Applied Mathematical Sciences, Vol. 8, 2014, no. 90, 4469 - 4496 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46415 Application of the Local Markov Approximation Method for the Analysis of Information Processes Processing Algorithms with Unknown Discontinuous Parameters O. V. Chernoyarov, Sai Si Thu Min, A. V. Salnikova Department of Radio Engineering Devices National Research University “Moscow Power Engineering Institute” Moscow, Russia B. I. Shakhtarin Department of Self-contained Information and Control Systems Bauman Moscow State University Moscow, Russia A. A. Artemenko Department of Bionics and Statistical Radio physics Lobachevsky State University of Nizhni Novgorod Nizhni Novgorod, Russia Copyright © 2014 O. V. Chernoyarov, Sai Si Thu Min, A. V. Salnikova, B. I. Shakhtarin and A. A. Artemenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We considered the local Markov approximation method for the definition of the precision characteristics of statistical analysis algorithms of information processes with unknown discontinuous parameters in the presence of Gaussian distortions 4470 O. V. Chernoyarov et al. and illustrated the use of the stated approach in practical applications for the analysis of operating efficiency of detectors and measurers of the quasi-determined and random signals. Keywords: Discontinuous signal parameter, maximum likelihood method, local Markov approximation method, signal detection and measuring characteristics, statistical modeling. 1 Introduction The problem of the statistical analysis of information processes under conditions of parametrical prior uncertainty has wide appendices in radio engineering, medicine, technical diagnostics, financial statistics etc. As is well known [1-3 et al.], the optimal (according to the maximum likelihood method) processing algorithm of the information process (signal) st, l0 with unknown parameter l0 deformed by random distortions generates output effect which is proportional to the functional of likelihood ratio (FLR) or its logarithm. Thus logarithm of FLR M l is a function of current value l of parameter l0 and is formed within all prior interval L1 , L 2 of its definition. As a result, it is possible to solve a task of signal detection against noise having compared maximum (supremum) of the logarithm of FLR M m sup M l to a threshold c chosen according to the accepted optimality criterion: 1 Mm c , (1) 0 or a task of measuring of information signal parameter l0 , having accepted as its estimate the position of the greatest maximum of solving statistics M l : l m arg sup M l . (2) l L1 , L 2 However, for the problem solution of the applicability of one or another processing algorithm it is not enough to define the degree of algorithm optimality. The final decision should be made on the basis of the concrete algorithm performance analysis with the assistance of the quantitative characteristics of its functioning. Besides, in the majority of real situations some of prior data can appear inexact, and real working conditions of devices can deviate from the established prior data. Working capacity of the synthesized processing algorithms under changed conditions can be estimated by the analysis of algorithms only. If the unknown parameter l0 is continuous [2] (i.e. logarithm of FLR is mean square differentiable at least twice), then characteristics of processing algorithm Application of the local Markov approximation method 4471 can be found by means of a small parameter method [3]. However, in some practical tasks more adequate description of real information processes can be specified with the assistance of discontinuous models [2, 4, 5 et al.]. In this case realizations of the logarithm of FLR will be non-differentiable with respect to current value of unknown parameter in any probability meaning. Consequently, it is not possible to calculate even the potential accuracy of the processing algorithm (for example, Cramer-Rao bound). The purpose of the present work is to illustrate a technique of the statistical analysis of the signals processing algorithms with unknown discontinuous parameters in the presence of random distortions. 2 Definition of Characteristics of Signal Detection and Discontinuous Parameter Estimation by the Local Markov Approximation Method When quasideterministic [1-3] or random (Gaussian) [4, 5] signal with unknown discontinuous parameter against Gaussian distortions is observed, the logarithm of FLR M l is Gaussian or asymptotically Gaussian (with increasing a signal-to-noise ratio (SNR)) random process [2, 3, 5]. We designate as Sl M l and N l M l M l signal and noise functions of the logarithm of FLR [2, 3], so Here M l Sl N l . (3) indicate averaging operation on all possible realizations of the observable data. According to [2, 5] for signal function the following approximation is valid Sl S0 max 0,1 l l0 S N , (4) where components S0 and SN characterize the accumulated (output) energy of a useful signal and distortions accordingly. Noise function N l , as well as M l , is Gaussian or asymptotically Gaussian centered random process which covariance function Bl1 , l2 N l1 N l2 under li l0 1 , i 1,2 can be presented in a kind [2, 5, 15] Bl1 , l2 2N max 0,1 l1 l2 S2 2N max0,1 min0, l1 l0 , l2 l0 max 0, l1 l0 , l2 l0 (5) 1 l1 l2 g min l1 l0 , l2 l0 , l1 l0 l2 l0 0 ; S2 l1 l0 l2 l0 0 ; 1 l1 l2 , where g S2 2N S2 . If max l1 l0 , l 2 l0 1 then Bl1 , l 2 2N max 0,1 l1 l 2 . (6) 4472 O. V. Chernoyarov et al. Values S2 , 2N describe dispersions of the process N l in signal l S and noise l N \ S areas of an interval of possible values of the parameter l0 accordingly. We understand subintervals S l 0 1, l 0 1 , N L1 , l 0 1 l 0 1, L 2 (7) to be signal and noise areas within which the signal