Stochastic flows: the foundation of multi-object filtering with point processes

Though largely unknown to the multi-target tracking community, some of the most significant developments in the application of point process theory to multi-object filtering are found in the literature published in the late Soviet Union, which culminated with a publication of Detection of Moving Objects in 1980. This major work is regularly brought up in relevant Russian-language publications, yet it is rarely contrasted to the works on multi-object filtering produced in the English-speaking world. In turn, many of the original results that led to it were translated into English at the time, but subsequently have been rarely, if ever, cited in the English language literature. In fact, very few researchers are aware of those translations, including, until recently, the authors of this article.

Highly advanced for their time, these Soviet developments do not now necessarily promise a breakthrough in multi-object filtering as a number of central results have since appeared in the western literature without reference to the originators of these works. For example, Achkasov's 1971 multi-target likelihood for point processes, and Bakut and Ivanchuk's 1976 method for propagation of the intensity function of a point process prior have both appeared. Nevertheless, a peculiar property of those results is that they have been systematically developed in a single coherent framework originating from statistical mechanics. The results of Achkasov and Bakut and Ivanchuk have inspired many practical implementations. Apparently these earlier results from the 1970s and 1980s in the Soviet literature were unknown outside the Soviet Union. The results were rediscovered in the West beginning in the late 1990s. To this day, this earlier Soviet literature remains very poorly known and is almost never referenced in the modern literature.

This project focusses on key results in the development of a theory of multi-object tracking and filtering with point processes from their roots in statistical mechanics through the Soviet literature. These form a coherent and sequential lineage in the chronology of methodological development of the theory of point processes, their representations, conditional estimation, and estimation of signals from a sequence of signals. (In the modern Western literature, signals are called targets and their estimates and called tracks.)

Daniel Clark, University of Southampton, UK

Alexey Narykov, University of Liverpool, UK

Roy Streit, Metron, USA

Topics

Point process modelling

Detection and estimation

Group target modelling

Multiple target tracking

Target classification

Optimal detection

Optimal energy allocation

Optimal sensor control

Continuous-time modelling

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Year

1961

1964

1965

1966

1967

1968

1969

1970

1971

1972

1974

1975

1976

1977

1983

1978

Author

Stratonovich

Bolshakov

Bakut

Zhulina

Ivanchuk

Vatollo

Minin

Latysh

Elagin

ABSTRACT

In many important practical cases the useful signal at the input to a receiver is in pulse form, in which case information is transmitted by the positions of pulses on the time axis. Apart from the useful signal, noise is also picked up which is a random function $\xi(t)$. After reception it is not the useful signal which is known, but the total signal $s(t)+\xi(t)=r(t)$. The exact position of the pulses is therefore unknown, and the pulse coordinates are only described by an a posteriori distribution on condition that $s(t)+\xi(t)=r(t)$. This conditional distribution can be found by the Bayes' formula. This makes it possible to draw conclusions regarding certain of the most probable values of the signal parameters.

REFERENCES

1. F. Woodward and I. Davis; The use of statistical methods in radar (Primeneniye statisticheskikh metodov $\mathrm{v}$ radiolokatsii). In book: The theory of electrical signal transmission in the presence of noise, ed. by N. A. Zheleznov, Foreign Literature Publishing House (1953).

2. P. I. Kuznetsov and R. L. Stratonovich; A note on the theory of correlated random points

3. R. L. Stratonovich; See article No. 37 in this symposium.

ABSTRACT

REFERENCES

2. P. I. Kuznetsov and R. L. Stratonovich; A note on the theory of correlated random points

3. R. L. Stratonovich; See article No. 37 in this symposium.

1.

R. L. Stratonovich,

Conditional distribution of correlated random points and the use of correlation for optimal selection of a pulse

Izvestiya AN SSSR, Energetika i Avtomatika, 1961, 2, 148.

1.

R. L. Stratonovich,

Conditional distribution of correlated random points and the use of correlation for optimal selection of a pulse

Izvestiya AN SSSR, Energetika i Avtomatika, 1961, 2, 148.

1.

R. L. Stratonovich,

Conditional distribution of correlated random points and the use of correlation for optimal selection of a pulse

Izvestiya AN SSSR, Energetika i Avtomatika, 1961, 2, 148.

1.

I.A. Bol'shakov , V. V. Vatollo and V , G. Latysh ,

Methods of joint detection and measurement of an unknown number of signals based on the theory of random points ,

Radio Engineering and Electronic Physics, Vol 4, 1964

ABSTRACT

REFERENCES

2. P. I. Kuznetsov and R. L. Stratonovich; A note on the theory of correlated random points

3. R. L. Stratonovich; See article No. 37 in this symposium.

ABSTRACT

The problem of observing (combined detection and measurement) an unknown number of signals received in the presence of noise is solved on the basis of concepts from the theory of solutions and the methods of the theory of correlated random points. The general structure of the solving circuit of a detecting and measuring instrument is explained by the use of a quadratic loss function.

REFERENCES

1. N. I. Nilsson, On the optimum range resolution of radar signals in noise, IRE Trans. Inform. Theory, IT-7, 1961, 4, 245.

2. R. L. Stratonovich, Conditional distribution of correlated random points and the use of correlation for optimal selection of a pulse Izvestiya AN SSSR, Energetika i Avtomatika, 1961, 2, 148.

3. I. A. Bol' shakov and V. G. Latysh, Selection of an unknown number of fluctuating signals in the presence of noise by the method of random-point theory, Radiotekhnika $i$ Elektronika, 1964, 9,3, 408 .

