model of fluctuations of parameters. ~
In concluding this secCion we will show graphically what gives optimal ~
spatial processing the advantage over non-optimal (primitive) processing.
I.et us consider a r.hannel with smooth fadeouts in time and frequency
but selective (homogeneous) fadeouts in space. The Cransfer function of
such a channel depends entirely on the spatial frequency H(w, t, wr) _
H (wr) .
Let us make a comparative analysis of Che two schemes. 'The first com-
putes and employs NR readings of the transfer function (2.124). The
second performs primitive spatial processing of signals and is constructed
on the assumption that the channel is descri.bed by a model of an ampli- ,
fier with a random amplif~cation factor H(w, t, wr) � H.
In the first case, the estimate of the signal in position 1 is computed
in the form
~.rr
:1 ( � f v.\ro r l
r:~lt. r):_R~~is~~~)~llr,c ' f
~ F. i I
where Hp is the estimate of reading p of the transfer function and ~mr
is the distance between readings on the axis of spatial frequencies.
16G
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~OR OFFICIAL USE ONLY
7'he expreseion of ~he lower boundgry oF probability of error for Chis
example followe from (4~3): sube~i~utiing N~ NR ~.ri (4.3) for optimal
proceaeing and N ~ 1 for primitiive proceaeing. Zt ig clear from ~hia
Chat Che euperiority of optimal spatial processing tio non-optimal
(primitive) increases ae the degree of selectiiv3~y of epatial fadeoure
increaees.
Now let us evaluate the effect of inaccurate measurement on the proba-
bil3ty of error in optiimal and primitive epace processing. Firse we
will make a general reinark. It follows from (4.14) tihat the intenaiCy ~
of additional additive noise cauaed by the inaccuracy of ineasurement is
equal to Che mean square of error of ineaeurement eT~ The optimal
~ linear estimatea minimize thia quantity. Any other eatimates give
larger values of' e~, and therefore there is additional noise of great
intensity. Thus, the effect o� inaccuracy of ineasurement on the prob-
ability of error will be greaCer in non-optimal proceseing than in
optimal~ Of courae, it should not be forgotten Chati ehia concluaion is
drawn for the upper boundary of probability of error, not for probability
iraelf. Tha question of how close probability of error is to its upper
boundary demanda addiCional investigation in each particular case.
Let us return to Che example under conaideration. With optimal proc-
esaing the mean square of xhe error of ineasurement of the 1-poeition
eignal on Che basis of (2.123) is written in the form
vR
,
I
( ~ N (m, u~p)
~ ~ 1 ~ s' ,vR ~ ` ~ -
o K s ,V (w, to,pl
v> ~2 ~S ~ ~ ~w~ ~~~r~
where SN(w) is the apectrum of the signal by which a channel lying in
band [-F, F) ia measured.
For primitive spatial proceasing on the basis of (2.164) we have
:~r
. ~
~i f ~ S1 N(c~, ~urn) d w,
:L o I Su ~w~ ~s
In writing the last formula we assumed, for the sake of determinacy,
that the primitive scheme estimates the transfer function at 0 spatial
frequency. Let us make a ~omparison of the two schemes in the case,
most advantageous for the optimal scheme, of equidimensional energy and -
amplitude spectra z_ a
N(cu, cu,) ` No: G(~~ ~ u~,) ~ ~o~ ~~S l~) F' ~ ~ St (w) E, 1 a~, 2.
Under these conditions, the intensity of supplementary additive noise
in primitive processing is (1~~�L~NG-1 times greater than in optimal
processing. ~
167
.FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
. ~
~OR OFFICIAL USE ONLY
The expreseion of the lower boundary of probabiliey of error for tihie
example follow~ from (4.3): substi~ue~ng N ~ NR in (4.3) for op~imal
proceseing and N~ 1 for primitive processing~ It is clear from Chis
thati tihe euperiority nf optimgl spatial procesaing to non-optiimal
(priml.tive) incre~eas as Che dagree of selectivity of epetial fadeoute
' increaees.
Now 1et us evaluate the e�fect of inaccuratie meaeurement on the proba-
bility of error in opti3mal and primitive epace process3ng. Fireti we
wi11 make a general remark. Ie followe from (4.14) that the intensity
of additional additive noise caused by the inaccuracy of ineasurement is
equal to ~he mean equare of error of ineaeurement The opCimal
linear estimates minimize thia quanCity. Any othar estimatee give
larger values of eZ, an~ therefore tihere ie additional noiae of great
intensiCy. Thus, the effect of inaccuracy of ineasurement on the prob-
ab311ty of error will be greater 3n non-opeimal proceseing Chan in
optimal. 0� course, it ahould not be forgotCen that thia conclusion ie
drawn for tihe upper boundary of probability of error, not for probability
itself. The quesCion o� how cloae probability of error is to its upper
boundary demande additional invesCigation in each particular case.
Let us return to Che example under conaideration. With optimal proc-
essing the mean aquare of the error of ineaeurement of the 1-poaition
signal on the basis of (2~123) is written in the form
,vR
.
a ~ ~ ~W ~ f~)~n~
e! - f ~ Sr (~~)~a ~ J u?,
~V(~o, ~~~,p)
~ o ~1'R fi: _,~~K, 2 ~ Su ( r,~) ~~'t' ~~~J~ a~~r1
where S~(w) is the apectrum of the signal by which a channel lying in
band (-F, FJ ie measured.
For primitive spatial procesaing on the basis of (2.164) we have
:~F
�i f ~ St N~~' ~~n) d ro,
:c o I Su ~~~'s
In wriCing the last formula we assumed, for the sake of determinacy,
that the primitive scheme estimates the tranafer function at d spatial
_ frequency. Let us make a comparison of the two schemes in Che case,
most advantageous for the optimal scheme, of equidimensional energy and
amplitude spectra � 2r ,
,V (c~~ ~ 41~) ~ Not G(~u . cur) = Go: ~~S (u~) F` ~ ~ S~ (u) E, l a l~ 2.
Under these conditiona, the intensity of supplementary additive noise
in primitive processing is (i~�~~NG-l times greater than in optimal
processing. ~
167
.FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
,
FOIt OT~'~ICIAL L'S~ ONLY
4.3 Characterietice of Dev~.ces �or Proceae~ng Space-Time Signals in a
Generalized Gausaian Channel (Smooth ~adeouCs)
To bring xhe results of calculaeing no~.se auppreesion more cloaely in
line with the phyaice of real channels it ~.e advisable Co coneider
separately Che particular c:aee of emooCh fadeouCs. In thie case the chan-
nel ie repr~seneed by a model of a ser~.es combina~ion of a deCermi.nistic
spac~-filCer and an ampli�~er with a random, complex amplification �actor.
Correapondingly, the optimal processi.ng devic~ for aignals in each trans-
mitted poeition ie a one-channel device. The proceasing a~gorithm for '
space-time aignal~ in a chAnnel with smooth fadeouts follows as a par-
ticular case from the general algoriChms of Chapter 3 where N a 1.
Ner~ we will conaider the characCeristica of ~pace-time procesaing
devices in a channel wiCh generalized Gaussian statistics.
Becaus~ we are investigating a generalized Gausaian model of a channel,
iC ia advisable to compute Che noiae suppresaion of the optimal algo-
rithm in this channel (3.5~). The probability of error computed in
this case ahould be treated as the lower boundary of probability of
error for the given channel. Algorithm (3.45) is invarianC to the s~a-
tiatics of fadeouta in a generalized Gaussian channel with a regular -
signal component; of course, it is inferior Co the optimal Bayea algo-
rithm (3.56). However, the corresponding energy loss is a fraction of
a decibel and in practice both algorithms provide the same probability ;
. of error. We will make our comparison of optimal algorithms against
the non-coherent procesaing algorithms which have become widespread and
follow from the formula (3.82) where ~1 = 1 for the problem of detection ,
and diacrimination of M-orthogonal signals and against linear algorithm ~
(3.91) in considering the problem of distinguishing two oppoaite aignals.
We witi conaider three typea of inessage-carrying signals separately:
signals corresponding to the detection problem (dual signals); signals
in a group of number M which are orthogonal in the amplified sense
[104J; opposite aignals.
In addition, we will solve the problem of computing noise suppression
for dual signals of arbitrary form and on this basis find the optimal
dual system of signals in a generalized Gaussian channel.
The working characteriatics of an op~imal detector. ltao types of errors
are possible in a detector that works according to algorithm (3.45):
the false alarm and missing a signal. To determine the probabilities of
these errors it is convenient to convert expression (3.12) to the form
_
v' : 4). ~4. ~7~
The quantities V and V are normal and independenC. When the incoming
oscillation has a usable signal these quantities have the following
parameters
d1~ (v) = n?,~ 2E 1-!- 2hX `2E 1-~' 2hy
No 21tX ~ A!~ = m'y 1/ No 2h~ 4.18
~ )
p(y) ~ 2hX . D(V) = 2h y. 168
~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~OR OFFZCZAL USE ONLY
In the abeence of a ueable eignal -
'lh ? ~ j
M~ {V m,r Na 2Ax ~ 1�~- 2hx~ ' M~ {l7} ~m; 2h y( I�}~ 21iy) '
4,19
2hx 2l~y ~ I
~ t~? ~ D {V} ~ -
, l -h 2hx 1 2h~ .
The probabiliCies of a falea alarm and miasing a aignal ahould be calcu- ~
lated according to the formulas [89~ for integral function F(A, B, C, D).
In the general case we may wriCe
Pnc F(i1 ~ B~~'~ ~ C~-> ~ p(-1)~ Pnr's F~A ~ dl ~l ~ ~ pl~F)~, (4 . ~0)
The parameters presenC in fo~rmulas (4.20) are written as follows
1~i~ 1
~ ~ ~ l~ -f- ~1 9') ' -
6 s- 2~~ h~ ~
1 ~ l ~ ( ~ -t- q')
q~ (1 q~) 1-~- a cos: ~r ~ 1-~- 9~) ~ 1-~- a=) 7~' h' sin~ ~Fr~
_~1_ c ~ ~C ~~+9t~~~+~>>.~..2,~~
~ .
1( l-~- 9') (1-F~ -f- 2h' J
2h~ ~ 1"~� Q; q!~ l~ ~ Y�~'f 2h~
r
~1z h~
~ c-13 arc tg t8 4'v~ , + ~ (1 + 9') ( ~
2h~ �
c~+~~c~+at~
IR't)~1 C ~ ~ C~ t~)]~ = y~ ~1 ~ t ~~~(h~-~- 9') ~cos' Tv -F~
~ ( I 9~) ( ~ -I- ~s1 sin~ ~v~1 �
( ~-1- 9') ( ~-f' 2~~ h'
2h'
D(+1 ~ arc t6 ~t~ I ( ~ -f- ( ~ -h 4') , ,
I ( ~ ( ~ 4') _
I 2h'
. (4.211
Parameter ~ is expressed by the formula
h~ = No ~mX + m y �s �y ~ � (4 .22)
~ As follows from formula (4.21), whe,xe h2 � 1 the following inequalities
are fulfilled: B(-) = 1 and C(-) = 0. The probability of a false alarm
in this case is determined by the relation
Pnr = exP ro). . . . ~ (a. 23
169
~'OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
FAIt O~FICIAL US~ ONLY ~ ~
~ i ,~.~-j~~
~
from which iC ie not difficult tio compute ehe threshold
m,.,... ~npnt~ (d,~~j
Using resulta from [89]~ the expreeeion for the probabili.ty of miesing
a eignal can be ob~ained in the form
(~/Pnr1(l -i- ~ I �4- q~l
pur 9 ~ ~~I ~ 25)
nJi~ ~ exp ~4 ~1~~ ~ (cos~ ~Fa �'s� sin~ 7'~,1~
6
We will analyze the chararterietica aftier conaideration of the quality
indexea of. a non-coherent detector.
The working characterietics of a non-coherent detector. We will detarmine
the probab ili.ties of errora in a detector working acc~rding to algorithm
(3.82). In the absence of a usable signal V and are distributed nor-
mally with a zero mean and disperaions equal Co ENp/2. Therefore, Che
modulus ~ 1~ v2.H v~ is dietributed according to the Rayleigh law. The
probability of exceeding the threahold in the absence of a signal (Che
probability of a false alarm) ia determined as follows
pnr ~ cxp m), (4.2G) -
from which iC ie poesible to find threshold level w which insures that
the given probability of a false alarm will not be exceeded:
_ m c+ - In Pat� (4.27)
In the presence of a usable signal at the input of thQ receiver V and V
will be distributed normally ae before with parametere M{V} ~ MxE, M{V} �
MyE:
D{V}~~2� ~i-}�2h~~; D{1')~~2� ~1 ; hb~.
In this case the modulus has a fo~ar-narameter distribution. The proba-
bility of correct detection (or misstng the target) ia determined by
n means of the integral function
- Pur � F(A, B, C, D). (4,28)
In the case under consideration A, B, C, and D are determined by the
formulas :
2h~ ~1
A~- 2lnl/Pnr ~ g_ '+1~-+-~�l(1~'q~) ~
~ 2h~ � 21i~
~ ~~l?+~')(~-+�y=) ~`l~~ ~`l(I.~_y~l
i~=y~
L'=~~.q:~ ~=~v�
In calculations of noise suppression the formulas in [89] may be very
useful.
170
~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~'OR OF~ICZAL USE ONLY
Analogouely to (4~25) for the doma~.n of emgii errors and ema11 q2 we may
use the approximate notaCion
Pur In I /Pnr) ~ ~ -i- ~1( I qt) ~ .29)
2k~ ~~xp I~~~~~ (cos~ q+p a� sinQ Tv~J
I. .
It is easy to observe that expreseion (4.25) and (4~29) coinc3de~ We
wi11 analyze the effecC of channel parametere on tihe working charac-
teriatics of a non-coheYent detector and ~hen optimal de~ector in the
aCipulated conditiona. It is apparent from (4.15) and (4.24) that the
p2obability of miaeing a aignal decreasea exponentially with growth in
9�
Where the channel tranefer coefficient doea not have a regular part a
deepening of aeynimetry by orthogonal component (decrease in B2) in-
creases the probability of a mies. The exisCence of asymmetry can pro-
vide a gain in Che probability of a miss if a weakly fluctuating com-
. ponent of the tranafer coefficient (B2 � 1)has an easential average
(q2 > 0, = 0). The working characteriatica of detectors figured by
the formulas and tablea in [11, 49, and 89] are represented in Figure
4.2 below (the dotted linea ahow the curvea of optimal detection and
the solid linea are non-coherent detection). A comparison ahowa that
,u' ~a' ~c~J ~o~ ~o~
~ ~ ~
i
_ ~
? ~
y~ - - - - -
~
\ I
~ ~ ~~y~ I
Figure 4.2. Working _ ~ r .r__
Characteriatics of a Non- ~A'� I
Coherent Detector and an ~ n�~
~ \ , ~ r : ~
Optimal Detector in a ~ ~ � ~o, ~ ~ '
Channel with Smooth Fade- ~ \ ~
outs. (Solid line is ~ ~r `'�r
~~'q
non-coherent processing; ~ ~ ~
\ \ � ~ ~ ,
broken line is optimal ~ ~ ~
processing. ) f l� y:1~;,,:v��io -
Q:.~,~l;~,P~,~~, ~
the loss of information on phase does not reduce the quality of signal
detection significantly. The energy loss is zero decibels for a Ray-
leigh distribution of amplitudea because the true distribution of phases
is uniform and reaches a maximum (about 1.4 decibels) for the ideal
channel (phase diatribution is a Delta function). In the intermediate
domain of change in parameters, the energy loss is almost intangible
(fractions of a decibel), but it increases slightly with greater asym-
metry in good channels (q2 � 0; �p = 0).