function (4) is distinct from S N (i.e., depends on true value of unknown signal parameter l0 ) or is equal to S N (i.e., does not depends on value l0 ). To start with we consider characteristics of the measurer (2). According to [3] the most general and full (in a probability sense) characteristic of the estimate l m is its conditional (under fixed l0 ) probability density w x l0 which can be written down in terms of w x l0 P0 w 0 x l0 1 P0 w a x l0 . (8) Here P0 P l m S (9) is the reliable-estimate probability, w 0 x l0 and w a x l0 are conditional probability densities of a reliable and anomalous estimate l m accordingly. As a reliable estimate we understand the estimate l m located within the interval S (7). If l m N , then, following [2, 3], we name such estimate and the corresponding estimation error as anomalous. The allowance of anomalous errors is necessary, if prior interval length m L 2 L1 of possible values of the parameter l0 is much greater than the range of the reliable estimate interval S [2, 3], i.e. m 1 . (10) 2 Let us assume that output (power) SNR z for the algorithm (2) is sufficiently great: z 2 Sl0 S N 2 N 2 l0 S02 S2 1 . (11) Then, according to [2], the estimate l m converges in mean square to the true value of the estimated parameter l0 with increasing z 2 . Thereupon, for the definition of the characteristics of the reliable estimate l m under z 2 1 it is enough to investigate the behavior of the functional M l in the small neighborhood L of point l l0 : L l0 , l0 so that l m L . Here 1 is some value, limiting the neighborhood of the point l l0 . Taking the last remark into account, we introduce the functional l M l M x S , l, x L (12) Application of the local Markov approximation method and present the distribution function F0 x l 0 x L1 4473 w x l 0 dx of the reliable estimate l m in terms of F0 x l 0 Pl m x P max M l max M l P max l max l . (13) l x l x l x l x The probability P max l max l in Eq. (13) can be found with the help l x l x of the two-dimensional distribution function F2 u, , x or probability density w 2 u, , x of absolute maxima of the functional (12): u F2 u, , x w 2 u , , x du d P max l u, max l . l x l x 00 (14) Here it has been taken into account that max l 0 by definition. Really, comparing Eqs. (13) and (14) we get F0 x l 0 u 0 0 0 F2 u, , x u w 2 u, , x ddu du . (15) u According to Eqs. (4)-(6), (12), l is Gaussian or asymptotically Gaussian random process with covariance function in form of B l1 , l2 l1 l1 l2 l2 min l1 x , l2 x , l1 x l2 x 0 ; 2 g l1 x l2 x 0 . 0, (16) This implies 1) realizations of the process l in the intervals l0 , x , x, l0 are not correlated and therefore they are statistically independent, as being Gaussian (asymptotic Gaussian); 2) within each of the intervals l0 , x , x, l0 conditions of the Doob’s theorem in the wording [6] are satisfied, so l is Markov random process of diffusion type. Under l x the drift K1 and diffusion K 2 coefficients of the process l are determined as K 1 lim l l l l l 0 l z , l l0 ; z , l l0 ; (17) l l l 2 K 2 lim l 2 g. l Making use of the asymptotical statistical independence of the random process l values in intervals l0 , x , x, l0 we now rewrite the probability l 0 4474 O. V. Chernoyarov et al. (14) in a form of F2 u, , x P1x u P2 x . (18) Here P1x u P max l u , P2 x P max l . Then taking into l x l x account Eq. (15) the distribution function F0 x l0 of the reliable estimate l m can be expressed as F0 x l 0 P2x u dP1x u , x l0 , l0 . (19) 0 We calculate the probabilities P1x u and P2 x , using Markov properties of the process l . For this purpose we introduce the random process l l , 0 (20) within the interval l x, l0 and express the probability P2 x as follows P2 x P l P l 0 w 2 x y, l 0 dy , 0 . (21) x l l0 x l l0 0 Here w 2 x y, l0 is the probability density that the process l , beginning at the moment l x from the value x , will reach the value l 0 y by the moment l l0 , and, at the same time, over the interval x, l0 the process l lies within an interval 0, . According to Eq. (20) the process l , as well as l , is Markov random process of diffusion type with drift coefficient K 1 K1 and diffusion coefficient K 2 K 2 where K1 and K 2 are defined from Eqs. (17). Then the probability density w 2 x y, l0 can be found from the solution of the direct Fokker-Planck-Kolmogorov equation [7, 8] w 2 x y, l 1 2 K 2 w 2x y, l 0 (22) K1 w 2 x y, l l y 2 y 2 with starting condition w 2 x y, l l x x (23) and boundary conditions w 2 x y, l y0 0, w 2 x y, l y 0. (24) Solving Eq. (22), we single out two cases. 1) x l0 The given condition means that drift and diffusion coefficients remain constants over the interval x, l0 , whereupon Eq. (22) can be rewritten as follows Application of the local Markov approximation method 4475 w 2 x y, l w 2 x y, l K 2 2 w 2 x y, l . (25) K1 l y 2 y 2 The solution of Eq. (25) can be received by the method of characteristic function [7, 8]. Taking into account the starting condition (23), we have as a result K1 l x y2 1 ( 0) w 2 x y, l exp . 2 K 2 l x 2K 2 l x or substituting an explicit form of coefficients K1 and K 2 from Eq. (17) w (20x) y, l z l x y 2 1 exp . 22 g l x 22 g l x (26) The upper index “0” of the probability density (26) means that boundary conditions (24), while solving Eq. (25), were not imposed so far. In order to find the solution for Eq. (25) with boundary conditions, we use a reflection method with sign inversion [7, 8]. According to this method, the solution of Eq. (25), with an absorbing barrier arranged in point y C (that corresponds to the condition w 2 x y, l y C 0 ), can be written down in a form of w 2 x y, l w (20x) y, l w (20x) 2C y, l . (27) Then, substituting Eq. (26) in Eq. (27) and believing, according to Eq. (24), that C 0 we finally receive z l x y 2 1 w 2 x y, l exp 22 g l x 22 g l x (28) z l x y 2 2z exp exp . 