4. D. Middleton, An Introduction to Statistical Communication Theory, N. Y., 1960.

2.

I.A. Bol'shakov and V.G. Latysh

Separating an Unknown Number of Fluctuating Signals from Noise on the Basis of the Theory of Random Points,

Vol 3, 1964 Radio Engineering and Electronic Physics

ABSTRACT

A statistical study is made of a system of n-fluctuating impulsive signals depending on a parameter. With the aid of the theory of random points, operators are found for the formulation of the a posteriori probability density of an indeterminate number of indicated parameters depending on a postesiori data and the obtained sample of a mixture of signals and noise.

REFERENCES

1. R. L. Stratonovich, Conditional distribution of correlational random points and the use of correlations for optimal separation impulse signal from noises, Izvestiya AN SSSR, Energetika i Avtomatika, 1961, 2, 148.

2. R. L. Stratonovich, Selected Problems in the Theory of Fluctuations in Radio Engineering, Sovetskoye Radio, 1961.

3. P. I. Kuznetsov and R. L. Stratonovich, Mathematical theory of correlational random points, Izvestiya AN SSSR, Ser. Matemat., 1956, 20, 167.

4. V. I. Smirnov, Course in Higher Mathematics, IV, GITTL, 1953.

5. P. M. Woodward, Probability and Information Theory with Applications to Radar, Sovetskoye Radio, 1955.

6. L. A. Vaynshteyn and V.D. Zubakov, Separation of Signals on a Background of Random Noise, Sovetskoye Radio, 1960.

Submitted January 30,1963

3.

Bol'shakov, I. A.

Determination of Parameters of Groups of a Random Number of Signals under Noise Conditions.

Engineering Cybernetics, Vol. 3, 1964

ABSTRACT

With the use of an apparatus of correlated random points, the a posteriori probability is formulated for the value of the parameter of a group consisting of a random number of statistically identical signals. The operator and characteristics of statistical evaluation of the parameter group are given. The a posteriori characteristics of the set of parameters of statistically identical groups are obtained where these parameters are random themselves On the basis of risk theory the optimal operator is constructed for simultaneous detection of groups and determination of their parameters.

REFERENCES

1. P. I. Kuznetsov and R. L. Stratonovich, Mathematical Theory of Correlated Random Points, Izvestiya AN SSSR, Ser. Matem. , $20,167,1956$.

2. R. L. Stratonovich, Conditional Distribution of Correlated Random Points and the Use of Correlation for Optimum Separation of a Pulse from Noise, Izvestiya AN SSSR, OTN, Energetika i Avtomatika, No. 2, 1961 .

3. D. Middleton, Introduction to Statistical Communication Theory, Vol. II, Sovetskoye Radio, 1962.

4. N. J. Nilsson, On the Optimum Range Resolution of Radar Signals in Noise, IRE Trans. Information Theory, IT-7, No. 4, 245 , Oct. 1961 .

5. A. A. Kuriksha, Optimum Use of Space-Time Signals, Radiotekhnika i Elektronika, VIII, No. 4,1963 .

4.

BOLSHAKOV, IA, GF BUGACHEV, and VV VATOLLO.

"Distinguishing signal parameters during extraction from noise(Optimum filter meter for discriminating varying parameters of similar signals received in unique mixture against noise background)."

RADIO ENGINEERING AND ELECTRONIC PHYSICS 9 (1964): 158-166.

ABSTRACT

The problem of optimal Bayes filtration of l randomly varying parameters of similar signals received in a single mixture in the presence of noise using a method similar to that described in [1] is solved. An optimal combined filtering and measuring apparatus contains a special device distinguishing parameters, according to their type of coding, in the mixture and the character of variation in time.

REFERENCES

1. I. A. Bol'shakov and V. G. Repin, Problems in Nonlinear Filtration. I. The case of one parameter, Avtomatika i Telemekhanika, $1961,22,4,466$.

2. D. Middleton, An Introduction of Statistical Communication Theory (Transl. from English McGraw-Hill Book Co., Inc., New York, 1960), Sovetskoye Radio, 1962.

3. L. S. Gutkin, Theory of Optimual Methods of Radio Reception in Fluctuation Noise, GEI, 4. S. K. Fal' 1961 .

4. S. K. Fal'kovich, Reception of Radar Signals in Fluctuation Noise, Sovetskoye Radio, 1961.

7.

BOLSHAKOV, IA.

"Extraction of an unknown number of signals having parameters in the form of markov processes in the presence of noise(Separation from noise of unknown number of signals with variable Markovian parameters)."

Radio Engineering and Electronic Physics 10 (1965): 173-180.

ABSTRACT

With the use of the method of correlated random points the a posteriori probability of the values of the parameters of a random number of signals belonging to different classes and inside classes which are statistically indistinguishable, is formulated. On the basis of the solution, an optimal operator of simultaneous detection and measurement of the parameters of signals of several classes and the signals of one class against a background of others is constructed. Regular and fluctuating signals in white noise are investigated.

REFERENCES

1. R. L. Stratonovich, Conditional distribution of correlated random points in the use of correlation for optimal separation of the pulse signal from noise, Izvestiya AN SSSR, Energetika i Avtomatika, No. 2, 1961.

2. I. A. Bol'shakov and V. G. Latysh, Procedures for separation on the basis of the theory of random points of an unknown number of fluctuating signals from noise, Radiotekhnika i Elektronika, IX, No. 3, 1964 .