171
,FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
.
~FOR OFFICIAL USE ONLY ~
7'he elight dectiease in noie~ suppression combined wieh aimplificarion
of prac~ical realization and tihe receiver's invariance in relation to
channel parameters makes the non-coherent methnd of simp~.e detection in
a channel with emooCh fadeouts preferable to the atrictly optimal me~hod.
The probabiliCy of error in a eyaCem of M-orthogonal signals wiCh optimal
proceasing. To calculate the probabilities of errors it ia convenient to ,
convert algorithm (3.56) to the �orm ~ ,
G~ > Og ~ g C' K�t~ (4 .30) i
where , ~
OleVj-{-Vj. ~
The quantitiea v~ and V1 are distribuCed normally. They are mutually in-
dependent owing to the orthogonality of the signals in the amplified
senae, and have theae parametere ,
`nE 1 2hx ~ ~~E 1 2h'~ ~ .
,1f~ {l'~} =,m.r I! ~`~u ,?~iX ' A1i {l~~} =~my :W L'hy ' -
u
b {Vr} 2hx, D tVt} _ ?h'y .
~ , /~2F. I
Ali { t . m,r ~ No L'h~ ~ l ~ :Ra. ' ~ ~ 1.311 ;
x ~ ~
.11~ { l'c} m1~ ~ / N ``I?': ( I :"i., > ' I
� U u ~
'~h2 2k"
D {1'~} " ; O {l;s} ^ , '
1 2h,~c ' ~ � ?h'y j ;
The probability of error is found from the relation ~
p ~ ~11, {l - (1 - F (Gll~ai-i ~�1.3�21
In this forniula the quantity F(G1) is the probability that the random ~
quantity G~ will exceed a certain random level G1. The averaging in
(4.32) is done according to G1.
Observing that in the general case the distribution of moduli of G1 and
Gg is a generalized Gausa3an distribution, we may, following [89],write
the formula for the probability of error in a binary system (M = 2) as
follows
ao m!Z~ !l~ d'n v:k V~
;
b2
~ _ { - .
P=~~ ~tl ki Q ~ i ~
i)ai Jbi ~a~~bk 1
n:_o k:-u
r'~aiJ: ~~i~l ~ f `~~~ut ~'Li~--a;--bj,l
s ~~p - ~r,
li-~~ ~ r1-'r~ L -~1-=.=~ ~
ll.= ~ �i ~i) ~ u~'~ ~'=~1 l
X !0 1 ~ 1 ~ ~4 .33)
' 172
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~FOR OFFICIAL USE ONLY
where
r~ ~ ~ I-~- 2h r~;
~~7 (~b1~_{b'r,)~:::.~'~i t~n}~~ L~ r ~,11~ {I~~'}._~:.~~~! .
l' l~ {V~~} ~ � L) {l~p} . ~ _ {I'~,} ~ {'~'n} ~
' ~2 r~ U{Np?---U{V~,} ~ P~ 1~2.
~ D{Vp}-{-U(l'n}
Familiar [49] particular cases follow from formula (4.33). Analyzing
parameters (4.31), of the quantity G which is in the right part of
formula (4.30), it is not difficult to obaerve that the diatribution of
the right part becom~ an x2 distribution with two degrees of freedom
when the condition h2y, h2X � 1 ia met. This makes it possible to ob-
tain [89] an approximate fo;,mula for the probability of error
. 2h3
. . _ _
exp - k9' (1 -f- t + ( ~ �i- (1 9') x
2~~ 21i'
p x ~ l~k-I-~ CM -1 ~ ~ k ~ ~ ~a) (1 9')
k=I 2lig lI 21t- l
_ [~+k (1-}-~')(l1'9')Jll+k (1-f-~')(1-f-9')J
2~~ li=
X~2 si n~ ~Fr-f- 1-~ (1 r_ ~t) (1 9') cos"- q~a `
2/r" ~a
1 + k ( l -1- (1 -f- q~)
. ~~.aa>
Calculations show that for small values of q2(q2 < 3), it is possible to
use formula (4.34) for practically any h~ 3 5. The explanation for this
_ is that the true diatribution of the right part of (4.3U) (Gg) in tinis
. case is very close to the approximating x2 distribution.
For the most interesting domain of small errors it is not difficult from
(4.34) to obtain
..1 ~ ( l q2). ~ . 2 _ _ _ h~_~
1
P = ~lt- S eXp - 9 ( 2~~ (cos= Tr -I- si n2 ~PN) ~ ~ ~t . (4 .35)
k=� I
It is interesting to observe that an expression analogous to (4.35) can
be obtained tor the domain of small errora in a non-coherent receiver
of signals of the type under consideration. It follows easily from the
: general expression obtained in [49]
173
,
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
6
1 ~
,FOR OFFICIAL USE ONLY
k~~' li~ ~'~~~'fi~
~ ~ rxn~ . ~ ~ ~ _ _ ~0~ D=vi -~-Vz-V3-Y4~ (4.44)
where B is the quadratic form of Gaussian variables.
The characteristic function of the quadratic form of Gaussian variables
is known [114].
176
~'0~ OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~FOR OF~ICZAL US~ ONLY
1 y
exp ~ ~I(K`"~ i'~riKK~)~~~ ~Il}
H t u) - - - ~W....~.., w,.... ~
n ~ ~ ! . 1 :r;h'r~ ~17i ~ ~1.1i~)
In thie case M~ {V }~.s the matrix��column of average values of vari-
ables; K is the matrix of covariationa of quantitiea Vk; I is the uni~ary
matrix of Che eame order ae K; Q is the matrix in quadratic form~
The staCis~ical parameters of Che variablee V~ can be computed eae3ly
~89~~
. In the general case it is more convenient to calculate the probabiliCy
of error using charactieristic function (4.45) following ~he Cechnique
described in [89]~ 7'he calcula~ed curves are shown in ~igure 4.5 below.
io � _ 'n.' _ ,n' _ ~ ` _ . _ _ ra ~ ~ ~n'
. ~ _ ~n
``~ti~ ~ ' ' ,
;D~ ' __i. .
` ;
~ ~ rL: ' , a 1
\ \ I \ ~
? ~ '
~ ~ ~y ~~~~.�jt' .1--f
10'r ~ ~ . ; --c-;. _ . _ . _
~ ~ �t ~ o~~~~ a o ~
Figure 4.5. The Probability ~ ~ .
of Error in a Binar S eCem ` C.'~~~s'�4~~.i+�q~
y y .c , ~---L____r.~.
of Non-Orthogonal Signals. , ~ ~ ~ ~,~?r~~.v, K, a o,
qRr;~r'n,ir .
~ ' ' Q~~
y- o; a-:o ~ i~~ J , t~~~ 4
l0~4 P , \ _ _ .)~l'_r ' +
4=1r/tiql; ~ ~
� P-o, ~i--qy ~ I -
~ ~
s ~ ~
~0~ t ' \ j
Q�l:frl~.~�D; ~ ~ \
l0` Q'1i f li .~"~f ~ I I\\
P
~r~.t:~�~ qr..~�-h YP�0
If there is no asymmetry by orthogonal components (B2 = 1), we arrive at
the known [142] result
` s i s
P�= Q~a~~ bc) - 2 b rep r- �~t 2 b c 1!~ (abc') ~ (a.46)
~ /
~
i~~
FOR OFFICIAL USE ONLY _
~
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~ ;
~
FOR OFFICIAL USE ONLY ~
' a ~ ~ ~ ~'y ' , b ,,c ~ ~1..~ ~ 1
Y 1-- ~ 1-- �'J~~ ~
(4.4y)
q' (12~~/c 4'?1 t i... _ 2 t i_ k~t) h~
c~
4Tiy ( ~ � ~~11( ~ 'I' q') ~ ~ I h' q'
To find t.e optimal form of the ei~nals we will inveatigaCe (4.46) to
the extremum according to a and a. The necessary condit~one for the
existence of the minimum can be written as followa:
~p aa~ ~r o. ~~p a~~ o, ta,~~a>
ae~ a
~ ' aa~ a ~
Having obaerved that
a'c~ G'ca 1 I oQ (ac, bc)
exp 2 ~ ~o fabcl) ac Jac ' h1.40)
\ .
~we will write Che necessary conditione of Che extremum as ~
~0 b 84Q (ac, hc) b dQ (nc, bc1 I
Jnc ~ 2cc d (ncj' ~ ( ~ ~ :a'ic1 ) auc r ~ .
Bac Jac 1~.50) ,
i
J~ Jl' ,
The differential equation atanding first in (4.50) is solved in an ele- ;
mentary manner, but it'yields values of ac and bc which can never be
achieved because they are outside the domain of definition of these
quantities. By studying (4.50), it is not difficult to show that the
coefficient of mutual correlation of a takes two optimal values with the
existence of a regular component in the channel a=-1 and a= 0 ~
= 0) and one value a= 0(~ = 0) in ~
the absence of a regular component. Calculations show that this situ-
ation persists in a channel with asymmetry by orthogonal components
also.
The investigation we have made allowa us to state that in a channel _
with smooth fadeouts and a regular component, the system with opposite
signals is optimal up to certain threshold values of the signal/r,.oise
ration h
n,P , but for larger values the system with orthogonal signals
is optimal.
The threshold value h~P can be determined from the condition of
equality of probabil~ties of error in these two systems.
` P~~_ ~ 1)=P~%=o)~ ~a.s~>
_ The expression for the probability of error in a system with opposite
signals has this form [49;.
178
~
FOR OFFICIAL USE ONLY
1
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
,
FOR OFFICIAL U3~ ONLY
~
' ...~2y~ h� ~ I ~ ~ ( y~'... j (~~~~s~ y~i~ I � (1~ ~lil'i j~.
p~ I--~L ~.i (i~i ~
.
� ~ `I~~' _ 21t~
. ~ ~ 9>> ~ ~ .--;~1; ~ C ? ~ N~?1.
. ~4 , 5'l)
The probability af error ~.n a system of orthogonal aignals wae determined -
above (4.33). A comparative analyais o� �ormulas (4.33) and (4~52) ahowe
that in good channels a sysCem with opposiCe signals rema3.na op~imal
throughout the domain of error Chat is o� practical inCereat. A eignifi-
cant feature of the syseem with opposite signals is Chat it hae an irre-
ducible (ae h~ increaees) probabillty of error whose value is determined
by the channel parameters [49]. This makea it posaible to determine the
threahold value h no,. in channele that are far from ideal bue have a
. regular component, uaing the relaCion -
~ ( ~ ~ ~ 4') exP I - Q4 ~ ~1~ (cos~ ~r -I- ~'sin~ mr1,
ynop r' / , ~
~ ~ ~ ~4.b3)
~ ~ 1 - m ~ 1 / q' - (cos~ 9'a ~'s i n~ TP), ~
V
The range of variation in threshold value h~- is very great. For ex- -
ample, in a Rice channel (B2 = 1, q~~= 2) h~- = 10, and in a gener-
alized channel with good statistics (q2 = 2, ~p � d, B2 � 0.1) h2~pp =
5�103. A rationally deaigned communications system should change the
appearance of Che signals used depending on the state of inean atatis-
tical parameters of the channel.
4.4. Characteristics of Detection of Space-Time Signals (G~neralized
Gauasian Statistics)
~ Let us determine the probability of a false alarm and missed signal for
the algorithms of op~imal and suboptimal space-time signal proceasing.
Optimal processing. To calculate the working characteristics, we will
represenr algorithm (3.45) in the form
N (4 .54)
� 2
G~ J l/k-~ l~k W.
k=l
The quantities Vk, Vk, k= 1, N have Gaussian distributions and are sta-
tistically independent. When there is a usable signal in the observed
oscillations these quantiti~s have the parameters:
` ~ ~~~~k 1
,11~ {l'k~ ~ n?.c~ 1~''~k l~ ~`A_' ~ ( +
xk
~ ~h:~ (4. 55)
,111 {l~k} = myk j~2dk V � 4k ,
`~~~yk
D {Vk} _ 2hXk: ~ {~~k} - ?hl~k' -
179
`FOR OFFICIAL USE ONLY -
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~'��;i;~ I
F0R OFFICIAL USE ONLY ~
~
In the abeence of a ueable eigngl Che parametere wi~.l be '
� I ` L~II;k ~ ~
J1~~ A,} r. Itl.tk V 1d4 :~h~k l~ ~.i. 2h,ik ~ ~
1 ?h"
tVA~ y!Ilyk ~~:dk 2h~ ~ .~~N~~ ' ~a.~6)
s'
Yk r ~ t~k
2hxR 2huk
1~(Vk} ~ ,t~~k}=
1�}- 21ixk t~ 2h~k ' I
~
In ~he general case, iC is convenient to calculate the probability of a ~
false alarm and miased signal using a characteriatic function, which is
eaey to compute for both the preaence of a usable signal OG~+~ (i u)
and for the abaence of a ueable aignal OG~'~ (i u). Theae characteristic :
functiona are dete~cmined by the identical expressiona:
' - i uhtl {Vk} i uA4~ {Vk}
e~p
~v f 1--~ ~ 2�u ~ t~uo ~~A}
c ~ n ,
A~ i ir k ' ~1 i 2uU {Vk}J [1 - i 2uD {l?k}j ~ ~ ;
_
in which the value of parametere from (4.55) should be subatituted te ~
compute 0~~+~ (i u) and the parametere ~rom (4.56) to compute 0~~'~(i u). ~
The.probabilities are expressed through the corresponding characCeristic
functions - ~ ~ ~
~ l?~ ~ (i tr1
P:i~ = I- 2:c i u ~~u~o di~, ;
~
} I 1.;~~) ~
' ~ Nu'`'~ (i u)
i u~n ~
Pur'= ~ C du.
?;t i tt
It is convenienC to make the numerical calculations for formulas (4.58)
using Che methodology and algorithms of [89].
Let us consider the domain of s?^~11 errors. Analyzing the expressions
included in (4.56), we observe ~chat where h2xk, h2yk � 1 the mathe-
matical expectations of quantities Vk and Vk become close to 0 and their
dispersions tend toward 1. Thus, the distribution of the random quantity
G in the abaence of a usable signal contracts toward an X2-distribution
with 2N degrees of freedz>m. This makes it possible to express the proba-
bility of a false alarm by a known [109] relation
,v-i
I wR
PnT = ~~V 1)I I' (t~t, .\'I ~ . I Y .
L: S
� sio
180
~'OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~FOIt OFFICIAL US~ ONLY
The threshold ].evel wp eh~e is op~imal according tio ~he Neuman-Peareon
criCerion ahould be computed from equation (4.59)~ Looking a~ exprea-
eione (4.55), we may observe that in the doma~.n h2x, h2y� 1 the dis-
persione of quantitiea Vk and Vk asaume larger valuea~ This makea it
posaible [89] Co obtain eimple expressione for the probabi.lity of miseing
a eignal .
cuo N 1 ~ ~ ~k/ ~ ~ ~ ~k~ ~k 1 ~ t" ~;k~
- Pua L2 ~i~ ~ :h'' c~p - X
k=+1 k ~k . k
a,li0
;C ~COSp Ta k'~' (~k Sltt~ t(p k~ ~ 1 ~
A whole series of particular reaults can be obtained from formulae (4.59)
and (4.60) with different asaumptione about ~he model of fluctua~iona of
channel parameters.
