JPRS ID: 8597 USSR REPORT GEOPHYSICS, ASTRONOMY AND SPACE--PROCEEDINGS OF THE 6TH ALL UNION SEMINAR ON STATISTICAL HYDROACAUSTICS

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ANO ' ~--PR1 f I NGS
OF THE 6TH ALL-UNION SEMINAR ON STATISTICAL
31 JULY 1979 HYDROACOUSTICS (FOUO 2l79) i OF 3
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JPRS L/8597
31 July 1979
F'Ult UFNIC'IAL liSM: ONI.1'
USSR Report
GEOPHYSICS, ASTRONOMY AND SPACE
(FOUO 2/79)
Proceedings of the 6th All-Union Seminar _
on Statistical Hydroacoustics
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J!?I25 L/8597
31 July 1979
USSR REPORT
GEOPHYSICS, ASTRONOMY AND SPACE
(FOUO 2/79)
PROCEED I NGS OF THE 6TH ALL-LtJ I OPl SEM I NAR
ON STATISTICAL HYDROACOUSTICS
Novosibirsk TRUDY SHESTOY VSE50XUZN0I 5HKOLY-SEMINARA PO GIDRO-
AKU5TIKE in Russian 1975 pp 3-24, 33-44, 54-73, 91-96, 109-136,
140-153, 220-269, 277-294, 298-365, 372-378
CONTENTS
PAGE
Modeling in Statistical Hydroacoustics
1
(V. V. O1'shevskiy)

Some Mathematical Aspects of Modeling in StatisCical
Hydroacoustics
Using an Electronic Computer
23
(A. A. Kaptyug, V. V. 01'shevskiy)

Digital Modeling of the Response Function for Comple
x Signals Using a
Fast Fourier Transform Algorithm
'
36
vov)
(I. B. Vaysman, K. P. L

Digital Modeling of Sea Reverberation
41
(V. V. 01'shevskiy, V. A. Panfilov)

Digital Modeling of a Sample Set oi a Nonstationary
Random Process
(Ye. V. Kirillov, et al.)

49
Use of Multivariate Statistical Analysis Methods in
Hydroacoustic
Diagnosis
56
(V. M. Levin, et al.)

FFT Equivalent of the Wiener-Khinchin The�-,rem far a
Nonhomogeneous
Nonstationary Random tJave Field
62
(V. A. Geranin)

- a
[III - USSR - 21J
S&T FOUO]
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COIi OFFt C[ AL lf tSli UNI,I'
CONTENTS (Continued) pnge
Influence of. Discretir.irion and Quantization on Che CharacCerisCica
of a Uigital Quadrature-Correlation betector -
(K. B. Krukovskiy-Sinevich, V. V. Mikhaylovskiy) 66
' 5ynthesis of HydroacousCic Signals in Che Region of 5trong Correlation -
of the Velocity-Lag Uncertainty Funcrion
(K. B. Krukovskiy-Sinevich) 71
Uncertainty FuncCions of 5ome Types of Complex Signals
(A. A. Belousov, eC al.) 90
Evaluation of Che Pulse Characteristic Curve of a Hydroacoustic
Channel
(V. I. Paderno, I. R. Romanovskaya) 99
Spectrum of Sea Reverberation as a Nonstationary Etandom Process
(V. A. Geranin, et al.) 106
Optimum DeCection of Multiray Signals _
111
(N. G. Gatkin, et al.)
Spectral-Correlation Analysis of an Antenna Situated in a Nonhomc.:n-
eous Nonstationary Hydroacoustic Field
(E. A. Artemenko, et al.) 120
Detection of Signals and SpaCial Localization of Their Sources on Che
Basis of Spectral Analysis -
(V. I. Chaykovskiy) 128
One Method for Determining the Coordinates of a Local Noise Field
Source
(A. M. Derzhavin, et al.) 137
Investigation of the Noise Immunity of a Standard Lletection Cttannel
in the Recept3on of a Two-Component Signal With a Narrow-Band Noise
Component
(V. M. P'yanov) 141
Detection of Signals With Unknown Parameters on a Reverberation `
Background _
(G. S. Nakhma~son, V. V. Pavlov) 149
Stabilization of a False Alarm in Hydroacoustic Detection Channels
(S. N. Gerasimenko, et al.) 156
Suboptimum Detection of Hydroacoustic Echo Signals on an Electronic
Computer
(B. P. Brezhnev, et al.) 167
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FOlt OFFT.CI:AL lt;;l; c)NI,Y
CON7'EN'l'S (Cantinued)
page
Itcrvlrw uf Aclapratton Methods in Stutisricul Nydroacouyric I'rnUlema
(G. N. klelonoztiko, V. V. O1'shevskiy)
174
.
AlgoriChms for Processing Sonar Information Under a Priori Uncertninty
- Conditinns
(Yu. Ye. 5idorov)
195
Nonparametric MeChocts Por Che Processing of Hydroacoustic Information
(N. G. Gatkin, eC al.)
203
Possibilities of ApplicaCion of Methods of Nonparamerric SrueisCicg
in Sonar
(F. P. Tarasenko)
220
_ Algorithms for Signal Recognition With Incomplete a Priori Information
-
(V. P. Vagin, V. D. Petukhov)
222
Optimum Detection and Differentiation of Signals Under Conditions of
Restricted a Priori Information
(L. G. KrasnYY)
225
Sequential Procedure of IdenCification in Hydroacoustic Research
(G. S. Lbov, V. I. Kotyukov)
235
5uppression of Side Lobes of an Antenna DirecCional Diagram by a
- Ntethod Based on Temporal Change of Aperture Size
(D. K. Solov'yev, P. Ya. Krasinskiy)
241
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a
UDC 534.232.2
MODGLiNG iN STATISTICAL HYDROACOUSTICS
Novosibirsk TRUbY SIIESmUY VSE50YUZNOY SNKOLY-SEMINARA PO STATISZ'ICHESKUY
GIDROAKU5TIKF in Russian 1975 pp 3-24
[Article by V. V. 01'shevskiy]
_ [Text] 1. Preliminary Comments. At the present time in different fields
o� science and eechnology extensive use is already being made of physical
and electronic modeling of di�ferent kinds of processes, fields and sys-
tems (�or example, see the books [1-4j, as well as the bibliographies to
these works). The basic tendency in this direction is the increasingly
broader use of digital computers in solving modeling problems. 'I'his is
relaeed to the considerable progress both in the creaCion of compuCers
and in Cheir mathematical support.
Relatively few studies have been published in the field of hydroacoustics
directly ccncerned with modeling problems. However, in the transactions
oC the five All-Union Seminar Schools on Statistical Hydroacoustics (1969-
1973) a scientific basis was essentinlly laid for a serious and intensive
development of this direcCion [5-59], and all the moreso in adjacent
f.ields, in particular, in statistical radioengineering, {n the creation
of inethods for the modeling of processes and systems, in which consider-
able progress has been attained.
Before discussing the peculiarities of digital modeling in statistical tly-
droacoustics, the subject to which this study is for the most part devoted,
- we wi11 examine possible methods for modeling in this field (see Fig. 1),
_ and specif.ically, hydrophysical and electronic modeling.
0
Fiydrophysical modeling. This is a method for reproducing real research ob-
Jects and the hydroacoustic conditions for their observation at a consid-
erably lesser scale wiCh adherence to the principle of similarity of phys-
ical phenomena.
Thus, in hycirophysical modeZing use is made of the similarity of physical
phenomena transpiring under natural conditions and in the model. An import-
ant positive property of physical mode:ing is the possibility of obtaining
a greater volume of experimental dara wtth monitoring of the conditions for '
1
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carrying out experl,ments. In hydrophysical modeling there are two pos-
sibLliCies of cnrrying out investigati,ons, to wiC:
mudeling under arti.ficially cxeaeed and monitorable conditiony (batli,
hasin);
mndeting under monitorAble candiC3ons (fluviul waCer body, lake, coaytal
marine regiott).
1'u;tpo hN:uttit, c:lc uo
uui;wifipuuntil"I~~1
I~or,WI Hpvnaunu fao)tenupon auiio
~1 11110 i t::yC'l'L'- I'NttP0'41tyCTN-
~cccx~~x oG"c1c� vuCttHx o6�uic-
:o: u ~ici;yc- 1roe n i(oumpo)r
crbuiuiu CUJ{t iititl)yoMux yc-
U:IULI!1:C ~1 Y,U1f_ JIOllNfIX
Tf�''';;~l;yxuu.e 3 4
;inuxTpu111106 VNtun~IP0101110
2
dlItINCi10'NNO X!1-
10110nHlioliauno
IM'COpNCTNK
UN60P04116qC QII-
I'NilpOUIty0TN48C
CAMOJ1011 rNApo�-
xtx npuucccos
'utycmn40c0Ix
t naleN
npul{occon H no
5
noA
1,1ouenNpovaimo
7
clcTOM
06p3Q0T
'
:sl rtiIq110a1cyaTH
4CCKOl1
~IHpU(1-
�
ua0tH
Fig. 1, Different modeling methods in statistical hydroacoustics
KEY:
l.. Hydrophysical modeling
2. Electronic modeling
3. Modeling of hydroacoustic ob3ects under artificially created
and monitorable conditions
4. Modeling of hydroacoustic ob3ects under monitorable conditions
5. Compiitation of characteristics of hydroacousric processes and
f ields
6. Modeling of sample sets of hydroacousCic processes and fields
7. Modeling of systems for the processing of hydroacoustic data
The principal difficulty in interpreting the results of hydrophysical mnd-
eling of hydroacoustic objects is related to the influence of numerous
scale effects which can substantially change the physical picture of the
distribution, absorption and scattering of acoustic waves in comparison wiCh
natural conditions.
_ Electronic modeling. This is a method for the reproduction of real research
objects, the hydroacoustic conditions for their observation and systems for
the processing af hydroacoustic information using electronic devices (ana-
log, digital and combined). Thus, in electronic modeling use is made of
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~
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the mathemaCical simtlariL�y of descripCion of phenomena tranepiring under
naCural conditions and duplicated in models. In elecCronic model.ing L-here
are three research directions, Co wiC:
compuCntion of the characteristics of hydroucoustic processes and fields
(nunterical meChods for solution of problems),;
modeling of sample seCs of records of hydroacoustic processea and fields;
modelictg of systems for ttie processing of hydroacousCic informaCion.
rltt imporrant advanrage of electron:Lc modeling is the possibility of repeat-
ed reproduction of the invesCigated objects in a broad range of Cheir ob-
servation conditions and also Che possLbiliCy of checking differenC 'typo-
theses. 'Che principal difficulty in interpreCing the resulCs of electronic
mode.ling is relaeed to the possibility of an incompiete allowance for the
ptiysical properties of real research objects and the possible ambj.guity of
ttie choice of their mathematical models.
Now we w.tll examine some peculiarities of digital modeling of hydroacoustic
processes and systems for the processing o� hydroacoustic informat:Lon,leav-
ing outside our attention etie following unconditionally imporranC problems:
the relationship of hydrophysical and electronic modeling;
t}ie role o� analog, digital and combined modeling methods; �
the relationship of numerical and analytical methods for constructing
models of hydroacoustic processes and fields.

These problems are o� considerable theoretical and practical interest and ~
merit individual discussion.
11ie Problem of Choir;e of a Mathematical Model.
Ttie choice of a stochastic model of a hydroacoustic process is the central
problem in the digital modeling of hydroacoustic processes and fields. The
success of the modeling as a whole is dependent on the adequacy of the adopt-
ed model to the real investigated ob3ects and on how constructive it is,
that is, how simple and productive.
I'irst we will give a review of studies [5-59] in relation to the formulation
_ of such models of }iydroacoustic processes and fields.
We will begin with some definitions of basic concepts.
In accordance with the established concepts [5, 29, 30, 46, 50], by a sto- ~
cfiastic model of a hydroacoustic process is meant its mathematical repre-
sentation, which makes it possible to compute (or postulate) the stochastic
characteristics of the process, important in the problem to be solved. A
stochastic model is an idealized real research object and.is formulated
on the basis of the following representation:
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X N '/T7i l/, (f), 1 1 a R
wiiere X(r) [s the inveatigaCed process, mA is the formaClon operaCor, L(L)
are elementary random processes whose stochnstic characteristics are stip-
ulnted and can be physically inCerpreCed. _
ASSUnie that d(~/m) is the sCochastic characteristic of Che process X(t)
under the condiLion of adoption of ita modcl m; then the representation (1)
musC make it possible Co derive Che equation: -
(1)
where 11m is the formation operator for the stochaeCic characteristic
00e/m) from the characterisCics-)W n'i(~) describing the properCies of
- the elementary processes gi(t), e and are the corresponding arguments
(time, frequency, level, eCc.).
The set of models
merl, (3)
whicli the researcher has forms a thesaurus of models M and each of these
models is such that
ma f9p(1/4 �16'(e, �~zn), a1 fiP� (4)
whereP Op(L'/m) is a stochastic characteristic corresponding to the model
m; a are parameters which are determined by hydroacoustic conditions
(condYtions of propagation, aCtenuation, scattering, etc.); Ap is para-
- meter space ap; P(P, ap) is the 3oint distribut~on of probabilities of
~he number p of classes of characteristics BP( /m) and their parameters
a p ; N is the number of classes.
'Chus, we assume that each mo~el' m corresponds to a parameCric set of sto-
chastic characteristics e( , ap/m), p= 1,N.
Among ttie studies which we are reviewing, ire can note the following two
groups.
The fl.rst group includes studies [11-15, 19, 20, 24, 28, 37, 42, 52-54] in "
which the authors develop wave models of hydroacousCic processes and -
fields; the level of stochastic description is limited by the determina-
tion of the correlation and spectral characteristics (except for studies
114-481, in which an attempt is made to seek the characteristic function-
als of the hydroacoustic fields). F
The second group includes studies [5-10, 16-18, 21, 22, 23, 25-27, 29-36,
38-41, 43-47, 49, 50, 55-59], in which the authors develop phenomenolog-
ical models of hydroacoustic processes and fields; in many cases the level
4
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oL l�.he stochastic description here is not limited to the correlation-
spectral characteristics, but attains the level of determination of the
dlstr[butions Eur probabilities, moment functions and semi-invariants
~
of higher orders than the correlatinn characteristics.
Our comparison of the methods for constructing wave and phetiomc.nological -
stoctlastic models shows that wave models, beinq physically adequately
sound, make it possible to attain only the correlatian-spectral descrip-
tion oE t}le processes and fields, whereas the phenomenological mndels,
although they are more tormal than physical, In principle make it pos-
sible to obtain the distribution of probabilities, but in not a11 cases
are they adequately well physically interpretable.
_ The thesaurus of models the set M-- can be divided into two sets Ml
and M2, such that
~
and
M= (5)

irr, E !�1, E (6)
M1 is a set oE so-called ideal models for which a full stochastic descrip- _
tion was given, Mz ;~,s a set of so-called working models for which a par-
tial stochastic desc'ription was given. Naturally, the researcher never
knows the ideal models ml; they correspond to an exhaustive determina-
tion of the properties of the investigated object. At the same time, the
working models are incomplete, only partially describing the objects, but
a they must be sufficiently adequate to the real investigated objects, and
what is especially important, adequately simple with respect to solution
of the modeling problem.
We introduce, much the same as was done in [29, 60] , the two character-
istics:
measure of adequacy of the working model relative to the ml model:
(/77 m,) '',P~9(1~1177J, (7)
cost function
9( emr ( 8)
- which characterizes the expenditures associated with modeling'of the ran-
dom process in accordance with the model m2. In typical cases of examina-
tion of models of hydroacoustic processes there is the following tendency:
with an increase in Pg (ml, m2) the C(m2) value usually decreases.
� As already noted, the researcher never knows the ideal (true) model. Ac-
cordingly, it is natural to replace the model ml by another the model
m2 6 M2CM21 which would satisfy the condition
(9)
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,
A k, mr") G ap ;
(9)
,
ef)* is a sufficiently sma11 value. According to the terminology in [ 601
the model which replaces the true model ml with aia accuracy adequate for
practical requirements is called Lhe meta model. The meta model, to be
sure, must be modelable and easily interpreted.
In the light of the introduced functions (7) and (8) and the meta model
(9) the following two types of problems arise in the choice of the best
(optimum) wo rking models opt m2 for modeling purposes.
The first type of problems is formulated in the following way:
restrictions on the cost of modeling are introduced
(10)
(me J < ~ C , .
here �
m,`G
the optimum mode1 opt rn2 is selected proceeding from the condition:
qPf mZ = 0?9 infPa ~mr m< ~ (12)
1171 E Me c
Thus, the optimum model opt m2, in accordance with (10)-(12), is a model
which corresponds to the maximum adzquacy of the meta model m2 and takes
into account limitations on cost (complexity) of modeling.
The second type of problems is formulated in the following way: -
limitations are introduced an the d*gree of adequacy of the working _
models m2 relative to the meta madel m2
"P e (mz, m; J< Qp, . (ls)
here
mf s MpC MZ ;
_ (14)
The optimum model opt m2 is selected proceeding from the condition
Gof in, = aa.9 inf C,(1771). (15)
= mz e / ~t;~
Thus, the optimum model opt m2, in accordance with (13)-(15), is such a
model which corresponds to the minimum cost (complexity) of modeling and
takes into account limitations on the degree of its adequacy to the meta
model.
Figure 2 gives a geometrical interpretation of the choice of the optimum
models corresponding to the two considered types of problems.
6
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Fon orFicrni. Ls: oNLY
Mr