3. I. A. Bol'shakov, V. V. Vatollo and V. G. Latysh, Procedures for simultaneous detection in measurement of an unknown number of signals based on the theory of random points, ibid, IX, No, 4, 1964 .

4. I. A. Bol'shakov, Determination of the parameters of groups of a random number of signals under noise conditions, Izvestiya AN SSSR, Tekhnicheskaya Kibernetika, No. 3 , 1964. Available in English in Engineering Cybernetics, same date, p. 75.

5. N. N. Bogolyubov, Problems in Dynamic Theory in Statistical Physics, Gostekhizdat, 1946.

9.

LATYSH, VG.

"SEPARATING GROUPS OF UNRESOLVED SIGNALS FROM NOISE USING RANDOM POINT THEORY."

ENGINEERING CYBERNETICS 2 (1967): 42.

ABSTRACT

In the development of [1] the problem of determination of the coordinates $\lambda_i$ of a random number of single-type signals is solved. A set of signals forms one or several groups which are characterized by general group parameters $\mu$, and the set is used in a unique mixture with noise. The coordinates $\lambda_i$ (for many groups also the parameters $\mu_j$ ) are subject to the laws of random points [2]. The optimal operators for formation of the a posteriori characteristics are obtained and the principles of their operation are explained.

RE FERENCES

1. I. A. Bol'shakov, Determination of group parameters from a random number of signals under noise conditions, Izvestiya AN SSSR, Otd. Tekhn. N. , Tekhnicheskaya, Kibernetika, 1964, No. 3 .

2. R. L. Stratonovich, Conditional distribution of correlated random points and the ube of correlation for optimal separation of a pulse noise Izvestiya AN SSSR OTN, Energetika 1 Avtomatika, No. 2, 1961.

3. D. Middleton, Introduction to Statistical Communication Theory, II, Sovetskoye Radio, 1962.

4. I. A. Bol'shakov, V. V. Vatollo and V. G. Latysh, Procedures for simultaneous detection and measurement of an unknown number of signals based on random points theory, Radiotekhnika 1 Elektronika, IX, No. 4, 1964.

5. R. L. Stratonovich, Application of Markov process theory for optimal filtration of signals, Radiotekhnika i Elektronika, V, No. 11 , 1960.

6. I. A. Bol'shakov and V. G. Repin, Problems in nonlinear filtration, Avtomatika $i$ Telemekhanika, Vol. 22, No. 4, 1961.

ABSTRACT

The joint problem of detecting and measuring varying parameters of an unknown number of statistically indistinguishable signals in the presence of noise is solved. At each specified instant of time, the parameters form a system of random points $[1,2]$ and their motion obeys Markov laws. An optimal operator is given for detection measurement, and some examples are discussed.

REFERENCES

1. P. I. Kuznetsov and R. L. Stratonovich, Izvestiya AN SSSR, Ser. Matem. 1956, 20,167 .

2. R. L. Stratonovich, Izvestiya AN SSSR, Otdel tekhnich. nauk, Energetika i Avtomatika, 1961, $\underline{2}, 148$.

3. I. A. Bol'shakov, V.V. Vatollo and V. G. Latysh, Radiotekhnika i Elektronika, 1964, $9,4,563$.

4. R. L. Stratonovich, Radiotekhnika i Elektronika, 1960, $\underline{5}, 11,1751$.

5. P. A. Bakut et al., Problems in the Statistical Theory of Radar, (In Russian), Part 2 , Sovetskoye Radio, 1964.

6. N. N. Bogolyubov, Problems of the Dynamic Theory in Statistical Physics, (In Russian), GITTL, 1946.

7. R. L. Stratonovich, Selected Problems of the Theory of Fluctuation in Radio Engineering, (In Russian), Sovetskoye Radio, 1961.

8. N. I. Nilsson, IRE Trans. Inform. Theory, 1961, IT - $7,4,245$.

9. N. Uaks, Improving the Signal-to-Noise Ratio and the Statistics of the Families of Signal Traces, (In Russian), Voprosy radiolikatsionnoi tekhniki, (Problems of Radio Engineering), 1956,1 (31).

5.

BOLSHAKOV IA.

"DETECTION AND MEASUREMENT OF PARAMETERS OF A STOCHASTIC NUMBER OF SIGNALS BELONGING TO DIFFERENT CLASSES."

ENGINEERING CYBERNETICS 4 (1965): 28.

Article

6.

BOLSHAKOV, IA.

“Determination of the inner group structure of one and many groups of a random number of signals(Problem solving- determination of coordinates of random number of single-type signals).”

1965. (1965): 148-157. Engineering Cybernetics

8.

GRIGORYEV AL, and BOLSHAKOV IA.

"SEPARATION OF PULSE FLOW FROM NOISE. 1. MATHEMATICAL APPARATUS."

ENGINEERING CYBERNETICS 5 (1966): 50.

18.

ACHKASOV, YS.

"Measurement of Trajectory Parameters from Tracks in a Noise Background."

Engineering Cybernetics 10.1 (1972): 151-157.

27.

N. A. Ivanchuk,

“Analysis of the Efficiency of Some Controlled Search Procedures”,

Probl. Peredachi Inf., 19:1 (1983), 30–39; Problems Inform. Transmission, 19:1 (1983), 25–3

26.

IVANCHUK, NA.

"POSTERIORI ANALYSIS OF RANDOM FLOWS OF OBJECTS IN THE CASE OF REPEATED OBSERVATIONS."