Aa a numerical example let us conaider a apatial model of a channel wiCh
fadeouta ~hat are non-selective in time and frequency. The number N in
formulas (4.59) and (4.60) is determined from the relation N~ NR =
~R~Pcor + 1] (one apatial coordinate r is being considered). The aurge
characteriatic in this case is repreaenCed in the form
. . -
cr, ~~i a c~~, c~.si)
where g(r) is a random complex-valued function of the spatial coordinaCe.
We asaume that the real and imaginary parts of function g(r) have normed
correlation functians of the type
. ~r--r'I
Rg(r-r')=exp~-- l. (4.62)
\ P~~ur 1
The results of calculations of the probability of error using formulas
(4.58)-(4.60) for the channel model under consideration are shown in
~ Figure 4.6 below.
,o� ~u? ~ot ru' ,c~ r:~s ;~s
~ ' n'
i. .
Fi ure 4 . 6 . ' ,1. Paf � ~o�'
g Working Charac- , 4= -4~: P- r
teristics of Optima1Detection ~ +
in a Channel With Space- jo~~ __I_`
Selective Fadeouts. N.~
, _.~.__.-L - - - ~
~ I N-2
~ ~c~'' ~ -
/9'' _~h `
181
FOR OFFICI.AL USE ONLY -
-i
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
i t
~
FOR OFFZCIAL US~ ONLX
Let us coneider one more epatial model, a homogeneous channel in terms
of epace with emooCh ~adeoute by �requency and se].ec~ive fadeoues in tiime~
In thie case~ as wae ahown in Chapter 3, optiimal spatial proceseing is
accomplished by x~arrowly directed anLennas witih dtr.ectivity diagrams of
the type ain d/,9. The number of auch anCenx~ae ie NR~ Optimal rim~ proc-
easing ie accnmpl~.ahed by a multichannel acheme with NT channels. Thus,
in thia case N- N NR in formulae (4.58)-(4.60). .
It follows from formulas (4.59) and (4.60).that ; ~
,~,rj k , I '
~ g ~
i~lit ~N~~ N t I)! I, ~~r'~~~ ~ ~._w ~ ~ ~
r~
R~~
r R ~\R NT
~~o ~ ~ a~~~ ~ ~r~~ - ~
Pn{, :s W _ ~
~N~ N~ r~i a..i "~~;k ~n~
~I(k l I ~t~~ e t' r �
- Y CX(1 - u` ~COS- ~P iR 1~i4 Siir t(i,i~~)
`I'ik
5imple reasoning allowa us to reach the conclusion that identicai exprea-
sions for probability of error may correspond to mathematical channel ~
modela that differ from a physical point of view. This occura when the ,
values of parameters of processes of fluctuation that differ from a physi-
cal point of view coincide in the models (for example, the nature of sig- ~
nal fadeouts in time for one channel model may be identical to the nature
of fadeouta in space for another model). In par~icular, formulas (4.63)
will describe the probability of error for a channel that ia homogeneoua ,
by frequency with amooth apatial fadenuts (~n this case the parameter ~
NR x NS should be uaed in these formulas). Thia feature, which taay be
called the reversibility feature, is typical not only of detection
units but of all other units for processing space-time signals in chan-
nels with selective fadeout.
Suboptimal processing. Generally speaking, it is more complex to inves-
tigate suboptimal algorithms in a stochastic channel than optimal ones.
The chief reaeon is the posaible appearance of a statistical relation- -
ship between particular paths. The second difficulty is that there is
~ust one optimal algorithm, but a set of suboptimal algorithms may be
proposed and this gives the investigator of suboptimal algorithms the
difficult ~ob of selecting the ob3ect of investigation. Here we will + ~
' consider several algorithms for processing space signals that have worked
well in ~ractice or show definite promise. The analysis of suboptimal
schemes is done in thqse channels where they actually provide processing
that is close to optimal according to definite variables (space, frequency,
or time). If a unit realizes spatial scattering, fo~ example, by means
of narrowly directed antennas, the spatial paths are considered inde-
pendent in a~alysis if. the opposite is not stipulated. This does not
at all mean that the general formulas for error probabiiiCy obtained in
this section cannot be used in the Gase of statistically dependent
paths. In fact, when investigating the probability of error (during -
182
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY
both deCect~.on and discrimination of aignals), we are always dealing
; with ~he quadraCic form of random quan~ities
i' Q ~ ~ uppVqirp, (4,64)
k~ p
where akp are coefficients.
We know (5] that thera ie alwayc, a poesibillty of reducing ~he quadratic
form to a canonical type
Q ~ ~ a~;Vk ~kUk, (~1, 65)
k
where ak and Bk are coefficienta correaponding to a new syatem of coordi-
nate and U and V aLe variabl�ea correaponding to the trans�ormed syste~ of
coordinates. , f,
Lack of correlation among the variablea of the tranaforms in form (4.65)
ie achieved by an appropriate choice of conversion from (4.64) to (4.65).
In ~he Gauaeian case, lack of correlation is identical to statiatical in-
dependence. Therefore, in the particular case, the probgbility of quad-
ratic form (4.64) exceeding~a certain level through which the probability
of error is calculated in any system may be computed through the corre-
sponding probability for form (4.65) after substituting the concrete
parameter values. Applied to correlat~ed paths of propagation, aome com-
putationa of probability of error are contained in works [6, 46, 104].
Let us pasa on to a consideration of specific algorithms. We will con-
sider a detector that contain?~ a set of narrowly directed antennas
(spatial processing), a delay line (processing by frequency), and a
filter coordinated with the tranamitted aignal (processing in time) and
working on the algorithm
_ NF N9
rj o~~ ~/ik-~. ~ik ~ W, j4.6G)
k=lt~i
where -
k
s t - -1 ~
_ ~ck e(sin ^/SJ - i n) d U J~ Z~r, o~ dr. -
~~k J i1/S1 - i n o s(f- kl
\ F1
The variables Vik and Vik are Gaussian independent quantitiea. When
the oscillation being analyzed contains a usable signal these quantities
have the parameters
A1~ {~'!k} ~ mxJk y2~k~ A9i t~tk} rnyiR Y 2dp~ 1
) (4.61)
D{i'~k} = I-1- 2hX k?~r: D{V,k? 1-?hyi k l~r, ~
. ~
183
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~OR OFFICZAL U5~ ONLY
If Chere is no ueable eignal
~4~~ (Vtk ~ r" l1~~ {~Ik} 0~ U {~~Ik~ U jk} ~4,~8~ '
Parame~er uT in formula (4.67) ie determined from the relation [105]
r \
' ~~r~, ~ J (I-- ~ l~i~lT)dT~ (d,69) !
0
where Bh(T)
is a normed correlation funCeion of the channel characteriat:ic according !
to Che time variable.
From (4.69) it is not difficult Co show Chat 0< uT < 1. Parameter uT
characeerizes the rate of fadeout. The greaCer this rate is, the
smaller quanCit uT will be. For the exponential correlation function :
m T T) the expression for uT takea the form
Rh(T) eXP ~ . ~o~
- ~~T ~~,/tu~r)'~ (~iYicup--- I �~-c"TrT~ror~. 1�1,iQ)
It can be seen from (4.68) that in the absence of a usable signal the
quantiCy G has an X2- distribution with 2N~A ~ degrees of freedom.
This makes it possible to write the expression for the probability of
a false alarm as .~~r.~�N-i [ _
1 ~
P,r ~Nr~ 1)! ~ ~co, r~'~' ~VA - 1) ~ 1:! ' 1~1,7t)
~
The numerical calcu).~tions of the probability of missing a signal .
should be done in the same way as done in the case of the optimal scheme
for formula (4.58), using the parameters (4.67).
Let us consider the case of identic~l, on the average, paths of propa-
gaCion nr.~rx ma, n~~rk iny. i:= 1, A'e ;
Qrfk r cTC; ~~ik Q~ ~ k ~1 , ~'T .
I
In ths presence of a usable signal the random quantity G has a non-
central semisymmetrical X2 -distribution with 2NFNe degrees of freedom
[89]. The probability of a mias for even values of NFN ~ is determined
from the relation '
~
Pnr ~ F cu I-~- B; 2 � arc tg q`p ~ ~tl a'n^Jb^ X
\ ~ ? ~~a b~ ' / Rn J:n
r,=p
184
~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
~ roR o~riczAL vs~ ortr~Y
' NA
pa ..1. p7 ~ m= _ :
~ �-1 _
' I -I� r'
~C ~;tp ~ r t~i ~~'2 \r
, ~ ~,y~ ~ ~~l -I'b~) ~ ~
/u 1 ~~~`1~,:W~~'-1 ~ (4.i?)
~
where a aeries of parametera depending on quantitiea (4.67) has been in-
troduced: , U il'~ D ~1'i
D-... U{t-};n;1 h - ~
n ~I'1 U ~r
1/' ~v~`n,~,',ii,
~v~ j~'~ ,t~F,-~~e.i~'-,'ii~i
n~ ,
j' D (l') D (1') ~
I~h'h~N~h1~f~')-~- N~~A'er11~ (l')
li = - (4.'73)
~l D (V) D (I~) ~
It is convenienC Co use formula (4.72) for calculations where R� 1.
LPt us consider the domain of small errora. Using Che same methodology
as used for the optimal scheme, we obCain the expresaion for the proba-
bility of a miss in the form
G1 NF ti,9 _I N~ ~1'e ~ 1-~- ~;k~ ~ 1-f- 9ik~
~~r~ F d n n :
(h~~ ~V~ - I~I ~~~T ~N N k=i r,~ 2~;k ~~~k
= ( ~ )
~ - 9ik n ik /COS~ q~P k-;- ~k sin1 Q'r k~ . (4 .74)
`~Pik NT l
A comparison of formulas (4.74) and (4.60) shows that the superiority of
the optimal scheme to the suboptimal one we have considered begina to
tell at a high rate of fadeouts T/T~or � 1�
Under conditiona of a high rate of fadeouCa, the optimal detector pro-
vides NT times greater multiplicity of scattering than the suboptimal one. ~
But if the rate of fadeouta is low T/TCOr ' 1, the optimal detector
has no apparent advantages in noise suppression over the suboptimal one,
but it is much more complex Co realize and demand a much greater volume
of a prior data for construction. When developing systems for detection
of spatial aignals (in particular in radio astronomy), it would be very
useful to use data on the rate of fluctuati.ons of reflected signals to
aubstantiate the choice of the particular method of space-time proc-
esaing.
185
~'OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
a.
FOR OFFIC?:AL USE ONLY
It can be eeen from expreasion (.4~74) that in the absence of a regular
component (q2~k ~ 0, i~ 1, N k~ 1, Nfi~ the selectivity of fadeouta
in time leade to an increase in Che probability of a miss in the aub-
opCimal detector. The loae in error probability caused by channel selec-
~iviCy in Cime in the domain of small errora is ION~N~ ~.g 1/uT decibels
and increasea as the number of branches of dispersion by frequency and
in space increases. Let us recall that in the optimal detector channel c
selectivity in time always improvea the quality of the detector regard-
lesa of the statiaCics of fluctuations.
4
In channels with a regular component time selectivity may result in an
improvement in the working characteristics of a suboptimal detector. It
is not difficult to show from (4.74) that the gain from time selectivity
wi~h respect to probability of a mias appears wfien the condition
Ne Np n
9(k Nlk~ ~cos~~f~, k-i- ~ik sin~ Tp k~ ~ j,,F ~,19 �T ~n( ~ l(4.75)
i./ ~ ~~(k ~ ` ~~T \ ~~r l
Iu~ k�~I ~
is met, and amounts to
Ne N~ ~ / F e r
~ 9(k l ~ plk~ (COS~ ~r k 'f' ~~k SIII~ ~e k~ 10 N ~ T~ . Ig ~T .7G)
,~~~k l I~
~
r._i k~i
decibels; in good channels it may reach considerable magnitude. -
Let us consider one more algorithm of suboptimal detection that r~:alizes
p artial dispersion and ignores selectivity in time and in space. 'In ex-
plicit form the algorithm is written as follows
NF .
~ _ yk~, ~,k a (4.77)
k=I
where -
R T s ~l
Vk r r Fe /
~ ~k J.I z(t, r) k dldi.
1 1
0 0 g t- ~ 1
cl
It is not difficult to show that in this case the proba,bility of a false
alarm, which defines threshold w, is written with the formula
Nt -I
n
!'n r - ~ V~ ~ I' ~t,>, iVf I~ - e"cu ~ . j4.78)
~~-�u
The expression for the probability of a miss for the domain of small
errors has the form -
186
~
FOR OFFICIAL USE ~NLY -
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02/49: CIA-RDP82-44850R000100064410-3
_ ,
~ FOIt OFFICIAL USE ONLY
F'iip - ~J v~ ~ ~ I . i. pk~ l ~ ' ~~k~ _ ~i'
~ ~ ~~tr - 1 z Rr
' k~,~ ')pk ~~k �~k ~l
~cos~ ~p k ~k sl tt~ ~'r k~ ~ (~1.7J)
~ .
where the Parameter �
, _ r~ _
~~1~T T R(~ r~, T I I R I an ~ P) d T d p. . 80)
ool 1\ 1
The normed correlaCion function of the channel by the time and apace vari- `
able Bh(T,' p) may be spatially divisible. Then the parameter
� rtT = �R~ �r , (~1, 81)
The parameter uT was defined above (G.69); uR is determined in similar
fashion.
In conclusion, let us consider an algorithm that is a spatial analog of
time processing Uy the Kostas scheme. To make it more grapY~ic we will
consider a channel with fadeouts that are non-selective in frequency and
time but selective in space. The processing algorithm is determined by
the expression R - ~
N
. G~ u~ Vk'~' ~Ip ~ u~,~ (4.82)
k~l
vk (k-}~1) Ar T S t
wh~re l G J- J ~ Z~t ~ r) {S (t) J dt~r, e r_ RINR .
k~r 0 -
Where there is a usable signal in the input oscillation the parameters
of quantitiea Vk and Vk will be:
2~ . _ . 2E
Ml {Nk} _~x 1' 1VR !V , M(yk) = my 1/ NR N~
o I o
(4.83)
. 2QY E . 2Q~ E . .
D{ Vk} ~ 1.-}- R ~ p(Yk} = 1-;- R ~
N No ~ N ~Ne
where
2NR
�R= R o~ (~-~r)Rh(P)dP� (4.84)
187
gOR OFFICIAL USE ONLY ~
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVE~ FOR RELEASE: 2007/02/09: CIA-R~P82-00850R0009 000600'I O-3
~ . .S~

6 JUNE i979 CFOUO~ 3 QF 3
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~ ~ FOEt n~~ICIAL US~ ONLY ~
In the absence of ~ u~~ble ~ignel the qugntiey G hae an xz-dietributidn
~ with 2N~ degrees of freedom.
Congid~ring th~ Cermg in (4.82) to be independ~nt (ehie i~ gpproxim~t~ly
fulfilled where ~r pcor~~ We Come tio the conclu~ion that the probabi~ity
nf a mig~ is d~terminad by an expreeeion gnalogou~ tn (4.72). ~or CA1Ct1-
letic~ng iti is essential to eubstituCe the parameCer N~ in (4.72) ingtead
of the product NFN e, and use parameter (4.g3) in (4.73).
Analogou~ly ro (4.74), for the domain of small errore we ob:ain ~
Nk~-~ , ~ a)
nur ~ ----~-N~- - ~ I, k ~ y-~~Q'`k (cosl ~'r in" V'r), ~ (4. ~5)
(N 1)! � I2i~,` , ~ _
\ Na /
where - N ~ ~mx my -3- n~ uy ) ,
u
Zn an analogous wa;+ iC is poesible [110~ to formulate the optimal problem
and determine the va,lue of NR which ins~res minimum probability of error,
but with a rigorous a~g~Froach thia investigation encountera serious diffi-
cultiea. For thic? reason, it is wiae to choose NR according to the ob- �
- vious coneiderations adopted above NR =~~/pcor + 1).
