Optimal-speed algorithms for localization of random pulsed point sources generating super short pulses

Optimal-speed al gorithms for localization of rando m pulsed point sources generating super short pulses A.L.Reznik, A.A.Soloviev, A.V.Torgov Institute of Automation and Electrometr y , Sibe rian Branch, Russ ian Academ y of Sciences, Academician Koptyug ave. 1, Novosibirsk, Russia, 630090 Corresponding author : A.L.Reznik E-mail: reznik@iae.nsk.su Abstract. The time-optimal technique of spatial localization of the random pulsed- point source that has the uniform distribution density on search i nterval and ind icating itself by generation of the instant impulses (delta f unctions ) at random ti me points is developed. Localization is carried out by means of the receiver h aving view wind ow fr eely reconstructed in time. The created algorithm s ar e generalized t o t he search carried out by system consisting of several receiv ers. Introduction Usually on e of two t ypes of a location – passive or the active is applied to search for unknown signal sources [1-3]. The passive location is b ased on reception of own radiation of the object. In case of the active location the search system radiates own probing signal and receives its reflection from the object. D epending on parameters of the received signal coordinat es (characteristic) of the tar get are defined. In this paper we consider al gorithms of time-optimal search fo r Poisson sources indicating themselves by generation puls es at random t imepoints. The major feature which is essentially distinguishing our research from the passive and active location methods mentioned above is that the searched source becomes i tself apparent by means of generation of the i nfinitely short ( i nstant aneous ) de lta-pulses. Time- optimal search al gorithm should, generally, meet one of two requirements: either to minimiz e the total searc h effort necessary to detect the object; or to maximiz e the full detection probability in the presence o f a limited search effort. We understand the pulsed-point s ource as an object with negligible small angular dimensions ( a mathematical point) that placed randoml y on the x axis with a priori probabilit y densit y function f ( x ) and emitting infinitel y short pulses ( δ -functions) with a Poisson intensit y λ . Thus, the t ime interval between pul ses is random variable t with an exponential densit y function h ( t ) = λ exp ( - λ t ). The search for the object is performed b y a recording device, whos e « viewing window » is capable to b e reconstructed in an y w ay in time. The pulse i s fix ed if the active object produced the pulse is in t he "viewing window" of the recording device. Otherwise, the pulse is considered to be missed. As the pulse is registered, the window narrows, and the source positi on becomes more precise. It is required to se arch for the sou rce during the mi nimum (st atistically) time wit h an ε accurac y. In §1 the general proble m definition of localization of a random pulsed source is given and th e complexit y of its analytica l decision in case of source’s a rbitrary distributi on on a search interval is shown. In the s ame s ection the strategies of search, optimum in certain classes of search algorithms are c alculated. All tasks considered in §1 belong to localization of the only source of impulses by means of the onl y receiver. In §2 the problem of c onsecutive localization of several pulsed sources i s considered. It is supposed that localization of each of them is carried out by means of one receiver, but, unlike §1 , it is considered that all s ources are uniforml y distributed on a search inte rval. For construction optimum search strategy are used not only tradit ional numerical algorithms, but also program algorithms of carr ying out s ymbolic calculations and analytical transformations which are seldom used in scientific practice. In §3 t he problem of creation of the strategy of localization of a random pulsed source, optimum on tim e, when the recepti on system consist s of several receivers is solve d. As well as §2, it is supposed that the unknown source has uniform distribution on a searc h interval. 1. Programs and algor ithms for optimal-speed searc h of single pulsed poin t object (a) One- step search algorithm s. Introducing the binary f unction u ( x , t ) describing t he viewing window at time t , we obtain the relation for t he average time from the beginning of the search to recording the fir st pulse [3]: ∫ ∫ ∫ ∞ ∞ ∞       − = 0 0 0 . ) ) , ( exp( ) , ( ) ( ξ ξ λ λ τ d x u t x u x tf dx dt (1) For the random a p riori distribution of the source on the axis x , the c onstruction even one-step (ending after registering the first pulse) optimal-tim e search proc edure causes considerable difficulties. In one- step periodic search al gori thms, the relative load φ ( x ) per point x (i.e., the relative tim e the point i s present in the viewing window) remains c onstant throughout the search time. With this approach, the task is to find the function that minim izes t he average search tim e ∫ = dx x x f ) ( ) ( 1 ϕ λ τ (2) provided that ∫ = , ) ( ε ϕ dx x (2a) . 1 ) ( 0 ≤ ≤ x ϕ (2b) Using the method of undetermined Lagrange multipliers, we search for the function φ ( x ) that m inimizes the expression dx x x x f ∫       + ) ( ) ( ) ( µϕ ϕ . Further, differentiating with respect to φ and taking into account the constraint (2a ), we get ∫ = dx x f x f x ) ( ) ( ) ( ε ϕ . (3) If at t he same time the condition (2 b) i s not violated, then the f unction (3) is a solution of the formulated extreme problem. If there are such areas of x where the solution φ ( x )>1, then inside of these areas it is necessary to set φ ( x )=1, and for t he remaining p oints it is required t o recalculate the undefined multiplier µ under already changed conditions (2) and (2a). After this, as an opti m al search strategy, any binary function u ( x , t ) can be chosen that satisfies the relations ∫ ∫ = = . ) ( ) , ( ; ) , ( t x d x u dx t x u ϕ ξ ξ ε In the general case, the construction of the optimal (not ne cessarily per iodic) one- step search algorithm for an unknown Poisson source is c onnected with fi nd ing such a function φ ( x,t ) – the relative load on the point x at a time t that minim izes the average search time ∫ ∫ ∫ − = t d x x dxf dt 0 ) ) , ( exp( ) ( ξ ξ ϕ λ τ (4) provided that ∫ = ε ϕ dx t x ) , ( for any t , (4a) . 