z
1112?
)
MI=~Mz\~h(Z UNiz )
M2 = Nl~ MZ U h11)
_ Fig. 2. Geometr.ical interpretation of choice of optimum model in modeling -
- of sample records of hydroacoustic processes.
_ a) case of limitation on cost of modeling; -
b) case of limitation on degree of adequacy of model. 3. Review of Specific Types of Models of Hydroacoustic Processes
_ Now we will examine some phenomenological stochastic models typical for
- the representation of hydroacoustic processes [5, 29] . Assume that X(t)
_ is the investigated random process 1~ i(t), i= N are elementary pro-
- cesses (determined, quasidetermined or random) with known characteristics.
- Parametric models. These can describe echo signals in sonar and siqnals
in hydroacoustic communication. These models have the form:
X({J= (16)
_
where ~ is the totality of random parameters with stipulated distribu-
tions of probabilities. -
Constructive models. These can describe the totality of signals and noise,
signals with additive and multiplicative interactions, and for these models
the following representation is correct:
_ XW . iti (E;(o t j (E)) , (17)
where & is the symbol of interaction of elementary processes. -
Discrete canonical models. These can describe signals with multiray prop-
agation, sea reverberation and some types of underwater noise. These models
- have the form:
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Fox oFFICIAL U5E nNLY
~
x(r)
K~U
(18)
where ak anc! N.-+re random valueg.
integral canotiical models. mhese can degcribe signals propagating in a
water mediwn, scat*_ered signals and echo signale. mhese modeis have tihe
form:
xrf) . iare' t')t�9)
whcre a(t, t') is same random, or determined, or mixed function.
bifferential mndels. These can describe siqnals propagating in a water
me6ium or echa siqnals. mhese models can be of the two following types:
c ~
~'l~J r ~ O,T ~ ~t~f , (20)
(21)
where lie and Ga are random values.
.
( 1 ~i1J,tONHhr. ti~~~(.9[i/ib/~
~ r:~yCCObl1 ~T:.~CiI3E)::~E ~ 6~~HUU~~ht
~.~~r~2~~i~e o.4occa � 6
� No~nntii~Hr~uo c;;y~t~HUt:o
, 2 uo::1yuu1j c
r V.
~ 8
I':zpaoHHVOCxeo npoLkoccu
3
v
a
r
w
'+~11l1~:1LC11 C ~33::N4HOV.
q~r;,G,, u:;0;1 M TU::y4(!N ~o Suoop ajjejcu0HTapnux npo-
YuJUiI yuCC0B
C t7
fOr
~o
A j 9
A;t&^~IQANVOCHNO ~a~
I 11~101~1.1:J i15 . ~
t,R
y' F~ ~
.4 7
1 _
Fig. 3. Classification of elementary processes ~ i(t) for constcvetion of
models of investigated processes.
Key on followinq page
8
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FOk UFFiCIAI, U5L oNLY
Y.rY :
1. Standard processes
2. Gaugsian stationary random prdceggea
3. tiarmonic processes
4. rulses with different envelope and current phase
5. Aperiodic processes
6. Additive-multiplicative interaction of standard processes
7. 5tandard r.andom values
U. Independent randnm values with different one-dimensinnal distribu-
tions
5. Choice of elemcneary processes
I H)onp 7110u011T~ap11ux npot01cc0n 1 I
~
Ytnor,vHHO 2
c,

14 o
.
.41 CU
1f1J
f Ct0
..r~...~
' ~ f
ft u:
Cyu+iipunauao 3
x u�L
vL
�
~
r
�
tU 1:
ti;~
h
w
(.!.ICCIUIINO DD DPoYONH
=M F i4
4
1 U~
.
~
1' QI :r
.~I
lil lf~~
Gt
"
~
L O
'
.

"
"
~ .