Formula (4.85), in particular, corresponds to a device for apatial proc-
essing of optical signals built in the form of a lattice of NR photo de-
tectors.
4.5. The Probability of Error in Discrimination of Orthogonal Signals -
(Generalized Gaussian Statistics)
We will consider data tranemiasion ueing M signals which are orthogonal
in the amplified sense under coaditions of selective fadeouts. For such
signals these relatioas mvst be fulfilled:
T R T H
o~ stk i) sak (I. r) dtdr ~ J s~k (l ~ r) s~k ~l dtdi 0: (a .bGl
k=1. N: g. 1=-1, Af: gy=1.
Let ua laok in turn ar optimal aad suboptimal processing algorithms.
Optimal processing. To calculate the probability of error we will write
algorithm (3.56) in the form
G~>G~. g=i. :~1, g~1~ (1.b7)
where
N
G/ = ` Vf4'{' V~4. (4.liH) -
k~l
188
,FOR OFFICIAL USE ONLY
~ I
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
.FOR O1~FICIAL USE ONLY
The random qu~n~itii~~ g~~ t~ 1~ -M ~r~ the qusdrgtic forme of Geuegian
variables. The componentie of the quadratic form are statistically inde-
pendeat. Aseuming thaC the osciil~tion under analyeis contains en 1-
poeition eignal~ it ie not dtfficult to compute the mathematical expec-
- tiations and dispereions c~f quadratic-form variablea
. _
h~i t~~1RZ ==?n,r~, ti"~-'dtk l~ 2h2 ,i~ ~
xlk
~ I -i�:h~~ ~
,1f~ (l't~} -3: ryt~k ~/=d~ 1 u k ~
~~lk
!){l'~A}:-21t~lk; /){V~k) -2ltyl~�
A1 I'. 1 ~;�:h~R~ (4.59)
. ~,k} n~.~~R ( 2~lrk ~
s ~z ~
~~r~k ~~i~4
,of~ {1'~~3 ?r~ ~ l/r~ `'hE~a4
M+~k ~ ~ ~sk 2h~ak Y ~h~~t4 ~
~ ~
~ 2h~Bk ~ ~
dt~~:t~~1-~-2h ~D{V~~)=._..:^1? .
at4 1 ' ~~`4
In the geaeral case, the probability of error should be calculated by
nwnerical methods using a characteristic function of the quadratic form
of Gauasian variables.
To simplify the formulas, we will assume in what follows that the eaer iea
of differerit sigaals in identical patha are idenCical h~~ �
h~1k '~~k~
g, 1 s 1, M. For a binary ayatem of signals (M ! 2) in a Rayleigh chaanel
it is not difficult to obtain '
v . `hk +
P=~-~ .
N m{~ ~4.90)
k+~t
( /1k 2, n ( /t~ - � /I~~~
/qY~
The graphs of probability of error calculated in formula (4.90) for a
channel aith smooth fadeouts in time and by frequency and selective fade- _
outa in space, described by a proceas with exponential correlation, are
ahoWn in Figure 4.7 below.
- 189 ~
~'OR OFPICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~OR OFFICIAL US~ ONLY
~ 1P~ l0~ fDt f0~ ~!D~ fbr fOr 10~
M~2r h~
N~l i
1!
!0'~ ~0; ,
~ 4
~
ID'r
Figure 4.7. The Probability ~ ~
~-~'~'�t
of Error in DiscriminaCion of \
Orthogonal Signals (M ~ 2) ~o'' `
in a Channel with Space- i\
SelecCive Fadeouta. ~p~ q:P~n~~ _
Optimal Procesaing; - - ~ ,
- Non-Coherent Proceeaing
NR � 2� ~p�~ -
~ ~ ~ i ~
~ : ~ .
s P ` _1._.
_ _ ~ ~
A2alyzing (4.89), we aee that vhen the signal/noiae ratio growa h2xk~
h yk the average values of the quadratic-form components disappear
and the dispersioae are evened out~ that is, the distribution of quan-
tity G becomes an X2-distribution with 2N degre~,e of freedom. Using
the result from [89], we will write the formula E'or the probability of
error in the indicated domain of values of the signal/noise ratio as
follows -
y ~~'~'~4~~~~~yk~ 9R~~'{'F'R~
p C n exp - ~4 (cos� Q,,~ ~t sin Q~ k~ X
,t:t 1hR
M-I n t~ t) ~N-}- g--1)I
x ~ (~~+I C~M-1 ~ ~~n (~+NJ ~N - i ) . (4.91)
AQ~ ~aG
where the coefficienta cg are determined in [132J.
A number of interesting formulas for calculating the probability of
error in particular cases follow from formula (4.91). For example, for
a binary syatem (M = 2) the expresaion ror the probability of error has
the form - _ ~y
p = y . (4.3:)
- n =h~ ~k aP 9k 'i' ~cos= 4'r R�L a,~t sin! Grk~
~ ~ rt' ~ I 9R~ =~4
A simple comparison of the formulas obtained in this section for proba-
bility of error during signal discrimination and the characteristics of
the optimal detector shows that the atructure of the formulas is the
190
FOB OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~ FOIt OFFICTAL USE ONLY
same. Therefore, all qualitative conclueions drawn above concerning the
effect of a etochaetic channel on the characterisCice of opCimal detec-
tion aleo epply to the caee of diecriminating M eignal~. The formulg~
for the probability of error in discriminating M signals for particular
channel modele may be writCen ~ust as thoae for the problem of deCection.
SubopCimal procesaing during signal diecrimination. Lee us coneider the
' charact~rietics of a uniC for proceasing during eignal diecriminaCion.
The uniC ia composed of a set of narrowly directed antennas, a delay line,
and a aet of filCers coordinated with the transmitted signals, working on
algorithm (4.87), and compuCing the quantitiea
- NF .
~ (4.93)
Gt ~ ~ ` ~irk ~~irk ~
where k"1 t"~
k
~~trc e s1r. (0/b~ i:c) T S~ ~ ~
~ ~ L I - d ~ ~ : (l, ~t) dt. (4.~J~i)
1~~;, ~ ,~;5~ i :t ,
-6 u (r ~ ~ 1
\ /
The variablea of quadratic forma Gl and Gg are mutually independent
Gaussian random quantitiea with the_parametera
:E ~~f
.Ul {Vlik} ~ m.~R ~ ~1'a ~ .1lN{V~tk} r= m1lk ;Yo ~ ~
D~V JtR} =1-}- -"l+~tk �r~ ~{~~l/k} ` 211yfk I~T I ~1.95)
~11~ {Val4~ = ~1~ {~rk~ ~ r t~'trk} = D {l~~rk) = i � 1
Noting that the quantity G is distributed according to the X2 law, we
will use the result from [~9] and obtain an expreasion for the probability
of error in discrimination of M aignals in the following form
M--I v n~ NF Ne -I~
~ p= v(- 1)"'}.~ Gn!-~ ~ 1)~ n"k Y.
n=1 k-0
4~M
" ~ P- z him cos~ 4v rm
~ �t' 9rm
exp
2hT~m�r
f A ~
~ ~ a} nn ~i+nP ' ~~-~~i ~~~+q?m~
a P~~~ r~i 2,,~ a~~,r
1-i-np I-~- ~~~'a~,~~~~'9m~ ~ x-�
t
191
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~
;
,
' FOR OFFICIAL USE ONLY '
_
np 1-}-m~ liin~sins 7r tri
qrm
- ?h?m !,r ^ 1
n p i
~ ~ (�1. G
' ~r~n) l ' ~~r+~
n
2hiM NT i.
. ~ p !~~a
t n I ~ ~ ) ~
~ T Nfnl ial p-~~
where the quantity uT is determined.by formula (4.96). If there ie no ~
as etry by oxthogonal components ia any of the paths (BZim s 1, i=
1, N'; m= l,~N
) ancl the paths are idenCical on the average (which is,
of course, an idealization), iC is not difficult from formula (4.96) ~to
obtain
a~-~i - _ _ i _
p~~~.Y Cn -1 'v~ Ne X
n:~a 4~T
n~ ~ I-}� n 1- as �
n l ~'~--1~ ;
n N~' ti~ q~l~s (1 r~Nf Ne k) '
c X
x exp ri 9=) nh' �r t~ k C(NF 1Ve ) ,
- �r
I ~ r ~
r ,F~ I- a, .~'F 11'e ,
~.hn~ n'
~ - l.
N~ Ny Q3k~ ,
,
(1-i- q' !i~ }?T )I(1.-{- n1 i ~-i- 4t~ nh= �T j
where 1F1(a, B, Y) fs a degenerated hypergeometric function.
For large M calculations formulas (4.96) and (4.97) become complicated.
In this case it is advisable to uae asymptotic formula (4.37), which
makes it poasible ta reduce the problem of discrimination formally to a
d~tectioa problem. In this case the probability of error should be cal- _
culated by the formulas given in the preced~[ng section for probability of
a missed signal. The threshold w included in (4.37) for calculating the
probabilit}~ of error in a ayatem of M orthogonal signals should be de-
termined from the equation ~
192
�FOR OFFICIAI. USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY
~ r n~F ,~~A i~ , . ~.~~-.n
(,i~~`;~~" i)1
or
r?~F n~~ _.i
e ~ rl ~ "'q~i,~,tl-II , (a.93)
L~
p=0
Where the number of paths is aufficiently large it ia advisable to use ~
Graham-Charlier aeries to compuCe the probability of error [64, 109~.
For laxge signal/noiae ratios, it follows from formula 4.96 that
NF ,v~ : ~ ~ ~ . ~a ~
~ ~fm~ \ 1"~' 9frn~ ~x ~ 9im irn x
P� 1 1 P -
m~t t~~ 21ttm ~lml~r `~~lm ~lT
hf-1 n~ N~ Ne fi T
X~cosz 9'r mt -f- ~i sin~ ~'v tm~ 1)"~~ ~�~_i ~kn- ~k-{�N ~v ~ X
1~
n 3l ' k*~0
x ~,V~ ,NA k - I )1 ,
l'1 � 99)
(NF~Ve- I)1
Expression (4.99) is very cloae in atructure to the formula for the
probability of error in optimal processing (4.92) and for the proba-
bility of a mise in the optimal (4.60) and suboptimal (4.74) detectore. '
Everything that has been said about the effect of channel parametera on
the probabilities can be repeated for the probability of error under
consideration here wiCh optimal discrimination of M signals. Speci-
fically, the algorithm (4.93) under conaideration affords an energy
gain resulting from time-selective fadeouts where the channel has a
regular component and condition (4.75) is met. The values of the en~rgy ~
gain are determined by formula (4.76).
For small signal/noise ratios the aelectivity of fac?eouts in time leads
to an energy loss, as�can be seen from (4.96), but its values are low
in channela with a regular component. A compari,son of the formulas for
- the probability of error shows that in channels with time selectiviCy
optimal processing has a grea+t advantage over the processing we are now
considering, which does not take account of the selective nature of
fadeouts in time. This advantage increases as the probability of error
~ decreases and depends on the statistical properties of the channel,
reaching its maximum value in channela that are close to Rayleigh
channels. Thus, in a Rayleigh ~hannel wiCh fadeouts that are non-
selective in apace and frequency, the correaponding energy gain where
p~ 10"4, M= 2, NT = 3(the exponential correlation function Bh(T)) is
15 decibels; in a Rice channel where q2 = 2 it.is six decibela, and in
a sub-Rayleigh channel where B2 = 0.1 it is just five decibels. The
physical explanation is that selectivity of fadeouts in time plays a
very small role in good channela (q2 � 1). Optimal procesaing ap-
proachea linear, whose superiority to non-coherent (auboptimal)
193
y FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~~'`~rr
,FOR OFFICIAL USE ONLY ` '
proceseing in an ideal channel q2 for the signal syetem under con-
aideraCion ia on the order of three decibels~ Yn poor channela q2 ~ 0
and b2 � 1 an r~ptimal processing device, as �ollowe from algorithm
' (3,45), aceually proceases one quadrature component of the aignal be-
cause the aecond almost always takes a zero ~!alue. But the component
which ie being proceased also has a high probabiliCy of asauming zero ~
values during an interval of analysis of length T. In this case it be-
comea ineffective to "grab" and sum noncorrelated segments of the ob-
aerved field, as is done in the optimal proceesing device, for virtualty
all aegmenCa will carry close to zero energy. It is sufficient to proc- ~
= ess the eignal through an eneire interval of analysis of duration T. !
Some curves of the probability of error in optimal and euboptimal proc-
esaing are given in Figure 4.7 above. '
Aa already noted above, space, time, and frequency variables are equiva-
lent within the framework of the approach adopted here to constructing
field proceasing algorithms. Therefore, it is possible to suggest a
number of algorithms for suboptimal processing of fields that are cloae
to the one under consideration and constructed by ignoring fadeout se-
lectivity for one of the variablea (time in Che algorithm under con-
' aideration) and considering selectivity of fadeouts in the others (for
example, space and frequency).
The analysis of characteristics conducted above can easily be trans-
ferred fr.om the suboptimal algorithm considered to otHer algorithms of
the same class.
A conclusion common to all suboptimal algorithms of this class is that
as the power of the aelectivity considered (number of branches of dis- ~
persion) increases the energy gain of the c~ptimal algorithm over the
suboptimal ones decreases in the domain of ~arge error (hT) and in-
creases inthe domain of small errors (large hZ) .
Let us go on to consider a.suboptimal processing algorithm that
realizes the Kostas idea. For a channel that is non-selective by fre-
quency and time but has fadeouts that are selective in space, the random
quantities Vlk � ~lk~ Which are included in processing algorithm (4.87),
have the parameters: _
r
Mt tv1i:} ntx 1~ ^_E(,VR'No. r11i {Pfk} = r,ry "_E;.~'K No; _
2ax E �R ~o~ E E~R .100)
D(Vtk} = 1-}- N~ ~Ve . D{V = 1'~ A'R No ~
where uR is determined by formula (4.84).
The quadratic form Gg included in (4.87) has an X2-distribution. The
expression for the probability of error is written by a form.ula analogous
to (4.96) where uT is replaced by uR and N~ by NR in this formula and
NF = 1, h2~ h2~NR. For Che domain of small errors it is not difficult
to obtain = 2
194
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~F'OR OFFICIAL U513 ONLY
tt
N ~~'~;~~i~~~~~'9i)N~t _ -_~i~l -4-~i)
p~ n._..~ _ ~xp - -~ros~ 4'n t'I- ~i sin1 Tr 1~ .
r~,~ 2~r 0. . (~.109)
4r
~('a~ , ~
The Gauasian random quantities Vk, Vk, k= 1, N are mutually independent
and have the parameters , ~
?E ~�k ~nxk
.lf~ {l'~) - D {~'A} No 1 ^h~k ~
~
, ^E ~�k nluk ~
.11 ~ { ~'k} _ D {Y'k} - No 1 ? .
l+4
� ` 196
~OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
,FOR OFF7CIAL USE ONLY ~
~ In ~hie caeg the Gausaian quantity I nas the paramr~tera
N n
t._ ~ 2~ ~'k /11_~.k .1. llt~rk
~ ~~~Q~
nt~ ~i~ - n{~} ~ e~' ,~~o i~;� zr~xk ~ i~;. 2i~~k c )
. The expreseion for the probability of error in a procesaing device
working on algorithm (4~108) is described quite eimply as the �unction
of disCribution of the linear form of the indepen,dent Gausaian variables
and hae Che form
-N -
p~ 2 ~"~D 211k Qk COS~ ~P k
~ i 4 y Y 2 r'}�
An~ 1 -r qk hk ~k
(~'~'a~)(1 ~'~k)
siit'i mr k
- -
2h' ~~1.1 I I))
' l ~ ~k~ l ~ Qk~ .