1 ) , ( 0 ≤ ≤ t x ϕ (4b) To simplify further calculations, we i ntroduce a f unction ∫ = t d x t x 0 ) , ( ) , ( ξ ξ ϕ α corresponding to the total time spent by the point x in the viewing window from the beginning of the search to time t . To take int o account t he constraints (4a ) and (4b), we introduce th e undeterm ined Lagrange multiplier µ ( t ). Then the problem of finding the optimal strategy reduces to finding the function α ( x , t ) that minimizes the functional [ ] ∫ ∫ + − ) , ( ) ( ) ( ) , ( exp( t x t x f t x dx dt α µ λα (5) provided that ∫ ∞ ∞ − = , ) , ( t dx t x ε α (5a) . ) , ( 0 t t x ≤ ≤ α (5b) The solution of this variational problem is the function          < ≤ ≤ < = , ) ( ) ( ln 1 , ; ) ( ) ( ln 1 0 , ) ( ) ( ln 1 ; 0 ) ( ) ( ln 1 , 0 ) , ( t x f t t t t x f t x f t x f t x µ λ λ µ λ λ µ λ λ µ λ λ α (6) where µ ( t ) is determined from the relation (5a). The optimal search strategy u ( x , t ) must belong to the class of binary functions. It is set by the equations ∫ ∫ = = t dx t x u t x d x u 0 . ) , ( ); , ( ) , ( ε α ξ ξ The use of op tim al search a lgorithms i n pr actice faces certain difficulties. The fact is bot h proposed algorithms of optimal o ne-step search, when a p riori density functi on of the signal sour ce differs f rom the constant, can not be physically realized (as a r ule) by moving an 1-connecte d viewing window wit h a width ε . Therefore, in real search procedures, one-step procedure is advisable to do in according to the following scheme. Preliminarily, the inte rval (0 ,L ) is divided i nto ε -step disc retes, and a priori given density f ( x ) is "stepwise" approximated on each of them. In this case, ε i s c onsidered to be a sufficiently sm all so that the variation of the function f ( x ) within one discrete c an be neglected. Search (i n accordance with the optimal one - step detection procedure) should begin only with t he highest "p eak " density ins pection, then after a time t 1 the window is alternately set to the areas under the two hi ghest "peaks", then after time t 2 , further inspection is done for t hree secti ons, a nd s o on. Al l the moments t i are exactly determined by the described finding method for optimal function α ( x , t ). It shoul d be not ed that discussed search plan assumes that t he intensit y of the s ource λ is known in advance. If suc h a pr iori information is n ot available, a per iodic procedure can be rec omm ended tha t does not depend on this intensity. In accordance with it, the i ntegrals of the density f ( x ) i n each of t he discrete must be calculated. If t here are m discretes, and its squares-integrals are P 1 , P 2 ,…, P m , then the viewing window should cyclically "run t hrough" all the di scretes with r elative load ∑ = = = m j m j P j P i i 1 ) ,..., 1 ( β , respectively. These values β i ar e easily obtained if we again apply the method of undetermined Lagrange multipliers to minim ize the average search time ) ... )( 1 ( 2 2 1 1 m m P P P β β β λ τ + + + = (7) provided that . 1 ... 1 = + + β β m (8) (b) Multistep search algorithms. If we are not limited to one-step procedures, but consider the search algorithm as a multi-step process (t hat ends after n -th pulse registration), then the optimal strategy should deliver a minimum to the functional , ) ) ,.. . , , ( exp( ) ,..., , , ( ... ) ( 1 1 1 1 1 1 0 1 1 1 1                           ∑ ∑ − × = ∏ ∫ ∑ ∫ ∫ ∫ ∑ = − − = ∞ = = − = k l t t l l l l m m l l k n k k l m m l m m d t t x u t t t x u dt t x dxf ξ ξ λ λ τ (9) provided that ∫ = − ε dx t t t x u n n ) ,..., , , ( 1 1 . (10) Here U i ( x , t , t 1 ,… t i -1 ) is the search strategy at the i -th step provided that t he i ntervals between the first ( i -1) pulses wer e t 1 , t 2 ,…, t i -1 res pectively. In the general case, to find the optim al strategy u ( x , t ) that minimizes the functional (9) is not possible. At the same time, for an important special ca se, f ( x ) =const , the analytic solution is rather simple. Let      ∉ ∈ = ), , 0 ( , 0 ) , 0 ( , 1 ) ( L x L x L x f i.e. there is no a priori information about placement of th e source within the interval (0, L ). Obviously, in the f irst step, the search load sh ould be equally distributed between all the points ) , 0 ( L x ∈ . It is possibl e to make such a load, f or example, by te levision sca nning of t he whole interval ( 0, L ) by t he apertu re with width l 1 (to avoid e dge effects, we consider the end of the interval closed to its b eginning, so, a ci rcle with length L is sca nned instead of the interval). When registeri ng a p ulse, the search co ntinues inside the window wit h width l 1 using another a perture with width l 2 . If we discus s a n n -step search, then at th e l ast step s canning it is carried out b y the aperture with width ε (this is dictated b y the c onditions of the task). Then the a verage search time is ). ... )( 1 ( 1 2 1 1 ε λ τ − + + + = n l l l l L (11) For a fixed n , it's possible to fi nd optimal values that minimize t he expression (11): . ) ( ... 1 1 2 1 1 n n L l l l l L ε ε = = = = − (12) Then the a verage time of the opti mal n -step search is . ) )( ( 1 n opt n L n ε λ τ = (13) Now ( from the ex pression (13)) we can find t he optimal number of s teps n minimizes the a verage search time. Since the f unction xa 1 /x for a >1 has only one minimum point ( x =ln a ), the opti mal valu e n opt is always either entier (ln( L / ε )) or entier (ln( L / ε ))+1. When ∞ → ε L , it is possible to c onsider n opt ) ln( ε L ≅ . Therefo re, we have asymptotic relati ons: . ... 