ENGINEERING CYBERNETICS 16.5 (1978): 103-109.

ABSTRACT

The mathematical apparatus of the generating function is developed for the solution of the problem of separation from noise of pulse flows with a limited aftereffect (recovery flows). The conditional generating functions are introduced; the integral relationships and expressions are found for them in terms of the unconditional functions. On the basis of this analysis, multidimensional functions of the flow density and the generating functions of the number of losses are investigated. A number of examples are studied.

REFERENCES

1. A. Ya. Khinchin, Operations with Respect to the Mathematical Queueing Theory, Fizmatgiz, 1963.

2. I. N. Kovalenko, Queueing Theory, Probability Theory, 1963, from the series, Scientific Results, Nauka, 1965.

3. V. Feller, Introduction to Probability Theory and Its Application, Mir, 1964 .

4. M. S. Bartlett, Introduction to the Theory of Random Processes, IL, 1958.

5. V. Smith, Repair theory and problems adjacent to it, Matematika, Vol. 5, No. 3, 1961 (A Collection of Translations). from noise, Energetika $i$ Avtomatika, No. 2, 1961 . points, Izvestiya AN SSSR, Ser. Matem. , Vol. $20,1956$.

8. R. L. Stratonovich, Selected Problems in Fluctuation Theory in Radio Engineering, Sovetskoye Radio, 1961.

ABSTRACT

On the basis of the apparatus of random point theory a posteriori characteristics of groups of unresolved signals are constructed. Using the decision theory, the operator for detection and measurement of the parameters of these groups and the number of signals in them is studied. As an example, regular and fluctuating signals are investigated in white noise.

REFERENCES

1. I. A. Bol'shakov, Definition of the group parameters from a random number of signals under noise conditions, Izvestiya AN SSSR, Tekhnicheskaya Kibernetika, No. 3, 1964.

2. I. A. Bol'shakov, Definition of the intragroup structure of one and many groups of a random number of signals, Ibid., No. 3,1965 .

3. P. I. Kuznetsov and R. L. Stratonovich, The mathematical theory of correlated random points, Izvestiya AN SSSR, Ser. Matem., No. $20,1956$.

4. R. L. Stratonovich, Conditional distribution of correlated random points and the use of correlations for optimal separation of pulse signals from noise, Lzvestiya AN SSSR, Energetika i Avtomatika, No. 2, 1961.

5. A. Ya. Khinchin, Works in Mathematical Queueing Theory, Fizmatgiz, 1963.

6. I A. Bol'shakov, V. V. Vatollo and V. G. Latysh, Methods for joint detection and measurement of an unknown number of signals based on random point theory, Radiotekhnika i Elektronika, Vol. 9, No, 4, 1964.

7. I. A. Bol'shakov and V. G. Latysh, The problem of separation of an unknown number of fluctuating signals from noise based on random point theory, Ibid., Vol. 9 , No. 3,1964 .

ABSTRACT

The problem of the optimal distribution of limited energy for determining the number of single-type signals is solved on the basis of Bayes' theorem and nonclassical variational methods. The signals are taken as a single mixture against a background of intense noise, and their parameters (time delay) form a random flow. The minimization of the average risk of the energy distribution strategy reduces to the problem of the maximization with limits of a generally nonlinear functional determined by the flow density and the type of signals. Examples are given of fixed and fluctuating signals and a numerical estimate is made of the gain.

REFERENCES

1. I. A. Bol'shakov, V. G. Latysh and Yu. N. Minin, Determining the number of signals forming a random flow against a background of noise, Izvestiya AN SSSR, Tekhnicheskaya Kibernetika, 1968,3 .

2. V.P. Perov, The optimal energy distribution for multiple searches, of given region, Izvestiya AN SSSR, Energetika i Avtomatika, 1962, 6 .

3. I. N. Kuznetsov, The problem of the optimal distribution of limited means, Izvestiya AN SSSR, Tekhnicheskaya Kibernetika, 1966,5 .

4. R. Bellman, I. Gliksberg and O. Gross, Certain Problems in Mathematical Control Theory [Russian translation], IL, 1962 .

5. P. A. Bakut and others, Problems in the Statistical Theory of Radio-location, Vol. 1, Sovetskoye Radio, 1963.

6. D. Middleton, Introduction to Statistical Communication Theory, McGraw-Hill Book Co., N. Y., 1960.

ABSTRACT

Statistical decision and random stream theory are used to solve the problem of the optimal number of signals receivable in a unified mixture with noise. The nature and characteristics of the optimal operator are examined. Some types of stream formed by signals of determinate and fluctuating form are considered as examples.

RE FERE NCES

1. I. A. Bol'shakov, V.V. Vatollo and V, G. Latysh, Methods for joint detection and measurement of an unknown number of signals based on theory of random points, Radiotekhnika i Elektronika, No. 4,1964 .

2. P.A. Bakut et al., Topics in Statistical Theory of Radar, Sovetskoye Radio, Vol. I, 1963; Vol II, 1964.

3. D. Middleton, Introduction to Statistical Communication Theory, MeGraw-Hill Book Co., N. Y., 1960.

4. P.I. Kuznetsov and R. L. Stratonovich, The mathematical theory of correlated frandom points, Izvestiya AN SSSR, Seriya matem., Vol. $20,1956$.