22 g l x 2g Using Eq. (28) in Eq. (21), where l l0 , and having carried out integration, for probability P2 x we get z l 0 x 2z z l 0 x P2 x . (29) exp 2 g 2 g l 0 x 2 g l 0 x Here x 1 2 x exp t 2 2dt is probability integral [9]. 2) x l0 In this case, according to Eq. (17), within the interval x, l0 the drift coefficient K1 varies stepwise in a point l l0 . Thereupon, we divide an interval x, l0 into two subintervals: x, l0 and l0 ,l0 . As it has been shown above, the solution of Eq. (22) with starting and boundary conditions (23), 4476 O. V. Chernoyarov et al. (24) under l x, l 0 ( K1 z , K 2 2 g ) is of the form ~ y,l w 2x z l x y 2 1 exp 22 g l x 22 g l x (30) . For the definition of the probability density w 2 x y, l at the moment l l0 , we find the solution of Eq. (22) within the interval l l0 ,l0 , where K1 z , K 2 2 g , the boundary conditions are defined from Eqs. (24), and it is necessary to choose the probability density (30) at the moment l l0 , as the starting condition. Then, following [7, 8] we receive 2 1 ~ y , l exp y K1 l l0 y w 2 x y, l w 2x 0 2 K 2 l l0 2K 2 l l0 0 z l x y 2 2z exp exp 22 g l x 2g y K1 l l0 y 2 2 y K1 exp exp 2 K 2 l l0 K2 ~ y, l or substituting explicit expressions of K , K , w w 2 x y, l 1 22 g l l0 l0 x 1 0 2 2x dy 0 z l x y 2 0 exp 22 g l0 x 2z z l0 x y 2 exp exp 22 g l0 x 2 g y z l l y 2 0 (31) exp 22 g l l0 y z l l0 y 2 2 y z exp exp 22 g l l0 2g dy . Now using Eq. (31) in Eq. (21) at l l0 and carrying out integration on a variable y, we have for probability P2 x P2 x exp z 2 l0 x 22 g 2 2 g l0 x 0 z y y 2 exp exp 2 g 22 g l0 x y 2 z y exp 2 yz z y dy . exp 2 g 2 g 22 g l0 x 2 g (32) Here the index «’» of the integration variable is omitted. In consequence of the symmetry of the signal function Sl (4) of functional Application of the local Markov approximation method 4477 M l with respect to the point l l0 and stationarity of its noise function N l (5), it is easy to establish that the probability P1x u is connected to the probability P2 x u by a relation P1x u P2 2l0 x u . Then, with Eqs. (29), (32) in mind, it can be written down with x l0 as z x l 0 u 2uz z x l 0 u P1x u (33) exp 2 g 2 g x l 0 2 g x l 0 and with x l0 as exp z 2 x l0 22 g P1x u 22 g x l0 0 z u y u y 2 exp exp 2 g 22 g x l0 u y 2 z y exp 2 yz z y dy . exp 2g 2 g 22 g x l0 2 g (34) Eqs. (29), (32)-(34) allow us to find the distribution function F0 x l0 of the reliable estimate l m . So, if x l0 , then, by substituting Eqs. (32), (33) in Eq. (19), we get F0 x l0 Pm l0 x , (35) where u y 2 z u y 1 z2v Pm v exp exp exp 2 g 2v 22 g 0 0 2 g 2 v 2 g u y 2 z y 2 yz z y 2z exp exp 2 g 2 g 2 g 2 v 2 g 2 g 2uz z v u exp 2 g 2 g v z v u 2 2 exp v 22 g v dudy . (36) When x l0 , in order to find the distribution function F0 x l0 , we use the expression F0 x l 0 1 P1x u dP2 x u , (37) 0 which is received from Eq. (19) by its integration by parts. Then, substitution of probabilities P2 x u (29), P1x u (34) in Eq. (37) leads us to (38) F0 x l0 1 Pm x l0 . Uniting Eqs. (35) and (38), we can definitively write down 4478 O. V. Chernoyarov et al. Pm l 0 x , l0 x l0 ; F0 x l 0 (39) 1 Pm l 0 x , l 0 x l 0 . Let us consider behavior of the distribution function F0 x l0 (39) under z 2 . In Eq. (36) we use the asymptotic formula for probability integral [9]: x x 1 exp x 2 2 2 x . (40) and neglect higher-order infinitesimal terms compared with z. Then, for function (36), the following approximation is valid z u y 2z z2v Pm v exp exp 2g 22 g 0 0 2v 2 g 3 u y 2 u y 2 2 yz exp dudy exp 1 exp 2 g 2 v 2 g 2 v 2 g or after integration operation completed: 4z 2 v 5 2z 2 v v 3 Pm v 1 z 2 exp 2 g 2 2g 2 g (41) v 2z 2 v z2v exp 1 3z . 2 g 2 g 2 2 g According to Eq. (41), the function Pm v is distinct from zero in a small neighborhood of the point v 0 . Similarly [2] it allows to extend approximation (39), (41) to the total number axis, sacrificing no accuracy: Pm l 0 x , x l 0 ; F0 x l 0 (42) 1 Pm l 0 x , l 0 x . We use Eq. (42) for distribution function F0 x l0 with finite but large z. Then conditional probability density w 0 x l0 , bias b 0 l m l0 and variance V0 l m l0 of a reliable estimate l m are determined as w 0 x l0 4z 2 x l 0 d 2z 2 F0 x l0 3 exp 2g dx 2g x l0 1 3z 2g z x l0 2g b 0 l m l 0 l m l 0 1 , x , , (43) x l0 w 0 x l0 dx 0 , (44) Application of the local Markov approximation method V0 l m l0 l m l0 2 x l0 2 4479 w 0 x l0 dx 132 g 2 8z 4 . (45) The accuracy of formulas (41)-(45) increases with z. Now let us calculate the probability P0 (9) of the reliable estimate l m . For this purpose we introduce the functional M l M l S N and the random variables H N sup M l , lN H S sup M l , (46) lS where S N , S , N are defined from Eqs. (4), (7). Then the probability P0 (9) can be presented as P0 P H S H N . (47) According to Eqs. (5), (6), the correlation time of the random process N l (and M l too) does not exceed 1. Thus, if the condition (10) is satisfied, then the occurrence probability of the greatest maximum of functional M l within the interval l 0 2, l 0 1 l 0 1, l 0 2 can be neglected in comparison with the occurrence probability of the greatest maximum of M l within the interval L1 , l0 2 l0 2, L 2 . So, as the functional M l is Gaussian (asymptotic Gaussian), then, therefore, the values of M l and also the random variables H N , H S (46), in the intervals L1 , l0 2 l0 2, L 2 and l 0 1, l 0 1 are approximately statistically independent. We designate FN PH N N , FS PH S S (48) as distribution functions of random variables H N N and H S S accordingly. Owing to statistical independence of H N and H S , likewise in Eq. (19), for probability P0 (47) of the reliable estimate l m we can write down P0 FN dFS r , (49) where r S N and integration is conducted for all possible values κ. Let us find probabilities FN and FS . According to Eqs. (3), (4), (48) FN P M l N PN l N , l L1 , l0 1 l0 1, L 2 . (50) 4480 O. V. Chernoyarov et al. Here N l is Gaussian (asymptotically Gaussian) stationary random process with zero mathematical expectation and covariance function (6). Then on the basis of the results [2, 10] for the function FN (50) the following approximation can be carried out m 2 , 1; exp exp 2 FN (51) 2 1. 