1 2 1 1 e l l l l L n = = = = − ε (14) ). ln( ) ( ) )( ( 1 ε λ ε λ τ L e L n opt n opt opt n = = (15) So, a multi -step procedure b rings a gain in comparison wit h a one-step search (for a one-step search in the case u nder consi deration, the aver age detection time is opt τ 1 = L / λε ), and this gain increases unlimit edly with the ratio L / ε incre asing. Now we can co mpare the constructed optimal search proced ure with some simplified algorit hms. For e xample, if the sea rch i s organized accordin g to t he p rinciple of a dich otomy , then the average time of the object search is ). ln( ) 2 ln 2 ( ) ( log ) 2 ( 2 2 ε λ ε λ τ L L = = (16) Dichotomous searc h has (in comparison with the optimal procedure) a sm all ( % 6 ≈ ) loss in t ime. Trichoto mic search i s even closer to t he optimal procedure: initial i nterval (0 , L ) is divid ed into three subintervals , the n the subinter val wher e impulse is fixed, in t urn, i s also divided into three subintervals, etc. This procedure loses to the optimal procedure only %. 4 , 0 ) 3 ln 3 ( ≈ − e e It is natural to expect that in the case of arbitrariest a priori distribution f ( x ) the m ulti -step s earch procedure (in c omparison wi th on e-step se arch) can bring a significa nt gain in time, especially for large values of L / ε . Since m inimizati on of the functional (9) under the c onstraint (10) in each con crete case i s very diffi cult problem, multi - step periodic se arch procedure s eems t o be more real from the practical realization point of view. In the first step, the interval (0, L ) is divided into three parts (three parts are chosen because for a uniform dis tribution f ( x ) this procedure is c losest to the optimal one). Then th e values P 1 ,P 2 ,P 3 are calculated ∫ ∫ ∫ = = = 3 2 3 3 2 3 2 3 0 1 . ) ( ; ) ( ; ) ( L L L L L dx x f P dx x f P dx x f P For a ny time interval t ∆ , viewin g window with width L/ 3 wil l periodically "run through" all t hree sections in such a wa y that , 3 2 1 t t t t ∆ = ∆ + ∆ + ∆ where i i t t β = ∆ ∆ is the re lative viewi ng wi ndow presence time on each of the subi ntervals (0 , L/ 3) , ( L/ 3 , 2 L/ 3) , (2 L/ 3 ,L ) . When the puls e is registered, t he search pr ocedure continues similarly on th e subinterval (segment) whe re the pulse is fi xed (i.e., this segment is di vided again into three parts, the coefficien ts β 1 , β 2 , β 3 are r ecalculated, et c.). At the first step their values are equal ∑ = = 3 1 ) 3 , 2 , 1 ( j j i i P P . T he un iversality of the propo sed proced ure is also in the fact that its realization does not requir e a priori information on the source intensity. 2. Symbolic-analytic and numerical algor ithms in the proble m of optimal multip urpose search In.§2 we invest igate an alytic and numeric al algorithms for opti mal multip urpose s earch. T he main problem consi dered here, can be f ormulated as follows. There are n point objects on the segment (0, L ). There i s no a priori i nformation about t heir location, so we can assume th at n points are uniformly distributed o n the segment ( 0, L ). Each object at ran dom moment s of t ime produces short pulses (d elta functio ns), the pauses between them have an exponential distribution with the parameter λ . As befor e, the pulse is fix ed if th e active o bject i nitiated the pulse i s in t he viewi ng windo w of the recording d evice. It is necessary to loc alize all sour ces with ε a ccuracy in the mini mum t ime. Gener ally sp eaking, the construction of an op timal multipurpose search strategy can be reduced to a se quential procedure, wh en at the first step the localization of one of the n sources is carried out with required accu racy, th en one of t he ( n -1) remaining sour ces is searched for, etc. Two different approach es are possible here: either all "previous steps" accumulated information is u sed, or t here is no such an accumulation . T he use of previously accumulated info rmation makes se arch algorithms and t he scanning system i tself more complicated , because in this case it has to have an opportunity to s tore and to process inter mediate data. The algorithms con sidered b elow a re c onstructed u nder the assumption that the recordin g equipment ha s no memory, s o t he problem d escribed above reduces t o finding t he optimal s earch strategy fo r o ne of n point objects with an accuracy ε . The search procedure for each of the n steps, in turn, can als o be considered as a multi-step pr ocess. Since initiall y the re i s no a priori informati on ab out the locati on of n sources withi n the segment (0, L ), in the first step the search eff ort shoul d be equally distribut ed betwe en all th e po ints ) , 0 ( L x ∈ . You c an imple ment such a sche me, f or e xample, by scan ning the whole segment by a perture of some width l 1 . T hen ( after registe ring t he pulse ), the search continues inside t he selected window l 1 by a nother aperture of width l 2 , etc. At the last step ( the number of su ch step s depends on n, ε , L ) to e nsure required localizatio n accuracy, the search is carried o ut by the aperture of width ε . The t ask now is to determine t he optimal number of scanning steps for the given values n , ε and L the width of the scanning viewing window fo r each of them. After finding one of the objects , the valu e n is reduced by one , and the sear ch is repeated for n = n -1, etc. We i ntroduce th e function g ( x ) – the density distribution of the aperture center at the t ime of t he first pulse re gistration. If we ass ume that n sources are located at points x 1 , x 2 ,… x n , and the scann ing i s carried out by the aperture of width l , t hen ∑ = − + × − − = n i i i x l x l x x nl x g 1 ] ) 5 . 