a~vy.oimoro S
6

7
i ujc~~;acn
L__.
Fig. 4.
Classification of operators m~ of formation of
investigated pro-
cesses.
KEY'1.
Set of elementary processes
2.
Multiplication
3.
Summation
4.
Time shift
5.
Change in time scale
6.
Additive-multiplicative interact
ion of transformed elementary pro-
cesses
-
7.
Investigated process
We exami
ned very simple models of random
processes typical for hydroacous-
tics. To
be sure, the need can arise for
constructing combined models. In
this cas
e an analysis of the stochastic
characteristics
is more complex
than in
a study of very simple models of
the type (16)-
(21).
9
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FOIt nFFICIAL USE ONLY
YKrn
at~
Q ~.t,~... 4i~ n
A'0 U
, ~
. �as
D~rltn~c~tiiiw.at)a)` ~
1
~ _
v ~ ~ -
~ t.~ :,o co dp ~oo ll
x,
rh ~ n
G). c
Fig. 5. Sample records o� nonstationary random process.
KEY� a. Gaussian stationary process;
b. Modulatinq function;
c. Sample ensemble of nonstationary random process.
4. Digital Modeling of Hydroacoustic Random Processes
The next step after developing and adopting a stochastic model of a random
process is the choice of a method (algorithm) for its modelinq. As follows
from the preceding section, the stipulation of a stochastic model X(t) in
accordance with (1) means to stfpulate the stochastic characteristias of
elementary random processes - i(t), i= 1,N, and also the operator ,
m% of formation of X(t) from ~i(t). Very simple phenomenoloqiGal mod- ,
els were determined by expressions (16) and (21). Thus, the pr.oblf.m first
arises of the choice of types of elementary processes # i(t) whi.:h in
accordance with (2) are described by the stochastic characteristics n'i
(X). Figure 3 shows the classification of the considered elementary pro-
cesses, which are formed from standard processes and standard random values.
We note that the elementary processes must have the property of physical
and systemic interpretability [64] . The elementary processes are very
simple processes which have the mentioned properties. However, the stan-
dard processes have only a very simple stochastic structure, their prop-
erties must be completely known, and finally, their aodeling with any
stipulated accuracy must be constructive.
Proceeding from the representation of the models (16) and (21), the oper-
ators for formation of mt of the investiqated processes X(t) can be
classified in accordance with the diagram in Fig. 4.
10
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z2(n
4s
� ,zo 40ko ~I-o
~~~ioo fit
~ .

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i'Ok UCFIc:IAL U5P, nNLY
~
~ I.
, ' 1; , ~~;'`~~4;,;'L-.~,V,~.~;n "~,,b�~~ ti
~ ~ ~ ' ~ ~ ~ I~ ~ lii ~1�
~ I
~ ~_tl`f, u ~1�ki~j , ~
1~1'; i �
:illli
~1~~~ f(�,~ .
4�!~'S�
t t ~ t 1~ i .
'i I pr%1~!
~ ~11~~ ;y . ~`,u~~~�3a ;t
J~+ ` W.
~ ;~it~ 'J

~
�''�i ~~1/
Fig. 6. Sample records of sea reverberation.
The principal operators here correspond to the physical transformations
_ of hydroacoustic processes durinq radiation, propagation, scattering
and reflection of acoustic waves in the water medium.
There is basis for assuming that using the elementary processes and their
transformation operators, considered above, it is possfble to model the
following hydroacoustic processes:
direct aignals, propagating under conditfons of influence of refraction,
reflection and scattering from the boundaries of the water mediwn;
echo signals from different sounded objects;
sea reverberation;
acoustic noise of the carriers of hydroacoustic systems;
acoustic noise of the sea.
11
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T, , I uuruuvlcuuc
11U:,uii;1 II 1joau C-
C.1
1
..._...r.~
Lu I! u1iTuoUTuuC
x 11+;1MI?qMuM
I(!U.tUCC1
4
FOIt UFFICIAL U5E dNLY
AjaulnITU luAo LutfojwuuuN au-
L11()DUI1!I1'11 UWbO Caf,101lL JMAJIIM=
41U41lWX pC;L'INJu ~ uNN
4411 2 ' 3
CTTricTuVIncicnri
,,yut4HOUn~~u ouCluta nuciu(oti.
111:!)lN4H,f1 IlOPO 11P040CC9
~
5 ' 6
CuilIIQ11Nd llC pDNTHOCTNON YHp3XTOhItCTNKN CD
C.ITNCTA4CCtt01 U�tiNlSDjI
Fig. 7. Structural diagram of modelinq and statistical measurements.
~ .
'1. Stochastic models of process
2. Algorithm for modeling of sample records
3. Sample ensemble of records
4. Stochastic characteristics of process
5. Values of difference functional
6. Statistical evaluation of sample process
7. Comparison of stochastic characteristic with statistical evaluation
Now we will examine two examples of the modeling of random processes spe-
cific for hydroacoustics.
The first example relates to so-called perfodically modulated noise. A
noise model of such a type relates to the constructive type (17) and has
:
the followinq form (10, 611
m., sin w. 1) (211)
where ~ is a Gaussian stationary random process, mp is the coefficient
of intensity of amplitude modulation, wp is mo&latfon frequency. Such
a model is characteristfc, for example, for the noise observed in the
course of acoustic cavitation (10). In a discrete form, suitable for dig-
ital modelinq, model (211) is written in the following way:
� ,l'(n) -[1. M. s~n(nr.~, nf~I~(nJ, (22)
' 12
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1-40[Z t)FFLC [AI. USL ONLY
where /_It is the interval of discrete readings, n is the current number
of the reading of the pracess witih tiime. F'igure 5 JI1bW5 sample records
of a pz�ocess of type (zz), mndeled on an electironic computer nf the F3ESM-
6 type using the meehnd described in (61).
Ila,i,lotcvt"HOCTb , Kolioauutf oG' du
i:UtUm1.3Oll;1N1100 un 21400p04flux uau-
;;a,; ~ uax
1 , 2
11CTu4NOC2'L :,ar,,1-
u11ri nuI'uNTuusTuu
Y ipa,crcI,RcsRIc 3n
ii(J'1tT tpljux lipouOc
con
4
Loau11ce3Tftwp WI-
ru~~~Tk uu;,o,icipo-
nau~x
. '
KOHOVIIO0 4NCJI0
nv0opotiilux poa-
iIN214110 8
uPoncTaunauNO-
uNGop04HOro tipo
Ouaa
3
,uncKppTHU9
nD npeuuut
6
KDANrG8pUN9
tln yponpa
9
Fig. B. Classification of possible reasons for the appearance of errors
- in modeling and statistical measurements.
ICEY :
1. Inadequacy of realized model
2. F'inite volume of sample data
3. RepresenCation of sample process
4. Inaccuracy in stiipulatinq stochastic characterfstics of element-
ary processes
5. Finfte number of readinqs in time
6. Time discretization
7. Inadequate modeling alqorithm
8. Finite number of sample records
9. Level quantization
The second example is related to the modeling of sea reverberation. The
reverberation model relates to discrete canonical models of type (18) and
has the following form (5, 50, 621
:
NIU
r~fJ~E'a~it; e(f-t,l, (23)
;.u
13
FOK OFFICIAL USE ONLY
;co~~u~~HOn wHCno
orcuOl~og ~o ~po-
N011N
5
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N'Ok ni'FtCIAL U5C ONLY
where ai are the random amplieudes of the elementary scattered gignals,
N(t) is their random number, f(ti) is a function charactierizinq the decreage
in signals with distiance durinq their propagatiinn, c(t) is the emitted gig-
na1, tii ig the random momenti of arrival of the i-th aiqnal ae the observa-
tidn point. ,
mhe ai8arete �orm of the record of the model (23) has the following form:
,vW (24)
0 o;1ji~;
i�i!
where n is the number di the reading, ni is a random value.
On the aegumpefon ehat N(n) is a pofgson random value, the mean value beinq
dependenti on time ithat is, ott n), ai is also a randam value, and ni ig dis-
trfbuted in some interval, a BESM-6 computer was used ~indelfng eample
records (the modelinq method wan described in (62) . An exar.r')te of such
records is shown tn Fig. 6.
Thus, digitial modeling makes it possib].e to obtain sample recordg for a
broad class nf hydroaroustic random proceases stipulated in the form af
stochastic models, the basis for which is the totality of standard and e1-
ementary processes. .
5. Seatistical Measurements in Modeling
After the optimum model npt m2 has been adopted and the sample records fxk(t)}
of the random process X(t), corresponding to thig model, have been modeled,
it is necessnry to detezmine the adequacy of the realized model m2 and the
model opt m2.
F'igure 7 is a structural diagram of modelinq and determination of the ade-
quacy of the characteristic of the modeled process and its stochastic
model (scheme of statistical measurements). in accordance with this scheme,
on the one hand we have the model opt m2 of the random process X(t) and its
stochastic characteristic B( r/opt m2), and on the other hand, the model-
ed sample process X(t) =f xk(t)j and the statistical evaluation ~(e
which is obtained by the processing of its sample records xk(t).
In order to bp able to make sound decisions concerninq the effectiveness of
the algorithms and the quality of mcdelinq of the sample ensemble X(t) of
the random processes X(t) it is necessaxy to introduce the quantitative
measure of the difference betweeq the stochastia characteristic B(fr/opt m2)
and ies statistical evaluation 9(1). We Will denote this difference func-
tion in the form
,l', (qa %M,), .9& l/1 (25)
14
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Foit 014,1C [AL 115L c1NLY
where .f' is the operaLor for formation di the dffference functidn, in gen-
- eral having the sane senge ag in formula (7)1 npt mZ is the optimum model
which is adopted in modeling (see section 2, mz is tihe evaluation of the
working madelAuging the rosults of mdcleling, tihat is, uging ttie stiatiigtical
_ eaaluation 6) (,I) of thc 5tochastic charaCteristic B(I'/opt m2) .
mhe reasons for the appcaYatlCe of errorg in staeistical measurementg can be
classified as (see t1ig. g): inaclequacy of the gtochagtic model realized in
mndcling, the finite volwne of gample data, and finally, the discrere repre-
sentation of the sample proCess.
A quantitative analysis of the mentinned reasons fnr the appearance of model-
ing errors is an importanti ehenretical and practical problem which still
must be sdlved.
mogether wieh the value of the error fn stiatistical measuY-ements pp(opt mz,
m2) it is natural to introduce the function
C1'0~e).1,
(2G)
which characterizes the expenditiures in stiatiistical measurements and in
determining the evaluation of the realized wnrking model m2.
Much as was done in section 2, we note two types of problems in choosing the
nptimum evaluation of the model m2 on tihe basis of examination of the pro-
cedure of statistical mcasurements.
mhe first eype of prnblems is formuleted in the following way:
limitations are introduced on the cost of the sCatistical measurements
A ~ (27)
Co (m,`)
p~
where < N(n)> is the mean value for the number of acattered signals forming
the reverberations at the n-th moment of the reading, that is, in the in- -
terval (n,4 t (n + 1) A t);
for the aki geminormal law with the probability density �
ezP ~ 2 a a rq .1 ' -

where da(n) is the dispersion;
42
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VdEt dFCtCIAt, US E C1NLY
I'cir cXki Iq a univerqal 1nw witlA ehe pr.obabiliCy dengity
W(cc) m ! , ct ~ !n ,
`I'he nnnsratinnhry prnpereies of reverbeYatiort were modeled usinq definite
lnws nf ctinnge nf the parameCerg < N(tt)> , dn(n) and the funcCinn f(tt).
Ag the emiCeed qignal we uged a segmene nf a sine Curve wiCh a rectangular
envelnpe
C(n).. 5j-rr j4,d,ot) ~e- lf, (g)
nnd in this Gdmptiter experimenC iC wttg xsgumed ChaC cc,bT - lOJI�
_ Figure Z ghnws typical sample recnrdg nf reverberatidn obCnined uaing n
- IICSM-6 compuCer.
Ttie uge of tlie modeled reverberarion, Caking inCo accnunt iCs "cnrri.er"
Erequency, as shown in Fig. 2, is inconvenient due to the necegsiey for
making a greaC number of readings in Cime (in Chis cage eacti recurd con-
sisted nf 500 readingg). A considernbly more eCnnomiCal represenCnCinn nf
reverberatian is obeained thrnugh its quadraCure componente:
FI/1J . F~(~J cos(nw,ot)-~(n)sr'n (,1&), ct)., (10)
~ where p tn) (n) tos
,cK Ec ~n~
. .
are the "cosine" (CK) and "sine" (SK) quadrature components ciE t}le lc-th re-
vcrberntion record, Ek(n) and 4! k(n) are its envelope ttnd current phase.
The quadrature components FCK(n) and FSK(n) of the initial process Fk(n)
were obtained by taking only those readings which differed from one another
by the value
rn. - 2Z
.
. otw, � (12)
We note that the conversion from reverberation records to the quadrature
components reduces the total nurober of readinga in each record by n faetor
of 10.
Figure 3 gives the quadrature components of reverberation corresponding to
iCs sample records, represenCed in Fig. 2.
In the next stage of modeling after obtaining the sample records it is nec-
essary to confirm the correspondence between the adopted stochastic model
of reverberation to the statistical evaluations of its characteristics [1]. ~
We will examine one of such stochastic characteristics the autocorrela-
tion coefficient
43
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FOR nFFICIAL U5E ONLY
N
0
u
a~
H
d
M
-W
ro
N
N
H
~l
N
H
d
~
~
Vl
W
O
00
a
~
n~
~
W
O
~
O~0
t{1
~
b
~
~
~
~
u
~
w
u
~
~
00
W
44
FOR OFFICIAL USE ONLY
9
-H
~
H
N
H
Gl
~
H
O
~
M
H
t
A
o ~
~
a
~b
N
q
0!
~
N
R1
O 0
O tA
44
~
v
~
O G
l
-i .~L cE
1
N
~
U~
.G
C:
u bH
14 cn W
u~ 11
D
~
~
uv
w
$4
$4 o m
w
a~
w $4 w
u
~
�
�
0
0
cd N u
rl rI ~
~o"
b�'~
cn~

~WU

AWk+
~
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rnR nFrr.clnL usE ntvLY
~ .
F rnJ
~
2,0 . . �
a. ,~~~!~,~~~~~~~;~i~~',~~~lj~ 4ti

~o r � :
~
~(n) . '
1!."Jo
~~1~'~ ~ ' ~~~I~1, ~1'~�,~~~1'I~~~Y~~ 1~1 ~ ' '
-10 . r1~ � . , .
i,o A~~ ' . .
+..~r~!�ri~~~~l~f~;;~''�~'Ij't~~~' ~ K
-1,0 ~i { ~ ~ �
~ .
, U fG0 tAD 3J0 AOJ ~ su0 IL
Fig. 2. Sample reverberation records F
k(n) with carrier
frequency taken into account.
45
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FOit OF'FTCIAL USE ONLY
A
F�n
n
n.
Fig. 3. Sample records FCK(n) and FgK(n) of quad-
rature components of reverberation.
46
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~ , , ,
. ' ~ � .
,

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rdti nFFtCIAL U5E ONLY
< F'(/j)
< F �M1 > > t (13)
where n ig the tnieial number nf the rettding, ~L m 0,...~ T/A C is the nnr-
re.tnCinn gliiEe. IC is icnowci (2) ChaC for the adnpCed reverberariom model.
~,~;4. 1 ~,~n) C ~n,,~ ) . .
(14)
9' P 1
For a.n emitted signal of Cypp (9) on the bnsis of (14) we find that
0, ~zl cOS (z rve Q t ) . (15 )
~
As n qtaCistlcal evaluaCion of the correlation coefficient Rr( 2) we used
the expression:
. n ~ ~ (n) ~ (Rr7) (16)
ho ~ � ' ^ 
r (n)
~c� ~ where M is the number of sample records nf the modeled reverberaCidn.
r
_ Fig. 4. Computed reverberation correlation coefficient (solid curve) and
its statistical evaluation (circles) for sample set with M= 60 and M= 50.
Figiire 4 shocos the rheoretical values of the correlation coefficient Rg( Z)
computed using formula (15) and the statistical evaluations AF( Z) obtained
t,y the processing of a sample reverberation set in accordance with (16). It
fol.lows from the comparison that at the correlation level the characteris-
tics of the sample set correspond to the adopted mathematical model of re-
verberation.
BIBLIOGRAPHY
1. 01'shevskiy, V. V., "Modeling in Statistical Hydroacoustics," in Chis
collection of papers.
- 47
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FdR dFFICTAI. USE dNLY
2. 01'yhevskiy, V. V., 5TATISTZCHESKIYE SVOYSTVA MOItSKOY itEVERBERATSI2
(Stutiseictil I'ropereies of Sea ReverberaCi.on), Ixd-vo "Nauka," 1965.
3. Middletdn, D., "A 5eaCistica1 Theory of lteverberation and Similarity
n� rirgr-Order Sr.netered Fielda," Parte I, II, TEEE, Vol 13, No 3,
1067; Pxrrs III, SV, IEEE, Vol 18, No 1, 1972.
COpYRIGNT: Notice Not Available
5303
C50: 8144/0938
48
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rOR OFI'ICIAL U5L ONLY
UDC 681.3:519.2
DIGITAL MODELING OF A SAMPLC 5ET OF A NONSTATIONARY RANDOM P1tOCESS
Novosibirsk TRUDY SHE5TOY VSESOYUZNOY SHKOLY-SEMINARA PO STATISTTCHESKOY
GIDItOAKUSTIKE in Russian 1975 pp 66-73 ,
[Article by Ye. V. Kirillov, V. V. 01'shevskiy and Ye. A. Savinov]
[TexC] Many types of random processes in hydroacoustics are nonstationary.
- Such processes include, for example, cavitation noise, sea reverberation,
some Cypes of direcC hydroacouseic signals, echo signals, eCc. (for ex-
ample, seP 1-3). One of the simplest models of a nonstationary random pro-
cess )G(t) is a multiplicative constructive model of the type [3, 4] _
- XW=j(Q(t)
- where S (t) is a stationary random process, f(t) is a known (determined) _
function.
In this case any k-th record of the /XX(t) process Y,(t) is determined as
(2) -
where 5 X(t) is a record of the stationary component of the process
- x(t) � { x, N~
in modeling on an electronic computer we deal with discrete models, each
record of which 'x x(n) is represented in the form:
f (ry) , n' Ab (3)
where n= 1,2,..., is the number of the current reading, the riumber of read-
ings is a power of the number 2. 6 t is the time discretization interval.
The algorithm for the modeling of sample records
X. (ry) = f (ry) Jw (a), (4)
is represented in the structural diagram in Fig. 1. -
In accordance with this model, the initial mass of data used is the mass
of independent numbers ~111 1, which is then subjected to so-called current
weight summation [5]
49
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Fdlt dFFIC1AL USE tlNLY
9 a. ~~-i
(5) ow
wliere ai are weightinf; coeffirients whoge valueg nre geleCted dn the bggig
Of the rehuired stuchastic: model nE the procegs (n), N is the number df
the summed indepenJene valueq ~
qpu;~'nMwa CROn,!/4'eR
~V."C~~~~' GVNN NuUSA~QH~~
K~~nerrNYll. A
_ c,,YV~a~:,f ~;y~;, ~
A C
o.nNpo neper+NO A
.a~rUQ g )ceHUe n x (n) E
~ f~A~ uN f(h~ j(h~ Bbl6OpOuNbl@
- peaAulakuu.
~
Fig. 1. Structurni dingram of modeling of ggmple recnrdg Xk(n) of nnngCa-
tionnry rnndom process
KEY :
A) ['rngrnm for generaeion of independenC random numberg (gi)
B) F'ormation of function f(n)
C) Moving summation...
b) Multiplication...
E) Sample records
'1'iirn an electrnnic cnmpuCer is used in modeling the function f(n), and then,
In accordance with (4) and (5), we obtain sample records of the nongtation-
riry process =
^ N A x~ f (n)Z Cu ~fN.~ � (6)
681
In the described computer experiment a BESM-6 computer was used in modeling
_ [tie Kj numbers conforming to a Gaussian distribution with a zero mean value
;ind a uniCary dispersion and the values of all the ai coefficients were
assumed equal to unity. Such ai m 1 values correspond to the following
_ autocorrelation functian of the ergodic process k (n)
- Bj(ti) =dfC N'), N ~ c7,
, wherc dk is the dispersion of the process.
Ttie distribution of the probabilities ~(n) should be Gaussian.
As the f(n) functions we used such
_ f ~~,,J , f �r�.s~a(l~:Z,y) (8)
No '
50
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Poli oFF1C IAL ust. ONt,Y
where m in the mddulation itttettgiCy coeEficient, Np in the ciumber of rend-
ingg of thc Eunction in the period of Che modulgting harmonic oticlllation,
and ,i lsd
eXP('
(9)
wliere Nf iH thr number of reddingq in ehe timc intervnl in which f(n) de-
creases by a fnrtor of e.
Tl~e charaCterigtic yamplc recordg jGk(n) of Che nonstationary pracess X(n)
are repregented in Figureg 2-4. Figures 2and 3 give Che sample records of
a periodical.ly nonstaticmary random proCegs
A
x4 /I)) s Cl # 171 SG/f ( n' N'FCL (10)
r ~itt!�~
c'�I
zind CiKure 4stiows a rccurd of gn aperiodicglly nongtationary procegg
xx~/) = e,w'Pf)~a i~n�i (11)
Now we will examine the correlation charaeCeristics of the modeled randoro
PrnceKS ,L' (n), whose records are deCermined in accordance wiCh (3) and (8)
X., ~ry)~ CI fm.Sir~(n No )JJK(a), (12)
where ~ X(n) is ,7 record of the ergodic process k(n) with the corretation
Eunction (7). Accnrding to Che classificaCion of the stochastic character-
istics of hydroacousCic random processes 3 we should consider three rypes
of correlation functions, ro wit: t is Che current (in this case n-current)
correlntion funcCion ~
- E~ 01 z) = < xIC (4) 21C (ni z) > -z
q,~ Q..~
,K (13)
and k is the current correlation function
SP
z ) ' Zti 00 x,C (/P' z) = ~Nn M-G xK 071) zIt (41 'Z~, � ~'1 � Zpol (14)
~..w, 4$ P .
is the mean correlation Euncrion
(z) = 44,00 ~''R(no, t)> Q~y .~r~n).X,e(~ LJ (15) -
+ o. ~ oo ~
K.�t.rp
1{ere IiX(n,Z ) characterizes the nonstationary properties of the random pro-
cess, BX(k,Z ) characterizes its inhomogeneous properties, and SX( 7,) char-
acterizes the properties of the process as a whole.
First we will examine the n-current correlation function. According to (12)
and (13) %
tim s ~Gtm~+n(R No )x(t+m:in~n.t)N.}~fKlrUfK(nrtl
Q--.aK--4' (16)
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1
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& 6
�
.

~
~ ~
n

I
16
.
~
_
o
h
`1
~
~ w � �F c,;,~ J Ob 14 T
After aubatiCution into (1) _
[/,(f -2.) �
and simple Crigonometric transformations we obtain:
r _ U ~r~=~(, fA~{Jrosl.S~t'~~� ~ Jms~'S~~f, ~lJ~/f * -
r
J~ sin~'tp~f- rl.7df)~t (2)
r �
`(~A~~f "n~f~(f1' S~�Jcbsl'f~'~f �rJJd~ - .
- � r~/!lllcarl'~fl~J, f?,I iin111lj . zJ,J-leJ ~ 1w .
In accordance with expression (2) the model for the digital detector was
formulated using the following algorithm:
)
41 (~l) `l ~.L A(N K1C01~~! N' P. J'~~Y7N'Y-jV
' N�:./ ?
N K). r~~�sinlr( N'r" iv ~~J~ .
. r.~ .
' N�i~I
e6 iy K/ � Si/7lSO(~ rJ+ S o~ COS~SP! N K N~>1
%Y K>C0.SlSO( %Y K/ + ~J � S%nl~N q' - /7
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where U('Gi) ig Che discrdte vaLue of Che deCecCor ouCpuC effect; N ig
Che nuJer of aignal discretization intezyale, i~ 11 2$...N.
Ar ~ o)Utlr~GlOi y
a b.
k--
~ h
r>2 cK
, g
7> STK
' 6 7 > so ~'~r
,
~
Ut l'p/uj l01
b). c"
u r> c u...,..
~ T ~Sfk
~ !0 Z'r IY
(TJ/Utf01 %
d

T>JOZ'x
w
Fig. 1. Influence of quantization pro^edure on magnitude of side labes of
processed signal: a) LFM-detector witn weight processing by Hemming method
in case nf a fixed Carget; b) LFM-detector with weight procesaing by the
Hemming method with Afadd � 0.25fdeviation; c) LFM-detector withouC weight
processing; d) detector with quadratic frequency modulation.
r
Fig. 2. Output effect of PRIS filter. Fig. 3. Output effect of digital an-
alog of PRIS filter with inversely
proportional weight processing.
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N is the number of yigna1 diHCreeizaeion intexvals, i- 1, 2,...N.
Ttie prngram for deCerniining the ittpuC and reference signalg provi.ded fnr
rhe uge nf a f1oating decim,il pqit1C Wirtt six decimal places. k'or mndeling
the level qunnrizaeion prnr_eduxe the xeferenCe eignals wcre rnunded off,
takink into accnunC the required qugnCixnCion inCerval. Ilie resulCg of Clie
modeling, preseneed in Fig. 1, make ie poseible tn evelucte ChN dependence
df C}ie inCrease in the relatiive leveY of the secnndnry side lobeg with a
decrease in the number o� C}ie quanCixatinn inCervals. Fipure 1a shows tlie
' r.haracterisCics of a detecCor Qnr n LFM yignal with weighCing procesging by
the Nemming methnd �or a fixed targee for differettt coe�Eicients of signnl
coroplexiCy q; Figure 1b shnws similar dependences �or a moving earget,
where 1Jftinnp1er = 0�25 f nf. rhe devinCion. Figureg lc nnd ld make ie png- -
sible en evn luaCe Ctie influence of the quantizaeinn procedure for a LFM
signal wittinut weigtiting proceseing and a signal wit}i quadratic ni with the
-
_ r.otnplexiey q- 15. In an investigation of the influence of level quantir.a-
tion for quadrat:ic FM and for Ft o� signals with boppler diseortions (Fig-
ures lb-],d) there are de�iniCe methodological difficulties atCriburable to
the facC that quantizArion leads to a differenk change in the levele U2(ti )
with differene fi. In Chese cases an evaluation was mnde of the mnximum level
of the si,de lobes for 'G> M'Gk, where 1, is the interval oE sCrong correla-
rion of the undistorCed eignal, M is a natural number. As can be seen From
the curves shown in Fig. 1, a decrease in the number of quantization in-
tervcils to 16/8 �or positive values of the reference signals and 8 for nega-
tive values does not lead ro significanC distortions of the processed signal.
It must be Caken inCo AccounC that the number of quantizntion interva'ls for
the input signal is selected on the basis of the required dynamic range.
The distortions in the shape of the output signal caused by uniform Cime
cliscretization were e:camined in considerable detail in the periodic liter-
ature. For example, the author of [2] determined the mean square error in
distortion of form for digital detectors, and in (3] for discrete-analog
(letecCors. Considerably lesser attention has been devoted Co the errors caus-
ed by discretization, the frequency of which is also a function of time, al-
though the use of such a procedure makes iC possible to simplify the digital
processor [4]. For example, the author of [5] proposed the analog device -
PRIS, afilter constituting a delay line with "branches." In order to com- !
pensate the energy losses caused by a decrease in the number of "branches,"
the author proposes that there be a corresponding weighring processing of
the signal before the summator. The value of each particular weighting co-
efficient is directly proportional to the lag introduced by the delay line
between a particular and adjacent branches.
A discrete variant of such a device would be a quadrature-correlation detec-
tar in which the input oscillation would experience discretization with a
constant frequency and all the samples of the reference signal, other than
the maximum values for each half-period of the modulating function, would