It can be seen from formula (4.110) that if the regular component is
misaing from the field received q2k = 0, k~ 1, N, the system with op-
posite signals becomes unworkable because the probability of error is
1/2 for any signal/noise ratio. Where there are paths of propagation _
with asymmetry in diapersions of quadrature components B2k ~ 1, the
~ probability of error depends strongly on the phases � k of the regular
components in these patha. If there is no aeymmetry ~(Rice fadeouts),
the phases of the regular components do not affect the probability of
error. Analysis of expression (4.110) shows thaC a typical feature of
the sysCem with opposite signals is a minimum probabiliCy of error
that cannot be reduced with growth in the signal/noise ratio. Assum-
ing h2k in (4.110) we obtain this value for the irreducible proba-
bility of error:
. -
~
p''~ _ 1 ~T~ ~ Qk k ~ct~5= rfr ; ~1~ sin= Tr+t) . (�1.11 I )
_ k_' ~}k
Calculations show that the values of p�� in channels with selective ~
fadeouts and fairly good statistics (high values q2k, k= 1, N) may
be very, very small. For example, in the channel described by the
delay line model with two branches NS = 2 with smooth fadeouts in time ~
NT = 1 and selective fadeouts in space NR = 2(exponential correlation
function) with Rice statistics q2k = 2, k= 1.2, the limiting proba-
bility of error is of the order 10'6. This level of noise suppression
can already be reached approximately where 10 where it is possible
to sw3.tch approximately to (4.111) to calculate the probability of
error from 4.110). As the degree of channel selectivity increases the
values of the limiting probability of error dropped sharply and may
reach vanishing small values even for a weakly expressed regular
197
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
,
FOR pFFICIAL USE ONLY `
' .
cumponent of Ci.ie tranafer funcCion of a etochastic channel. These proper-
ties uf a system wi.th opposite signals leave no doubC of�Che advisabiliCy
of tranemiCting information by oppoaite aignale in stochastic channels.
IC ie a ma~or $dvan~age that optimal proceaeing of opposite aignals is
linear, Certain curvea of probabillty o� error calculated from formula
(4.110) are shown in Figure 8 below. The dot-daeh line ahows the char-
acteristics of a syatem with a test pulse (lower boundary) calculated
according to formula (4.3). .
i
' JD� lD~ 10t !OJ !0~ ~Qf JD~ !O?
____.~_.__h= ~
Q~1~ fi~/cN;l;ll=1 ~
10'~ -
~
Figure 8. Probability of 10~? - y ~ ~
Error in Discrimination \
of Two Oppoaite Signals , ~ ~ ,
in a Channel with SmooCh !o' - -
Fadeouts: Without
a TesC Signal; - - ~0�4 - - `
wiCh a Test Signal; ~ ~
Ideal ChanneX. I N~4' 2\
~o-s ~ . - \ - -
, I
. ~p�6 P I. _ ~ -
Suboptimal proceasing. Let us determine the probability of error when
opposite signals are processed by algorithm (3.91). The probability :
o� error is determined by the probability of fulfillment of the in-
equality H r R
No ~ J J 2~~ ~ Intxks~k i) -f- myksik Il, rj dtdi > 0 (4. ! 12)
k=! 0 0
on the assumption that signal s2(t) is contained in the observed oscil-
lation. This probability is easily found in the form
. .
~ N 2 ~
P= 1 1- m ~j 2ltk 9k _
2 l`f' 9k 21tk ~~k CoS= ~r k-~- Sin"- ri'r k)
k-~ 1-{-
~ ~1-;-~k~~1-~-9k~
(4.113)
198
?FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL USE ONLY
`
Comparing (4.113) and (4.110), we aee tha~ the errore coincide in a
channel without asymmetry of expressiona for probabilities of e~ror.
This is natural because algorithm (3.91) is optimal in ~he g3ven case~
If there is asymm~etry, of courae, the optiimal algorithm is superior to,
the suboptimal one and tihis superiority increases as the aeymmetry
grows. The value of the ~.imiting probability of error obtained from
(4.113) where hae the form
N
n~�� 2 ~-~n }j yk~~-~-~k) ~ ~ . ~a,i~a~
k~~ ~kcos�cpak+sin~9'rR _
The independent variable o� limiting probability of error in the optimal
- proceasing device exceeds the correaponding value of the independent
variable in (4.114) by the quantity
N 9k ~ 2~k~ ~cos1 ~Pn k~I- ~k sin' V'~~k~
~
r~ 10 Ig k~'N ~k . (a.l 15)
I
~~k ~ l I. ~k cos~ ~p,, i,. . s i n= ~rr k
k=>I
The degree of superiority of the optimal algorithm to suboptimal ones
can be seen most graphically by considering a channel with identical
average paChs of propagaCion. For such a channel it follows from (4.115)
that ~ - -
q~ 101~ ~j (cos~~~,-~-~~sin=q+r1(~2cus=~P�;�sin~~,,l. (�1,I1G1
Ar.alysis of formula (4.116) shows that optimal processing permita a
sharp improvement in noise suppression in channels with non-zero regu-
lar parts of both quadrature components (~P ~ 0, �P #~r/2).
To complete our consideration of the characteristics of optimal and sub-
optimal algorithuns for processing space-time signals in channels with
generalized Gaussian statistics, let us review the advantages afforded
by optimal processing. The benefits of such processing compared to sub-
optimal processing come chiefly from the fact that it makes it posaible
to organize N-multiple accumulation in a channel with selectivity of.
degree N. �
The non-correlated nature of all N branches which give rise to statis-
tical independence in the Gaussian case under consideration is achieved
by selecting a channel model based on the Karunen-Loew expansion. Usi.ng
any other coordinate functions of a discrete channel model leads to the
appearance of dependence among the N branches of the receiving device
and the efficiency of accumulation will be lower. An alternative in sub-
optimal processing is to choose the number of independent branches
N' < N, which ia often done in practice.
199
~FOR OFFICIAL USE ONLY
~
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
i:. ~
, ;
FOR OFFICIAL U5E ONLY
'
A second advantage of optimal procesaing is the fact that it permita the
~ bes~ processing of eignals in each of the N branchea (~his queation was
i.nvestigated in adequate deCai.l during consideration of a channel with
emooCh fadeoutis). '
Speaking only of apatial procesaing, we may say Chat the optimal algorith~
poinCs o~t the forma of antenna directivity diagrams thaC allow organiza-
tion of accumulation for N independent branches.
Let us use examplea to eatimate the effect of non-optimality of antenna
- directivity diagrama on noise auppression for a channel with amooCh fade- '
outs in time and frequency, buC selective in epace. ~
5uppose the discrete mddel of the channel has the form
NR
It (r) = y Itp ~+a (r) ~ (4 . I ! 7)
P=~
where the functions {~p(r)} form an orthonormalized system but are not
Karunen-Loew functions. ,
The magnitudes of covariance of expansion coordinatea (separately for
each quadrature component) are det~rmined by the relations
R..R . ~ ,
~kp - 1 ~ Bx ~ r~) ~Ck ~fr ~r') drdi~; I
p p i
R R ~ (~1. I19) ,
~kp-~J Hylr--r')V'~lrlc~,,lr')drdr'; I
0. o ~
k= 1~ NR: p==~~ ,1'R. ,
We assume the quadrature components are non-coherent: Bkp = 0, k, p,
1, NR.
Suppose, for example, Chat coordinatea hp in expansion (4.117) are equi-
distant (distance of ~ r) readings for the space variable r(which cor-
responds to reception at narrowly directed antennas with a diagram of
the type sin f1/~}). Then the magnitudes of covariation of the coordi-
nates with different indexes are determined, on the basis af (4.118),
by the relations : ~~kp R., (lk - pl ~ r); I~RP cc By I(k ~1) (.t. I I~}
Thus, for the exponential correlation functions:
/ .1r ~
' 13k~~-=~~xerpl --�-~k-p~l~ I~kp-oucep(---~k-Y~). (1.IY0)
\ ('~tnr / \ ~KnD
The existence of a correlation among individual branches affects the
probability of error differently depending on the sy~tem of signals and
statistics of fadeout in the channel [6, 46].
200
~'OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFT.CIAL USE ONLY
_ ~
Let us review some examples of pxactical intereati.
l. For a Rayleigh channel with Cwo identica7, on tihe average, paCha
_ (N ~ 2) where ortihogonal eignals are uaed (M m 2), we write, on the
baeis of [6],the following
2
p~, ~!i'~)~ ( I- ~ k I1) 41i~_.. f~ ~4,121)
where
~ ~ I/ ~ ~kp~llkk~~ 'T' \ I.ikr Ilkk~1 . ' (4.
It can be seen from (4.121) that the correlations of aignals in the
branchea are reflected in the domain of large aignal/noise ratios. If
the correlated coefficientis are described by expresaion (4.120),
when ~r/pcor � 0.5, from (4.122) we will receive IRI= 0.85. Under
these conditions the energy loss owing ta non-optimal special procesaing
where h~ � 1 will be about 2,5 decibels.
2. Let us consider a system of opposite aignals working on algorithm
(4.108). Where there is a correlation among particular components, in '
~ addition Co parameters (4.109) the covariation quantities below wi11
also characterize the linear form (4.108):
2 N I~~k~~A BkP 1
8 {Vkvp} = ~ `
1~ "I" 2IIAk/ ~I ~IIIAD/ ' r
~a , i 2s~
c
2 I~VkVp QkP
B{l'kVp} = ~V0 ~ k~i~P~
y~ ~.ir.�~ c ~ + ~;,~P~
The probability of error wlll still be determined by the probability of
fulfillment of inequality (4.108). Quantity I is Gaussian with an
average value and dispersion equal to, respectively:
R ~ ~
2E vk mzk m~~k
~~'~f'=~j N 2h2 + i -~~n2 1 ~
0 1 1" rk yk
k =l
� R
D{l} N~E ~'k ntXk + m~k ?E l~~k~�o ~ (4. I?~4)
y Np 1-f- 211ak 1-}- 211~k ~ No
Bkp!ll.i~;nlXp I3kp llt~kJ/IJP
Y ~i ~l 2h~~r/ ~I �j�'~II?k~ ~ l~ ~1 ^lt~kl -I. ~i~'~r~
, The quadratic form in (4.124) is negatively determinate, which follows
from the property of the correlation function [64). Therefore, dis-
persion (4.124) is always at least as great as dispersion (4.109).
Zoi
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
1'
~rOEt O~~ICIAL U5~ O~iLY
~rom thig it tg rl~ar Chat the probabiltCy of error in Che caee under
coneideration is alwaya greater than it ie in oprimal epntiai proc-
- esein~. '.Che genernl exprea~ion for the probability of error witt~ non-
opCin?~1 proceeeing is wriCten in the form
' ~ C A.
o~= x ' i'u-{i; ' ~ '
wherp M1{I} and D{If ar� determined by formul8.e (4~124). ,
, Ie ie egey to determine th~ limitiing probability of error in the form
n
`i (~R ~ ~ ' ~CUS~ ~~v R ~1R sl nt 4~v k~
L - _
1 ~k
~~n _ 2 ~ ~ ~ w~. ~....._....._._~::---Y...,.~.~...~~.~...`..'__
N
yk ~ 1. ~cos~'('o R!�~k s i n~ V'r k~
~ ~ ~R
N~ 9k ~ ` ~k~ i Q p ~ 1 ~o ~ ..__._v..
X 2 ~y dkv - z_-- cos~ mo r: ~z cas= Te n r_..
0 Ga; g n 1~~ l. (4 .129)
N
where Gr ~ ti} ~
i?~�
x~~
The disperaions of variables of the quadrature form G1 and Gg have the -
form
203
~ ~FOR OPPICIAL USE ONLY
~
t.;
v~.
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
i
- FOtt ON~ICIAL USE ONLY '
r
D {trr~} ~ U {~w} :
~h~~,
~ j,~ (4.1a0)
d{l'QA~ ~ d I~BR? ~ 1~- h ,
tR
Noting that where h2 gk-~, g~ 1~ N Che qu~drature form r is disCri-
buted according to CFie law X~ and 2N de~re~of freedom an~ uein~ the
result of ~89~~ we vrite an aeympCoeie expres~ion for ehe probability
of error ,
~ 4 N-�i R tN -1)
p_ ~
~ edX~ (1 11R4~~ r~l-~ ~ ~~y--tt~v?~Vt,,R+ill11- , (d.l1~j~ -
L
R:, ~t 7 r~,~ a- o
For the binary eygtem M~ 2 it follows from (4.131) that
v ~
r.~ r.;`'-~ n-!- - r,a.;, ~a. i:s7i~
azti~
Aesuming ,,:~k i-n~R!" it ie poesible to investigate the degr~e
of increase in the probability of error de~ending on the increase in .
the depth of fadeout (decrease of Nk~ k~ 1, N). Comparing expressiong
(4.131) and (4.92), we come to the conclueion that logarithmically nor- ~
mal statistics yield higher probabilities of error than Reyleigh eta-
tieticR s+here the conditiong �z4~ j~na~� are fulfilled.
Let us go on with our coneideration of suboptimal processing that does
- not take account of ttre mean statisCical parameters of channel fluctu-
_ ationg, aseuming that the quantities F1 included in (3.45) are de-
termfned from the relation
Fl = V~ ~ ~i~ ~R. ~d.133)
~e
. R~1
The algorithm for discrimination of M signals may be written in the
form (4.119), and the dispersions of the variables of the quadrature
forms G1 and Gg are determined by the formulas
D t~'t4) ~ D{ti'tt} a 1;=hr?~: D{V~c3 ~ ~ t~'ct} ~ 1. (~t.13~ )
It is not difficult to show that in the domain of high signal/noise
ratios the expression for the probability of error in the sub-
optimal device uader coneideratioa coincides aith the correaponding
expresaion for an optimal device (4.131). This makes it possibte to
applq the coaclusioa that the characteriatics of cohereat and non-
coherent diecrimination of signals are idenCical �or large signal/noise
ratios to a non-Caussian chaaael.
204
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
� ~OR OI~FICIAL U3E ONLY
- For arbitrery eignaZ/noiee ratioe ~he probab3litiee of error ehould be
determined by campueer calculation. We will give the formula for prob-
ability of ~rror for ~ ~in~i~-pt~tih ehgnn~i
a"i "
t" ~R I kh~ ( In y d' ~s d
p ~ ~ ~ ~ k '~~aa~ ~ p~p ~ k ~ Ys ^ ~ . (~1,135~,
~~t V x o .ax Y
Figure 4.9 ehowe a graph calcuZatad 8ccording to formula (4.135) where
M� 2. The dotted line shot~rre the corresponding curve for a channei with
Gauaeian staeietice (a ltayleigh channelj, and the dot-dASh line ie for
a channel Without fadeoute. Comparing them alinwe us to eetimaC~a how
much the characteriatics of discriminatioe change with the changa from
Gaueeian to non-Gauesian etatietice.
l ' ?I' ~0 ~ ?1 ~
_ t
' ~ .._.__i
~
, ( i
w~~
~
_ Figure 4.9. Probability of Error ~e�~ r ~
when Discriminating Orthogonal ~
Signals (N = 2) in a Non-Gauseian
Channel: Rayleigh Channel; w ~
Channel Without FadeouCs. ~ ~ ~
~
~D' I ~'L~r '
~
~ r
~ p g~-'qr.s`~~ ~ ! ~
The analysi~ ae ~~ave made ahows that Where the values of the parameter
~x2 change., logarithaoically normal fadeouts cover a broad clase of
channels from channele cloee to the ideal (where ~x2 0) to channela
of the sub-Rayleigh type ~ az ~ 3 ~n +n~ .