0 [( 1 )] 5 . 0 ( [ 1 1 ) ( , (17) where    > ≤ = 0 , 1 , 0 , 0 ] [ 1 x x x – unit ste p function (Heaviside ste p function ) . Further in the text, we'll n eed knowl edge of the nu mber o f obj ects distrib ution in the aperture at the moment of r ecording the pulse. To find this distribution, it's required to ca lculate P n ( k , l ) – th e p robability that will be exactly k ( k =1,2,…, n ) sources i n the aperture at the time of pulse registration, including the object generated the pulse. Since the realization of the s cheme with uniform search effort l eads to undesirabl e edge effects, then in order t o make the subsequ ent presentation str icter, we ass ume that search interval is th e circle with circumfer ence L . Moreover, wit hout l oss of generality we a ssume that L =1, and 1 0 ≤ < l . Let t he o bjects be l ocated a t points x 1 , x 2 ,… x n . Assuming t he point x 1 to be the origin, we rank (put in order) all other sourc es moving fro m t he point x 1 =0 clockwise. The prob ability P n ( k , l ) c onsists of the probabiliti es that, at th e moment of p ulse fix ation, either the aperture co ntains po ints x 1 , x 2 ,… x k , or x 2 , x 3 ,… x k+ 1 and so o n. T he number of s uch c ombinations is n . If we assume that at the moment of pulse registrati on there are exactly k objects in the aperture, then taking i nto account (17) { } , ... 1 ; ; min ... 2 )! 1 ( ) , ( 2 1 1 ∫ ∫ + − − − × × − = + + n k k k k n dx dx l x x x x l D l k n l k P (18) where the area D is given by the syst em of inequalities:        < > − + − ≤ − < < < < < + + . , ) 1 ( , 1 , 1 ... 0 1 1 3 2 l x l x x x x x x x x k n k n k k n (19) The doubling of t he coefficie nt in r elation (18) is taking pl ace because inequality n k k x x x − ≤ − + 1 1 i s included i n t he c onstraint syste m (1 9), t he geometric meanin g of wh ich l ies i n th e fa ct th at one of the intervals (formed as a result of r andom thr owings of n po ints into a circle) is g reat er t han t he other interval. Since the pro bability of such an event is 1/2, then th e cofactor 2 appears in the expression P n ( k , l ) (note that i n the case k = n the ineq uality n k k x x x − ≤ − + 1 1 turns i nto the identity inequality, so it is simply excluded from the system (19), a nd doubling cofactor removes fro m (18)). The integral (18) over th e are a (19 ) can be represented as the sum of three integrals that correspon d (in form) to the expressions for computer analytic calculations d escribed in [4]: . . .. ) 1 ( ... ... ) ( ... ... ) ( ... )! 1 ( 2 ) , ( 2 1 3 2 1 2 2 1      − + − + −      + − − = ∫ ∫ ∫ ∫ ∫ ∫ + + n n k n k k n k n dx dx x l x D dx dx x x D dx dx x l D l n k l k P (20) Here, the followi ng systems of linear inequ alities cor respond t o re gions D 1 , D 2 , D 3 with o ne free parameter l :            > − + − + > − < > − + − < − = < < < < = = + + + + + , 0 2 1 , 0 , , ) 1 ( , 1 , 1 ... 0 1 1 1 1 1 3 2 1 1 l x x x l x l x l x x x x x x x x x x D n k k k k n k n k k n n (21)            > − + − > − < > − + − < − = < < < < = = + + + + , 0 1 , 0 , , ) 1 ( , 1 , 1 ... 0 1 1 1 1 3 2 1 2 l x x x l l x l x x x x x x x x x x D n k k k n k n k k n n (22)            > + − + − > + − + − − < > − + − < − = < < < < = = + + + + . 0 1 , 0 2 1 , , ) 1 ( , 1 , 1 ... 0 1 1 1 1 3 2 1 3 l x x l x x x l x l x x x x x x x x x x D n k n k k k n k n k k n n (23) Computer c alculations carried out for fixed values n and k ha ve shown that in the general case k n k n l l k n l k P − − −         − − = ) 1 ( 1 1 ) , ( 1 . (24) In fact, thi s means that we are dealing wi th a "shifted" Bern oulli scheme: ( n -1) inde pendent tests are carried out w ith t he probabilit y of " success" p = l a nd the prob ability of "failure" q= 1 - l , an d f ormula (24) corresp onds to the prob ability that there w ill be exactly ( k -1) " successes". Now, when a pr ogram- calculate d expression for p robability P n (k,l) is k nown, we can also construct a p roof that computer analytic cal culations are correct. Since the scanning of the unit circ le (i .e. circle whose circumference is 1) by arc - aper ture of l ength l creates a uniform search effort over all points t hat form circumfer ence of t he circle, the pr obability the initiator of t he pulse is a fixed source x i ( i =1,2,..., n ) is the same for all obje cts and is equal to 1/ n . Therefore, by the formula of total probability ∑ ∑ = = = = = n i n i n i n i n i n A l k P A l k P n A l k P A P l k P 1 1 1 ) / , ( ) / , ( 1 ) / , ( ) ( ) , ( , (25) where the event A i means that t he initiat or of the registered impuls e was the object x i . Th e c onditional probability P n ( k , l / A 1 ) can be interpreted as follows: all so urces except the first are "locked", and at the moment o f pu lse registering the nu mber of objects in t he a perture, including the first one, i s co unted. Since the re maining ( n - 1) sources have a uniform d istributi on on the circle , the p robability fo r an y of them to get into the aperture is l , and the pr obability that the aperture will contain exactly ( k -1) objects (except the first) is just described by t he relation (24). From ( 24), it can be shown tha t if we seque ntially scan a unit circle by the apert ure l 1 ( l 1 <1) a nd then (after p ulse registration) the circle of circumference l 1 (this circle is t he arc l 1 converte d) is scanned b y another apert ure l 2 ( l 2 < l 1 ), the n the prob ability th at at the m oment o f se cond p ulse re gistration, t here will be e xactly k objects i n the apertu re i s t he same a s the proba bility t hat the ap erture will contain exactly k objects if i nitial unit circle i s scanned at once b y the aperture of width l 2 . In fact, the pr obability that k objects will get into the aperture of width l 2 during a doubl e scan is , ) 1 ( 1 1 ) 1 ( ) ( 1 1 ) 1 ( ) ( 1 1 1 1 1 ) 1 ( 1 1 ) , ( ) , ( 2 1 2 1 2 1 0 1 2 1 