be equal to zero. The amplitudes of the remaining samples should be direct-
ly proportional to the time interval between the corresponding zeroes of
the modulating function. It is evidenC that for such A detector of the LFM
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gignnl the number of mulCiplicarinn nperations in each quadrature ch~unel
ig reduced by a facedr of approximately 3. Modgling 3,ndicaeed thge the nut-
put ef�ecr of such a detectox has a conei.derable inexeage itt the seeondary
gide lobes (Fig. 2). In gceualiey, att i,ncreasa in eome semples at the ex-
penge of nthexg seemingly 1eads Cn al'conAtrintion" o� the reference aignal
energy ed individtial digcreee intervalg and gccordingly Co an increase in
the CXl)gg-correlaeion ineerval in the caee of suf�iciently greae dplgy times.
Proceeding on this bagig, in order Co decrease the level of the side lobes _
iC is desixable rn decrease the uemaining samples, noC increase them. Fig-
ure 3 shnws the outputi effect for the dereceor of a LFM signal for which
the levcls of the remaining samples are inversel.y proportional Co the
time interval beCween the corresponding zeroea of the modulating funcCion.
A comparison of F'igures 2 and 3 shows Chat such a representation of the ref-
erence signal makes possible a considerable decrease in the level of the
side l.obes of the processed signal. However, it must be Caken into account
that such weighting processing leads Co a decrease in the noise immunity of
the detecCor. There�ore in the deaigning of devices of auch a type the op-
Citqum eechnical system solution must be sought as a compromise beCween the
menCioned factors.
- BIBLIOGRAPHY
1. Cook, C., Bernfeld, M., RADIOLOKATSIONNYYE SIGNALY (Radar Signals), Mos-
cow, "Sov. Radio," 1971.
2. Chaykovskiy, V. I., KVANTWANNYA TA INTERPOLYATSIYA SIGNALIV V IMPUL'S-
rTIKII SISTEMAKH (QuantizaCion and Interpolation of Signals in Pulsed
Radioelectronic Systems), Kiev, "Tekhnika," 1966.
3. Tsykin, I. A., DISKRETNO-ANALOGOVYYE METODY OPTIMAL'NOY OBRABOTKI SICr
NALOV (Discrete-Analog Methods for the Optimum Processing of Signals),
Moscow, RADIOTEKHNIKA (Radio Engineering), No 2, 1969.
4. Gol'd, B., Rider, C., TSIFROVAYA OSRABOTKA SIGNALOV (DigiCal Processing
of Signals), Moscow, "Sov. Radio," 1973.
5. Atzeni, MazotCi, "New Procedure for Discretization and its Use for the
Synthesis of Linear Transversal Filters," Moscow, ZARUBEZHNAYA RADIOEL-
EKTRONIKA (Foreign Radioelectronics), No 8, 1972.
COPYRIGHT: Notice Not Available
5303
~ CSO: 8144/0938
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UDC 681.883
SYNTHESIS 0F HYDROACOUSTIC SIGNALS IN THE RE-GION OF STRONG CORRELATION OF
THE VELOCITY-LAG UNCERTAINTY FUNCTION
Novosibirsk TRUDY SHESTOY VSESOYU2NOY SHKOLY-SEMTNARA PO STATISTICHESKOY
GIDROAKUSTIKE in Russiatt 1975 pp 117-136
[ArCicle by K. B. Krukovskiy-Sinevich]
[Text] 411. Introduction. In connection with Che complication of the tac-
tical problems Co be solved by modern echo sounding systems, Che problem
of the synthesis of complex signals with a stipulaCed uncertainty (ambig-
uity) function during recent years has been devol�ed considerable atten-
= rion. In particular, we should mention sCudies [1-5]. However, the menCion-
ed studies are characterized by definite limiCationa which are caused to
a considerable degree by their radar directivity. Radioelectronic sonar
methods are characterized by Che following relationships:
the velocities of the targets are extremely sma11 in comparison with
the velocity of signal propagation;
the required time resolution (lag) is considerably less than 1/fDopl
where fDop is the Doppler shift of the carrier.
In these cases, as a rule, it is possible to use a simplified form of the
two-dimensional velocity-lag uncertainty function proposed by Woodward:
9'~ S t ~ ex z~Tf t dt ~
~'ft,f J f s(t-~) Pl ~ (1)
-w
(jon = Dop] where s(t) is a complex signal (complex envelope).
For an uncertainty function of the type (1) it is possible to develop meth-
ods for signal synthesis in a quite general form. Use is made of the con-
cept of distance in some generalized space and a signal having the uncer-
tainty function closest to the stipu].ated function is determined [2, 5].
. In addition, if it is assumed that the uncertainty function is described by
(1), it is also possible to formulate the conditions under which it can be
applied (2), (5). However, it appears that known methods of signal synthe-
sis using a simplified uncertainty function are still far from that which
could be used extensively in the planning of echo sounding systems.
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i
1
We wi11 notiyij1 only tlie twu tnosC importanC rensons fox guch a siCuation.
If ik is unI~n wn whether the stipulated uncertainty funetion is practicnble,
it iy imposqible to gtittrantee a high accurACy in the approximaCion Co it;
exiqCing metliads inake iC possible Co evaluaCe only the mean aquare approx-
imntion (2). For pracricgl purposes, however, it ia important Co enaure a
unifnrm npproximation, whicti, :in particular, guaranteea an admissible lev-
el of the secondary maxima of the uncertainty function.
L On the other hand, the evaluation of the pracCicability of the uncerCainCy
_ function involves solution of un essenCially nonlinear integral equation
(2), (5). At the present time we know of no e�fective methods for soluCion
of equaCions of such a eype, which puCs in doubt the practical poasibility
of checking the uncertainty function using the practicality criterion.
In the synthesis of sonar signals the siCuarion is aggravated Co a seill
greater degree. In this case these methods are essentially inapplicable
since in the case of sonar detection of targets:
there are relatively great ratios of targeC velociCy Co the veloaity of
propagation af an acoustic signal in the water (up to 0.01-0.02);
the extenC of the targets is great, as a result of which the required
time resolution (lag) is considerably less than 1/fDop�
Therefore, the synthesis of sonar signals must be based on the concept of
a generalized uncertainty function. At the present Cime there are no gen-
eral methods for the synthesis of signals for the generalized uncertainty
_ function. There has been adequate solution only for the problem of synChe-
sis of a signal invariant to the Doppler transform of the spectrum (6), (7),
(8), (9), (10). Such a signal ensures the worst velocity resolution with a
stipulated timP (lag) resolution.
Sie will show that there is a definite possibility of applying the Sussman
method to the generalized uncertainty function.
Adhering to Sussman, we will wriCe the sought-for signal in the form of the
- sum
(2)
where fk(t) form a full orthogonal system of base functions.
We will assume that series (2) converges uniformly. Then the generalized ;
uncertainty function for a signal with a unit energy will be equal to the
sum of an also convergent double series
~ ~~Z,Of)"rS(t+Z'JS~lt~~~d)Jllt=~~~S`ss,n~P~m~z~d~. (3) -
/
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Here
a
d

VIs the ri;dial component of target velociCy relaCive to Che aonar; C is
the velocity of prnpngation of a aignal in rhe medium.
Assuming rhat there is some signal ?f`(C) which corresponds to a stipulaCed
uncertiaintiy function Itp(ti, o-) and ehat for this signal the seriea
d"(t) - S'~'~ fti(t) , (s)
convexges uniformly, we will represent I2p('L,oc) in the form
Aoo ft-, d ) � Z &Oicm ~'X.m ft; CG ) , (6)
fl?
In accoxdance wiCh (1.3) and (1.6) the closeness coefficienC will be equal
to
c(z d 0 M 17 ~ ATmnp "S,'5m'P"p ~ (7)
where jf~alm (T, 4P~p0? Gr ) ll~Z'QG~f
Then the synthesis problem, according to the ideas of Sussman, is reduced
to the selection of the coefficients Sk, maximizing c( C,oC) under the con-
dition
(9)
[Such normalization makes sense if with a definite degree of approximation
we consider the volume of a body of uncertainty to be independent of the
signal.]
Since expression (1.7) can be reduced to the form
C (z, cX ) � ,Pe s ~c sen , (10)

where �
- B,c~7 � ~ G~ ~ Pnp A ~tmnP .
(11)
it forms a quadratic form relative to [symbol omitted]. Thus, synthesis of
a signal on the basis of the general3,zed uncertainty function, the same as
~ on the basis o.f the simplified uncertainty function, is reduced to the
problem of maximizing the quadratic form under normalization conditions.
lIowever, synthesis of the generalized uncertainty function on the general
basis of its envelope has not been considered at all in the known liCera-
ture.
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Thus, the solution of the pxoblem of si,gnal synehesis on Che basis nf the
generalized uncertainty �uncCian i,s extremely complex and aC the presenC
time we do nat know any means which could lead Co construcCive reaults.
idevertheleas, in developing new sonar syatems it ia an urgent matter to
synthesize complex signals having a definite (and not the worst or best)
accuracy in measuring range and velocity. Tt is well known Chat this reso-
lution is almost entirely determined by the extent of the region of strong
correlation of the uncertainty function. Therefore, in this sCudy the re-
quirement on correspondence between the stipulated and syntheaized uncer-
tainty �unction ie limited to the region close to the main maximum. In ad-
dition, it was assumed thnC the synthesizablQ signal is related to a clasg
of signals of the type
u(t) p ltl a T, (12)
t -r~-Q`-t~�-- . . . . .
where
The following argumentation can be cized in support of choice of precisely
this class.
On the one hand, the advanCages of phase-modulated signals are well known.
On the other hand, U(t) of type (12) is a quite general form of registry
of a signal with intrapulse FM. In actuality, a continuous function, in
this case the FM law, in accordance with the Weierstrass theorem, can
with an accuracy as great as desired, be approximated by a power-law poly-
nomial.
Here it should only be added that for a11 practical purposes in real elec-
tric circuits there cannot be signals with a discontinuiCy of the insCant-
aneous phase, despite the fact that in a number of cases it is convenient
to examine precisely such a model.
_ Taking inCo account the cited restrictions, we solve the problem of syn-
thesis of complex signals on the basis of the uncertainty function.
The selected approach makes it possible to carry the formulated problem to
a successful solution and develop a method acceptable for engineering prac-
tice and necessary in the planning of optimized sonar systems using complex
signals. This result seems ill-suited for practical application because
the difficulties arising here of a computational and fundamental character
are substantialiy greater than in the case of a simplified uncertainty
function.
p
In particular, we note the complexity in computing the
(6). This difficulty is attributable to the fact thaL�
is not orthogonal, since during movement of the target
in the time scale of the echo signal. It is known that
nonorthogonality the determination of the coefficients
tion of a system of linear equations.
74
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expansion coefficients
the base f0 km('t, oC)
there is a change
in the case of
is reduced to --olu-
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If the signal is nf greae cnmplexi.Cy, as indicgted by an ad.ditiona1 Anal-
ysis, the number o� equations ie great and in a number of cases the re-
sulCing system is slightly conditionaJ.. 7.'he sYight condiCionality 1eade
ro considerable errors in comput3,ttg Rkm, $nd according].y, Ca alow uc-
curacy in approximaCing RQ(G, oC.).
Moreover, as a resulC of the extremely prolonged computations in connection
with the unauccessful choice of Rp(-G,o/-) the aynthesized signal cnn be
~ unsuccessful in practical use. [For example, the peak power of such a
signal will considerably exceed the mean.] Tn this case the Sussman method
does not give an anawer to the question Aa how to change Rp ( t,a) ao C11aC
the signal propereies wi11 change in the necessary direcrion.
Finally, we will menrion what in our opinion is the greaCest weakness of
such a method. Since the system of functions Vkm(-G,oc.) for the general-
ized uncertainty function is not orthogonal, in this case the assertion
of completeness o� the sysCem (Pkm(-G, ot) loses sc tse.
Moreover, iC follows from the demonstration of the completeness of the base
functions, cited in the study by Sussman, thaC evidently for the generaliz-
ed uncertainty function the system 9km(t ot) in a general case is incom-
plete. Our attempts ro demonstrate the reverse were not crowned with suc-
cess. If the system (Pkm('G, a) is incomplete, the errors in approxima-
tivn of RQ( -6, a) by the series (6) can be so great that the closeness
coefficient ceases to be a criCerion of the smallness of the mean square
error in approximation. [The latter comment also pertains Co extremely
wide-band signals when the volume of the body of uncertaintiy is essential-
ly dependent on the signal energy spectrum [14]].
As correctly pointed out by D. Ye. Vakman [5], equal difficulties are en-
countered in the synthesis of the Woodward uncertainCy function, stipulat-
ed only in absolute value. In particular, it is unclear how to solve the
nonlinear integial equaCion determining the optimum phase of the uncer-
tainty function.

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#2. 5ynthesis of Signal With Unambiguous Reading of Lag and Velocity
Among Che signals having n sCipulated extent of the region of atrong cor-
relaCion in velocity-range coordinates it is of particular intereat Co
examine signals allowing an unambiguiCy of both range and velocity in the
- xeading. As is well known, for such signals the maximum value of the un-
cextainty funcrion or its envelope, regardleas of the velociCy of mot�ion
- of the target, corresponds ro the relative lag of the reference and echo
signa.l, equal ro zero. Here as a simplification we will ].imit ourselves
to an examination o� a vezy simple correlation detector when the dependence
- of the use�ul output si,gnal on velocity and range is determined by the cor-
- zesponding secCion of the body of uncertainty.
_ Proceeding on the basis of the requiremenCs on unambiguity, using (12) we
we find the maximum value of the useful signal aC the output [as a sim- -
plification the attenuation of an echo signal during propagation and re- _
flection was assumed equal to 0.] -
U� fcosf2t,t*dv),t+Z: Q:t'+t`]dt+ ~
~ -r 0' (13) -
+ 2 fCOSf cIuJ,t (l,*a)'qjt']dt, � -
-T
where ~
da2~
V is the radial component of velocity of a targeC relative to the sonar; C is
is the velocity of signal propagaCion; al = iJ p is tlie carrier frequency,
which is assumed to be stipulated.
The first term with 6)pT >1 can be neglected. Then for ~ a ~l6 1 -
= T ~
ul (00 ' 2 , f COS[c!cJ,t ill: t(14)
-r
We wi11 examine how it is possible to ensure the necessary constancy U1(0Z)
during movemenC of the target. A Crivial soluCion of this problem is choice
of the signal parameters from the condition
,7,
~ a>, d a; t`1 �-3 (15)

u1,T� ~ (16)
for '
0 1iS

. ~ .AJ=~' ,ffhl~o)li.l~tlh. t-e.Jnl~jdr:dt~clt=
. .
-w ~
Ar ' T~.I~ff lt~J~r
~
'~s, ~~~1' I sx~~~l=~~ y A. = f s/a1~ _
,2 ~SL=~u~~r S~ r~~ Sx
21~.-~ { 7- r _
f f S,y lu1~1: S` rw) t~ Sx lcJ) I: fl
T
lde substitute into (14) the optimum solution for Sh (ul) found in (13). Af-
ter some transformations we obtain
y~
.
F�t~'is (15)
x
It can be seen from (15) that in the absence of noise the mean square
error in evaluating the pulse characteristic of the medium is equal to 0.
Now we will examine some statistical properties of the process x(t) in
selecting a filter with a frequency characteristic in the form (13), tak-
ing into account that in this case z(t) is the best evaluation of the
pulse reaction of the mediwn.
Taking into account that the mean noise value is equal to zero, we will
- write
1y{2 =,lJ~~f)~,~, (t,~x(f-ta.-tf)dt,df, _
.
f C4
= t sb /w~ �Sjo [a) �S'x aled = (16)
ra) eJ'"It q~(J ' .
~M' . w .
- ~Dl31f)} = M~.~ t~tlJ 1.? 0] . .
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Using (16), from (4), after similar Cransformations of Che process, em-
ploying its representation in the frequency region, we obta:Ln
t~
(17
z~f ~ s (td)Ir Nh l0)I e'�m d' =
~ )
d(d
_L
J
from (16) and (17) iC can be seen that when Hn(cJ 0
f',f s,r jw) e-'we ccJ =hy
a
f 2M} p , -
that is, with the spectral density of Che noise being equal to zero, the
 evaluation of the pulse reaction of the medium is unbiased with a zero
dispersion. With an increase in the spectral density of the noise there
is an increase in the bias of the evaluation and its dispersion.
We note in conclusion that in the solution of the formulated problem we
employed the so-called sCructural approach in which it was necessary to
_ select the best processing system in a particular class. By varying the
structure of the system within the framework of a given class of linear
systems we found a discrete system optimum from the point of view of the
. selected criterion.
- An obvious advantage of the structural approach is that in its use it is
usually sufficient to have only a partial description (stipulation) of
the processes.
An obvious shortcoming of this method is that frequently it is impossible -
to say whether the structure has been correctly selected. At first glance
it may seem that one of the ways to choose the most suitable structure is
to assume that the sought-for structure is an arbitrary nonlinear system
with time-variable parameters. In other words, the class of structures
selected is so broad that it takes in all the possible systems. The dif-
ficulty here is that there is no convenient mathematical approach, for
~ example, such as the faltung integral (Duhamel integral) for expressing
the output voltage of a nonlinear system through the voltage across its
input. Precisely for that reason we have limited ourselves to an examina-
tion of a class of linear systems. However, we feel that in the future it
will be possible to apply the results to the case of the pulse character-
istic of a channel with time-variable parameters.
BIBLIOGRAPHY
1. Levin, B. R., TEORETICHESKIYE OSNOVY STATIST.iCHESKOY RADIOTEKHNIKI
_ (1'heoretical Fundamentals of Statistical Radio Engineering), Moscow,
"Sov. Radio," Vol 1, 1969.
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_ 2. Akhiyezer, N. I., LCKTSTI PO VARIATSIONNOMU I5CHISLENIYU (LecCurng nn -
ehe CAlculus oE Variarions), Moscow, 1955. -
COPYRIGHT: Notice Not Available
5303 -
CSO: 8144/0938 ~
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,
~
FOEt dFFICIAL USE nNLY
UDC 621.317.757
SpECTRUM dF SLA ItEVERliERATI0N A5 e1 NON5TATIONARY EtANDOM PRUCES5 ,
NoVOsibirgk TItUbY "HE5TOY V5E50YUZNOY 5HKOLY-5EMINA1tA PO 5TATISTICNESKOY
GIUit0AKU5TIKE in Etussian 1975 pp 220-224
(Areicle by V. A. Geranin, A. N. Prodeus and B. I. Shotskiy]
(Texe) An investigation of the correlaCion properties of seu reverberntion
stiows thaC in the case of an arbiCrary signEil duration and obgervation
time (lJ the reverberatinn at Che outpuC of an acoustic antenna does not
belong Cd a clasg of random proceases reducible to a sta[ionary class.
The auehor of (11 investigated a symmetrized reverberation correlation func- -
tion and found its approximate spectrum. T'he auChor used a determination of
the specCrum of a nonstationgry random procese from Bendat and Pirsol. A
weak side of this determinat.ion of the spectrum is the impassibility of
its measurement and the difficulty of a physical inCerpretation. BoCh Chese
weaknesses are attributable to the fact that the tJ argument of the Bendat-
Pirsol spectrum is not identical to the argument of the spectral function of
the analyzed process.
In this connection it is of Cheoretical and practical interest to seek an
analytical Pxpressinn of the reverberation spectrum in the S. Ya. Rayevskiy
definition [2].
This study is Che next step in the plan for investigating the spectral-cor-
relation structure of sea reverberation as a nonatationary random process.
Tlie conditions under which the assumptions adopted in compuCing the spectrum
ir. [1J are justified were formulated. A general expression for Che spectrum
in the S. Ya. Rayevskiy determination of volume reverberation was rigo:ously
derived. The spectra in tonal and f-m signals were found.
The complex envelope of the symmetrized correlation function of reverbera-
tion
A'ft',r)' ~ ~
�d .
Y
They can be ineerpreeed as the autocorr,eletinn functinn, spectrum, frequency
correlttCion funceion and tlie functinn of Che difference argumpnrs for Cime
and frequency respectively.
Here
. . , .
. I-(QJ .C1'a, ~ CtJQ~y~ GJ~ G1i J�~ N~C~1~Al~ ~W~j~~ w~ ~J� (14)
~ . .
wy/fdf
(is)
vr~~'%x~Y'~if~'~v~x�uxiy~vyi.3~~~/~~1, z~y
0tv~xx,xy,n;'rs.'ry,wJ' Sw('r;,'ry,w; ~',.�.?'x,~c-y�xy, w..QJ c16>
is Che complex conjugation symbol.
We will express the autocorrelation function
c17>
- the Ra,yevskiy spectrum
,ei~ t0 >t'�``� (18)
the frequency correlation function
Ty (W/Iw,/ < Sy~Cf.fi/ SY ~WI/~I (17)
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the functinn of the difference argwnenea for titne and frequettcy
I . - ? : e� . a . (Zo)
t~ 1.~..
~
o"w1d~~"rCJ~w~
4v 02, 4'~ w ~tU, tJ..2)~
(zl)
(zz)
the anCenna rpspdnse u(C) Chrough ehe specergl-Gnrrelntidn c1�rgceerigCics
nf ehe fie1d tP (x, y, z, C) in which it ig Eiieugted.
Ag tlie iniCial expresgiong iC is cnnvenient en uge:
~.~~W.", y, x, ~~W~x, y dx, dt, (23)
;
S~w~' ~~j n" J~~s~, ~w'a,~ , w~aY, c~,~) R
_w)cu,�e/ceR'alely a'a, (24)
.
~ � (25)
cu'ay,
We obtain analytical expresgions relating the apectral-correlation characeer-
istics of antenna response nnd the field affecting it by combining (23),
(24) and (25). �
For example, from (23), with (10) taken into account, it follows that:
n�
~t)O~~, dm, dx,dv (26)
On the basis of (25) and (12) we obtain
(w'a, , w'ay,u'% w~dX, waj;, wJ~
.Aycar,~ , w'c~,', w'a~ , w _ w; -c~; -(w, �a,~I, - fc.,~ -~.>J~
� p.~p/~(,, t,Jf 'y~ day daJ'dw, o',Y; ri,y dr.,(27)
Multiplying (23) and (25), with (11) taken into account, we obtain
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~
e. r~� t.~� J/l~.1~~ lw-d,~, ~~y, r~; y, L.~ ~
� S~ ~a~~t',~,~Et'y,�r�~;�Cw�~y,~~',y,~',yt',~.~/,~~E�41fw~~� (28)
� dx a`tr p'ata daty v''a' a'cv
It fnllnwn from (23), (24) and (11) ehat
(ev� 9.0 (aa. , G'~ty, r.,; .'~y y,,~r, � y,,, a (29)

~ I 7
i Ct1 Q~ i1J CZ' - t~ ~
N I ri ~~~�~~Ji xi f/i Y 1~~Q1~t O~Q' ti'ed'~aff4d~, O~ol~
Muleip]ying (24) and (25), with (12) tgken into gccount, we nbtain
~~W~~� (n rfI/IIIIf''v(a'a:,a'~�'y,~~ w'~ii~~`xr~wlt
'W., lww,,a'ay,wux,a'ct~.-aS-w;-(*}�w), -PW �wlJ" (30)
� P.~p~~w, - w,~~.~w,ra�f da~,; doty dw'a'a~* day da'a~a~
Expregginns (24) and (12) gre reduced to.the expregeion
y(~~ t/ , . f~ w at,~, a~ily. G~i et~ er,~. a' dr. A1 J
W W ) � l~~~ / y ~ , ~ , w ~ ~ �
, iWCwa{~way~w�al,;~cv�ay~_w;_w;-(w,-w~ -~w,�~J~w..a.., (31)
ea,; a4~ry dw' o'vr,; v'cy
In nrder to obtain the expression 2e 'G )we employ the expreasion
K�(Ttl=Z' ,r-t%X,y,x,t�e) .
(32)
� Kw(vt, u� ux, z; f,; .x; y, x, s1,'rJdx dv; P'Palv '
With (13) and (18) or (23) taken into gccount, on the basis of (20) we ob-
tain: ~
ae~: n~-n ~I~~~~~~~y ~X:. Xr. /1: u,, t/y,
/1-11.Iill,iMy,!/.1, Pir)'XaJx (33)
- K;''Cx
Using formula (27) we obtain the known expression for the dispersion of re-
sponse of a nonparametric inertialess antenna for a homs)geneous stationary
field through the field indicatrix and the antenna directional diagram
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~at"r~~, ; tl~~,p (U, 6), rv~1AYjysin llrcu d; ~ J~nll~lnBJ~:
,v;7tl a'ved' Vw (34)
Antenna nrray. lJe will examine a dne-dimengidnal array wieh Cfie inCervnl
d oriener.d along the x-axis. The weigheing funcCidn of an elemwne 1J( (n * E)
21, t - f , tJ ig the antenna retipnnge ed un effeCt in Che fdrm
S lx -
and we have (n) �Eyley ~f,C(,7/exl,
(?�c~~ldr -.f Un reXIoxJ, (38)
(39)
where Che typeg of funetional dependences are determined by equation (6).
('I'he cdm�lex frequency characteristic of a diacrete antenna was uged in
the studies of Yu. Z. 5hlipchenko.)
The analytical expressions for the autocorrelaCion function, specCruro, and
fxequency correlation function for an antenna arrgy are discrete equiva-
lents of (10), (11) and (12).
tJe will toatch the x-gxis of the rectnngular coordinaCe eystem with (an
arbitrarily nrientcd) antenna in such a wgy thet the origin of the reading
coincides with one of the receivera. Then
O, O, f-ltJ~ln4x, U, t JdU' (40)
or
4:J'w 4i;dx,+wl
+~a. . (41)
and
S" ~wl'~.~~~ (wd,~, w/r'~uial,~,-t~; -w/w'`d~t,~ d~;
(42)
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where yj'(x, y, z, t) ig the T(x, y, z, t) field in the Gdnrdinate gystem
relgted td the nntenna
St' d x, l~JdY ) !rJ 11~04, ~ (43)
Avl r.1 vt y + /Ii,r d tu109.1~04 +/I�.ty Ile x~//;v tv, .Z M d~f'� 0~r t� Jf (44)
Here (n11, n12, 1113) and (nzl, n22, n23) nre the Cosineg of the gnglea fdrm-
ed by the old axeg x xnd y with the gxeg of the new codrdinate eygtem; xp,
Yp, zp nre the Codrdinnees di the nld oriQin df coordinateg in the new cd-
drdinaee syseeta.
'I'he regponge cdrreigtidn funnrinn cgn be computed using expregsion (40) dr
expresgion (41) or the expreggions (40) and (41).
Accordingly, we dbtain:

~~~~t~ ~ - r~,....~~~Y'~:~v,, ~%~~,�r~.'n~o~~ oa,rR-.~/� (45) -
4.~,:1X.
, u
/~'`~�(~4! /Wt
J r
� , , . , �~:'a (46)
- � ;~i --(t~ . w)1,rP~~ ~..ft-
~
an t nr
? Oo i
, J -
Guided by models of digital spectrum analyzers using fast F'ourier trans-
form algorithms [7 the 'G parameter can be evaluated at tens of micro- -
seconds. In those cases when this speed is inadequate, it is possible to
recommend use of digital synthesizers of the Fourier coefficients (8],
making it possible to increase the speed 'G to a few microseconds. Finally,
in order to have analyzers with the speed 'G of the order of fractions of
a microsecond it is necessary to use a discrete-analog spectral anal.ysis
method [9]. A peculiarity of the discrete-analog spectral analysis method
is a discrete representation of the investigated signals and an analog
processing of the principal functional transformations constituting the
content of the Fourier transform algorithms. Such a combination makes it
possible to organize a simultaneous determination of the required number
of complex Fourier coefficients at the rate of information receipt.
The possibility of synthesis of an optimum detector on the basis of spectral
- processing indicated above is based on the existence of a spectral (rig-
orously equivalent to the temporal) description of operation of a linear
optimum filter or a correlation detector. It is natural to assume that the
spectral models exist not only for linear systems for the processing of
signals (time functions), but also for systems for the processing of dis-
tributions (space coordinate functions). An example of such a system is
a linear (one-dimensional) antenna system or its discrete equivalent
an equidistant antenna array. An analysis of the spectral model of a lin-
ear antenna makes it possible to clarify a number of useful peculiarities,
_ whose use makes it convenient to synthesize a diagram-forming unit or a
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unit for controlling the directional properties, whoaQ functiioning 3g
basQd on the spectral processing o� in�ormation. The use o� the spectra1
processing method ensures, ae in tihe case of synthesig of an optimum detec-
tor, dQfinitQ technical advatttagos o� the considered systeme.
The functiinning of a band antenna with independent elementis is �ormally re-
duccd to the integration nf the instantaneous values along its aperturea,
taktng into accounti the distribution of the sennitivity of antenna elements
G(x). In this case, using the generalized coordinates u- Tr sin ~t P- ZX/,A t '
N= 2L/,\  where N is the wavelength of the excited oscillationst x is the
space coordinatet Y is distancet y is the angle of incidenca of tihe wave
�ronts 2L is antenna lengtih, the nutput signal is represented in the �orm
"
Ir; u; t) f AP u; 0 Cr (p) dP (10)
As �ollows �rom (10), for a point source of harnionic oscillations situated
in the distant zone, the dependence of the complex amplitude of the output
signal F(u) on the angle o� incidence, ar what is the same, the complex
directional diagram, with an accuracy to a constant �actor is determined
as a complexly conjugate F'ourier transform of the distribution of sensitivity
of elements in the system G(p) [10]
N
F(u) �,frol e,yPW",
~
Expression (11) is a symbolic description of the spectral model of an an-
tenna which can serve as the basfs for developing a method for the electric
control of the directional diagram or organizing a multiray antenna on
the basis of a fixed linear base. In actuality, it follows from (11) that -
for moving a directional diagrazn formed by an antenna with the spatial sen-
sitivity G(p) by the value L1u it is necessary to use the sensitivity (re-
sponse) distribution G Q,u(p)
(12)
. G.u G(P) e
since , v
.w
G(P) e' (13)
' N This situation is a direct consequence of the theorem known as the "movement
theorem" in Fourier transform theory. Thus, in accordance with the modeling
of functioning of a linear band antenna (10), for movement of the diagram
it is necessary to carry out integration of the exci.tations in the aperture
with a new weight. The output signal of the antenna is represented by the
integral
f(u-au; z; t) _A? ~z; u; t~ G!P) e~14PdP c14)
It follows from expression (14) that for movement of the directional dia- _
gram by the angle L1u it is necessary to use a diagram-forming unit which
at the "frequency" L1 u accomplishes a Fourier transform of the spatial
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continuum af the elementary exr,itatinng in the antienna aperture. The
procedure for moving the clirectinnal diagram of an antenna with a st.ipul-
ared distiribution d� response of ieg elemdntis G(p) is thuy reduced ta etle
procedure of carrying outi a spatial Fourier trangform ati e gtiipulated -
angular Lrequency and ehe diagratn-forming uniti is representecl in the
form of an analyzer nf the SpeCtrUI11 of space harmonics.
Applicabla tn a band linear gygeem not having branches along its aperture,
it is virtually impossible td carry nut the spatial Fnurier transform prn-
cedure. However, it is quitie easily carried out for a discrote equiva].ent
of such a system an equidiseant antenna array.
For a discrete equivalene of a linear antenna with a distance between the
elementary detectors c1 equa1 tn half tihe wavelength A , tihe procedure nf an
integral F'ourier transform in the space conrdinate is rpplaced by the pro-
cedure of discretie inertialess weighted summatiion of the sample values
of the continuum o� pertiurbatiions
R ~-ti
fru-au/ z; t) A~t=; u; t~~~,~~~�~.~,K (15)
MI tiV
where Ak is the instantaneous value of excitation of the k-th antenna
array element, G(k) is the response of the k-th element. Then the procedure
of directional diagram movement can be considered a discrete i'ourier trans-
form procedure weighted in accordance with the distribution law for excit-
ation in the space of antenna array elements
~Iwx
f jr/ - arr; 2 ; ,'`'~l G�, tJe (16)
A
where Rk is the excitation of the k-th element of the antenna array, weight-
ed with the coefficient G(k)
(17)
- Expression ,16) is the algorithm for functioning of the diagram-forming
unit accomplishing inertialess summation of the instantaneous values of
the elementary excitations Rk( 2; u; t), weighted by the complex coeffic-
ients exr[-je uk].Such a procedure, as noted, coincides with the procedure
for forming of the complex Fourier coefficient for the discrete distribu-
tion of real readings Rk at the frequency L1 u and can be reduced to two
independent weighted surnmation pracedures
.A..v
~e ~ j(u- e,i; a; f~l - k,. uu ~t�a r~ (18) -
. �,v
,r�.~
l.rnl fj;ret:.; ~,t11=2 ~ t!'nXau (19)
. A ..~r
Each of the weighted summation procedures (18) and (19) is easily realized
using 2N + 1 channel analog summatora in a bipolar operational amplifier
- with a set of weighting resistors stipulated for the required 'A u. Varia-
tions of these resistors, matched with the required law of change L1u,
make it possible to carry out plane or discrete measurement of the angle -
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o� mdvemene of th,i directional di,agram.
In drder to orgaitize a mtlJ.tiray synLem with 2Q *1 independent direCtiional
diagrams it is convenient tr> use as the muleipole cliagram-forming
scheme a maerix weighting tmiti formed as a regulti df the para11e1 cutting-
in oE 2Q + 1 pairs of 2N + 1 channel analoq summators. Itt thfs case eaCh
pair o� summators contains a deffnite set of waigheing resiseors ensuring
formation of the n-th pareial directinnal diagram wieh d stipulaeed move-
ment 4un. Tf the pareial diagrams Con�ozm to the eondition o� uniform
(with the interval 6u) movemonti of the generalized dngle coordinate
ncl, neu (20)
the weight coe�ficientin are determined by the values
~~K ~21~
s^e n UUItNACI ~ In~ n D,l,...~ Qi ~KI �O,l~..,N
a
din nKa u
with formation of the matricesl) CnkII andllSnkll , each with the dimension- -
ality (2Q + 1) x(2N + 1). in this case the algorithm for functioning of
, the muleipole diagram-forming scheme, realizing the processing of 2N+1
signals from the outputs of a lfnear antenna array for the formatiion of
2Q+1 partial directional diagrams,can be wrieten in the form of a matrix
product, separately for the real and fictitious componen's of the cnmplex
directional diagrams:
.
BRef"U-IIC,"BVIRtI (ZZ)
uYm f�MS"?X#RJ
where 11 Rkil is the columnar matrix of perturbations at the output of the
antenna array channels.
'Che procedures for the formation of partial directional diagrams for a
multiray antenna array, represented by expressions (22), are convenient-
ly carried out using two scalar matrix inertialess weighting schemes with
the dimensionality (2Q+1) x(2N+1), in essence representing a complex dis-
crQte-analog space harmonics spectrum analyzer [11] .
3 A diagram-forming unit of the type considered above for each space frequency
(movement angle) auk has two real outputs, representing the real Refn (u -
L1u;t ; t) and fictitious Im fn(u - Ad u; 2it) components of the harmonic to
be analyzed. This circumstance in a number of cases makes it possible to
extract additional information on the parameters of the received signal.
tiowever, in the case of formation of the directional diagram of interest
to us the presence of two separate reception channels is an undesirable
factor, since for each of the channels the directional diagram is repre-
sented in the form of identical lobes symmetrically crossed at the anqle
+ p uk. As a result, the amplitude directional diagrams for each output have
an uncertainty with respect to the sign or direction of movement of the
main maximum. In order to eliminate this uncertainty it is necessary to
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Fori oFric.'TAL usE oNi,Y
proceed from a complex diaqram to itg modulug, which ig eagily achievod
with the quadraeure summaeion of the oueput signals di Lhe c:hanttels for
each space hartndnic. Tho rc,i-ned arnplitucle direCtional diayran, atl otlier
conditiong being equal (number of U5ed deteCticJrg, distributi.on ot respdnge,
angle of movement and frequency of the received oscillations), i.s Gom-
pletely equivalent to the directional diaqratn of an antenna array witti sig-
nal phaging. A distingui.shing characterigtfC and advantage of the Elpentrdl
methdd fdr the forming of directivity is the use of resistor group5 aq thc
regulating elemenes fnstead of the phase inverters used in classical an-
tenna arrays, which is manifested particularly significantly in the plann-
ing and use of antennas with a directional diagram controllable by proyram,
since it ensures imprdvement of the weight and size characteristics of
the clfagrwm-forming units and the cenvenience of eheir eie-in to diyital
control unfts.
We ndte in conclusion ehat the effectiveness of the 5peCtral models and data
processing procedures indicar_ed in ehe examples of cletierminatinn of Che
cross-correlaeion coefficienes fnr signals and the space harmonics spertrum
is retained in the solution of a number of other problems of independent
interest and falling outside tihe framework of tihis study.
BI$L24GF2APHY
1. Gutkin, L. 5., TEORIYA OPTIMAL'NYKH METObOV RADIOPRIYEM11 PRI FLYUK'CUATS-
IONNYKH POMEKHIIKH (Theory of Optimum Methods of itadio Reception in Cttses
- of FluctuaGing Noise), Moscow, Sovradio, 1972.
2. Carpentier, M. H., "Le filtrage dans le detection," L'ONDE ELECTRIQUE,
Vol 51, November, 1971.
3, Van-Tris, G., TEORIYA OBNARUZHENIYA, OTSENOK Y MObtILYATSII (Theory of
Detection, Evaluations and Modulation), Moscow, Sovradio, 1972.
4. Fergusson, M., "Communication at Low Data Rates-Spectral Analysis Re-
ceiver," IEEE TRd1NS. COMMUNICATION AND TECNNbLOGY, V. COM-16, No 5,
October 1968.
5. Williams, J. R., Ricker, G. G., "Signal Detectability Performance of
Optimum Fourier Receiver," IEEE TRANS. AUDIO AND ELECTROACOUSTIC5, Vol
AU-20, No 4, October 1972.
6. Echard, J. D., IIoorstyn, R. R., "Digital Filtering for Radar Signal
Processinq Application," IEEE TRANS. AUDIO AND ELECTROACOUSTICS, Vol
AU-20, No 1, March 1972.
7. Bergland, G. D., "Fast Fourier Transform Hardware Implementations A
Siirvey," IEEE TRANS. AUDIO AND ELECTROACOUSTICS, Vol AU-17, No 2, Jan
~ 1969.
135
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Fox dFFictAL UsE urn,Y
B. Chaykovskiy, V. I "nigcreta�Analog Spectrum Analyzerg on ehe Basia
of Matrix Weighting Devices," TItUDY VII V3ESOYUxNOGO SIMPOZIUMA "N1ETObY
pl2Eb5mAVLENYYA T APPAI2ATUI2N'YY ANALIZ SLUCHAYNYKH pRtYrSESSOV I PhLEY"
(mransactinns df ehe 5eventh All-tJnion Symposium "Methods for the Rep-
resentatinn and instrumential Analysts of Ranclom Procesges and Fields"),
Leningrad, 1974.
Cbi'YitIGHT: Ndrice Ndt Availgble
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FOIt nFFICIAL USE dNLY
UDC.534.6
ONE METt10D FOit UETERMINING THE CnORUINATES 0F A LOCAL NOISE rICLll SOUItCC
Novnsibirsk mRUDY SHE5TOY VSE50YUZNOY SHKOLY-SEMINARA PO 5TATISTICHESKOY
GIDROAKU5TIKE in Russian 1975 pp 251-254
[ArCicle by A. M. Derzhavin, L. A. Bespalov, 0. L. Sokolov, 0. Yu. Boren-
sheeyn and A. G. Strochilo]
[Text] A determination of the coordinates of 1oca1 sources on the basis
of the noise field created by them is an extremely timely problem in atat-
istical acoustics.
The considered method is among the rangefinder-difference methods for the
deCermination of coordinaCes. The basis for the method is the use of a
sufficiently great number of nondirectional detectors situated on a
straight line parallel to Che selected axis of Cartesian coordinates.
We will use the notation yl, y2,�,. ,yn to designate the vertical coordin-
ates o� these detectors and y to denote Che coordinate of the noise aource.
For implementing the method i[ is necessary to satisfy the condition
yi,< y S vn
It is obvious thaC for estimaCing the coordinates of the source y it is
sufficient only to esCablish the number of the detector closest to the
source.
This can be done using the maximum difference of the distance from the
noise source to the first detector and accordingly to each of the remaining
detectors.
From the set of ineasured distance differences p Ri/i = 2, 3,...,n/ we
select
[1 /C/� _ ~(iC, - K[ ~i % ~nsax ~ O ~
(1)
where R1, Ri are the distances from the source to the first and i-th detec-
tors.
.137..
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41
In drder td et]CimaCe the disCance dif�eretices in the case of noiae sig-
nn1s it is moge commnn tn measure the time 1ng nf the Crngs-correlation
funcCinn -
Tiie condiei.ong for che applicabiliCy of cnrrelgtian analysis fnr acnustiC
prnbtemg are analyzed in derail in [1]. it ig shown, for example, ehat
Clte use df correlntion analyeis ig preferable only in ehoge casee when the
width of the signnl spectrum is adequgCe fur the forming nf a poinCed cnr-
relation �unction.
Thig condieinn fipplicable to the described meChnd tg expresged in the fol-
lnwing way
Af t i 3 /Tl
~i,:~~ ~2~
where 6 f ia the width of the energy spectrum of the noige signal, Pl'i(L),
Rl,i+t('C) are the maxima of the cross-correlation functions of the signals
from the first and accnrdingly from the i-th and from the (i+1)-st detec-
tors, t i i+l ig the eime lag between the extrema of the cross-correla-
tion functions Rl'i(t) and R1,i+l(ti), m is the error coefficienC fnr
evnluaCion of the correlation coefficient.
A preliminary evaluation using formula (2) shows thaC with lag durntions
'~i, i+l of the order of msec, being uaual for the correlation analysis of
acoustic signals, the width of the spectrum must exceed hundreds of cpa.
Expression (2) does not take into account the nonstationary nature of real
noise signals. We will esCimate the error in measuring the correlation
function caused by the nonstationary staCe. Since the use of discrete
methods is characteristic for developing the correlation analysis approach,
we made an analysis of the error in the example of ineasurement of the auto-
correlation funcCion of a signal quantized by level and time.
For generality we will assume Chat the quantization levels can be variable
but limited in value, Chat is
-A m,zd < -0
Similarly, for time intervals the following is correct
We will also assume that the noise process X(t) has a zero mathematical ex-
pectaCion and the covariation function BX(tl, t2).
As an evaluation of the covariation function of the noise process X(t) we
will take a random value determined by the expression
BXIt~ ~~t+nnt)�.~(t+cfaat). (3)
.~.i
where N is the number of observation points in the process x(t), obtained
as a result of quantization of the X(t) process by levels and time; d t
is the time interval between adjacent observations.
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Fdtt tlFrICIAL USE ONLY
For cach C we use the ndCaCidn (ej m max, L Ci,< tii
nue en the facr rhue we hnve the ittequaliCy -
~c l~2 p ` u., >
we dbeain the exrresaion '
_ x~t) ~~'~~t>) ~ d'(!t~) s z. c4>
'Chen iC ig pngeible Cn wriCe
x(1E,) - x(t) - (tt) -tJx" rtJ t o(t - W) cs>
Froro the expressions (4) and (5) we obtain gn expresgion for the qunnCized
process
x(t), x(e) + d'(N) - (t - Ct]J.rt (t) { e (t - Ct1)
(6)
Using (6), it is possible eo write bn expreasion for the producr of
values
of the quantized process at the momenta in time tl and e2, separgted
interval 'L, and Chen, proceeding to the mathematical expectaCion we
by the
final-
ly obCnin < Bx{t`,~
- 'ae t t ) at,
a - (6r ~t~ +C7'xod f
(7)
Using (7) it is possible to find the mathematical expectation of the
i
eval-
uation
Z (t, 'G, p t, N), and then replacing the sums by the correspond-
ing integrals and assuming T a N AC, we arrive at the expreasion
,c r
/ All8r rt2; 4t, N) 4 T J Bz (t+12, t tZ'+ z)de +
+ d~ //_aBt~f~L t-o+ t`4Z'ir3 ll~z' i~
' l x a
0
(