~
P
o r s m a l l v
a lu e s o f ~x 2, exp a
n d ing t he func t ion exp - z t�z, in ( 4. 1 3 5)
into a Taylor series relative to point (-~X it is poseible to re-
ceive a cenvenient calculation formula for the probability of error
~M ~ 2~
. ~ _.a3
p~ l txp C.- i e' x}. ~ M.13ti?
It can be aeen from (4.126) that the nature of decreaee in the proba-
bility oF error relative to the signal/noise ratio is exponential~
ahich means that the chaanel is close to ideul.
205
c~R OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
;7
FOR OI~FICZAL U3E ONLY
In conc]~uaion, iee u~ coneider suboptim~l proces~i.n~ b~s~d on r~place-
ment of the modei of a muitipaee channel by a eingle-pase adaptiv~ model
(autoeeiection). We aill aseume that the quantiCy F1 inciuded in the �
proc~~~ing ~igorithm i~ computed from the relation
I 1' nR ~(~I f4 Ik~ ~ Id . I;17I
that i~, the path with maximum po~?er of the trenefer coefficient ~hk ~2
ie seiected as the workin path. The distribution of the modulue of
~he transfer coefficient ~hk) we assume eo be logarithmically normai~
and Che pathe are taken to be idenLical on the averaga. Neglecting the
inexectneas of estimates of the power of the transfer coefficiente of
parCicular paehe, we wili determine the probability of errors from the
relation -
4~1 !
p~~~l�-'~)I+~k,t1-1_~~~pf ~.%~r,-1~,s~n~i(1~dY~ Id,l~B)
L
4-=-1 . ~
, 4 . ~ ~
tv) ~ CxP ~ ~n ~ uz ) ~r ~'?1' -i tlz ~ , ~ a . ~:3n?
, ~2:toxY =tlx , ~x ,
Calculatfona by formula (4.138) shoa that the echeme vith autoselection
in a channel aith non-GauBeian etatietica, ~uat as With the optimal
acheme, ~~~uree a dacrease in the probability of error inversely
proportional to the N degree of this ratio for large eignal/noiee ratioe
and large ~X2. Por amall values of the parameter dx~dx~ 3 ina:~,
the curvee of probability of error are exponential (the channel is clo9e
to ideal) and the effectiveneee of autoselection ia lo~r. �
~ * ~
We have analyzed the quality of optimal and suboptimal algorithms for
proceasing fielde carrying digital information. The generalized
Geuasian probabiliatic model of a channe~. as the most Wideapread in
practice and, moreover, the one orith the beet approximating capabilities
in highly diverse situatione, Was used nast.
We investigated different channela ~+ith non-seYective and selpctive
fadeouts. The problems t~f detection gnd discrimination of signals
Were considered aeparately. It ~ras demonatrated that the probability
of error depends aignificaatly on the statietica of fadeouta in the
channel. The beet chaanel aiLl be the one in ~+hich ~reakly fluctuating
quadrature components have clearly expreased ragular components (an
aeymmetric chanael close to ideai).
206 -
.POR OPPICIAL US~ ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~OR OFF?CIAL U5E ONLY
A typical feaCure of the binary eyeCem with oppos~.Ce aignals during inde-
pendent reception of charactere is Chg exietence o� a limieing proba-
~ biliCy of error Chae i~ irr~duCible with growth in the eigngl/ndiee
ratio. ~towever, the values of thie limieing probability are very ama11
in good channels. For example, in a non-aelective channel where
q2 ~ 2, g2 � 0, and ~p ~ 0 we ha~~ p�D � 10'6. An opCimal binary eyetem
of signals was conatructed in a channel with non-selective fadeoute and
we determined the threshold eignal/noiee raCio at which the syetem of
oppoaite eignals loses ite optimal featurea and the aystem of orthogonal
aignals acquiree them.
It wae ahown with concrete examples ~hat the uae of optimal direct3vity
diagrams makee it possible to greatily improve noise suppression in com-
parieon with the meehode of epatial proceesing of signals used ~xten-
sively in practice at the present time.
The use of auboptimal algorithms shows thar where the model of the chan-
nel is intelligently choaen, they insure error probability valuea that
are almoat as good ae thoae of optimal algorithms and they are much
simpler to real~ze. The eystem with teet aignals ie more effective in
a channel with fadeouta that are smoath in time where it ia capable of
insuring an energy gain of u~ to three decibels compared Co a eyatem
without test signals.
We reviewed certain modifications of the ideas of autoselection with
application to space-zime aignals and demonstrated that the use of thie
procedure for euboptimal procesaing is ~uatified in many cases (both
for Gauasian and non-Gauseian fadeout statistics).
Our investigation of the asymptotic behavior of error probabilities
(where E/No showed that independently of channel atatiatics, con-
sideration and use of channel selectivity in time, apace, and frequency
makes it posaible to achieve an accumulation effect. In thia case, the
probability of error diminiahes as a quantity inverse to the eignal/
noiae ratio to Che extent of Che selectivity in question (E/No)R.
Conclusion
This book reviewed the general principles of constructing optimal and
auboptimal signal proceasing devices in stochaetic apace-time communi-
cation channels.
Optimal proceseing in this case was based primarily on obtaining renew-
able estimates of the coordinates of channel characteriatics and was
oriented in its realization part to the technology of space-time filtra- ~
tioa accompliahed by both the classical techniquea of dispersed recep-
tion and by techniquea based on holographic principles.
It should be kept in miad that practical realization of many of the
procesaing algorithms inveatigated in this book depends greatly on the
207
' ROR OPFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
,r
FOR OFFICIAL USE ONLY ' ~
_ progrese of integr8~ted technology that characterizes tha developmenC
of electronice in our day.
In concLuding this book the authora~acknowledge thaC many questions o�
intereaC in coneCructing e�fective digiCal information r.ransmiseion sy~-
tems in etochastic epac~-time channels proved to be outiaide our frame-
work. Among them are optimization of the communicatiiona syetem as a
whole by finding optimal apace-time proceasing operators not only in re-
ceprion but also in tranemisaion; the effectiveneas of use of a feedback
channel in space-Cime channels; aelecring codea with due regard for thP ;
apecific features of the apace-time channel; assesaing the difficulCies
of realization and noise euppresaion of aystema for transmieaion of
diacrete a~eseages by meane of aimple signals that do not satiafy tihe
conditions of aeparaCion of paths; application of deciaion feedback in
construcCing optimal and auboptimal signal processing devices in a gpace-
time channel; investigation of the prospecta for non-linear filtration
in proceaeing apace-Cima signals; procesaing for specific distributions
of noiae fielda, and othera.
The authors hope that Cheir book will stimulate Che intereat of a broad
range of specialiats in the problems of apace-time signal procesaing,
including intereat in solving the problema we have formulaC2d here.
208
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
. F
- FOR OFFICIAL USE ONLY
,
~
1
1.
Appendix l;
The linear estimate of coordinate x is sou~sht in the form (2.11)
rR . _
x~A ~ f z(!, rl~~�(!, r)dfar-!.8 ~Il,l,l)
00
or in symbolic (operator) form _
x:_n~a, t;.___ ~ ~rt.~.~)
All further reasoning relies on the reaultr~ of work [75]. The expres-
sion for the conditional risk function witil a fixed atate of the eati-
mated (centered) parameter in operator forin is written as follows .
~ (x~ V') _ (R V~1 ~ (4' S' z). . ~Tl. I .3)
Operator R~ is determined by Che correlation function of noise
B n,~ t~ t' ~ r~ r' , r R - . _
� n.~.a
R~- J f Bn~l~ r~)~'~~~~ ~~)dl'dr'. (
00
Operator S and, correspondingly, its con~ugate operator S* are determined
by the apectrum of the transmitted signal. The expression for average
riak can be wriCten, averaging (II.1.3) the parameter being estimated by
the distribution of probabilitiea wl(x):
r _ ~ ~ (X ~ ~I'1 w't (xl dx. (il .1.5)
: Substituting (II.1.3) in (II.1.5) we obtain
r l~l') lR V'~ ~l') (~A.~ lT - S� ~'1. - S' V 1) ~ ~Il. l.6)
where ~is a self-con~ugated, negatively determinate linear operator
defined y the correlation function B�(t, t', r, r',
209
~FOR OFFICIAL USE ONLY
~
1
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
9
. . . . ~ . ~i~K~
`FOR OFFICIAL US~ ONLY ,
r ,
~~t~kV~ ~~i~':s J\~~ 0~~~~~l~k~dXt~ I f~P~~~ b~ r~~y~~~~~~~i~:~
~
;C dtdt'd ~ ~ b' didr' , (Il, I .7) ' -
It can be ahown that the minimum average riek ia achieved where ~ -
~(R S ~i~,~ s.~-~ s ~h,c~p, ([l. I~ K
~
Uaing (II.1.8) in (II.1.1) and adding the known mean mX we obtain tihe !
, optimal 1in~ear estimate in the form ~
.N
= ( ~a S ~Dr S')~'' s ~a~, Z~ R~~e.~. S cDx S�)~' ~ ~n. ~ .s~
Moving from the operator fora~of writing to conventional form it is not ~ ~
difficult Co aee that the optimal linear estimate coincidea with the
Bayea eatimste of a Gauasian coordinate in a setting of Gausaian noiae
(2.96).
The linear estimate (II.1.9) was obtained without conaCraints on the
form of eignals tranamitted.
.
S
21U ~
~'UR OFFICIAL USE ONLY ~
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~'OR OFFICIAL USE ONLY
Appendix 2
We wi11 find the average value of the funcCion
. ~ . . _
~cX~-=~Ya -~-~~dk�~�k , dk>o, ~r~.~~.i~
,k:y I ~
Suppoae quantitiee, xk, k~ 1, N in (II.2.1) are distributed by arbitrary
laws and are statiatically independent. The average value we are seek-
ing is written in the form
N .
F " 1 1 ~k� (i I,'~,'~)
where k-~
~ ~
`~k xk
~k = e ~i 1�~'kl dxk� ((t.2.31~
We will consider the domain dk � 1. Using the asymptotic formula from
[17] to estimate an inCegral of type (II.2.3), asauming that the necea-
sary conditiona are fulfilled, we obtain
, . - .
~k V t�~�k - o) ~tt.2.a)
and correapondingly
.v
~ Yak "''.Y~��~� ~n.~.~> .
k=1
We will consider two examples.
1. The quantities xk, k= 1, N are Gausaian with parametera Mk and Q2k.
Then - -
2
4'i ~a'k = = 1 e~nk :ok ~I7.2.6).
Y~~~~k
211
~`OR OFFICIAL USE ONLY -
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL U5E ONLY ~;".Yi' '
~ ~ ~ 1
and
,v ~ ~
, ~ n , _,~~k :nk
~ r~ . C ~ If~l~l.l)
2N k-ul ~k'~R
, ,
2. The quantitiea xk, k~ l, N are distributed according to bimodal
i laws (1.49) with parametera u, ~x2k� Then ;
~
. . . ~z . _
~xk L30) ~ ey~~~'~ ~k , (11~2.N) '
2rc
and
z
F ~ N ~ ~ -~~k~ oxk
- ~ 2 ~n
dk e ! ~ (t1.2.9)
� kml
~
1
' 1
�
,FOR OFFICIAL USE ONLY
,
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
~
FOR OFFICIAL USE ONLY
_ ~
BIBLIOGRAPHY
1. Al'perC, Ya. L., "Rasprpstraneniye Radiovoln i Ionosfer,a" [The Propa-
gation of Radio Waves and the Ionosphere], Moscow, Izd-vo AN SSSR,
1960, 480 pp. =
2. Alekaeyev, A. I., eC al, "Teoriya i Primeneniye Paevdosluchaynykh
Signalov" [The Theory and Application of Paeudorandom Signals], Moacow,
_ Nauka, 1969, 367 pp.
3. Amiantov, I. N., "Izbrannyye Voprosy StaCi$ticheakoy Teorii Svyazi"
[Selected Isaues of the SCatistical Theory~of Co~unicationa], Moacow,
Sovetekoye Raclio, 1972, 416 pp. _
4. Amosov, A. A., and Kolpakov, V. V., `'The Kalman-Busaey Receiver for
Linear SCochastic Channels with Distributed StaCes" VOPROSY RADIOELEK-
TRONIKI: SERIYA TPS 1973, No 8, pp 49-56.
5. Anderaon, T., "Vvedeniye v Mnogomernyy Statisticheskiy Analiz" [Intro-
duction to MulCidimensional Statistical Analysis], Moscow, GIFML,
1963, 500 pp,
6. Andronov, A. A., and Fink, L. M., "Peredacha Soobshcheniy po
Parallel'nym Kanalam" [firansmission of Messages on Parallel Channels],
Moscow, Sovetskoye Radio, 1972, 406 pp.
7. Voskresenskiy, D. I. (editor), "Antenny i Ustroystva SVCh" [Antennas
and Superhigh Frequency Devices], Moscow, SoveCakoye Radio, 1972,
318 pp.
8. Arsenin, V. Ya., and Ivanov, V. V., "Solving Certain Integral Equationa
of the First Type of Convolution by the Regularization Method" ZHURNAL
VYCHISLITEL'NAYA MATEMATIKA I MATEMATICHESKAYA FIZIKA 1968, Vol 8, No 2,
pp 310-321.
9. Arsenin, V. Ya., and Ivanov, V. V., "On Optimal Regularization" DAN SSSR
1968, Vol 182, No 1, pp 9-15.
10. Tartakovskiy, G. P. (editor), "Voprosy Statisticheskoy Teorii Rs3io-
lokatsii" [Questions of the Statiatical Theory of RadarJ, Moscow,
Vol 1, 424 pp, 1963; Vol 2, 1,079 pp, 1964.
11. Bark, L. S., Bol'shev, L. N., and Kuznetsov, P. N., "Tablitsy
Raspredeleniya Releya-Raisa" [Tables of Rayleigh-F.ice DistributionJ,
Izd. VTs AN SSSR, 1964, 196 pp. _
213
~'OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
run yrrt~t~ u,~ uri~i ~y,,
1 ' ~r, ~,~h
12. Baea, F. G., and Fuks, I. M., "Rasseyaniye Vo1n na Stat3atiicheaki
Nerovnoy PoverkhnosCi" [Diaperaion of Wavea on a Statistically Un-
even Surface], Moscow, Nauka, 1972, 424 pp.
13. Bochkarev, V. A., Klovskiy, D~ D., and Soyfer, V. A., "Optimal Recep-
Cion o� Signals in Channels with Frequencq-Time 5electivity" RADZO- '
TEKHNIKA 197~., Vol XXVI, No 2, pp 36-44.
1G. Bykov, V. V., "Tsifrovoye Modelirovaniye v Statieticheskoy Radio-
tekhnike" [Digital Model~.Y~g in Statistical Radio Engineering), Moscow,
Sovetakoye Radio, 1971, 326 pp. ;
15. Buslenko, N. P~, and Golenko, D. I., "Metod Statiaticheakikh IspyCaniy
'SI~ [The 'SI~ID' Method of StatiaCical Testing], Moscow, Nauka, ~
1962, 331.pp.