2 1 1 2 1 2 1 1 2 1 1 1 1 2 1 k n k i k n i k n i k i n k i n k i k k i k i n i n k i n k i i n l l i n l l l i k n l i n l l l i n k n l k n l l l l k i l l i n l l k P l i P − − − − − = − − − = − − − − − = = −         − − = = − −         −         − − = − −         − −         − − = =         −                 − − −         − − = ∑ ∑ ∑ ∑ (26) So, it does not depend o n wh at apert ure was use d for the first scan. Using the relations (2 4) and (26 ), we find the aver age time of t he objects n l ocalization with a given accuracy ε , ass um ing that the search is carried out with the help of s everal steps by co nsistently nar rowed apertures l 1 , l 2 ,…, l m -1 , ε . T o do this, we first calcula te the average search time at the i -th step: . ) 1 ( 1 ) /( ) 1 ( )) / ( ( ) / ( ) 1 ( 1 1 1 1 1 1 1 1 1 1 1 1 1 i n i n k i k n i k i i i n k i i k n i k i i l n l nl l l k n l l k l l l l k n λ λ λ τ − = − − − − − = − − − − − − − = −         = = −         − − = ∑ ∑ (27) The average s earch ti me f or one of the so urces n i s t he sum of the avera ge s earch ti mes a t each step, therefore ∑ = − − − = m i i n i l l n 1 1 / ) ) 1 ( 1 ( 1 λ τ , (28) here l 0 =l; l m = ε . Th e pr oblem of opt imal s earch has now reduc ed to fi nding m , l 1 , l 2 ,…, l m -1 t hat minimize expression (28). For a fixed value m , the optimal values l 1 , l 2 ,…, l m -1 can be calculated from the system of equations ) 1 , ..., 2 , 1 ( , ) 1 ( 1 ) 1 ( 1 1 2 1 − = − − − = − − + m i l l nl l n i n i i i (29) that is obtained by equating all the partial derivatives of the expression (28) to zero. The system of equations (29 ) was solved on computer f or fixed values m . T he optimal m values (i.e. the number o f scanning steps) and the corresp onding values l 1 , l 2 ,…, l m -1 when the average loca lization time (28) re aches a minimum wer e calculated f or dif ferent values n and ε / L . These data are s umm arized in Ta ble 1. We can see fr om this ta ble t hat at high requirements to l ocalization ac curacy (when 1 / << L ε ), the average time of search o f the first of objects is pro portional to ln( ε / L ). Table 1. Parameters of the time-optimal search. Initial dat a: n - number o f impulse o bjects; L is the length of the segment on w hich the impulse objects are located; ε - required accuracy of localization; λ - Poisson generatio n intensity of eac h pulsed o bjects. Re sults of the pro gram calcu lation: m - the number o f scanning steps fo r the localiza tion o f the f irst objec t (fro m n) ; l i - the size of the scann ing ape rture at the i- th ste p (i = 1,....m); τ - the avera ge localization ti m e of t he first object . L / ε n m,l,T 2 3 5 10 30 5 0 m 2 2 1 1 1 1 L l / 1 0.26 0.24 0.1 0.1 0.1 0.1 L l / 2 0.1 0.1 - - - - 1 10 − τ λ 4.19 3.26 2.0 1.0 0.33 0.2 m 4 4 3 3 1 1 L l / 1 0.23 0.19 0.09 0.07 0.01 0.01 L l / 2 0.08 0.07 0.03 0.03 - - L l / 3 0.03 0.03 0.01 0.01 - - L l / 4 0.01 0.01 - - - - 2 10 − τ λ 10.22 9.02 7.55 5.73 3 .33 2.0 m 6 6 6 5 4 3 L l / 1 0.21 0.16 0.12 0.06 0.02 0.01 L l / 2 0.07 0.06 0.043 0.02 0.007 0.003 L l / 3 0.024 0.02 0 .016 0.007 0.003 0.001 L l / 4 0.008 0.007 0.00 6 0.003 0.001 - L l / 5 0.003 0.003 0.00 3 0.001 - - L l / 6 0.001 0.001 0.00 1 - - - 3 10 − τ λ 16.48 15.22 13.7 6 11.76 8.77 7.42 3. Systems with mul tiple receivers: optimal- speed algorithms for random pulsed point sources search Problems and algorithms f or the optimal s earch of random s ources that wi ll be discusse d in this paragra ph arise in many s cientific and t echnical fiel ds, in particular, in classical d isciplines as reliability theory and mathematical communica tion t heory. Si milar studies a re necessary for the development of methods for tech t roubleshooti ng, app earing i n a fo rm of th e al ternating equipment failure; i n astr ophysics these problems a re enco untered in th e search of pulsating r adiation sources; in modern s ections of computation al mathematics these methods are requ ired to develop algorithms for detecting low-contrast and s mall-sized obj ects on a erospace images, and , for ex ample, in signal theory, the same methods a re used to e stim ate the reliability of rand om fields and point images r egistration. In §1 and §2 opt imal search algorithms for p ulsed point sources were d escribed, which assumed t he using of a single recei ver with a tunable viewing window. In the presence of several receiving de vices, the average lo calization t ime can be greatly r educed. T he purpose of this pa ragraph 1.3 is to con struct t he time-optimal (in statist ical terms) localizatio n algorithms for ra ndomly locat ed pulsed point source that take i nto account t he number of rec eivers used a nd provide the require d localization accuracy. W ithin this study, it will be assu med that a priori informati on about the prob able lo cation of a random pulsed object inside of the search interval ( 0, L ) i s absent, i.e. that the pro bability de nsity of an unknown source on the x axis is given by the function      ∉ ∈ = . ) , 0 ( , 0 ) , 0 ( , 1 ) ( L x L x L x f The f irst questio n we need to answer is: "What ac curacy of localization can be achieved with a single-tact search procedure that is carried out with the h elp of n receiving devices and ends at the time of f irst p ulse generation by source?" m 9 8 8 7 6 6 L l / 1 0.24 0.15 0. 11 0.05 0.0 18 0.013 L l / 2 0.09 0.05 0. 04 0.017 0.006 0.005 L l / 3 0.03 0.02 0. 