0, Accuracy of Eq. (51) increases with m and . Let us pass to the probability FS definition. As it was noted above, when condition (11) is satisfied, it is possible to consider that the estimate l m is situated in the small -neighborhood of the point l l0 . Then the distribution function FS can be presented in a form of FS P 0 l 0 , Here l l0 ,l0 . 0 l l x l M l M l 0 S , 0 M l0 S N S 0 (52) (53) and l is defined from Eq. (12). As appears from Eq. (16), realizations of the random process 0 l are approximately statistically independent in the intervals l0 ,l0 and l0 ,l0 . Then for FS (52) we have FS P 0 l 0 P 0 l 0 . (54) l0 l l0 l0 l l0 According to Eqs. (2)-(5) and as the functional M l is Gaussian (asymptotical Gaussian), the random variable 0 (53) is Gaussian (asymptotically Gaussian) random value with mathematical expectation z and unit dispersion. We designate F1 P 0 l , F2 P 0 l , (55) l0 l l0 l0 l l0 and w 0 x exp x z 2 2 2 as probability density of the random variable 0 . Then the distribution function (54) can be expressed as FS F1 y F2 y w 0 y dy . (56) Let us find the probabilities F1 , F2 (55). For this purpose we intro- Application of the local Markov approximation method 4481 duce the random process 0 l 0 l , assigned to the interval l 0 ,l 0 and, by analogy to Eq. (21), we write down the probability F2 as F2 P 0 l 0 w 2 y, l 0 dy , l0 l l0 0 where w 2 y, l w 2 x y, l x l and w 2 x y, l 0 is the solution of Eq. (22) 0 with conditions (23), (24). It is easy to show that F2 P2 x x l , where P2 x is defined from Eq. (29). Then for function F2 we have 0 z 2 z z F2 (57) . exp 2 g 2 g 2 g Owing to symmetry of statistical properties (4), (5) of the functional M l (3) concerning a point l l0 , the probabilities F1 , F2 are connected by a parity F1 F2 . (58) As a result, taking into account Eqs. (56)-(58), for function FS (56), we find 2 z y y z 2 2z y z y FS exp exp dy . 2 g 2 g 2 2 2 g 1 (59) Let us consider the behavior of the function FS (59) for the case of the sufficiently great SNR (11). Similarly to Eq. (39), using asymptotic representation (40) of probability integral under z 2 and neglecting higher-order infinitesimal terms compared with z, we have 2 z 2 2z FS d , 1 exp 2 g exp 2 2 0 or after integration is carried out – 1 FS z 2 exp 2 z 2 2 z z z 1 exp 2 2 z 2 2z z z 2 1 . Here 2 2 g . (60) Substituting Eqs. (51), (60) in Eq. (49), for the probability P0 of the reliable estimate l m , we find 4482 O. V. Chernoyarov et al. mx 2 2 z 2 x 2 zx 2 z 2 exp P0 exp 2z exp exp 2 r 2 r 2 1 (61) 2 2 2x x 3 z x z 1 exp z z z 2 1 dx . r r r 2 Now we can write down the expressions for the characteristics of the signal parameter estimate l m with anomalous errors. According to [2, 3], when condition (10) is satisfied and M l (or N l ) is stationary Gaussian random process, for probability density w a x l 0 it is possible to use the approximation 1 m, x; w a x l 0 (62) 0, x. Substituting Eqs. (43), (61), (62) in Eq. (8), by analogy to Eqs. (44), (45) for conditional bias bl m l 0 and variance V l m l0 of the estimate l m with anomalous errors we have bl m l0 P0 b 0 l m l0 1 P0 b a l m l0 1 P0 b a l m l0 , V l m l 0 P0 V0 l m l 0 1 P0 Va l m l 0 . (63) Here b 0 l m l 0 , V0 l m l0 are defined from Eqs. (44), (45), and b a l m l0 L2 x l0 w a x l0 dx L 2 L1 2 l0 , L1 Va l m l0 L2 x l0 2 w a x l0 dx L22 L1L 2 L21 3 L 2 L1 l0 l02 L1 stay for the conditional bias and variance of the anomalous estimate l m , accordingly. Hence in general the estimate l m is conditionally biased. Accuracy of formulas (61), (63) increases with m (10), z (11). The results received above allow simply to write down expressions for detector (1) characteristics of the information signal st, l0 with unknown discontinuous parameter l0 against Gaussian distortions t . The type I (false alarm) and II (signal missing) error probabilities [1, 2, 5, 10] will be used by us as detection characteristics. We are limited to a practically important case, when the prior interval length m essentially exceeds the range of the reliable estimate interval S (7), i.e. the condition (10) is satisfied. Firstly, we believe that the useful signal st, l0 is absent. Then the false-alarm probability α can be presented in a form of Application of the local Markov approximation method P sup M l c x t t 1 FN c , lL1 , L 2 where FN c P sup M l c x t t . lL1 , L 2 If Eq. (10) holds then 4483 (64) M l m S N c S N FN c P lm Γ N , σN σN and, taking into account Eqs. (3), (4), (6), for function FN c the approximation (51) can be used with substitution of u c S N σ N (65) in place of κ. Therefore, for false-alarm probability we have 1 exp mu 2 exp u 2 2 , u 1 ; (66) u 1. 1, Now let us believe that the useful signal st, l0 is present on the detector input. Then the missing probability will be determined as M l m S N c S N . (67) β P sup M l c x t st, l0 t P σS σ S lL1,L 2 Using a condition of approximate statistical independence of random variables (46), for probability (67) we can write down FN u FS u r or when expressing functions (51), (60) in explicit form u r z 2 exp 2z 2 2 zz u r (68) u r z 1 exp2 2 z 2 2z z u r u r z 2 1 , exp mu 2 exp u 2 2 if u 1 , and 0 if u 1 . Accuracy of the formula (68) increases with u, m, z. Let us now consider the application of the Markov local approximation method for the definition of the accuracy characteristics of the concrete detectors and measurers of discontinuous signals. 