16. Vaynahteyn, L. A., and Zubakov, V. D., "Vydeleniye Signalov na Fone
Sluchaynykh Pomekh" [Identification of Signals in a SetCing of Random
Noise], Moecow, Sovetskoye Radio, 1960, 447 pp.
17. Vakman, D. Ye., "Slozhnyye Signaly i Pr~,ntsip Neopredelennosti v
Radiolokatsii" [Complex Signals and the Uncertainty Principle in ~
Radar], Moscow, Sovetskoye Radio, 1965, 304 pp. ~
18. Van Tris, G., "Teoriya Obnaruzheniya Otsenok i Modulyatsii" [Theory ~
of Detection of Estimates and Modulation], Moscow, Sovetskoye Radio,
1972, 744 pp.
19. Van Tris, G., "Applications of the Variable States Methods in the ~
Theory of DaCection" TIIER, 1970, Vol 58, No 5, pp 55-72.
~20. Vilenkin, S. Ya., and Dubenko, T. I., "Optimal Linear Estimates of
the MaChematical Expectation o� a Homogeneous Random Field"
TEKHNICHESKAYA KIBERNETIKA, AN SSSR, 1971, No 1, pp 134-141. '
21. VozenkrafC,,Dzh., "Sequential Reception in Communication over a Channel
Whose Parameters Change in Time," in the book "L~ktsii po Teorii Sistem
Svyazi" [Lectures on the Theory of Communications Systems], edited by
B. R. Levin, Moscow, Mir, 1964, pp 241-288. ~
22. Vozenkraft, Dzh., and Azhekobs, I., "Teoreticheskiye Osnovy T~khniki
Svyazi" [Theoretical Foundations of Communications Theory], edited by
R. L. Dubrushin, Moscow, Mir, 1969, 640 pp.
23. Volokhatyuk, V. A., Kochetkov, V. M., and Krasovskiy, R. R., "Voprosy
Opticheskoy Lokataii" [Issues of Optical Location], Moscow, Sovetskoye
Radio, 1971, 256 pp. ' -
24. V'yeno, Zh.-Sh., Smigil'skiy, P., and Ruays, A., "Opticheskaya
Golografiya, Razvitiye i Primeneniye" [Optical Holography: Development
and Application], Moscow, Mir, 1973, 212 pp.
214
. FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICTAL USE ONLY ,
_ ~
25. "CoherenC Optical bevicea for Generalized Spectral Analyaie of _
Images" AVTOMETRIYA 19~2, No 5, pp 3-9.
. 26. Gnedenko, V. V., "Kurs Teorii Veroyatnostay" [Course in Probability
Theory], Moscow, Nauka, 1965, 400 pp.
27. Golenko, D. I., "Modelirovaniye i Statietichesk~.y Analiz Psevdo-
sluchaynykh Chiael na EVM" [Modeling and Statistiical Analyais o� P~eudo-
random Numbera on the Computer], Moscow, Nauka, 1965, 227 pp.
28. Gold, B., and Reyder, K~, "Tsifrovaya Obrabotka Signalov" [Digital
Signal Procesaing], Moscow, Sovetekoye Radio, 1973, 368 pp~
~ 29. Gradahteyn, I. S., and Ryzhik, I. M., "Tablitsy Integralov, 5umm,
Ryadov i Proizvedeniy" [Tablea of Integrals, Sums, Seriea, and Pro-
ducts], Moscow, Fizmatgiz, 1962, 1,100 pp.
30. Gutkin, L. S., "Teoriya Optimal'nykh Meeodov Radiopriema prt
Fluktuatsionnykh Pomekhakh" [Theory of Optimal Methoda of Radio Recep-
tion with FlucCuating Noise], Moscow, Energiya, 1971, 487 pp. ~v
31. Davenport, V. B., and Rut, V. L., "Vvedeniye v Z'eoriyu Sluchaynykh
Signalov i Shumov" [Introduction to the .Theory of Random Signals and
Noise], Moscow, Inostrannaya Literatura, 1960, 468 pp (translated
from English under the editorship of R. L. Dobrushin).
- 32. Vvedenskiy, B. A., (editor), "Dal'neye Tropoafernoye Rasprostraneniye
na UKV" [Distant Tropospheric Propagation on UHF], Moacow, Sovetskoye
Radio, 1965, 115 pp. ~
33. Denisov, N. G., "Wave Diffraction..on a Chaotic Screen" IZVESTIYA
WZOV. RADIOFIZIKA 1961, Vol IV, No 4, pp 630-638.
34. Derusso, P., Roy, R., and Klouz, Ch., "Prostranstvo Sostoyaniy v
Teorii Upravleniya" [The Space of States in Control Theory~, Moscow,
Nauka, 1970, 620 pp.
35. Dzhenkins, G., and Vatts, D., "Spektral'nyy Analiz i Yego Prilozheniye"
[Spectral Analysis and Its Application], Moscow, Mir, 1972, No 2,
287 pp.
36. Ditoro, M., "Communications in Media with Dispersion in Time and by
Frequency" TIIEP 1968, Vol 56, No 10, pp 15-45.
37. "One Approach to the Problem of Machine Syathesis of Holograms," in
~ the book "Voprosy Postroyeniye Sistem Sbora i Obrabotki Dannykh"
[Issues of the Construction of Data Collection and Processing Systems],
Novosibirsk, 1973, pp 64-69.
38. "Integral'nyye Uravneniya S1~" [SMB Integral Equations], Moscow,
Nauka, 1966, 448 pp.
215
`FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR q1~FICIAL U3E ONLY ~
ra.,~~ '
, .
39. Zyuko~ A. C.~ "pomekhoue~oychivo~t' i Effekeiveoet' Sietem Svyazi"
(No~.e~ Suppreesion and the Eff ici~ncy of Communi~aCione Syeteme~
Mo~cow, 3vyaz'-izdat~ 1963~ 320 pp.
40. tteylat, T.~ "Kanaly e Parametrami, Izmenyeyuehchimieya vo Vremeni.
Lekteii po Teorii Sietem Svyazi" (Channele with Parameter� ~hat Change
in Tim~. Lectiuree dn the Theory of Communir.grions Syrtema~~ Mngc~w,
Mi,r, 1964 ~ pp 50-78.
41. Key18t~ T~~ "The Generating Proceee Method in Application to the _
Theory of Deeection and Eetimation" TIIER~ 1970~ Vol 58~ No S~ :
PP 82~99.
42. Kelman, R.~ and B'yusi, R., "New Reeults in Linear Filtretion 8nd
Prediction Theory" TEKHNICHESKAYA t~KHANIKA. SERIYA D 1961, Vol 83~
No 1~ pp 95-108. .
43. Kanareykin, D. V., at ai, "Polyari~gteiya Radiolakateionnykh
Signalov" [Polarization of Rader Signalsj, Moscow Snvetskoye Radio~
1966, 440 pp.
44. Kennedi, R., "Kanaly Svyazi e Zamiraniyami i Raeseyaniyem" ~Communi-
cation Channels with Fadeouts and Dispersion], Moacow, Sovetakoye
Radio, 1973, 302 pp.
45. Kennedi, R., "Intraduction to the Theory af Meseage Tranemieeion on
Optical Channela with Dispereion" TIIER 1910, Vol 58~ No 10, pp 264-
278.
46. Kirillav, N. Ye., "Pomekhouetoychivaya Peredach Soobehcheniy po
Lineymyrm Kanalam so Sluchayno M~eayayushchimisya Parametrami" ~Noise-
proof Message Tranamiasion on Linear Channels with Raadomly Chang-
iag Parametera], Moacow, Svyaz', 1971, 526 pp.
47. Kirillov, N. Ye., and Soyfer, V. A., "Space-Time CharacterigLice
of Linear Chan~ele aith Yariable �ParameCere" PIt~OBLLMY PEREDACt~I
INFORHATSII 1912, Vol VIII~ No 2, pp 40-46.
48. Klovakiy, D. D., "ConsCruction of Ideal Receivera of Sigaals with
Fadeouta on the Basis of Computers" TRUDY~LEIS 1959, No 6~43),
pp 40-46. , ,
49. Klovekiy, D. D., "Peredacha Diskretnykh Soobshcheniy po Radiokaaalam"
[Tranemisaion of Diacrete M~easages on Radio Channelsj, Moscow, Svyaz',
1969, 375 pp. ~
50. Klovekiy, D. D., and Klyzhenko, B. A., "Questione of the Phyeical
Subataatiation of a Generalized Gausaian Channel Model" TUIS 1971,
No 54, pp 54-63.
` 216� '
�FOR OFFICIAL USE ONLY '
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR pFFICIAL U9E ONLY
,
51~ Kiovakiy~ D~ D~, "Teoriya P~r~d~chi 3ignaiov" (Theory of 8ignai
~rnnamieaion~~ Mo~cow, Svyaz', i973, 37f~ pp.
52~ Klovekiy, D. D~, and $oyfer, V. A.~ "Optimai ProceaAing of Spac~- ~
Time Fielde in Channel~ with 9e].active FBdaoute" PROBLEMY PEREDACHI
INFORMATSII 1974~ Vol X, No 1~ pp 73-79.
53~ Kiovakiy, D~ D., ~nd NikolByev, 8. I., "Inzhenernnya R~gii~atoiye
Radioeekhnicheekikh 3khem (v 3ietemakh Peredachi DiekrBenykh
Soobehcheniy v Ueloviyakh Mezheimvoi'noy InCerf~rentsi,i)"[Engineer-
ing Rea~ization of ItadiotechnicaZ Schemee in 8yateme for Tranemiteing
Diecrete Meseagee with Inearw~ve Interference), Moecoa, Svyaz'~
1975~ 200 pp.
54. Kiovekiy, D~ D., And Soyfer, V. A.~ "Noiee Suppreeeion of a Broed-
Hand Syetem with Oppoeiea Signnis in Optimal Space-Time Proceesing"
RADIOT~KHNIKA I ELEKTRONtKA 1972~ Vol i7~ No 12, pp 2609-2611.
55. Koliinz, "Feaeibia Filters �or Converting a Random Procass iato
White Noisa and Their Reali~ation by the Variabla States Method"
TIIER Vo1 56, No 1~ pp 114-115.
56. Kondratenkov, G. S., "Obrabotka Informatsii Kogerentnymi
Opticheekimi Siatemaia" ~Proceesing Inforcmtion with Coherent Optical
Syeteme~~ Moacow~ Sovetekoye Radio, 1972, 206 pp.
57. Kopilovich~ L. Ye., "The Dietinguishability of Diatributione of the
Eavelope of Radio Signals" RADIOTEKHNIKA I ELEKTRONIKA 1966,
Vol XI, No Z.
58. Korablin, M. A., "Investigatioa of the Proceeaes of Statistical
Self-Tuning (Adaptation),"(Author's Abetract of Dissertation for the
Learned Degree of Caadidate of Technical Sciencee), Moacov, 1~NS,
1973, 16 pp.
59. Koatas, Dzh., "Carrying Capacity of Channele vith Fadeovts under
Conditions of Channels with Strong Noise" TIIER 1963, Vol S1, No 3,
pp 478-489.
60. Kotel'nikov, V. A., "Teoriya Potentsiai'noy Pomekhoustoychivosti"
_ (Theory of Potential Noise Suppreeeion], MoscoW, Goaenergoizdat,
1956, 152 pp.
61. Kronrod, M. A., Merzlyakov, I. 3., and Yaroslavekiy, L. P., "Experi-
menCs on Digital Holography" AVTOI~TRIYA 1972, No 6, pp 30-40.
62. Kuriksha~ A. A., "Optimal Use of Space-Time Signals" RADIOTEtC~1IKA
I LLEKTRONIK~ 1963, No 4, pp 352-563.
- 63. Kuznetsov, V. P., aad Levin, B. R., "Optimization of the Statistical
Adaptation Procedure" RADIOTEi~tIKN I ELIICTROI~tIItA 1971, No 1, Vol
XVI, pp 184-186.
217
~FOR OFFI~IAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL U3E ONLY ~ .
~
64. Levin, R., "Taor~eich~ek~.ye O~novy Steti~t~che~koy Radio-
Tekhniki" (Th~oret~cgl Founda~iott~ of 3tiAeiaticel R~dio 8ngineer-
ieg~, ttoscoa, Sovetekoye Rndio~ 1966~ i968~ Vol 1 and 2, 728 pp,
503 pp~ -
65. Lezin, Yu. 3~, "Optimal'nyye Fi'd'try i Naknpiteii impui'enykh
S~gnAlov" COptimal ~iltere and Accumul~~c?r~ ~f Pu~~~d Sign~l~~,
Moacow, 3ovetskoye Radio, 1969~ 447 pp.
66. Lipteer, R. Sh., and 3hiryayev~ A. M.~ "3tatiAeika Slucheynykh
Proreeesov (Neiineyneya Fii'tratsiya i Smazhuyye Voproey)"(Statie-
tice of Random Procesaes (Non-Linear Fiitration and Related QU~s-
tione)l,Moscow, Nauka, 1974~ 696 pp.
67. Lourene. R., and Stroben, Dzh.~ "Effacte Significant for Optical
Communicatione That Occur During Propagation of Light (Survey)"
TIIER,1970, Vo1 58, No 10, pp 130-153.
68. Middlton, D.~ "Vvedeniye v 5tatieCicheekuyu Teoriyu Svyazi" ~intro-
duction tu ehe Statist3cal Z'heory~ 6f Com~aunicatione~, Moecoa~
Sovetskoye Radio, 1961, 1962~ Vol 1 and 2~ 782 pp, 831 pp.
69. Midditon, D., "Multidimen~iaual Daeection and Identification of
Signals in Random Media" TIIER 1970, Vol 58, No 5, pp 100-111.
70. liinkovich, B. M., and Yakovlev, V. P., "Teoriya Sinteza Mtenn"
(Thaory of Aateana Synthesis~, MQacoa, Sovetekoye Radio, 1969,
294 pp.
71. Nezhevenko, Ye. S., Potaturkiu, 0. I., ,~d Tverdokhleb~ P. Ye.,
"Lin~ar Optical Systems for Performance of General Integral Trans-
forma" AVTOMETRIYA 1972, No 6, pp 88-90.
72. Obukhov, A. M., "Statietical Description of Continuous Fields"
TRUDY GEOFIZICHESKOGO INSTITUTA AN SSSR 1954, No 24 (15), No 3,
pp 3-42.
73. Okunev, Yu. B., and Yakovlev, L. A., "Shirokopolosnyye Sistemy Svyazi
- s Sostavaymi Signalami" [Broad-Band Communi:aCions SysCems with Com-
posite Signals], Moecoa, Svyaz', 1968, 167~pp.
74. Papulis, Yu. B., "Teoriya~�Sistem i Preobrazovaniy v Optike" (Theorv
of Syatems aad Conversiona in OP~ica], Moacow, Mir, 1971, 495 pp.
75. Petrov, A. P., "Eatimates of Linear Functionals in Solving Certain
Inverse Problems" ZHURNAI. VYCHISLITSL'NAYA MATEMATIKA I
MATEMATICHESItAYA FIZIKA 1967, Vol 7, No 3, pp 648-654.
76. Petrovich, N. T., and Razmakhnia, ~i. K., "Siatemy Svyazi s Shumo-
podobnymi Signalami" [Commimications Systems with Noiee-Like Signals~,
Mcacov, Sovetelcoye Radio, 1969, 232 pp.