014 0.006 0.002 0.0017 L l / 4 0.01 0.006 0.005 0.002 0.0008 0.0007 L l / 5 0.005 0.00 2 0.002 0.0008 0.0003 0.0003 L l / 6 0.002 0.00 08 0.0007 0.0003 0.0 001 0.0001 L l / 7 0.0007 0.0003 0.0003 0.00 01 - - L l / 8 0.0003 0.0001 0.0001 - - - L l / 9 0.0001 - - - - - 10 -4 τ λ 22.74 21.4 8 19.97 18.0 1 4.96 13.6 We pay special attention to t he fact t hat, according to the conditi ons of the problem, t he search proced ure ends not at t he moment o f fi rst pulse registration by the receivin g s ystem, but at the moment of random source's f irst puls e generation. This i s not the same, because i n principle t he rec eiv ing s ystem can work i n "missed-some-pulse s" mode (not registering all the generated pulses). Th us, i n si ngle-tact search, it is required that the system does n't miss any impulse. Obviousl y, in this case, the agg regate viewin g wind ow of all n re ceivers has to overlap the entire s earch i nterval ( 0, L ) at an y moment . Let N be th e n umber o f elementary s egments which t he interval (0, L ) sho uld be divided into (pulsed object is loc ated in one of such a se gm ents ). Generally speakin g, i t would be possible, for example, to put N equal to the number of receivers n a nd, di viding the search interval into n equal parts, assign to each of N = n segments theirs own available receiver. In such a s earch procedure , there is no problem to " link" the s ource to desired segment, since the detected pulse is alwa ys fi xed b y only one receiver. But, unfortu nately, such a simplified algorithm has an extremely low accur acy of localization n L = ε , and therefor e it is very far from opti mal. (a) Optimal single-tact search procedure . T he c onstruction of a single-tact search p rocedure f or n receiving device s, which really minimizes the localization error, will be carried out as follows. Eac h of the n r eceivers i s specified with the help of T able 2, the i -th l ine o f whi ch describes the monitori ng zone of t he receiver with the number i . Binar y variables x ij , N j n i , 1 ; , 1 = = for ming the ta ble (not e that the optimal value of the parameter N has yet to be determined) will have the values 0 or 1 according to the followin g rule: – if x ij =0, segment j does not enter the ob servation zone of receiver i ; – if x ij =1, the segment j enters the observati on zone of the recei ver i . Table 2. A two -dimensional a rray describing t he observ ation zo ne for each of n receiving system’s dev ices in a single-ta ct search procedure . Segments RECEIVERS 1 2 3 … N 1-th receiver x 11 x 12 x 13 x 1 N 2-th receiver x 21 x 22 x 23 x 2 N 3-th receiver x 31 x 32 x 33 x 3 N M M M . M M M i -th receiver x i 1 x i 2 x i 3 x iN M M M M M n -th receiver x n 1 x n 2 x n 3 x nN Thus, the row vector x i = ( x i 1 , x i 2 , …, x iN ) uniquely defines the receiver i , and the whole array ( x ij ) uniquely fully describes the receiving system. T he dynamic state of this s ystem i s charact erized b y the column vector r =( r 1 ,r 2 ,…,r n ) T , in which all binary ( i.e., posse ssing the value 0 or 1) variables r i , n i , 1 = ar e equal to zero throughout the t ime interval from th e beginni ng of the search to pul se’s registration moment. When the pulse is fixed , each of the varia bles r i goes (or not) into the state r i =1, dependin g on whether th e received pu lse was fixed by the i -th receiver o r not. The tas k is: by means of this changed col umn vector r to determine t he number of the segment whi ch has generate d the pulse. Three statements are fo rm ulat ed below t hat make the c onstruction of an opt imal single-tact s earch algorithm an a lmost obvious proce dure. Statement 1 . That the pr ocedure of single - tact search was a ble to define un ambiguously the n umber of the segment j , where the point pulse -initiator source is located , it is necessary and suff icient t hat all possible realization s of the vector r , characterizing the receiving system’s s tate at t he moment of pulse registratio n, differ one from another. Statement 2 . The maximal number of el ementary segments N max initial search interval (0, L ) may be divided into (this number N max , actually se ts the localization accurac y o f the pulsed so urce), is N max = 2 n -1, where n is the number of recei vers used (the total number of differe nt states of the vector r is 2 n , but t he system can not be in t he state r =0 when regist ering the pulse. If we try t o exceed the number N max = 2 n -1 it will bec ome impossible to restore unambiguous ly the number of a segment with a puls ed source). Statement 3 . It is e xpedient to c arry out c reation of the binary table ( x ij ) corresp onding to th e optimal algorithm for t he single-tact random point source localization, not by combining row vectors x i = ( x i1 ,x i2 ,…,x iN ), n i , 1 = , de scribing ea ch of the n receivers, but b y combining the c olumn vect or x j T = ( x 1 j ,x 2 j ,…,x nj ) T , N i , 1 = , characterizing th e receiving system’s dynamic res ponse to the pulse in the segment j . Considering statement s 1-3, t he optimal formati on of the table (x ij ) i s to choose one o f (2 n -1)! vari ants to pack two - dimensi onal array t hat inc ludes all nonzero realization of column vector x j T =(x 1j ,x 2j ,…,x nj ) T . Each binary column {x 1j ,x 2j ,…,x nj } T corre sponds to its number ∑ = − × = n i i n ij x j 1 2 from the range . 1 2 , 1 − = n j Note t hat the order of the columns x j T in th e formed ar ray (x ij ) can b e any. If t he columns o f array follow i n the increasing order ( by their " content"), then the matrix ( x ij ) descri bing su ch a r eceiving system, is presented by the Table 3. I n this case, the al gorithm to f ind t he s egment n umber with an unknown s ource is ver y simple: if t he dynamic stat e of the receiving system is descri bed by the vector r =(r 1 ,r 2 ,…,r n ) T , then th e pulsed source is located in t he segm ent wi th the number , 2 1 ∑ = − × = n i i n i r j and the set of scalar coordinates of the column vect or r is the binar y record of this number. The absolute accuracy of the single-tact loca lization pr ocedure of an unknown sou rce is ε =L /(2 n -1), a nd the average l ocalization time < τ > is equal to exp ected value of pause bet ween the pulses λ λ λ τ 1 ) exp( 0 0 = − = >= < ∫ ∫ ∞ ∞ dt t t h (t)dt t and does n ot depend on the number of rec eivers