5. R. L. Stratonovich, Conditional distribution of correlated random points and the use of correlations for optimal extraction of an impulsive from noise, Izvestiya AN SSSR, Energetika i Avtomatika, No. 2, 1961.

6. W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley and Sons, N. Y., 1950.

10.

Bol’shakov , L.A. , V.G. Latysh , Yu . N. Minin,

Determining the number of signals forming a random stream in a noise background,

Engineering Cybernetics, Vol. 3, 1968

11.

BOLSHAKOV IA, VG LATYSH, and YN MININ.

"OPTIMAL ENERGY DISTRIBUTION FOR DETERMINING A RANDOM NUMBER OF WEAK SIGNALS."

ENGINEERING CYBERNETICS 4 (1968): 45.

ABSTRACT

The problem is solved of filtering a multidimensional variable parameter, described by nonlinear differential equations, from the results of observations at discrete instants of time on independent random signals, constituting functions of the current values of the parameter. The assumption is made that a single observed signal gives the possibility of estimating the current value of the parameter with high accuracy. Optimal operations of smoothing the individual measurements of parameters in a trajectory are found in the form of recurrent formulas, connecting the smoothed estimates at the $n$-th and $n-1$ st instants. The question is considered of identifying repeated unit sightings according to membership in previously constructed trajectories. Equations are found characterizing the systematic and fluctuating measurement error. The results obtained are considered in application to the problem of radiolocation information processing.

REFERENCES

1. N. I. Nilsson, On the optimum range resolution of radar signals in noise, IRE Trans. Inform. Theory, IT-7, 1961, 4, 245 .

2. R. L. Stratonovich, Conditional distribution of correlated random points and the use of correlation for optimal selection of a pulse signal from noise, Izvestiya AN SSSR, Energetika i Avtomatika, 1961, 2, 148.

4. D. Middleton, An Introduction to Statistical Communication Theory, N. Y., 1960.

Submitted January 30, 1963

12.

ZHULINA, YV.

"MEASURING COORDINATES OF MANY OBJECTS."

ENGINEERING CYBERNETICS 3 (1969): 99.

ABSTRACT

Observation of random processes is considered in which the processes depend on unknown parameters and on parameters whose values can be chosen by the investigator. Optimal algorithms are sought for choosing these parameters in such a way that the observed process will contain maximum information about the unknown parameters. This problem is solved for the case in which the observed process is the sum of a white noise and a known function of the chosen and unknown parameters. As an example, we analyze the optimal control of the directivity of a one-channel radar in the location of a solitary target.

REFERENCES

1. R. Fano, Transmission of Information, Mir, 1965 .

2. R. L. Stratonovich, Conditional Markov Processes and Their Application to the Theory of Optimal Control, American-Elsevier, 1968.

3. I. A. Bol'shakov, V. G. Latysh and Yu. N. Minim, Optimal distribution of energy in the determination of a random number of weak signals, Engineering Cybernetics, 1968, No. 4 .

13.

BAKUT T, PA, YV ZHULINA, and KUZNETSOV SY.

"OPTIMAL CONTROL FOR OBTAINING MAXIMUM INFORMATION FROM OBSERVED RANDOM PROCESSES."

ENGINEERING CYBERNETICS 2 (1970): 353.

ABSTRACT

The optimal distribution of a finite amount of energy is found for a loss function that takes into account requirements for the detection and measurement of the parameters of a random number of signals of a single type. Limiting cases of high and low noise levels are considered improvements obtained are estimated for determinate and fluctuating signals. The proposed distribution strategies can be achieved in present-day control systems for parametric nonstationary fields.

REFERENCES

1. I. A. Bol'shakov, V. G. Latysh and Yu. N. Minin, Determination of the number of signals consitituting a random flow on a background of noise, Engineering Cybernetics, 1968, No. 3 .

2. I. A. Bol'shakov, V. G. Latysh and Yu. N. Minin, Optimal distribution of energy in the determination of a random number of weak signals, Engineering Cybernetics, 1968, No. 4 .

3. R. L. Stratonovich, Conditional distribution of correlated random points and use of correlations for optimal extraction of an impulse signal from noises, Izvestiya AN SSSR, Energetika i avtomatika, 1961, No. 2 .

4. I. A. Bol'shakov, Extraction of an unknown number of signals with parameters in the form of Markov processes from noises, Radio Engineering and Electronic Physics, 1965, 10, No. 2.

5. I. N. Kuznetsov, Problems of optimal distribution of limited means, Engineering Cybernetics, 1966, No. 5 .

6. R. Bellman, Dynamic Programming, Princeton University Press, 1957.

14.

MININ, YN.

"Optimal Energy Distribution in Extraction of a Random Number of Signals from Noise."

Engineering Cybernetics 1 (1970): 130.

ABSTRACT

A calculation is made of the a posteriori intensity of the flow of signals with random parameters defining trajectories of moving objects. The flow is assumed to be Poisson. The obsexvations are an additive mixture of signal points that have a known probability and random deviations from object position, and noise points which al so constitute a Poisson flow.

REFERENCES

1. I. A. Bol' shakov, Statisticheskiye problemy vydeleniya potoka signalov iz shumov (Statistical Problems of Extracting a Flow of Signals from Noises), Sovetskoye Radio Press, 1969.

2. N. Wax, Improvement of the signal/noise ratio and a statistic for families of traces of signals, Vopr. Radiolokatsionnoy Tekhn., 1956, No. 1 (13).

3. Sittler, The problem of optimum construction of traces from the data of radar observation. Zarubezhnaya Radioelektronika, 1966 , No. 6 .