216
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICiAt. U3E ONt,Y
~
~7~ Polyakov, 0. A~, "Id~nrif3eae~oa of ~ Lin@~r Dyn~mie Ob~ec~ by th~
Loca~. A~proximeC~on T~chniqua" AVTOMAT~KA I TBLRi~KNAN~KA ~971,
No 10, pp 154-~,64.
78. Prett~ V. K., "Lazerayye 8ietemy Svygzi" (Laeer Communicotiona 3ya-
tems~, Moscow, Svyaz', 1972, 232 pp.
79~ Progkie, Mili~r~ "Adaptive Receivera for Digitai Communicatione in
Channal~ w~,th Zn~ercharacter Interferenca" 2ARV882HNAYA RADIO-
SLBKTRONIKA 1970, No 2, pp 3-24.
80. Prosin, A~ V., "Toward e Th~ory of Radio Communicetion Channele with
Statiaticaily Rough Surfaeea" TRUDY CHETVSRTOGO KOLIAKVIUMA PO UKV
SVYAZI Budapest, Pt 1, 19709 pp 29/1-28/9.
81. Pugachav, V. 3~, "Teoriya 8iuchaynykh Funkt~iy i Yeye Primeneniya
k Zadacham Avtomaticheekogo Upravieniya" (The Thaory of Random Func-
tione and Ite Appiication to Che Probleme of Aueomatic Control~,
Moscow, PM,~1960, 883 pp.
82. Ramm, A. G., "Disringuiehing Rand~~m Fielde in a Noiae Background"
PROBLEMY PEREDACHI tNFORMATSII 1973, Vol 9~ No 3~ pp 22-35.
83. ~smm, A. G., "One Class of Integral Equations" UII~It~N`rSIAL'NYY~
URAVNENIYA 1973, Vol 9, No S, pg 931-941.
84. Rytov, S. M., "Vvedeniye v Statiatiche~k~yy Radiofi~iku" ~IntYOduction
to Statietical Radiophysica], Moaco~, Nauka, 1966, 404 pp.
~ 85. Savelova, T. I., and Tikhomirov, V. V., "Solving Integral Equations
of the Piret Type of Convolution in the i~tultidimenaional Case"
ZbURNAL VY(~iISLITEL'NOY MATLMATIKI I MATSt~'ATICHESKOY FIZIICI 19~3~
Vol 13, No 3, pp 555-563.
86. Savelova, T. I., "Solving Convolution-Type Equations aith an Impre-
ciaely Aesigned Nucle+~s by e~-~.. .',~G~:~r.~~~.iun Method" Z~iURNAL
VYCHISLITEL'NOY MATEMATIKI I MATEMATICHESKOY PIZIKI 1972, Vol 12~
No l, pp ~12-218.
87. Siforov, V. I., "The Carrying Capacity of Commuaications Chaaneis
aith Random Changea in Abeorption" RADIOTEEC~tIKA 1958, Vol 13,
No 5, pp 7-18.
88. Smolyaninov, V. M., et al, "Printeipy Otozhdeetvleniya Kanalov
Peredachi Sigaalov" [Principles of Equating Channels for Transmiseion
of Signals], Moecoar, Nauka, 1973, 115 pp.
89. Soyfer, V. A., "!lodeling a Generalized Gauasian Channel for Malysis
and Syathesie of Information Tranamiasion Syatems" (Author's Abatract
of Dieaertation for the Learned Degree of Candidate of Technical Sci-
ences), Leningrad, LEIS, 1971, 25 pp.
219
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
i+~R ONFIC~AL U9E Ot~fLY
~1f~~
c
i'
~
90~ 9cyfar, V. A., "MeaAUr~m~nt of tihe Space-Time Char~cteriet~c~ of
Linear Channaia with Di~paraion" RADiOTffi~tIKA ~9~3, Vo1 28, No 10~
pp 12-17 ~
9~~ 8oyfer, V. A., "The Optim~l Sinary Syetem.of 8lgnai~ in a Chaneei
wi~h Smoorh Padeouee" RAflIOTBt~H~tiKA 1972~ Voi 27~ No 4~ pp 97-98.
92. 8oroko, M., "O~novy Goiografii i Kogerantnoy Optiki" (Foundatione
of Hoiography anfl Coherent Opt~c~~, Moecow~ Nauka, i97i, 6i6 pp.
93. Soroko, L. M.~ and Stri$h~ A. P., "Spectral Conversiona by Computer"
OIY~I~ Dubna, i972~ 136 pp.
94. 3tratonovich, R. L., "Izbrannyye Voproey Fiuktuatsiy v Radiofizike"
~SelectQd IeOUea of F1ueCuations in Radio Phyeicsj, Moscow,
Sovetekoye Radio, 1961, 558 pp.
95. Straeonovich, R. L., "DeCaction and Seti~oation of Sigeale in Noise
When One or Soeh era~Non-Caues~an" TiIBR 1970, Voi 58, No S,
PP 73-82.
96. Serouk, Dah., "Opeical Compueatione" AVTOt~TRIYA 1973, No S, pp 4-12.
97. Tatarskiy, V. I., "Raeprostraneal.ye Voin v Turbulentnoy Atmosfere"
(Wave Propagation ifl a Turbulaut Atmoephare~, Moecoa, Nauka, 1968.
98. Tikhonov~ A. N., "Solving Incorrectly Stated Probleme and the
Regularlaation Method" DAN SSSR 1963, Vol 151, No 3~ pp 501-504.
99. Tikhono~?~ A. N., "Regu~arization of Incorrectly SCated Problems"
DAN SSSR 1963~ Vol 153, No 1, pp 49-52. -
100. Tikhoaov, V. I., "Staticheakaya Radiotekhnika" (Statietical Radio
EngineeringJ, MoBCOW, Sovetakoye Radio, 1966, 678 pp.
101. Turchin, V. F., "Solving e Pirat-Order Fredholm Equation in a Statis-
tical Aaeemblage of 3mooth PuACtious" ZHURNAL VYCHISLITEL'NOY
MATEMATIKI I MATBMATICEi8SK0Y FIZIKI 1967, Vol 7, No 6, pp 1270-1284.
102. Fal'kovich, S. Ye., "Prfem Radir~lokateionnykh Signalov na Fone
Fluktuateionnykh Pomekh" (Receptioa of Radar Signale in a Setting of
Pluctuating Noise~, Moscoa, Sovetakoye Radio, 1961~ 311 pp.
~03. Pal'kovich, S. Ye., "Otaenka Parametrov Signxla" (Estimation of
Signal Parameters], MoscoW, Sove~akoye Radio, ]~970, 332 pp.
104. Fink~ L. M., "Teoriya Peredachi Diakretnykh Soobshcheniy" [Theory of
Transmiesion of Discrete Messages], MoscoW, Sovetskoye Radio, 1970,
728 pp.
220
'FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR OFFICIAL U9E ONLY
~ ~
105. Fiekel'~hCe~?n, Ya. Z. ~"Race~Cion of Diecre~e Sign~ls w~.Ch lt~p~.d
and Unaven Changee in Che Parameters of the Communicatione
Channel" (Author'~ Abetract of Die~ertation for ehe Learned Degree
of CendidaCe of Technical 3ciences), Leningrad, LEIS~ 1967, 22 pp.
106. Franeon, M., "Golografiy~" [Hoingraphy~~ Maecow, Mir, 1972, 246 pp.
107. Fu, K., "Poeladovatiel'nyye Metody v 1taBpozn~vani3 Obrazov Z
Obuchenii Mashin" [3equential Methoda in Pattern ltecognition and
IneerucCing Machines], Moscow, Nauka, 1971, 225 pp.
108. Khab3bi, A., "Two-Dimeneional Bayee ~stimatiion of Images" TIIER
1972, Vol 60, No 7, pp 153-159.
109. Khvoroetenko, N. P., "Statiaticheakaya Teoriya Demodulyataii
DiekreCnykh Signalov" ~StaCistical Theory of Demodulation of Die-
crete S~.gnals), Moacow, Svyaz', 1968, 334 pp.
110. Kheletirom, K., Liu, I., and Gordon, Dzh., "Quantum Mechanical
Theory of C~mmunicaCions" TIIER 1970, Vol 58, No 10, pp 186-207.
111. Khoversten, Ye., Khardzher, R., and Khalme, S., "Theory of Com-
municationa in a Turbulent Atmoaphere" TIIER 1970, Vol 58, No 10,
pp 236-263.
112. Khomyakov, E. N., "On Deaigning Space-Time Syst~ns" RADIOTEICHNIKA
1969, Vol XXIV, No 9; pp 88-94.
113. Khurgin, Ya. I., and Yakovlev, t~. P., "~'initnyye Funktsii v Fizike
i Tekhnike" [Finite Functions in Physics and Engineering~, Moscow,
Nauka, 1971, 408 pp.
114. Tsikin, I. A., "One Method of Computing Integral Functiona of
' Probability Distributions" RADIOTEI~iIKA I ELEKTRONIKA 1968, Vol
XIII, No 10, pp 1887-1889.
115. Tsikin, I. A., "Discrete-Analog Methods of Optimal Signal Procesa-
ing" RADIOrEKI~NIKA 1969, Vol 24, No 2, pp 1-8.
116. Chernov, L. A., "Rasprostraneniye Voln W Srede eo Sluchaynymi
Neodnorodnostyami" [Propagation of Waves in a Medium with Random
_ Heterogeneities], Izd-vo AN SSSR, 1958, 159 pp.
117. Shakhgil'dyan, V. V., and Lokhvitskiy, M. S., "Metody Adaptivnogo
priema signalov" [Techniques of Adaptive Signal Reception], Moscow,
Svyaz', 1974, 158 pp.
118. Sheanoa, K., "Raboty po Teorii Informatsii k Kibernetike" [Works
on Information Theory arid Cybernetics], Moscow, Inostrannaya
Literatura, 1963, 829 pp.
aai -
FOR OFFICIAL "JSE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
FOR QFFICIAL U3E ONLY t
;
, -
~
119~ Shereme~'yev~ A. 0.~ "Seatistiicheskaya Teor~ya Lazernoy 5vyazi" '
[Seatiistical Theory of Laser Communicatione~, Moecow, Svyaz', 1 ,
1971, 264 pp. ` ~
~
120. Shestov, N. S., "Vydeleniye Opticheskikh Signalov na Fone Sluchaynykh
Pomekh" [IdenCifi.caCion of Optical Signals in a Setting of Random
Noise~, Moscow, Sovetekoye Radio, 1967, 347 pp.
121~ Shirman~ Ya. D., Szhatiye i Ra~reshayushchaya Spogobnoat' ,
Radiolokateionnykh Signalov" [Compreesion and tihe ResoluCion of ~
Radar Signals~, Moacow, Sovetekoye Radio, 1974, 360 pp. )
~ i
122. Yaglom, A. M., "Some Classas of Random Fielde in n-Dimenaional ; ;
Space Related to SCa~ionary Random Procesaes" TEORIYA VEROYATNOSTEY
I YEYE PRII~NENIY~: 1957, Vol 2, No 3, pp 292-338.
~
123. "Acoustical Holography," Plenum Presg, New York-London, 1969, Vol 1, , !
292 pp, 1970, Vol 2, 376 pp.
_ 124. Beckmann, P, and Sprizzichine, A., "The ScaCtering of ElecCromag-
netic Waves from Rough Surface," London, Pergamon, 1963, 503 pp.
125. Beckmann, P., "The Statiatical Distribution of the Amplitude and
Phase of a Multiply Scattered Field," CHECOSLOVENSKE ACADEMIE, VED, ,
v~, i96i, 41 pp.
126. Beckmann, P., "Statiatical Dietribution o� the AmpliCude and Phase
of a Multiply Scattered Field," USA (RADIO PFOPAGATION), 1962, ~
No 3, pp 231-240.
127. Bello, P. A., "Time Frequency Duality" IEEE TRANS., 1964, January,
Vol IT-10, No 1, pp 18-33.
128. Bello, P. A., "Characterization Randomly Time-Variant Linear.~
Channel" IEEE TRANS., 1963, December,.Vol CX-11, No 4, pp 360-393.
129. Golomb, M., "Approximation by Function of Fewer Variables," in the
book "On Numerical Approximation," edited by R. E. Madison, 1959,
The University of Wisconsin Press, pp 275-327.
130. Goodman, P., and Reswick, I. R., "Determination of System Charac-
teristica from Normal Operating Record" TRANS ASME 1956, February,
PP 259-271.
131. Kailath, T., "Measu~emeat of Time-Variant Communication Channels"
IRT TRANS. ON INF. THEORY 1969, May, Vol IT-15.
132. Linsey, W. C., "Error Probabilities for Rician Fading Multi-
channel Reception of Binary and N-ary Signals" IEEE TRANS. 1964,
October, Vol IT-10, No 4.
222
~FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3

APPROVED FOR RELEASE: 2007/02149: CIA-RDP82-44850R000100064410-3
,
~
F'OR QFFZCIAL U3~ ONLY ~
,
133. Lucky, R., "A Survey of Che Communicat~.on Theory LiCeraCure,
1968-1973" IEE~ TRANS, J. T., 1973, November, Vo~ 19~ No 6~ pp
i~i-~.~y.
134. Lohman, A. W., and Parie, D. H., "Binary ~raunhofer Hologramg by
Computer" APP. OPTIC. ~.967, No 5, pp 1.739-1751.
135. Middleton, D. A., "A Stiatietical Theory of Reverberation and
Similar ~iret-Or~er Scattiered Fielde" IEEE TRANS, 1968, July,
' Vo1 IT-13, pp 372-~414.
136. Musa, I. D., "DiscreCe Smoothing Filtiera for Correlaeed Noise,"
THE BELL S~tSTEM TECHNICAL JOURNAL 1963, September, Vol }~II, No 5,
pp 2121-zisi.
131. Nakagami, M., "On the Inteneity Distribution and ICs ApplicaCion
to Signal SCatics" RADIO SCIENCE JOU1tNAL OF RESEARCH NES (USNC-
URSI 1964, September, Vol 68D, pp 995-1003.
138. Price, P., and Green, P~ A., "Communication Technique for Multi-
paCh Channels" PIRE 1958, No 9, pp 555-5~3.
139. Root, W. L., "On the MeaeuremeaC and Use of Time-Variant Communica-
tion Channels" I&FORM. AND CONTROL 1965, Auguat, Vol 8, No 4,
pp 390-422.
140'.' S'~.epian, D., "Some Commenta on the Detection of Gaussian Signals
in Gaueaian Noise" IRE TRANS. 1958~ Vol IT-4, No 2, pp 65-68.
141. Turin, G. L., "On the Estimation in the Pxesence of Noise of the
Responae of a Random Linear Filter" IRE TRANS. 1957, March, Vol
IT-3, pp 5-10.
~ 142. Turin, G. L., "Error Probabilities for Binary Symmetrical Ideal
Reception Through Nonselective Slow Fading and Noise" PIRE 1958,
September, Vol 46, No 9, pp 1603-1319.
143. Tzafestas, S. G., and Nightigale, I. Ii., "Optimu~m Filtering, Smooth-
ing and Prediction in Liriear Distributed Parameter Systems" PR. IEE
~(London), 1968, Auguat, Vol 115, pp 1200-1212.
,
COPYRIGHT: Izdatel'atvo "Svyaz'," 1976
ry~
11~176 ,
CSO: 8144/0766 - END -
223 .
. FOR OFFICIAL USE ONLY` �
APPROVED FOR RELEASE: 2007/02/09: CIA-RDP82-00850R000100060010-3