3 Reception of the Quasideterministic Video Pulse with Unknown Appearance Time Let an additive mix of kind x t st, 0 n t t , is accessible to observation. Here t 0, T (69) 4484 O. V. Chernoyarov et al. 1 , x 1 2 , t 0 st, 0 aI (70) , I x 0 , x 1 2 , is the useful signal (video pulse) with amplitude a, duration τ and unknown appearance time 0 , while n t is Gaussian white noise with one-sided spectral density N 0 , and t are correlated distortions. As a model of correlated distortions, we choose the stationary centered Gaussian random process, possessing spectral density [4, 5, 11] G 2 I , where Ω – bandwidth, γ – value of spectral density (intensity) of the process t . In radio engineering appendices white noise n t describes internal instrument noises. And an unintentional (interburst) interference which has passed through input filter (preselector) of receiving device or barrage jamming can afford examples of random distortions t [11]. With the observable realization (69), it is necessary to estimate the parameter 0 , which values are from the prior interval 1 , 2 . It is assumed that the condition 0 1 2 2 2 T is satisfied, so the pulse (70) is always situated within the observation interval 0, T . If 4 1 , (71) then the logarithm of FLR can be written down as [12] M T 2a y 2 t dt N 0 N 0 0 N0 Here yt 2 x t dt 2 a 2 . (72) ln1 N0 N 0 x t h t t dt , and h t is the function which spectrum H satisfies to a condition H I . 2 In accordance with Eq. (72), the maximum likelihood estimate (MLE) m of the signal (70) appearance time is determined as λ m arg sup M 0 , 1 , 2 M 0 2 x t dt . (73) 2 Let us define characteristics of the estimate (73). For this purpose we put into consideration the dimentionless parameter l , designate l0 0 and, following Eq. (3), present the sufficient statistic M 0 (73) as the sum of signal Sl M 0 and noise N l M 0 M 0 functions: M 0 Sl N l . Carrying out averaging for all the possible realizations of the observable data (69) Application of the local Markov approximation method 4485 under fixed λ 0 , we find from Eq. (73) that the signal function Sl and covariance function of the noise function Bl1 , l2 N l1 N l2 are determined according to Eqs. (4)-(6) under S0 a , S N 0 , σ S2 σ 2N N 0 2 . Then Eqs. (63), with reference to conditional bias and variance of the normalized MLE l m λ m τ and with anomalous errors taken into account, have the appearance bl m l0 1 P0 L 2 L1 2 l0 , Vlm l0 P0 V0 l m l0 1 P0 L1L 2 where V0 l m l0 13 2z 4 , L22 L21 3 L2 L1 l0 l02 , (74) (75) 2 3z 2 exp mx exp x exp zx P0 2z exp 2 2 (76) 2 1 x 2z expz 5z 4x 2x 3zdx – conditional variance and probability of the reliable estimate l m , obtained on the basis of Eqs. (45), (61), z 2 2a 2 N 0 – output SNR (11) for algorithm (73), L1,2 1,2 , m L 2 L1 . For signal (70) detection, in accordance with Eqs. (1), (72) detector should compare a maximum of the functional M 0 (73) with the threshold c determined on the basis of accepted optimality criterion. It is easy to see that false-alarm and missing probabilities can be found from Eqs. (66), (68), where u c 2 N 0 , (77) 1 , r 1 , and m, z are defined the same as in Eq. (76). In particular, when u 1 for missing probability, we have exp mu 2 exp u 2 2 u z (78) 2 expz 3z 2 u u 2z exp2z 2z u u 3z . In order to establish the applicability range of asymptotically exact formulas (66), (74)-(78) the statistical computer modeling of the measuring and detection algorithms of the signal (70) with unknown appearance time was carried out. During the modeling for specified values μ (71), z, q ν N 0 and l0 samples ~ of the normalized functional M 0 l M 0 N 0 were formed within the interval L1 , L 2 with digitization step l 0.01 as follows K 1 max 1 qν α ~ M 0 l z max 0,1 l l0 Δ ~ν k k , 2 2Δ k K min (79) 4486 O. V. Chernoyarov et al. In Eq. (79) it is designated: K min int l l0 1 2 , K max int l l0 1 2 , int – integer part, ~ν k ~ν l0 kΔ – samples of l 0 k 1 ~ ~ ~ ~ the normalized process ~ t t N 0 , α k 2 n t d t – l 0 k independent Gaussian random numbers with characteristics α k 0 , α 2k 1 , ~ ~ ~ n t n t N 0 – normalized Gaussian white noise, t t – normalized time, and Δ – sampling step chosen so that mean square error of stepwise ~ t does approximation ~ν ~t ~ν k , l0 kΔ ~t l0 k 1 Δ of the process ~ not exceed 10 % [5, 13], i.e. ε 2 1 R ν Δ 2 0.1 . (80) ~ ~ ~ ~ t . Here R ν t sin 2 t 2 t is correlation coefficient of the process ~ The inequality (80) is satisfied, in particular, if 0.05 . (81) ~ ~ In this case neighboring samples ν k , ν k 1 have correlation coefficient R ν Δ 0.98 . ~ t were formed on the basis of the sequence Samples ~ν of the process ~ k of independent Gaussian random numbers by a moving summation method [5, 13]: ~ν q k ν 2 p 1 C jβ j k , j 0 Cj 1 sin2πμΔ j p , j p π 2Δ (82) where β j are independent Gaussian random numbers with zero mathematical expectations and unit dispersions. In the sum (82) number of summands was chosen proceeding from a condition [5, 13] σ 2~ν σ 2~ν k Here σ 2~ν μq ν , σ 2~ν k q ν 2 p 1 C2j σ 2~ν . (83) ~ are dispersions of the process ~ t and its j 0 formed samples accordingly, and 1 is the maximum allowed deviation of a dispersion of the generated sample from a dispersion of the modeled process. We are limited to 0.1 [5]. Then inequality (83) starts to be carried out, if p 103 . (84) Formation of Gaussian numbers α k , β j with parameters (0,1) was implemented from sequences of independent random numbers n , n , uniformly distributed within the interval [0,1], by the Cornish-Fisher method [14]: Application of the local Markov approximation method i Zi