15.

ACHKASOV, YS.

"FINDING TRAJECTORIES BY A POSTERIORI ANALYSIS OF FLOWS."

ENGINEERING CYBERNETICS 9.5 (1971): 919.

ABSTRACT

A study is made of the evolution of the a posteriori characteristics of random flows of resolvable signals under continuous observation. Continuous operators generating the a posteriori characteristics, and the block diagrams realizing them, are given for specific kinds of flows.

REFERENCES

1. I. A. 'Bol'shakov, Statisticheskiye problemy vydeleniya potoka signalov iz shuma (Statistical Problems of Extracting a Flow of Signals from a Noise), Sovetskoye Radio Press, 1969.

2. P. A. Bakut et al., Voprosy statisticheskoy teorii radiolokatsii (Questions in the Statistical Theory of Radar), Vol. II, Sovetskoye Radio Press, 1964 .

3. R. L. Stratonovich, Application of the theory of Markov processes for optimal filtration of signals, Radio Tekhn. i Elektron., 1960, $\underline{5}$, No. 11 .

4. R. L. Stratonovich, Uslovnyye protsessy Markova. Teoriya veroyatnostey i yeye primeneniya (Conditional Markov Processes. Theory of Probability and Its Applications), Vol. 5, No. 2, 1960.

16.

RAKOSHITS, VS.

"Continuous Observation of Random Flows of Resolvable Signals."

Engineering Cybernetics 9.1 (1971): 110-+.

ABSTRACT

Optimal control is considered of the scanning sequence of coordinate space cells in the search for an unknown number of objects. The efficiency of controlled search is estimated. The obtained results are applied to radar problems.

REFERENCES

1. P. A. Bakut, Yu.V. Zhulina, and S. Ye. Kuznetsov, Optimal control for obtaining maximum imformation from observed random processes, Tekhnicheskaya Kibernetika (Engineering Cybernetics), 1970 , No. 2.

2. R. Fano, Transmission of Information, A Statistical Theory of Communications, MIT Press and Wiley, New York, 1961 (Russian translation, Mir, 1965).

3. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, New York, 1957 (Russian translation, Mir, 1964).

17.

BAKUT, PA, and YV ZHULINA.

"OPTIMAL CONTROL OF CELL SCANNING SEQUENCE IN SEARCH FOR OBJECTS."

ENGINEERING CYBERNETICS 9.4 (1971): 740.

ABSTRACT

A calculation is made of the a posteriori intensity of the flow of signals with random parameters defining trajectories of moving objects. The flow is assumed to be Poisson. The observations are an additive mixture of signal points that have a known probability and random deviations from objection position, and noise points which also constitute a Poisson flow.

REFERENCES

1. Yu.S. Achkasov, Application of the theory of a posteriori analysis of fluxes to problems of detection of traces of trajectories, Tekhnicheskaya Kibernetika (Engrg. Cybernetics), 1971, No. 5 .

2. D. Middleton and R. Esposito, Simultaneous optimum detection and estimation of signals in noise. IEEE, Tr, on IT, Vol. IT-14, No. 3,1968 .

3. D. Middelton and R. Esposito, New results in the theory of simultaneous optimum detection of signals and estimation of their parameters in noise, Problemy Peredachi Informatsii, Vol. 6,1970 (Trans. from Engl.).

4. N. E. Nahi, Optimal Recursive Estimation with Uncertain Observation IEEE Tr on IT, Vol. IT-15, No. 4, 1969 .

5. Sittler, Problem of optimum construction of paths from data of radar observations, Zarubezhnaya Radioelektronika, 1966, No. 6, Sovetskoye Radio (Trans. from Engl.).

6. I. A. Bol'shakov, Statisticheskiye Problemy Vydeleniya Potoka Signalov iz Shumov (Statistical Problems of Extracting Flux of Signals from Noise), Sovetskoye Radio, 1969

7. R. Lee, Optimum Estimation, Identification and control, Wiley (Russian Translation Nauka, 1966).

8. Yu. V. Zhulina, Measurement of coordinates of many objects, Tekhnicheskaya Kibernetika (Engrg. Cybernetics), 1966, No. 3 .

ABSTRACT

On the basis of the information-theoretic approach criteria for the control of observations are obtained, which have the form of quadratic functionals in the case of strong interfering signals. The reception is carried out on an antenna of phased array type. The control of the observation reduces to the control of transmission of the probing signal.

REFERENCES

1. I. A. Bol'shakov, Statisticheskie problemy vydeleniya potoka signalov iz shuma (Statistical problems of extraction of flux of signals from noise), Sovetskoye Radio, 1969.

2. P. A. Bakut, Yu. V. Zhulina and S. E. Kuznetsov, Izv. AN SSSR, Tekhnicheskaya Kibernetika, 1970, 2 (Engineering Cybernetics), 2,1970 .

3. P. A. Bakut, Problemy peredachi informatsii (Problems of information transmission), 1971 , 1 .

4. R. Fano, Statistical communication theory, transl. from English, ed. R. L. Dobrushina, Mir, 1965.

5. P. A. Bakut, et al., Voprosy statisticheskoy teorii radiolokatsii (Problems of statistical theory of radar) I, Sovetskoye Radio, 1963.

19.

ELAGIN, VV.

"INFORMATION-THEORETICAL APPROACH TO OBSERVATION CONTROL PROBLEM IF THERE ARE MANY TARGETS AND STATIONARY RADIATION SOURCES."