T I
a~,
s
~
~2~~ ~~x(t~~)+6,~(f+r~~Jld~ + Z ~Bx(tfTt~rfT) -
- Bx ~t, t~z)~ �
Now the sought-for expression for the modulus of the bias ot evaluaCions
can be writCen at once:
n
~a B(t 2' o t N ~
2, /T'
S lL ~x~l~t2 tttt~)~~ -Bx~t~~tZ'~l* l
(9)
86,C (tt2, L`rzt2) p ,
f ~ttz, f~t rtf+
o
(rfz tfc~r).-~x(t,t{r)/.

+zr~B
-
~

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FdIt UFFICIAL U5E ONLY
Lxpregsion (g) mnkes ir pnssible to cdmpuCe the errdr in biaa of the eval-
uation of ttie CorrelaCinn funeeinn of rhp nnnaCaCionnry nniee signalg,
- quantized by level and eimp. In a specigl aase, aggwning Q-+ 0 and a-�--t, 0,
ie ig poggible Co compuee the bias of the evaluaefon of Che correlueidn
funceion of a nonquantized noige gignal. -
- Ag un example, we can cnngider g proceas of the Cype x(C) - ~P(t)�
z(e), where V (t) - e,8 C, z(C) ig a Gauggian procegg wiCh a zero mean and
the correlaeidn funcCinn -
~a ~t~ ~ e �~%rl
In thig CF1ge from fdrmula (9) we obtain
n
foBx(t,t,at,~V)J~/>!~/~$e~:,U..~, ,d.
~e~. l~+ y~di eyr~1+e=~r)
,
.qpT f~ + or~s eVt # 0-d1t
. . T r
The evalugeion of the bias error makea it possible to clarify the influence
of nonstationarity of the noise signals on the accuracy in measuring the
coordinates of local snurces of Che noiae field. It ahould be noCed that
this systematic error has a tendency to increase with an increase in the
' number of readings or an increase in integration time.
BIBLIOGRAPHY -
1. Broch, T., BRUEL AND KJAER, TECHNICAL REVIEW, 4, 3, 1970.
COPYRIGHT: Notice Not Available
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,
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- UUC 621.391.82:534.88*621.396
INVESTIGATION 0F mHE NOISE IMMUNITY 0F A STANUARD UETECTION CHANNEL tN TNE
; ItECEPTION OF ATJO-COMI'qNENT SIGNAL WITH A NAitltOW-BANn NdISE COMpONENT
- Nnvosibirsk TItUUY SHE5TdY VSESOYUZNOY SHKOLY-SEmINARA Pn STATI5TICHESKOY
GZUIt0E1KUSTIKE in Russinn 1975 pp 255-262
(ArCicle by V. M. P'yanov]
(Text) In some praceical applicaeions of hydroacoustics one mugC coneend
wieh the detection of a so-called two-compnnenC signal 82 (t), one of whoge
components s(t) is a.deeermined (or quasidetermined) componene, the second
gp(t) is Che noise componene (for example, gee [11). In this connecCion iC
can be of interege to invesCigate the noise immuniCy of n standard detec-
tion channel (SDC) with the reception of euch a signal. In this study it ie
propdsed that s(t) is a narrow-band radio pulse wieh a fixed amplitude A
gnd a random initial phase ,8, existing also in sp(t), in the interval
[tp, tp + Ts]. The noise component is a random, norna1 quasistationary narroW-
band process with a zero mathematical expectation and Che correlation func-
tion
ep (i, u>= G'~~ A M)ros w, (t.-
i- /-f"-! , /f-u/a r, (1)
tP (f, u)S r
T (2)
~ wPawer~w~
The power spectra sp(t) and s(t) differ only by a shifC along the frequency
axis by 41.
- A.s additive noise N(t) we used white noise with the correlation function
(3)
We will examine a SDC consisting of a preselector, quadratic detector (QD)
and low-frequency filter (LFF).
We will determine the signal, noise and signal-to-noise ratio at the SDC
output with the arrival of a mixture of the above-mentioned signal 82(t)
and noise at the channel input.
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FoR nFFtcinL usE nrtLY
it is annumed thaE the efEeetive eranapareney band af the proseieeeor ia
D1~j --0 Af t 44
[3 =eff(ecEive)] where A feff g and L1faff p in the effective aideh di the
gpeetral band a(t) and ap(t). The reapdnee of the preselectdr to the naige
and determinpd CnmpdnenCg nf the gignal gp(t) ig
In the presence of a aigngl aC the preseleeeor output ae dbtain
yr srlJ* ~,~l~. yN~{~ (4)
whpre 4 N(t) ia the regpnnse of the prngeiecenr tu ndige.
Aa the gignal C nt the ctiannel output we uge (2) the inrrement of the math-
ematieal pxppceaeidn of vnleage at the output, eauged by the presenGg of
a signal at the time of the reading tp *T. Althbugh usually ve se1ect T~
Tg, vp cite the signal and nnige values for a morp general cgge Whpn for
nne reason or andCher T4 Tg.
In thig case Q G. /
�'er)/"`r~~�~~ (s)
.
where az is the detection constant, h a(16) is the "impulse" transfer char-
acterigtic of the LFF
i
~ w . ~ /P (6)
9~T ;re ~ r; M= af, T; 'p+ T a t
As the noise 7T we use (2) the mean aqugre value of channel reaponse at the
reading time.
Arcordingly, ~�r
n~~//h~r ~r-t~~s ~E�r (f;E~d~ot~`~~ ~ ~ t � i ro ~ ~ ~ t ,
G (7)
Where 82 E 0(tl; t2) ie the autocorrelation function of the low-frequency
romponent of the process z,,(t) at the detector output.
Taking into account the nondependence and normality of the procesaes sP(t)
and N(t) and representing the the correlation funetions of the nolae com-
ponent and the noise at the preselectnr output in the form
8~, (f ~J=sta~, Iv~lt'~:�a~,Xf.,,4J ' (8)
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14
FOIt dFFYCIAI, USE dNLY
i
fifeor n number nf txyn8farmneinns we dbtain
t (9)
t!� d~ G~~~6/frz~~'f)cr'~;Jt', .
where r,~, , �
~f,r jf)hd j~� c; a'f,
.
(1o)
~r~3Q~ f~r NlDdaal~^. -
cvr I w, i r r'rV
The resU1ts nf camputation df the gignal-tn-nniee rgtin fnr a cage when
the LPF ig gn ideg1 integrator (II) and the preselector ig an individual
regnnance circuit (IRC) under the eondition ehaC AiJg = 0 are given in
Fig, i.
C
R
.y
4
3
.2
!
0
tr
Fig. 1.
It follovs from Fig. 1 that the signal-to-noise ratio increases with an
increase in but the rate of thie increase is dependent on r. In the
case of large Lr the gignal-to-noise ratio is virtually not dependent on
ji and on r and approaches r2 . It also follows from Fig. 1 that when
_ �fA1 there is a decrease in the signal-to-noise rat3o. With
M= SO the � 1 value is of the order of 15-20; a decrease (or increase)
in M cauges some decrease (or increase) in �l.
Thus, cases are possible when an increase in s p(t) or s(t) at the channel
output leads not to an increase, but a decreage in the signal-to-noise
ratio at the channel output; an increase in t or fL causes a greater in-
crease in noise than the signal at the output.
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rok orFzctAL usE nrtLY
In ehim c:onneceion the quesrion ariaeg nf hnw an increase in r or � ex-
prCm an influence on the reliabiliCy of deCecCion of a Cwn-COmpnnent eig-
na1. In nrder tn nbeain a rignrnus answer eo thig question iC is necegeary
eo determine the digtributinn law for the procees at the output and Chen
nn the bagig nf this law consCruct the detecCor deCecCion ahnracterisCic
curve (bCC). 1lowever, unfortunately, iC was noC posaible to deCermine the
precise diyeributinn luw for the procesa at the channel output. IC ia neces-
snry to be saCisfied wiCh different approximate representaCions.
If the prdcesg ne the SnC ouepue is uesumed to be normal, the DCC is deter- -
mincd by the parameterg (2]
< �
7 (12)
rs�
where C1'2uNo and G 2uZp is the disperaion of the process at the channel
output in Che gbsence and in the preaence o� a aignal aC the i.nput.
An increase in the dispersion at the output leads to a decrease in d2 and
accordingly to some increase (wiCh a given dl) in the probabiliCy of cor-
recC detection D. Thus, the influence of a decrease in dl with an increase
in S'or which was menCioned above, on the reliability of deCection can
to one degree or another be compensated by a decrease in d2.
The question arises of the possibiliCy of approximaCion, by a normal law,
of the process at the LFF for a two-component siQnal.
It is known that the norcnalizing effect of LFF is manifested only in the
cASe of a large M value. Therefore, for a two-component signal, to the
usual condition of normalizability M>>1 is added the additional condition
,A f ef f pT� 1, wtiere A f ef f p is the ef f ective width of the band sP (t) at
the preselector output.
However, for sp(t) there is satisfaction only of the first condition and
Cherefore the DCC,,constructed on the assumption of normality of the pro-
cess at the outpuC, may be inadequately precise. Assume for the IRC-QD-II
channel ehere is satisfaction of the condition M� 1 and the condition
Af ef f pT < 1 or Af ef f p< 2 A Fef f To where AFef f T is the II ef f ective
transmission band [2].
ldith satisfaction of these conditions the process at the LFF input (see
expression (11)) can be divided into two groups: one group (consisting
of several components), which with passage through the LFF is virtually
normalized, and the second group (consisting of one component sp2(t)), for
which passage through the LFF 'is virtually not related to the change in
the distribution. In this case the distribution of the process at the LFF
output is determined by the cQmposition of the normal and exponential
tlistributions; on the assumptior. of a nondependence of these distributions
the resultant one-dimensional distribution density of the process has the
form [3]
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FOR 0FFICIAL USE ONLY
ea~'
(13)
where T (x) is ehe Laplace ineegral; 6 ~ QJ/(~'2; al, 0 7are ehe parn- _
meters of the exponentinl and normgl initial diatributione.
We wi,l:l call (13) xn exponeneial-normal distributinn.
An equation for the deeeceor working characterigeiC (UWC) follows frum (13)
(14)
where T-1 is a function inverse of tj 3, ,Q are the diepersion nnd
the mathemaCical expectntion of the process aC the channel outpuC in the
absence of a signal ae the input; a is Che maChematical expecCation of a
group of components having a normal distribution; F is the probability of
a �alse alarm.
Idith a norroal distribution the DWC equation has the form [2]
(15)
Thus, an examination of the SDC noise irmmunity with the reception of a two-
cotqponent signal with a narrow-band noise componenC is carried out by suc-
cessive approximations:
a) zero approximaCion evaluation of noise immunity from the signal-to-
noise ratio;
b) first approximation determination of the DCC on the assumption of a
normal distribution at the ouCput;
c) secnnd approximation determination of the DCC on the assumpCion tha[
this distribution is an exponential-normal diatribution.
Some results of computations of the DCC in the case of normal and expon-
ential-normal distributions of the process are represented in Figures 2,
3 and 4, 5 respectively. A comparison of these curves indicates an insig-
nificant dif�erence between the DCCs in the first and second approximations.
The difference increases with an increase in 2r and when al= 0 disappears;
for,rJ 18
,pj -~M r G J / )
A_(TM1�, ~1��e ~ f q le-
Taking (16) into account, we obtain
t /U~ T r 1 e-~N ~
pM ,?ri~3i'-/JJj 1 (19) -
The dependence _ pM e
�~~i~l~G~~, .w + 2r~K(2M (20)
on M�or values.of the parameter A2/4 G n= 0, 0.1, 1, 10 and 00 (Q 2= 0)
is shown in Fig. 6. p
~
Fig. 6
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FOtt UFFICIAL USE dNLY
[t c:.tn be seen from Fig. 6 ChuC the oprimum vglue Mopt a d feff dpC'T o
0.5-0.7 with vireunlly nny relaeionships beeween Che deeermined and nniee
Compdnenty of the gignal.
'Chu,-4, the presence of the Cnnyidered noise component exerts virtually nn
tnfluenCe on the chnice of Che nptimum eraneminsion band of the fi1Cer,
whiCli is determined only by gignal duration.
BIBLIbGE2APHY
1. bl'shevskiy, V. V., 5TATI5TIC1-tESKIYE METObY V GIDROLOKAT5II (Seaeis-
ticul Methods in 5nnar), Izd-vo "Sudostrnyeniye," Lenitdrad, 1973.
2. Caekin, N. G., Geranin, V. A., 1(arnovskiy, M. I., Kragn,yy, L. G.,
, POMEKHOUSTOYC112VO5T' TIpOVOGU TFtAKTA OgNAItUZ1iENIYA SIGNALOV (Nnise
Immunity for a 5tnndard Signal betection Channel), Izd-vo "Tekhnika,"
- Kiev, 1971.
3. p'ynnov, V. M., "Cnmposition of Normal and Exponentinl Uistributions,"
VES'TNIK KPI, SEItIYA ItAbIOTEKHN?IKI IELEKTROAKUSTIKI (Herald of Kiev
polytechnic Institute, Series nn Rgdio Engineering and Electroncous-
Cics), No 8, 1971.
4. Catkin, N. G., P'yanov, V. M., "On Clioice of Che OpCimum Preselector
liand in'the Detectiion of a Two-ComponenC Signal," VESTNIK KPI, 5ERIYA
RAAIOTEKHNIKI IELEKTROAKUSTIKI, No 8, 1971. -
COPYRIGHT: NotiCe Not Available
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roK nFrrctnl, U5E ONLY
UllC 534.883.4 -
UETCCmION OF 5IGNAL5 WITH UNKNOtJN pARAMETEItS ON A ItEVERBEitATION BACKGROUNU
Novnsibirsk TItUUY 51125T0Y V58SOYUZNOY 5HKOLY-SEMINARA PO 5TATISTICHESKOY
GIUROAKUSTIKE in Ituasian 1975 pp 263-269
(Article by G. S. Nakhmnnsdtt gnd V. V. Pavlov)
[Text] In a gCudy of Che problemg involved 3n Che detection [1-3] nf sig-
nals agninse n noise background iC is usually aeaumed ehaC Chere ie a co-
incidence af Che pgrgmeters af the received and reference signals, which
clneg nnC geem possible in n real sieuatinn. A miamaCch nf parameCera can
lead eo an appreciable deterioration of Che detecCion characCeristics;
Cherefore, simultaneously witti detection iC is necessary to evaluaCe Che
signal parameters (4]. In Chis connection it is of inCeresC to examine Che
detecCion chnracCerisCics of a maximum probability detector, comparing
the output signal maximum with the threshold.
Characterigtics of Uetection of Signal With Random Amplitude and IniCial
Phase
Assume that an additive mixture arrives at the detector input
x(t) � s (t, e,, a,, V, ) + n�(tJ * ~Lrt) (1>
where S(l,> t
- K r uo Here the symbol 11 II denotes the norm or length of the vector. Thus, with
a suitable choice � the weighting vector can be made arbitrarily close to
the vector opt W. However, if it is assumed that the vectors ?(k), X(k+l), ~
obtained by successive measurements are statistically uncorrelated and
Gaussian (and this should be correct in practical cases when the time in-
terval between successive applications of the alqorithm is great in compar-
ison with the intervals o� correlation of the results of ineasurements),
convergence is ensured with less rigorous conditions. Specifically, if
_ the scalar component f-l- satistied_the inequality
0
In the aase of a single-regime adaptation procedure
x+ w
The mean vector itW, obtained as a result of use of both single- and two- -
regime adaptation procedures with a control signal, in a general case
will not be equal to the weighted vector, corresponding to the criterion
of a minimum of the mean square error opt W. This is correct even in those
cases when the control signal ideally reproduces the real signal �rom the
target g(k) = g(k). A bias ot the solution arises because the input sig-
nals of the processor usually contain not only a control signal, but also
the real signal from the target. The bias can be eliminated either by
carrying out the adaptation process in the course of intervals when there
is no real signal from the target or by means of use of a control signal
whose amplitude is small in comparison with the amplitude of the signals
actually received by the array.
The author of (8] examined the conditions for convergence of the algorithm
(10) for the vector of the coefficients in the case of an unknown cor-
relation structurie of the signal. As demonstrated in the study, the eval- -
uation (10) is unbiased. The rate of convergence and the dispersion of
the evaluation, as before, is determ3ned by the choice of the coefficient
-