Z 3i 3Z i , 20 N Zi 4487 12 N θ N i 1 n 0,5 , Nn1 (85) where i is one of sequences α k , β j , and n is sequence n , n corresponding to it. The number of summands N in the sum (85), following [14], was chosen equal to 5. Thus the mean square error of the step approximation (79) ~ of functional M 0 l realization does not exceed 5 %. ~ For each realization of M 0 l , generated by Eqs. (79), (82), (85), according to Eqs. (1), (2) the normalized estimate l m was determined, and the decision on presence or absence of a useful signal (70) in realization of the observable data was made. Also experimental detection and measuring characteristics were found. In Figs. 1-3 some results of statistical modeling are presented, where corresponding theoretical dependences are shown also. Each experimental value ~ was received as a result of processing of not less than 3 10 4 realizations M 0 l (79) under l0 L1 L 2 2 , L1 1 2 , L 2 m 1 2 , 50 , q ν 1 . Thus with probability of 0.9 confidence intervals boundaries deviate from experimental values no more than for 10...15 %. In Fig. 1 solid line represents dependence (74) of the normalized variance ~ Vl 12V l m l0 m 2 of estimate lm from SNR z, taking into account anomalous errors, if m 20 . Here analogous dependence (75) of the normalized ~ variance V0l 12V0 l m l0 m 2 of reliable MLE lm is also drawn by dashed ~ ~ line. The experimental values of estimate variances Vl , V0l are designated by squares and crosses accordingly. In Fig. 2 the theoretical dependence of false-alarm probability (66), where u is normalized threshold (77), is traced by solid line. The length of the reduced interval m (76) is taken equal to 20. By squares the experimental values of false-alarm probability are designated here. In Fig. 3 the theoretical dependence of missing probability (78) is plotted for m 20 . The threshold c was defined from Eqs. (66), (77) by Neumann-Pirson criterion, according to the specified level of false-alarm probability 0.01 . Experimental values of missing probability are designated by squares. 4488 O. V. Chernoyarov et al. Fig. 1. Normalized variance of appearance time estimate. Fig. 2. False-alarm probability. As follows from Fig. 1-3, theoretical dependences for probabilities α (66), (77), β (78) and variance V l m l0 (94) well approximate experimental data, if SNR z 0 , and theoretical dependence for V0 l m l0 (75), – if z 3 . Under z 3 theoretical dependence for V0 l m l0 (75) deviates from experimental data as formula (75) was received without considering finite length of the prior definition interval L1 , L 2 of the parameter l0 . Thereof, when conditional variance V0 l m l0 of the reliable MLE lm becomes commensurable or more than value m 2 12 , the error of Eq. (75) increases essentially. For more exact approximation of variance V0 l m l0 in the domain of small SNR z the following expression can be used V0 l m l0 minV0 max ,V0 l m l0 . (86) Here V0 max 1 3 is variance of the uniform random value in the interval S (7) of the reliable estimate lm . It should be also noted that with decreasing z, when SNR z 78 , the probability of anomalous errors Pa 1 P0 (76) considerably increases and verges towards 1. It leads to abrupt (in comparison with a case of a reliable estimate) increment of MLE lm variance. With increasing z, when z 78 , values of variances V l m l0 (74), V0 l m l0 (75) almost coincide, and the estimate of video pulse (70) appearance time becomes reliable with the probability close to 1. Application of the local Markov approximation method 4489 4 Reception of the Random Radio Pulse with Unknown Appearance Time Now let us present the useful signal st, 0 as a random radio pulse, which represents a realization segment of the stationary centered high-frequency random process t [4, 5] st, 0 t It 0 . (87) Here 0 – appearance time, τ – duration of the pulse and I x – unit duration indicator (70). We fit spectral density of the pulse substructure t as [4, 5] G D 1 I 1 I 1 , where ϑ – band center, 1 – bandwidth, and D – dispersion of the process t . As well as earlier, we approximate internal instrument noises by Gaussian white noise n t with one-sided spectral density N 0 . As a model of correlated distortions t , we choose the stationary centered Gaussian random process, possessing spectral density [4, 5, 11] G 2 I 2 I 2 , 2 1 . Here 2 – bandwidth, and γ – value of spectral density (intensity) of the process t . Let us consider that the signal duration τ is much greater than the correlation time of the process t (process t fluctuations are "fast"), then 1 1 2 1 . (88) Appearance time 0 of the signal (87) is unknown a priori and possesses values from a prior interval 1 , 2 . Meanwhile, observation interval boundaries 0, T satisfy to condition 0 1 2 2 2 T , i.e. the pulse (87) is always located within an interval 0, T . Synthesizing measurer of pulse (87) appearance time by a maximum likelihood method and in view of a condition (88), we write down the logarithm of FLR as follows [15] γ d d Mτ λ γ MT γ K 1 ln1 . Mλ μ ln1 N 0 γ N 0 γ d N 0 N0 γ N0 N0 (89) In Eq. (89) it is designated that: d 2D 1 , K T 2 1 , M τ λ 2 t dt , y12 2 T M T y 22 t dt , 0 (90) 4490 O. V. Chernoyarov et al. and y i t x t h i t t dt , i 1,2 , where h i t is the function which spectrum H i satisfies to a condition H i ω I ω Ω i I ω Ω i , 2 and x t st, 0 n t t is realization of the observable data. Then MLE λ m of the parameter 0 is determined as λ m arg sup M λ arg sup M τ λ . (91) λ Λ 1,Λ 2 λΛ 1,Λ 2 In order to calculate characteristics of the estimate λ m (91), following Eq. (3), we present the sufficient statistic M τ λ (90), using dimentionless values l , l0 0 , as the sum of signal Sl M τ λ and noise N l M τ λ M τ λ functions: M τ λ Sl N l . If condition (88) holds, then