RADIO ENGINEERING AND ELECTRONIC PHYSICS 19 (1974):

20.

Elagin, V. V., and P. A. Bakut.

"Space-time processing of signals from multiple targets and sources of interfering radiation."

RADIO ENGINEERING AND ELECTRONIC PHYSICS 19 (1974):.

REFERENCES

1. I. A. Bol'shakov, Statistical Problems in Extraction of a Train of Signals from Noise, Sovetskoye Radio, 1969.

2. P. A. Bakut, et al., Problems in Statistical Theory of Radar, 2, Sovetskoye Radio, 1964 .

ABSTRACT

The problem of controlling the flow of observed marks generated by the flow of some objects and background noise is considered. An algorithm for optimal control of the observed flow is proposed and analyzed based on the criterion of the maximum amount of information about the flow of objects.

REFERENCES

1. Bakut P. A. On the information-theoretic approach to the problems of statistical analysis. Problems of transmission of information, 1971, 7, 1, 51-57.

2. Baput P. A., Zhulina G.O. V., Kuznetsov S. E. On optimal control for po. obtaining maximum information when observing random processes. Izv. AH USSR, Techn. tibernetics, 1970, 2, 171-179.

3. Bakut P. A., Zhgulia GO. B. On the optimal control of the cell inspection order when searching for objects. Izv. USSR Academy of Sciences, Techn. cybernetics, 1971, 4, 172-179.

4. Boltsakov I. $A$. Statistical problems of extracting a stream of signals from noise. M., "Owls. radio", $1969.$.

21.

Zhulina, Yu V.

"Control of a Point Process of Observable Tokens Associated with the Detection of a Point Process of Objects and Measurement of Their Coordinates."

Problemy Peredachi Informatsii 11.1 (1975): 91-101.

ABSTRACT

An examination is made of the problem of calculating the a posteriori characteristics of a random flow of point objects. It is assumed that the process being observed has the form of a random set of resolved signals which is the superposition of independent random flows of signals generated by objects and of the flow of false signals. A method is proposed for calculating the a posteriori characteristics of the flow of is based on differentiation of the joint generating functional of the flows of objects and the signals being observed. For the particular case in which the a priori flow of objects is broken into a finite number of independent partial flows, a method is proposed for an approximate description of the a posteriori flow. This method enables us to obtain fairly simple and easily grasped results. An estimate of the errors incurred in the use of this method is derived. Examples are examined.

REFERENCES

1. I. A. Bol'shakov, Vydeleniye potoka signalov iz shuma (Extraction of a flow of signals from a noise), Izd-vo inostr. lit., 1956 .

2. J. L. Doob, Stochastic Processes, Wiley, New York, 1953.

3. Yu. S. Achkasov, Application of the theory of a posteriori analysis of flows to problems of detecting the traces of trajectories, Izv. AN SSSR, Tekhnicheskaya Kibernetika (Engineering Cybernetics), 1971, 9, No. 5 .

22.

Bakut, P. A., and N. A. Ivanchuk.

"CALCULATION OF A-POSTERIORI CHARACTERISTICS OF FLOW OF RESOLVED OBJECTS."

Engineering Cybernetics 14.6 (1976): 148-156.

ABSTRACT

Bayes algorithms are derived for secondary processing of information in the presence of interference. The observed signal is represented by a train of random points (marks). Effect of compensation of interference pulses during secondary processing is exposed and analyzed.

REFERENCES

1. Ya, D. Shirman (editor), Teoreticheskiye Osnovy Radiolokatsii (Theoretical Fundamentals of Radar), Sovetskoye Radio Press, 1970.

2. Widrow, Mantey, Griffith, Good. Adaptive Array Antennas, Proceedings IEEE, 1967, No. 12, (Russian translation).

3. V.S. Chernyak, Proceedings of the 6-th All-Union Conference on Theory of Coding and Information Transmission, part 6, Moscow-Tomsk, 1975 .

4. V.V. Yelagin, and P.A. Bakut, Radiotekhnika i Elektronika, 1974, 19, No. 2, p. 422 [Radio Engng. Electron. Phys., 19 , No. 2, p. 422 (1974)].

5. I. A. Bol'shakov, Statisticheskiye Problemy Vydeleniya Potoka Signaloviz Shuma (Statistical Problems Relating to Extraction of a Train of Signals from Noise), Sovetskoye Radio Press, 1969.

Submitted May 21, 1976

23.

Elagin, V. V., and N. A. Ivanchuk.

"Statistical analysis of algorithms of secondary processing in the presence of a flow of interfering signals."

Radiotekhnika i Elektronika 22 (1977): 1399-1405.

ABSTRACT

An optimal algorithm is found for control of statistical observation with minimization of the risk associated with displacement of points designed for servicing an unknown number of objects in coordinate space. A solution is sought in a simplified posing of the minimization of the "varying risk". The signal being observed is a random flow of point marks in the coordinate space.

REFERENCES

1. I. A. Bol'shakov, Statisticheskiye problemy vydeleniya potoka signalov iz shuma (Stati stical problems of extracting a flow of signals from a noise), Sovetskoye Radio, 1969.

2. V. I. Brikker and V.A. Khvorostov, An algorithm for optimal placement of servicing points, Izv. AN SSSR, Tekhnicheskaya Kibernetika (Engineering Cybernetics), 1969, 7, No. 6.