used. Table 3. A binary table co rresponding to the receiv ing system with monotonically increa sing numbers of pulse-initiato r segments. Segments RECEIVERS 1 2 3 … N max =2 n -1 1-th receiver 0 0 0 … 1 2-th receiver 0 0 0 … 1 3-th receiver 0 0 0 … 1 M M M M M M i -th receiver 0 0 0 … 1 M M M M M M ( n -1)-th receiver 1 1 … 1 n -th receiver 1 0 1 … 1 (b) Opti mal mul ti-step search using n receivers . The single-tact search procedure describ ed above shows how the se arch e ffort should be di stributed among the n receivers if it is re quired t o reac h th e highest accuracy of r andom pulsed sou rce loc alization. T he m ain moment wh ich will be directly used further is that the proced ure d escribed above is extremely co nstructive and completel y un am biguous, so in the case of an o ptimal di stribution of sea rch effort s, the pulsed obje ct is always locali zed with accuracy W /(2 n -1), where W is the integrate d viewing wi ndow uniting all n receivers used. Now, ta king these c onsiderations into ac count, we can begin to solve a more complicat ed t ask: it is necessary to constr uct time-optimal algorithm when the search o f a r andom pu lsed source has to be carrie d out by a system of n receiving devices with th e r equired loc alization a ccuracy ε . Exc ept sti pulated in advance loc alization a ccuracy, an important complicatin g factor in the formulat ed problem i s that the choice of the optimum sear ch procedure isn't li mited only by s ingle-tact localization al gorithms, and there is no require ment to fi nish the se arch at the time of the first pulse generation. Moreo ver, as will be shown below, even at rathe r low require ments to t he localization accuracy, the optimal algorithm is a multi-stage procedure (tra nsition from one s tage to other stage comes a t the moment when th e receiving system regist ers the next pulse) . At the same t ime, i t is q uite accep table t hat a part of the pulses generated by a ra ndom sour ce is not fi xed by the system. As for t he previous problem, it is ass umed that the localized point source has a uniform distributio n density on the interval (0 , L ). Passing to the s olution of the task, let us introduce a number of a dditional designations . T he symbol M will be the number of s tag e in the sea rch procedure, and the symbol W i will be the aggregate syste m viewing wind ow uniting n receivers at the i -th stage of the search (i n cases that do not allow d ouble interpretatio n, the same symbol wi ll mean not only th e w indow itself, but also its l inear dimension). Taking this into acco unt, the average t ime τ of any M -stage (n ot ne cessarily opt imal) se arch procedu re of a rand om source that guarantee s the localization accurac y ε will be             − + + − + − + × = − M n M n n W W W W W W W L 1 2 ... 1 2 1 2 1 1 3 2 2 1 1 λ τ . (30) ε ) 1 2 ( − = n M W . (3 0a) The integer parameter M and continuous variables M i W i , 1 , = , which d eliver the m inimum to the expression (30), need to be estimate d. It is taken into ac count in relation (30 ) t hat each subse quent ( i+ 1)- th stage of t he search procedure i ncludes scanning by the cumulative viewi ng window W i+ 1 withi n o ne of the se gm ents (2 n -1) that formed the viewing window W i on the previous sta ge, and ( i +1)-th sca nning stage is carried out inside of the segm ent whe re i -th pulse was fixed. For a fixe d M , t he optimal sizes of scanning viewin g wi ndows M i W i , 1 , = , for whic h average localizati on time (30) reache s a mini mum, have to s atisfy the system of equatio ns              =         + − × − = ∂ ∂ =         + − × − = ∂ ∂ =         + − × − = ∂ ∂ − − − . 0 1 ) 1 2 ( 1 ; 0 1 ) 1 2 ( 1 ; 0 1 ) 1 2 ( 1 2 1 2 1 3 2 2 1 2 2 2 1 0 1 M M M n M n n W W W W W W W W W W W W λ τ λ τ λ τ M (31) ε ) 1 2 ( − = n M W . (31a) . ) 1 2 ( 0 L W n − = (31b) The system of equa tions (31) is obtained by si mple equa ting to zero all the p artial derivat ives of average search ti me τ (30) , and th e relation (31 b) is intro duced f or the symmetrizatio n of the notati on. It is more c onvenient to break the f urther solution of the pr oblem int o t wo parts. Firs t, we find the number M opt of stages for the optim al lo calization proced ure, a nd then we calculat e all other parameters. Let us note that f or a fixed value of the inte ger parameter M the soluti on of the system (31) is M i L L W n i , 1 , ) / )( 1 2 ( M i = ×         − = ε . (32) Taking in to account that the dimensionless co efficients M n L i ) / ) ( 1 2 ( ε − can no t be more then 1 , we get the condition u nder which the solution (3) i s valid: M n L ) 1 2 ( 1 ) / ( − ≤ ε . (32a) Substituting (32) i nto (30), we obtain the expression f or the average time > < τ in the case of the M -stage localization procedure: M 1 n M L / M − − × = ) ( 1 2 1 ε λ τ . (33) Function ) 1 2 ( ) / ( ) ( 1 − = − n x L x x f λ ε , (34) which is a continuous ana logue of e xpression (33) , has onl y one l ocal minimum (this will b e used l ater), reached at the point ) / ln( mi n L x ε − = . Rewriting the condition (32a) in the equivalent form ) 1 2 ln( ) / ln( − − ≤ n L M ε , (35) we f ind that for any value n ≥ 2 (positi ve and integer), the number of M opt stages in t he optimal localizatio n procedure is the same as the maximum inte ger sati sfying th e constraint (35). He re we use the fact that t he functio n (34) more to the l eft of a po int ) / ln( mi n L x ε − = monotonously decreases. T hus, f or the values ε /L in the n eighborhood of th e point M n L ) 1 2 ( 1 ) / ( − = ε the optimal searc h procedure consists of M stages. For a complet e descri ption of the optimal lo calization algorithm it remains to find ou t under wh at require ments to the accuracy of localizati on (i.e. for wh at the values of ε /L ) transition from M -sta ge to ( M +1)-stage se arch o ccurs. It i s ob vious that at the tran sition point the