it can be shown [15] that the signal function S l is defined from Eq. (4) under S 0 D and S N τ E N E γ . (92) where E N N 0 Ω1 2π , E γ γΩ1 2π are signal band average powers of noise n t and hindrance t . Noise function N l is asymptotically (when 1 ) Gaussian random process [5] with a zero mathematical expectation and covariance function of a kind (5), (6) under [15] σ S2 τE N 1 q ν q 2 μ1 , σ 2N τE N 1 q ν 2 μ1 , (93) g q 2 2q ν q 1 q ν q 2 , q D E N , q ν γ N 0 . Therefore conditional bias and variance of the normalized MLE l m λ m τ (91), and in view of anomalous errors, can be found from Eqs. (63) as bl m l0 1 P0 L 2 L1 2 l0 , V l m l0 P0 V0 l m l0 1 P0 L22 L1L 2 L21 3 L 2 L1 l0 l02 , (94) where V0 l m l0 and P0 are conditional variance and probability of the reliable estimate l m calculated using Eqs. (45), (61) under g (93), z 2 μ1q 2 1 q ν q 2 , 21 q q 2 1 q 2 1 q q 2 , (95) r 1 q ν q 1 q ν , m L 2 L1 , L1,2 1,2 . For signal (87) detection, in accordance with Eqs. (1), (89), detector should compare a maximum of the functional M τ λ (90) with the threshold c, determined on the basis of the accepted optimality criterion. It is easy to see that Application of the local Markov approximation method 4491 false-alarm and missing probabilities can be found from Eqs. (66), (68), where u c S N σ N (96) and S N , σ N , z, ψ, r, m are defined from Eqs. (92), (93), (95). Experimental characteristics of the measurer and detector of the random pulse with unknown appearance time were found by methods of statistical computer modeling. For the reduction of the computational burden during formation of the sufficient statistic M τ λ samples, it was supposed that the bandlimitedness condition of a kind 1 for the process t is satisfied. It allows us to use representation of the function y1 t (90) through their low-frequency quadratures [13] and to form sufficient statistics M τ λ (90), as the sum of the two independent random processes: M τ λ M1 λ M 2 λ 2 , M j λ λ τ 2 y12j t dt , j 1,2 , (97) λ τ 2 y1 j t x j t h 0 t t dt , x j t s j t n j t ν j t . Here s j t ξ j t It λ 0 τ , ξ j t , n j t , j t are statistically independent centered Gaussian random processes with the spectral densities G ξ 0 ω 2πD Ω1 Iω Ω1 , N 0 and G ν 0 ω γIω Ω 2 respectively, while the spectrum H 0 of the function H 0 2 I 1 . h 0 t satisfies a condition During modeling within the interval L1 , L 2 with discretization step ~ ~ ~ t 0.05 1 in normalized time t t , samples ~ y jn ~ y1 j l0 nΔ t were ~ y1 j t y1 j t N 0 , j 1,2 formed of normalized random process realizations ~ ~ ~ ~ (90) possessing by the correlation coefficient R yj t sin πμ1 t πμ1 t . Thus, according to Eq. (80) mean square errors of step approximations ~ ~ ~ ~ ~ ~ y1 j t ~ y jn , l0 t t l0 n 1 t for continuous realizations ~ y1 j t did not exceed 10 %. With Eqs. (97) in mind, we can achieve the normalized ~ sufficient statistics M τ l M τ λ N 0 (90), presented as ~ N 1 t max ~ 2 ~ 2 ~ (98) M τ l y1n y2 n . 2 nN min ~ Here N min intl l0 1 2 Δ t , N max intl l0 1 2 ~t . Samples of processes ~ y jn , j 1,2 were generated in terms of the sequence of independent Gaussian random numbers by a moving summation method [13], as it is described in [15]: 4492 O. V. Chernoyarov et al. ~ y jn R max 1 r R min ~ 1 2 ξ jr H nr 1 2p n p 1 r n p ~ 1 q 1 β j mr , jr H mp 1 m 0 H nr1 r 11 2 1 s r 1 2 α js ~ν js . Δ 2 (99) 2 p 1 ~ν 1 q ν H 2 χ j m s . js π Δ 2 m 0 mp (100) Here R min max R , n p , R max minR , n p , R int1 2 1 , ~ ~ ξ jr ξ j l0 rΔ1 , ~ js ~ j l0 s 2 ; 1 , 2 are discretization steps of ~ ~ ~ normalized processes ξ j t ξ j t τ N 0 and ~ν j t ν j t τ N 0 accordingly, α jm , β jm , jm are independent Gaussian random numbers with zero mathematical expectations and unit i H mp sinπμ i Δ i m p m p , i 1,2 , 2 2 2 . dispersions, Discretization steps 1 , 2 in Eq. (99) are chosen, proceeding from a ~ condition of type (80), where R ν t are substituted by correlation coefficient ~ ~ ~ ~ ~ ~ ~ ~ R ξj t sin πμ1 t πμ1 t or R νj t sin 2 t 2 t of the process j t or ~ ~ j t accordingly. Then, by analogy Eq. (81), values Δ i , i 1,2 can be taken equal to Δ i 0.05 μ i . In the sums (100) number of summands corresponds to the value p p 103 . According to relation similar to Eq. (83), it provides a ~ relative deviation of the generated sample jr , ~ν js dispersions from the modeled process dispersions to be no more than 5 %, as in Eq. (84). Formation of independent Gaussian numbers with parameters (0,1) was carried out following Eqs. (85). In the issue the mean square error of step approximation of continuous ~ realization of functional M l received on the basis of Eqs. (98)-(100) did not exceed 10 % at a digitization step l 0.01 . In Figs. 4-6 some results of statistical modeling are presented, where corresponding theoretical dependences are also shown. Each experimental value ~ was received as a result of a processing of not less than 10 4 realizations M τ l (98) under l0 L1 L 2 2 , L1 1 2 , L 2 m 1 2 , 1 2 , q ν 0,5 . Thus, with probability of 0.9, confidence intervals boundaries deviate from experimental values no more than for 10...15 %. Application of the local Markov approximation method 4493 Fig. 3. Missing probability. Fig. 4. Normalized variance of appearance time estimate. Fig. 5. False-alarm probability. Fig. 6. Missing probability. In Fig. 4 solid lines represent dependences (94) of the normalized variance ~ Vl 12V l m l0 m 2 of estimate l m from parameter q (93) taking into account anomalous errors, if m 20 . Here analogous dependences (45), (93), (95) of the ~ normalized variance V0l 12V0 l m l0 m 2 of reliable MLE l m are also drawn by dashed lines. Curves 1 are calculated with 50 , 2 – 100, 3 – 200. The ~ ~ experimental values of estimate variances Vl , V0l are