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,
300 1 . . ANO ~
OF THE 6TH ALL-UNION SEMINAR ON STATISTICaL
31 JULY 1979 NYDROACOUSTICS (FOUO 2l79)
30
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5ummary
'I'iiis paper censtitutes a review of inethods for the adaptation of hydro-
acoustic syytems, for ehQ mont part, hydrdacoustic receiving antenna
arrays.
The article gives an analysis nf aigorir.hmg obtained by different authorn
fdr the adaptive optiimizatien d� antienna systiems intended for tihe c1eteC-
eion and discrimination nf a ugeful siqnal and tihe suppressinn of noige
signals under condStions of a priori ambiguitiy nr changing input effeces.
The eongidered $ystems enngigt of linpar, plane or spatial antenna arrays
and a proCessinq unit designed in the form of a sQt o� flolay lineg and
_ lead-affs, with wefghted sununation and aubsequenti averaging. The adaptive
algoriehms are usQd fnr iter+eive "ad3ustment" of the weighting coe��ic-
_ ients in accordance with the criterion o� minimum of the mean square
errur. Such dn antienna systeM can be used in frequency and direction fil-
tering. In the course of the work it changes the directional diagrcun
proper, the frequency characteristics or other parametiars by means of
- an internal feedback and thereby increases the response to the useful
signal and rQduces response to noise siqnals, asymptoticaily tending to an
optimum variane.
_ In mogt of the analyzed stiudieg it is postulated that the sources oF the
usefv.l signal ar.e localizpd in space and cttn be regarded as point sources. ;
The noise is cause9 either by point sources in the medium surrounding the
array elements nr by the thermal noise of the amplifiers. The useful siq-
nals and the signals �rom the lncal noise sources are propagated in space
as uniform plane waves. The space in which the antenna system functions
is linear; the influence of space on the siqnals is reduced only to their
time delay.
Most of the cited adaptive algorithms were developed for a case when the
direction to the source of the siqnal and its spectral characteristfcs -
or frequency bands are asyumed to be known a priori. However, the direc-
tion to the noise sources, the amount of noise, its spectral anrl correla-
tion characteristics may not be ):nown, the noise can change its position
and characteristics, can appear and disappear fn the course of reception.
Such a formulation is characteristic �or problems in underwater cammun-
irations. Available a priori iniormation on the useful signal makes it
possible for the antenna systF-.m to form its diagram in such a way that
its main lobe is directed toward the signal source and makes it possible
- to determine the width of the main lobe and the frequency characteristic
of the filter. With arrival of signals from directions other than the
stipulated direction, the system classifies these signals as noise sig-
nals and for:ns diagram zeroes in the direction of their arrival.
However, in the most extensive and important class of cases of practical
interest the direction to the source of the useful siqnal not only is
not known, but frequently must be determined. In this case the problem
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VOlt UF'FIOtAI, USC ON1.Y
arises of usinq adxptatiidn fdr geeking a signal and determining ita bear-
ing.
'1'he pxoblem is tio detece a signal in tihe field of digtiributed and 1oca1
nnina. Neirhex the dirr.ction of arrival of the eignals nor the direction ~
nf arrival of the ndise are known. iti is of interest to examine a class
of noise broader in compari.son witih point sourcag, with mnre complex spa-
tia1-tiemporal charactprigeics, fnr example, sea noise, reverberation noise
and nther types nf noise characteristic for hydroacoustic appliCationg.
ir is necessary to aynthesize such a spatial-frpquency filter which un
the bagis of use o� characteristic crieeria o� a useful signal (for Ex-
ample, gpeCtral, correlatiion and other properties) will ensure receptl.on
nf this signal and simultianeous suppression of a11 noise. This problem
in parti is dealt with in [14, 15]and elsewhere .
Notiatfong
uj(t) is the signal ar the output of the 3-th detector in the antenna array,
j - 1, 2, 3,...;
Ni (t) ) is the filter at the output of the j-th antenna array detector, the
array being designed in the form of a delay 1ine with I.. lead-affs;
Vj (t) is the signal at the filter output Hj (W)
;
is tihe summator;
Y1(t) is the signal at tihe output of the summator N;
G(W) is the po5t-summator filterj -
Y2(t) 3s the signal at the filter output G(cJ)j
S is the squarer;
Yg(t) is the signal at the output of the squarer S;
Oodt is the averager;
Z(t) is the signal at the output of the antenna filter;
0 is the delay time between filter lead-offs Hj (cJ);
xi(t) is the signal in the i-th lead-off of the processing unit, i= 1, 2
...,KL;
xi(k) is the signal in the i-th lead-off corresponding to the k-th adapta-
tion cycle;
ur. is a weiqhting coefficient by which the signal xi(t) is multiplied;
ni(k) is the tatal noise in the i-th lead-off in the k-th adaptation cycle;
si(k) fs the useful signal in the i-th lead-off in the k-th adaptation
f cle;
is the vector af the weighting coefficients wi;
X(k) is the vector of the signals xi(k);
ir(k) is the vector of useful signals si(k);
I'~(k) is the vector of noise ni(k);
_ g(k) is the signal from the target perceived at the origin of space coor-
dinates by a non-noise array element invariant relatfve to direction;
RSS is the correlation matrix of the useful signal;
RNN is the correlation matrix of noise;
Rxx is the correlation matrix of the process at the input of the processing
unit;
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S(k) is the errar in evaluatiing the signal g(k)s
6 Z is the mean square error Sn evaluati~ng the signal g(k)t
dpt W is the optimum weighting vectior, corresponding to the Wiener filterj
~t ig t~e vecror o� cross-correlation between the vector of tho observed
s~gnal X(k) and the signal from the targeti g(k)t
� is the intervai of the iteration procedure during adaptation;
d(k) is the "desirQd" signal or the signal which must be obtained at the
nutput of the procESSing unit;
MW(k) is the mean weighting vectort
RWW(k) is the weighted autocorrelation matirixj
a max is the maximum eigenvalue of the correlation matrix Rxxt
mI, mII is the number of adaptation cyclea in regime I and in regime II
respectively in the cage of a two-reqime adaptation procedurer
� I, � ri are constantis determining the magnitude of the interval in -
regimes I and iI respectiivelyj
Y is the amplitude of the oscillatiions at the output of the generator of
control signals;
Rss is the correlation matrix af the control signal;
Pg is the vectar af cross correlation between the vectar of the control
signal and the signal �rom the target.