3. R. Bellman, I. Glicksberg and O. Gross, Notes on matrix theory-VI, American Math. Monthly, Vol. $62,1955, \mathrm{pp} .571-752$.

4. P. A. Bakut and N.A. Ivanchuk, Calculation of the a posteriori characteristics of a random set of resolved objects in the case of point representation of an observed signal, Izv. AN SSSR, Tekhnicheskaya Kibernetika (Engineering Cybernetics), 1976,14 , No. 6 .

5. Y. V. Zhulina, Control of a flow of observed marks when one is observing a flow of objects and their coordinates are being measured, Problemy peredachi informatsii, 1975, 11, No. 1.

6. P. A. Bakut and Y. B. Zhulina, Optimal control of the order of examination of cells in the search for objects, Izv. AN SSSR, Tekhnicheskaya Kibernetika (Engineering Cybernetics), 1971, 9, No. 4 .

24.

ZHULINA, YV.

"CONTROL OF A STATISTICAL OBSERVATION WITH MINIMIZATION OF RISK ASSOCIATED WITH DISPLACEMENT OF SERVICING POINTS."

ENGINEERING CYBERNETICS 15.1 (1977): 115-125.

ABSTRACT

An optimal algorithm for detecting randomly appearing signals of random duration on a background of a stationary noise process and the equation of the mean risk for such an algorithm are examined. It is shown that the solution of the equation for the mean risk reduces in this case to solving a second-order differential equation. The differential equation is solved and graphs making it possible to estimate the value of the mean risk a number of specific cases are included.

REFERENCES

1. A. N. Shiryayev, Statisticheskiy posledovatel ${ }^{\dagger}$ ny, analiz. (Statistical sequential analy sis), Nauka, 1969.

2. R. L. Stratonovich, Uslovnyye markovskiye protsessy (Conditional Markov processes), Moscow State University Press, 1966.

3. R. Bellman, Dynamic Programming, Princeton University Press (in Russian transl.), 1960.

25.

Tikhomirova, IG, and PA Bakut.

"Detection of random appearing signals of random duration." Engineering Cybernetics 15.2 (1977): 114-118.

ABSTRACT

Four procedures for searching for signals in a multichannel system are considered - a uniform uncontrolled search and a controlled search according to the criteria of maximum a posteriori probability, maximum information increment and minimum current risk. Estimates of the main probabilistic characteristics of these procedures are found and their comparative analysis is carried out.

REFERENCES

1. Shiryaev A. N. On the theory of decision functions and control of the observation process by incomplete data. - B book: Trans. Third Prague Conference on Inform. Theory, Statistical Decision Functions. Random Processes. Prague, 1964, p. 131-203.

2. Wald A. Statistical decision functions. - In the book: Positional games. M.: Nauka, 1967, p. 300-522.

3. Stratonovich $R$. L. Conditional Markov processes and their application to the theory of optimal control. M.: Publishing House of Moscow University, 1966.

4. De Groot M. Optimal statistical solutions. Moscow: Mir, $1974 .$

5. Zigangirov K. III. One problem of optimal scanning.- Theor. prob. and its application., 1966, v. 11, no. 2, p. 333-338.

6. A. F. Terpugov and F. A. Shapiro, “Two-stage signal search in a multichannel system with channel ordering,” Izv. Academy of Sciences of the USSR. Technical Cybernetics, 1974, No. 2, p. 126-130.

7. P. A. Bakut, Yu. V. Zhulina, and S. E. Kuznetsov, “On optimal control for obtaining maximum information when observing random processes,” Izv. AH CCCP. Technical Cybernetics, 19-0, No. 2, p. 171-179.

8. Bakut I.. A. On the information-theoretic approach to statistical decision problems.- Probl. peredachi inform., 1971. v. 7, no. 1, p. $51-57.$

9. Bakut P.A., Zhulina G.O.V. On the optimal control of the cell inspection order in the search for objects, Izv. Academy of Sciences of the USSR. Technical cybernetics, 1971, no. 4, p. $172-179$.

10. Ivanchuk N. A., Tartakovskii $G$. P. Optimization of management of observations based on minimization of the current risk.-B book: Tp. IV Intern. sympos. according to the theory of infrm. Ch. I. Tez. report M.-L., 197b, p. 72-74.

11. N. A. Ivaniuk, Yu. VI Conf. on the theory of coding and transmission of information. Part VI. Tez. report Moscow - Tomsk, 1975, p. $46-51.$

12. Basharinov A. E., Fleishman B. S. Methods of statistical sequential analysis and their applications. M.: Sov. radigo, $1962.$

13. Spitzer F. Principles of random walk. Moscow: Mir, $1969 .$

14. Feller B. Introduction to probability theory and its applications. T. 2. M.: Mir, $1967 .$

Received $21. I V .1981$ After revision 10.XII.1981

ABSTRACT

It is shown that, if the a priori flow is the superposition of a Poisson and a Bernoulli flow, then the a posteriori flow can be approximately represented in the same form. Rules for converting the a priori characteristics of the flow into a posteriori characteristics are found.

REFERENCES

1. I. A. Bol'shakov, Vydeleniye potoka signalov iz shuma (The extraction of a flow of signals from a noise), Sovetskoye Radio, 1969 .

2. P. A. Bakut and N. A. Ivanchuk, Calculation of the a posteriori characteristics of a random set of resolved objects in the case of point representation of a signal being observed, Izv. AN SSSR, Tekhnicheskaya kibernetika (Engineering Cybernetics), 1976, 14 , No. 6.

Article