designated by squares, crosses, rhombuses and pluses, circles, triangles for 50 , 100 and 200 accordingly. In Fig. 5 the theoretical dependence of false-alarm probability (66), where u is normalized threshold (96), is traced by solid line. The length of the 4494 O. V. Chernoyarov et al. reduced interval m (95) is taken equal to 20. By squares, crosses and rhombuses the experimental values of false-alarm probability are designated for 50 , 100 and 200. Finally, in Fig. 6 the theoretical dependences of missing probability calculated, using Eqs. (68), (95), (96), are plotted for m 20 . Curve 1 corresponds to 50 , 2 – 100, 3 – 200. The threshold c was defined from Eqs. (66), (96) by Neumann-Pirson criterion according to the specified level of false-alarm probability 0.01 . Experimental values of missing probability are designated by squares, crosses and rhombuses for 50 , 100, 200 accordingly. As follows from Fig. 4-6, theoretical dependences for probabilities α (66), (96), β (68), (95), (96) and variances V0 l m l0 (45), (93), (95), V l m l0 (94) well approximate experimental data, if, at least, 50 , q 0.1 and z 11.5 . With decreasing q, when SNR z 3 4 , the probability of anomalous errors Pa 1 P0 (61), (95) considerably increases and verges towards 1. It leads to abrupt (in comparison with a case of a reliable estimate) increment of MLE l m variance. With increasing q, when z 3 4 , values of variances V0 l m l0 , V l m l0 almost coincide, and the estimate of pulse (87) appearance time becomes reliable with the probability close to 1. The divergence of theoretical and experimental dependences V0 l m l0 under z 1...1.5 is connected by that final length of an prior interval L1 , L 2 of possible values of parameter l0 was not considered upon receipt of expression (45). Thereof, when conditional variance of reliable MLE l m becomes commensurable or more than value m 2 12 , accuracy of the formula (45) essentially worsens. In order to improve rather the mean square error amount of an appearance time estimate under small SNR the transformation similar to Eq. (86) can be used. The deviation of theoretical dependences (45), (93), (95) (and, therefore, (94)) from corresponding experimental values is observed in case of great SNR z also, when q 2 3 . It is connected by that formulas (89), (92), (93) for functional M and its characteristics were found on the assumption that sizes of order of correlation time of the pulse (87) random substructure t are negligible. Hence, when MLE l m variance decreases to the size of order 2 , the miscalculation on formulas (45), (94), (95) becomes considerable. 5 Conclusion For operating efficiency definition of optimal (maximum-likelihood) receiving devices of signals with unknown discontinuous parameters a method based on approximation of the solving statistics increments by Markov random process Application of the local Markov approximation method 4495 (local Markov approximation method) can be used. With the help of the given approach the closed analytical expressions for characteristics can be found of detectors and measurers of discontinuous quasidetermined and Gaussian random signals, which well describe corresponding experimental data in a wide range of output signal-to-noise ratios. The received results make it possible theoretically to estimate practical application appropriateness of one or another processing algorithm of discontinuous signals in each specific case. Acknowledgements. The reported study was supported by Russian Foundation for Basic Research (research projects No. 13-08-00735a, 13-08-97538) and Russian Science Foundation (research project No. 14-29-00208). References [1] H.L. van Trees, Detection, Estimation, and Modulation Theory. Part I, Wiley, New York, 1971. [2] A.P. Trifonov, Yu.S. Shinakov, Joint Discrimination of Signals and Estimation of Their Parameters Against Background (in Russian), Radio i Svyaz', Moscow, 1986. [3] E.I. Kulikov, A.P. Trifonov, Estimation of Signal Parameters Against Hindrances (in Russian), Sovetskoe Radio, Moscow, 1978. [4] Applied Theory of Random Processes and Fields (in Russian) / Edited by K.K. Vasilyev, V.A. Omelychenko, Ulyanovsk State Technical University, Ulyanovks, 1995. [5] A.P. Trifonov, E.P. Nechaev, V.I. Parfenov, Detection of Stochastic Signals with Unknown Parameters (in Russian), Voronezh State University, Voronezh, 1991. [6] T. Kailath, Some integral equations with nonrational kernels, IEEE Transactions on Information Theory, 4 (1966), 442-447. [7] E.B. Dynkin, Theory of Markov Processes, Dover Publications Inc., New York, 2006. [8] V.I. Tikhonov, M.A. Mironov, Markov Processes (in Russian), Sovetskoe Radio, Moscow, 1977. [9] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards. Applied Mathematics Series 55, USA, 1964. [10] Signal Detection Theory (in Russian) / Edited by P.A. Bakut, Radio i Svyaz', Moscow, 1984. [11] V.D. Dobykin, A.I. Kupriyanov, V.G. Ponomarev, L.N. Shustov, Electronic Warfare. Digital Storing and Reproduction of Radio Signals and Electromagnetic Waves (in Russian), Vuzovskaya Kniga, Moscow, 2009. 4496 O. V. Chernoyarov et al. [12] O.V. Chernoyarov, The efficiency of reception of a random pulse signal with unknown parameters under the conditions of detuned duration. Telecommunications and Radio Engineering, 1 (2013), 1-23. [13] V.V. Bykov, Numerical Modeling in Statistical Radio Engineering (in Russian), Sovetskoe Radio, Moscow, 1971. [14] L. Devroye, Non-uniform Random Variate Generation. Springer-Verlag, 1986. [15] Information and Communication Systems and Technologies: Problems and Perspectives (in Russian) / Edited by A.V. Babkin, Polytechnic University Publisher, St.-Petersburg, 2007. Received: June 11, 2014

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