aver age time of t he M -stage search has t o be e qual t o the average ti me of the most "fast" ( M +1)- stage s earch: λ ε λ 1 ) / ( 1 2 1 M 1 + = − × − M L M n . (36) From her e we o btain the point where the optimal l ocalization algorithms ha ve a tr ansition from t he M - stage s earch strategy to ( M + 1)-stage one: M M n M M M M L       + × − = + → 1 ) 1 2 ( 1 ) / ( 1 ε . (37) The main calculation resu lts t hat syste matize t he p arameters of time-optimal l ocalization algorithms of objects that in random timepoi nts genera te instantaneous pulses (de lta function s) are pr esented in T able 4. These results are applicable to systems with several ( n ≥ 2) r eceiv ers. T he last li ne of the table contains t he parameters of the optimal search for the asym ptotic procedure, when the required localization a ccuracy ε /L tends to zero. Table 4. Parameters of the optimal search for the random pulsed source depending on the number of receivers n ( n ≥ 2) and the required loca lization acc uracy ε . ) / ( L ε (required localizatio n accuracy) opt M * opt m M m W , 1 , = (Viewing window s of the receiving sys tem at each of opt M stages of optimal sear ch) τ (average localization time) 1 ) / ( 1 2 1 < ≤ − L n ε 1 L W = 1 λ 1 1 2 1 ) / ( ) 1 2 ( 2 1 − ≤ ≤ − n n L ε 1 ε ε ) 1 2 ( ) / )( 1 2 ( 1 1 − = × − = n n L L W 1 ) / ( ) 1 2 ( 1 − − L n ε λ ) 1 2 ( 2 1 ) / ( ) 1 2 ( 1 2 − ≤ ≤ − n n L ε 2 L W = 1 L W n × − = 1 2 1 2 λ 2 2 2 2 ) 1 2 ( 1 ) / ( 3 2 ) 1 2 ( 1 − ≤ ≤ ≤       × − n n L ε 2 L L W n × − = 2 1 1 ) / )( 1 2 ( ε ε ε ) 1 2 ( ) / )( 1 2 ( 2 − = × − = n n L L W 2 1 ) / ( ) 1 2 ( 2 − − L n ε λ 2 2 3 3 2 ) 1 2 ( 1 ) / ( ) 1 2 ( 1       × − ≤ ≤ ≤ − n n L ε 3 L W = 1 L W n × − = 1 2 1 2 L W n × − = 2 3 ) 1 2 ( 1 λ 3 3 3 3 ) 1 2 ( 1 ) / ( 4 3 ) 1 2 ( 1 − ≤ ≤ ≤       × − n n L ε 3 L L W n × − = 3 1 1 ) / )( 1 2 ( ε L L W n × − = 3 2 2 ) / )( 1 2 ( ε ε ε ) 1 2 ( ) / )( 1 2 ( 3 − = × − = n n L L W 3 1 ) / ( ) 1 2 ( 3 − − L n ε λ M M M M 1 1 1 ) 1 2 ( 1 ) / ( ) 1 2 ( 1 − −       − − ≤ ≤ ≤ − M M n M n M M L ε M L W L W L W L W M n M m n m n × − = × − = × − = = − − 1 1 2 1 ) 1 2 ( 1 ... ) 1 2 ( 1 ... 1 2 1 λ M M n M M n L M M ) 1 2 ( 1 ) / ( 1 ) 1 2 ( 1 − ≤ ≤ ≤       + − ε M L L W M n × − = 1 1 ) / )( 1 2 ( ε L L W M n × − = 2 2 ) / )( 1 2 ( ε M L L W M m n m × − = ) / )( 1 2 ( ε M ε ε ) 1 2 ( ) / )( 1 2 ( − = × − = n M M n M L L W M n L M 1 ) / ( ) 1 2 ( − − ε λ M M M M 0 ) / ( → L ε 1 1 1 ) 1 2 ( ) / ( ) 1 2 ( − − − ∞ ∞ − ≤ ≤ ≤ − M n M n e L e ε ) 1 2 ln( ) / ln( − − ≈ ≈ ∞ n L M ε ε ε ) 1 2 ( ) 1 2 ( 1 2 ) 1 2 ( 1 ... ) 1 2 ( 1 ... 1 2 1 ) 1 2 ln( ) / ln( 1 1 2 1 − = × − − ≈ ≈ × − = × − = × − = = − − − − ∞ ∞ n L n n M n M m n m n L L W L W L W L W n ) 1 2 ln ( ) / ln( − − ≈ ≈ ∞ n L M λ ε λ * Optimal nu m ber o f stages for given localization accurac y Table 5. Opt imal search para meters of a rando m pulse d source fo r a syste m with o ne receiver ( n = 1 ). ) / ( L ε (required localizatio n accuracy) opt M * opt m M m W , 1 , = (Viewing windows of the receiving sy stem for each of opt M stages of optimal sear ch) τ (average localization time) 1 ) / ( 4 1 < ≤ L ε 1 ε = 1 W 1 ) / ( 1 − L ε λ 4 1 ) / ( 3 2 6 ≤ ≤       L ε 2 L L W × = 2 1 1 ) / ( ε ε ε = × = L L W ) / ( 2 2 1 ) / ( 2 − L ε λ 6 12 3 2 ) / ( 4 3       ≤ ≤       L ε 3 L L W × = 3 1 1 ) / ( ε L L W × = 3 2 2 ) / ( ε ε ε = × = L L W ) / ( 3 3 1 ) / ( 3 − L ε λ M M M M ) 1 ( 1 ( 1 ) / ( 1 ) − +       − ≤ ≤ ≤       + M M M M M M L M M ε M L L W M × = 1 1 ) / ( ε L L W M × = 2 2 ) / ( ε M L L W M m m × = ) / ( ε M ε ε = × = L L W M M M ) / ( M L M 1 ) / ( − ε λ M M M M 0 ) / ( → L ε ∞ − ≈ M e L ) / ( ε ) / ln( L M ε − ≈ ≈ ∞ L e L L W M × = × = − ∞ 1 1 1 ) / ( ε L e L L W M × = × = − ∞ 2 2 2 ) / ( ε M L e L L W m M m m × = × = − ∞ ) / ( ε M ε ε = × = × = ∞ ∞ − L L L e W M M ) / ( λ ε ε λ ε ε λ ε ) / ln( ) / ( ) / ln( ) / ( ) / ln( 1 1 L e L L L M L M − = = − ≈ ≈ ∞ − ∞ * Optimal n umber of stage s for give n localization ac curac y Conclusion The obtained results solve the problem of creation t he algorithms of time-optimal locali zation of random pulsed point sources, when such objects have a uniform distribution on the search interval (0, L ). The proposed search strategies open up the prospect to find the o ptimal l ocalization algorithms in cases when the pr obability density function of random puls ed source differs from uniform. Another interesting and poorly investigated direction of t he problem is th e co nstruction of optim al search procedures f ocused on the simultaneous localization of several random sources. The described algorithm s are partiall y presented in [5,6]. The authors are grateful to the Russian Foundation for Basic Research f or the support ( grants №№ 16-01- 00313, 13-01-00361) of scientific works, which are the basis for this publication. Bibliography 1. Potapov A.A., Sokolov A.V. Perspective methods of processing of r adar signals on the basis of f ractal and textural measures. Izvestiya o f RAS. Series Physical, 2 003, V. 67, № 1 2, p. 1775-1778 (in Russian) 2. Ho wland, P .E.: " A P assive Metric Rada r Usi ng the T ransmitter s of Opportunity", Int. Co nf. on Radar, P aris, France, Ma y 1994, p. 251 –256. 3. Mark R. B ell, "I nformation theor y and radar w avefor m design." IEEE Transaction s on In formation Theory, 3 9.5 (1993): p. 1578–1597 . 4. V. M. Efimov, A.A. Nester ov, A.L. Reznik. Algorithms of the o ptimal sear ch for light po int o bjects in speed. Avtometriya, 1 980, № 3, p .72-76 (in Russian). 5. A.L. Reznik. T he progra ms for analytical calcu lations in the pro blems of point ob ject localizatio n. Optoelectron ics, instrumentation and data pro cessing, 1991, № 6, p.2 1-24. 6. A. L. Rez nik, A.V. Tuzi kov, A. A. So loviev, A.V. T orgov. T ime-Optimal Algorith ms of Searching for Pulsed - Point Sources for Systems with Several D etectors. Optoelectron ics, instrumenta tion a nd data pro cessing, 2017, № 3, p. 3-11.