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Fon orrzcrni. usE oNLY
BIBLI OGItAPHX
= 1. Tgypkin, Yg. Z., AbAI'TAmSIYA I OBLUC118NIXC V AVTOMATICHESKIiCH SISTt;rt-
AKH (Adapration and Teaching in AuComaeic SygCems), Moscow, "Nnukn,"
' 1968.
-
2. Tsypkin, Yn. Z., OSNOVY rEORII OBUCHAYUSHCHIKHSYA STSTEM (FundumenCals
of the Theory of Tenching SysCems), Moscow, "Nauka," 1970.
3. StraConovich, G. A., pItINTglpy AIlApmTVNOGO PRIYE-MA (principleq nf Adgp-
Cive Recepeion), Moscow, "Sov. Radio," 1973.
4. Balnkrishnan, A. V., "Conciae Review of Some Moder..n Problems in the
Theory of Coromunications," TEORZYA SVYAZI (Communicatinns Theory),
trnnslated from English under ehe editorship of B. R. Levin, Moscow,
"Svyazl 1972.
5. Shgkhgil'dyan, V. V., Lokhvitskiy, M. S., METODY AAAPTIVNOGO PRIYEMA
5IGNALOV (Methods of Adaptiive Reception of Signals), Moscow, "Svyxz',"
1914.
6. Grif�iths, L. J., "Simple AdapCive Algorithm for the Processing of Sig-
nals of Antenna Axrays in Rea1 Time," TIIER, Vo1 57, No 10, pp 6-16,
19 69 .
7. Widrow, B., Mantey, A. I., Griffiths, L. J., Good, B. V., "AdapCive
Antenna Systems," T'IIER, Vol 55, No 12, pp 78-95, 1976.
8. Frost, S. L., "Algorithm for Linear ResCricted Processing of Signals in
an Adaptive Array," TIIER, Vol 60, No 8, pp 5-16, 1972.
9. Shor, S., "Adaptive Technique to DiscriminaCe Against Coherent Noise
in a Narrow-Band 5ystem," JASA, Vol 39, No 1, 1966.
10. Chang, J. H., Tuteur, F. B., "A New Class of Adaptive Array Processors,"
JASA, Vol 49, No 3, 1971.
11. Ligett, W. S., "Passive Sonar Processing for Noise With Unknown Cavari-
ance Structure," JASA, Vol 51, No 1, 1912.
12. Winkler, L. P., Schwartz, M., "Application of a State Variable Tech-
nique of the SNR of an Underwater Array Sub3ect to a Constraint on the
Supergain RaCio," IEEE CONF. ENG. OCEAN ENVIRON. REC., San Diego, Cal-
if., 1971, N. V. 1971.
13. Winkler, L. P., Schwartz, M., "Adaptive Nonlinear Optimization of the
Signal-to-Noise Ratio of an Array Subject to a Constraint," JASA, Vol.
52, No 1, 1972.
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Lq. 5am, Charles L.,"inElugnce o� Errors in Dntiermining the Ang1e og In-
cidanae on the c:haracrorigtice o� an Adaptive Array," TJIER, Vol 60,
No 8, 1972.
15. Riegler, Robert L., Compton, Ralph T.(Jr.), "Adaptive Antienna Array
_ for Nnise Suppression," TY2ER, Vol 61, No 6, pp 75-86, 1973.
16. O1'shevsk3y, V. V., "Adaptive Methods for Minimizing Errors in Stat-
istical Measurements," T VSESOYU2NYY STMPOZIUM "METODY PREDSTAVLENIYA
I APPARA'.CURNYY ANALIZ SLUCHAYNYKH PROTSE5SOV I POLEY" (F'irst Al1-Union
Symposium on "Methods of Representation and Inatrumental Analysis of
Random Processes and Fields"), Novosibirsk, Vol I, 1968.
17. OL'shevskiy, V. V., "Adaptive Methods for the Optimization of Measure-
ments o� Characteristics of Nonstationary Random Processes," II VSE-
SOYUZNYY SIMPOZIUM "METODY PFtEDSTAvLENIYA I APPARATURNYY ANALIZ
SLUCHAYNYKH PROTSESSOV I POLEY," Novosibirek, Vol II, 1969.
18. zufran, A. M., "Adaptive Methods �or Measuring the Current Coordinates
of Signal Sources. Part I," TRUDY VTOROY VSESOYUZNOY SHKOLY-SEMINARA
PO STATISTICHESKOY GIDROAKUSTIKE (Transactions of the Second All-Union
5eminar-5chool on Statistical Hyflroacoustics), Novosibirsk, 1970. Part
II, TEZISY DOKLADOV TRET'YEY VSESOYUZNOY SHKOLY-SEMINARA PO 5TATIST-
ICHESKOY GIDROAKUSTIKE (Summaries of Reports at the Third All-Union
Seminar-School on Statistical Hydroacoustics), 1971, Moscow, 1972.
19. Kurzenev, V. A., Perov, V. P., Shelomanova, S. S., TEZISY DOKLADOV
TRET'YEY VSESOYUZNOY SHKOLY-SEMINARA PO STATISTICHESKOY GIDROAKUSTIKE,
Moscow, 1972.
COPYRIGHT: Notice Not Available
5303
CSO: 8144/0938
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Fox OFricYAL usc otvLY
unc 621.391.2
ALGORITI-IlMS FOR PROCESSING SONAR 7NFORMATION iJNDER A PRIORI
UNCERTAINTY CONDITTONS
Novosibirsk TRUDY SHESTOY VSESOYUZNOY SHKOLY-SEMINARA PO STATISTICHESKOY
GIDROAKUSTIKE in Russian 1975 pp 320-327 .
[ArCicle by Yu. Ye. Sidorov)
[TexC] Under Che real conditions of sonar observaCion Che sCaCisCical
characCeristics of the reg3sCered processes vary in a broad range and
cannot be a priori completely known to Che observer. T}iis circumsrance
makes it necessury to sulve the problems involved in the statisCical. syn-
thesis of algorithms for the processing of sonar in�ormaCion under
a priori uncertainty condiCions.
The deCecCion algoriChms, Che algorithms for deCermining coordinntes (pri-
mary processing) and trajectory of movemenC of objects (secondary process-
ing) are based on use of the principles of nonbias [1, 2], invariance [1]
and the theory of rank crit-eria [1, 31 when checking complex statistical
liypotheses, which makes it possible to synthesize methods with a structure
which is extremely stable relative to real observation conditions.
I. Rule for Detection of Echo Signal
The detection of a signal ref lected from an ob3ecC in the form of a packet
of pulses (such a type of signal is very common and is created, for ex-
_ ample, with the use of a periodic source of the explosive type) must be
accomplished in two stages and in each of these stages there is an inde-
pendent optimization of the processing rules. The first stage is the stage
of binary quantization of the received oscillation, whereas in the second
stage there is an accumulation of quantized signals for the purpose of de-
tecting the packet. The rules for each stage are �ormulated as the rules
for checking Che hypothesis Ho of absence of a signal relative to the al-
_ ternative H1 of its presence.
The binary quantization rule is based on a comparison of oscillations ("con-
trast method) received from n(n >,,2) elementiry radar range resolution
sectors. An "inspection" of the distance is made by the sequsntial method
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FOEt 0FFICIAL USE nNLY
, on the n8sumpCion ChaC Che targeC cnn be situgted nnly in the 1asC, n-th,
oE rhe compared range secCors. Ttie rule doeg noC require u priori informtt-
tinn nn the di.etribution lnw �or the noiae background and uses rnnkg of in-
Itinl abservnCians. [By "targets" is meant aingle or group reflectinrs, such
as sea anlmalg, surface lnyer acatterera, bOCCom, etc.] It can be demon-
strneed Cl1AC this rule ig the uniformly moat cl.early expreased i.nvariant
rank ru1e in the form:
1 when Rp > C,
(1)
~(gO) a ~ when Ro m C,
0 when Ro < C,
where Rp is the r.ank of the value 1/", being a xeading of the envelope of
the oscillatinns in the n-Ch rnnge secCor in the J-th period of repetiCion
of the pulses, 3= 1,..., hj;m(Rp) is the probability of a decision in
�avor of the H1 hypothesis. The Chreshold number C and the probabil:ity a"
_ are deeermined unambiguously with respec+c to the sCipulaCed probability of
a fnlse alarm m l from the condiLion:
_ c~G; : E'~ !46/.�P~~.J (2)
Here the averaging of Ep is carried out using the distribution of the rank
Rp with Ho, which is equal to 1/n. This indicates that rule (1), (2) has
a constanC probability of false alarm wiCh any noise intensity and any
law of disCribution of the noise background.
Computation of the effectiveness of rules (1), (2) with the approximation
of noise by a normal law, being typical in the reception of an echo sig-
nal against a background of reverberation noise [5, 61, and with "harmon-
ious" fluctuations of pulses in conformiCy to the Rayleigh and Rice laws,
show ChaC the losses in the signal-to-noise ratio caused by a lack of know-
ledge of the disCribution law for the noise background and its intensity,
and the use of rank valuea instead of observed values, with an increase in
n decrease and when n> 10 for all practical purposes tend to zero.
The zule for the second stage is a randomized invariant rule (RIR) in the
form:
1 when X> M,
CP(x) _ L S when X = M,
- 0 when X = M,
cahere
and xi assume the values 111" or "0" with the unknown probabilities p and
1-pj and are solutions indicating the presence or absence of a signal re-
spectively, used in the first stage. The threshold M and the probability
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FOk nFFICIAL USL ON1.Y
S are deCermined on the bayiy of Ghe given resuleune prdbabiliCy of
a false alnrm from the condiCion
(4)
e1'C,~e~ ~/~-e~ ,W
where ch is the riumber of combinations of h re~.aCive to k.
It Eollnwe from condieion (4) that the numbers M and d' nre deeermined nnly
by the probabiliCy C06Z and are not dependent on the unknown probabi.liCies
pj, which tn the presence of g Cttrget do not remain constant with C-Lme ~
ns n resulC nf nonstationarity (in a general case) of the nol.se background
zlnd the reflected signal. Accordingly, the rule (3), (4) ia not dependent
_ on the a priori unknown noise intensity, on the distribuCion lAw for the
mixture of signal and noise and operabiLiCy in the case of a nonsCationAry
_ state of the observed process.
T'he effectiveness of the rule (3), (4) is determined by the inrensity func-
tion, which with identical pj = p has the form;
~'t /4�/! ~ w/ I ~ ~ ~II r
~ and increases with an increase in the number h of pulses in the packet. Due
to the fact that the algorithm for deeection of the packet is "two-peaked"
' and the dependence 0[2 = g(oc1) is known, it is posaible to construct a ~
family of curves for detection of the packet for different values 011, a 29
n, ti, by using which it is possible to selecC the becessary work "regime"
Eor a two-sCage deCector in general. Comparison of such a detector with
the Manna-Whitney detecCor, in which in the second stage use is made of
the sum of the weighCed rank statistics (which leads to a result close to
- opCimum), and not the adopted decisions themselves, shows that the losses
in the signal-to-noise ratio of the synthesized detector in comparison
wiCh the Manna-Whitney detector with an increase in the number of accumul-
ation cycles decreases monotonically and when h-.~130 do noC exceed 1.5 db ~
(in the comparison the noise was assumed Co be Gaussian and the signal
was assumed to fluctuaCe in conformity Co the Rayleigh law, AZ = 0.9,
04 2 = 10-2). It should be noted that the Manna-WhiCney deCec"or is con-
siderably more complex in its circuitry than the synChesized two-sCnge
deCector, which moreover can be constructed completely using components
from compuCer technology.
II. Rule for Determining Coordinates of Ob3ect [This rule was formulated
Jointly with L. A. Zhivotovs?:iy.]
The rule for determining coordinates is based on use of the information r
spatial-temporal relationships between the target and a group of sensors
at the time of adoption of a decision. The essence of the rule is as fol-
lows. Assume that the processing of information is accomplished simul-
taneously from Z sensors (in Z channels), Z= 1, 2,..., each of which has
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its ef�ecCive range. Uecidinng concerning the presence of a target are made
using dnra from Ctie ouepur of Clie ewo-SCage detectoti nf euch channel for
ear.h rnnge yecror. Assume ChaC in each of gny Cwo range distAnces Ctiere
iq one :suc:h deci.yion or ealuCinn. On the baeie of these decisiane, knnwing
Lhe r.ncr.cll.nrited of the seneorEi und the snuree, it is poseible eo CottsCruet
two position lines for the CflrgeC (in this case ellipseg), at whose in-
tersectinn beverul points nre f:ormed; a target cnn be gituaCed nt ench of
- these, Fnr ec,^}i of ehese "suspinious" pointis at the next distances it is
possible to determine in advance the range sec`ors in one of which the gig-
nal should Call if it was actually reflecCed frc,m the targer.. Geomerrically
Chis means that a11 ttie posiCion lines in the presence of a target should
inCergect aC one of the "suspicious" points, which also will determine its
coordinates. If the "suspect" sectors aC each of the distAnces are connecC-
ed by lines, we obtnin "paths" charncterizing the coordinates of possible
turget position. Thus, the problem of detQrmining coordtnates is reduced
to the choice o� a targeC "paCh" among a set oE "paths." For this purpose
we successively carry out tt paired comparison of any two "paths" for the
purpase of selecting the true path, whici in turn is compnred with Ctie new
path, etc., unCil all the patlis have been "sorted." As a resu].t, only n
single path should remnin, and this will be considered the true one. Geo-
meCrically this means that we have deCermined the poinC of intersection
of the greatest number of posiCion lines.
The rule for determination of coordingtea is the invariant uniformly sCrong-
_ est and has the form:
1 when
~X~ � when ~X~s' (5)
j =
0 when l
(6)
.~o
where Xi = 1 when x,1~~rSX= 0 with xWJ`drdX, (32)
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Where N*(e, 1) afid No*(t, I) are ehe m@an squjkre evaluationa of ineerfer-
@nce N(i, t), comput@d ia the preaence and abs@nca of a signal.
Since N(e, x) ia a faussian fiel8, eh@ eveluaeion@ Np*(t, i) are iinear
functionais aE u(e, ~C) and in accordance aith (30)-(31) 8re equal eoi
N,~~~~'f~~~~E*l~,i~~~'~f~~'B'~;f~lfo~rd~ji /A�a~~i (33)
y
where S�Yt Vx y~ ie solut3on of the intagrai equntion
` mt,; f~S~~rE, A PA: OW- Orr A i,:
or in operator �ornt we hav~
~0 NJ f � ofr /a '~,v -
(34)
Nere KN(t, ti; xt ~1) is a spatiai-t mnporai correlaCion functian N(t, 'A)o
Cp is the spectr8l density of white noise n(ro 1)9 I in a unit operator.
, From (33) N,'� N,'~- jW'~
hence 3 � N,'~ N,'�(! � va.
5ubstituting thie expression inCo (32), We have
(35)
F,(uf s-~. ,
where is the symbol for the acalar product in space JL x T. Ueing
the "generating proceas method" (11], it can be demonstrated that
(I- i~'~`� (R~~ I ~ KN~'! (36)
1'he aubsCitution of (36) into (35) gives
' Ft~/l! ~LI~d ~"~�~L~� .
1,0dfdf -;f yIsm, xmf jrf 0,1da.*, c37>
where ~(t, x) is solution of the equation
I+KN)Q'l. , �
The synChesized E-optimum detection algorithm coincides with the known
result, obtained earlier [12] from the probability ratio.
Itandom signal. In this case, in accordance with (30)-(31):
(38)
: (ul' I dr
AE .rrv/ , n (5)
where Q is the angle coordinate, Qp ia the direction of compensation, K
is the wave number, d is the array interval, 2N + 1 is the current number
of elemer.ts in Che array, T is the averaging period.
The mean value of volCage sums during the time T is given by the expression:
~
Fl4f P exp~j,rnd~s~~ Q- s~~ Qo J,J+ (6)
nr�a . .
+ z~ P exo~;rnd~Jin Q- s/nQo .7. .
where N - ~ = mp.
IJe will consider the amplitude distr.ibution in the array to b e uniform. By
turning the geometrical progression in formula (6) we obtain:
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l~0~'0 a ~u51 Rn"6'
Fig. 2

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Fox orFicrnL usE orn,Y
>>W51'
F(o) r (7)
. . Ji/1! ~Ji/J Q� Jlil Q~ ~J ~
Thus, we obtiain a resultant d3rectional diagram i.n ehe form of the product
of the direceional diagram of an array conalaCing of 2M - m/2 radiators
and a minimizing factnr in the form:
COJl~ ~ f rdiAQ - r/n Q� JJ (8)
IFI
2
Fig. G
IFI a6
-4i
�Ii
-H
-JS
� '
~
'
. ' �
Fig. 5
It is evident that in the neighborhood of Che po3nt at which the cosine fac-
tor becomes equal to zero a suppression zone will appear. The direction in
which the suppression zone is oriented is given by Che formula:
, ~~'~n)Q,: , (9 )
Q~ = oza sinz:s/~Q, + .
It is therefore clear that with one and the same number of cut-out elements
there will be formation of several suppression zones whose coordinates are
given by expression (9). With n= 0 the first suppression zone, closest to
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I? op,._...._.,.....,.~
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the main maximum, is formed 3,n the diagxam. The width of the yuppression
zone is dependene on iCs position rel,ative to the main maximum. The yup-
pression sector is narrowest near the main maximum of the directional din-
grttm and bxoadens duxing moyement of the suppression zone in the direction
from the main 1obe. k'igures 2 and 3 show the di.rectional d3agrams of rAys
with a ray width 1�, M = 101, compuCed on a digital compuCer. The suppres-
sion zone is symmerxic relative to the main maximum and with a change in
its direcCion is displaced accordingly by this same angle.
Now we will consider how the cutting-off of some of the elemenCs exerts an
inf luence on the width of the main lobe. Whereas b efore cutting-off of
some oF the elements the width of the main lobe was
aQ= o,88s ~ cas~~Q,). _ . (10)
after cutting-off of some of the elements the width of the main lobe becomes
equal to 0 4= o, 8as ~M A, cbrec~Q, )2 (11)
'['hus, with suppression of the side lobes the widCh of the directional dia-
gram is increased
aU�
a4 (12)
Since usually m< M, the m/2M - m value is small. It can Cherefore be seen
that controllable suppression of the side lobes exerts no appreciable in-
fluence on the width of the main maximum of the directional diagram, espec-
ially in the case of large suppression angles.
Now we will determine the anCenna concentration coefficient with a change
in aperture size with time and subsequent signal averaging. We use d max
to denote the maximum aperture size. Then if
� P~}~~
when /I/ v /p(dl/ '
when /I/
(13)
it is possible Co write ~M� 1
'~(~iI ex~�l K~~~n Q- sinQe~f
here _JPt t)dt is an amplitude distribution giving a directional dia-
gram e4uivalent to the directional diagram of an array with a change in
size of the aperture and subsequent signal averaging in time. tience, the
concentration coefficient for an antenna with suppression of the side
lobes can be computed using an equivalent amplitude distribution. For
the case of formation of one suppression zone in the directional diagram
of a linear equivalent array with a uniform amplitude distribution, when
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the diaeances berween adjACene elemeneg are multiples of 0.5 X,rhe concen-
trntion cnnffi.c;Lene will bg equa1 eo: Kn [2M - m]2/4M - 3 m, where M 3s
the number oC elpmeneg in the array, m is the number of cut-off elemenCg.
- I.ven in a c1He when the guppression zonp io c1ose Cd the main maximum of
che diroCtional diagrum, the cancentrnCion coefficienr ia iitsignificanrly
ltns ehan the conCpntration coefficient for an antenna without suppression
oE thE eide lobeq.
Nnw we will examine operation oE an gntenna wirh suppr~~sion of the side
tobe8 in ttie frequency ranga. The directiongl diagrgm of gn anrenna arrny
in the Erequenny range can be written in tha form (4):
~r F ,y
e;o! a.>.1dEJ~'fJ~' c15~
. ~
We intrnducp weighe filee-ring, varying wiCh time, into the channels of a
ltydroacoustic anCenna. In pgrticular, this can be accomplighed by periodic
cutting-in o� band filters wieh a distribution of limiting frequenciee in
the aperture. It cgn be seen from expression (8) thgC if the number of cut-
off plemenes is dependenC on frequency and the angle of compensation in ac-
cnrdgnce with the law:
c
m+ 0.9707 0, -J /17 G,) � (16)
wr obthtn a Eectnr having nsCable position of the minimum not dependent
nn frequency. The directional diagram of an antenna without stabilization
of the width of the main maximum and with the suppresaion of the side
lobes tn the frequency band will have the form:
(17)
Mt~~
Thus, we obtain a controllable zone o� a reduced level of the additional
lobes in the frequency range caused by a minimum nf the cosine factor.
Figures 4 and 5 show the directional diagrama of an array with degree noise
in a frequency range equal to one octave.
The mean intensity of antenna reception in the suppression sector is more
than 50 db lower than t'ae intensity of reception from the main direction
and is approximately 20-30 db lower than the intensity of reception in
this sector of the array with a uniform distribution of amplitude. The de-
scriheJ method for suppressing the aide field makes it possib le, relatively
simply, to control the position of the suppression zone both in the harmonic
signal and in the frequency range. In Chis connection there will be no re-
stricCions on the scanning speed.
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gxgLxOGRAPHY
1. Yampollekiy, V. G., LINEYNYXE ANTENNY S NIZKIM UItOVNEM nOKOVYKH LEP-
E3T'KOV (Linear Antannaa WiCh a Low Level of Che 3i.de I.obes), "Elek-
trt�vyaz'," No 4, 1965.
2. Yur'yev, A. N., "On Che Theory of Synthesis of Antennas Wieh n Minimum
Level of Side Radiatton,"RAbZOTEKHNZKA (Itadio Engineering), Vol 26,
No 11, 1971.
3. Voroblyev, Ye. A., "Pencil Beam Antennas With a Variable Aperture and
Signal proceesing AfCQr Frequency Conversion," IZVESTIYA VUZov, RAUIO-
TEKHNTKA (News of InsCitutione of Higher EducaCion, Rndio Engineering),
Vo1 VIII, No 1.
r
4. Smaryshev, M. D., NAPRAVLENNOST' GIDROAKU3TICHESKIKH ANTENN (DirecCiv-
ity of Nydroacoust3c Antennas), "Sudoatroyeniye," Leningrad, 1973.
COPYRTGHT: NotiCe Not Aval.lable
5303
CSOs 8144/0938
� END
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