LOCALIZATION OF MULTIPLE SOURCES USING TIME-DIFFERENCE OF ARRIVAL MEASUREMENTS

Transcription

1 LOCALIZATION OF MULTIPLE SOURCES USING TIME-DIFFERENCE OF ARRIVAL MEASUREMENTS Florian Meyer Aleandra Teei, and Moe Z Win Centre for Maritime Reearch and Experimentation, La Spezia, Italy Laboratory for Information and Deciion Sytem, Maachuett Intitute of Technology, Cambridge, MA, USA ABSTRACT Thi paper addree the problem of localizing an unknown number of tatic ource emitting unknown ignal from time-difference of arrival TDOA) meaurement Baed on the framework of random finite et and finite et tatitic, we formulate the Bayeian etimation problem and develop a particle-baed localization algorithm that overcome the challenge related to the highly non-linear TDOA meaurement model, the data aociation uncertainty, and the uncertainty in the number of ource to be localized Simulation reult confirm that the number of ource can be determined correctly and accurate location etimate can be obtained when the number of fale alarm i low and the probability of detection i high Index Term Time-difference of arrival, random finite et, multiobject etimation, finite et tatitic, equential Monte Carlo technique INTRODUCTION We addre the problem of localizing multiple ource from their electromagnetic or acoutic ignal, when the abolute time of tranmiion i unknown The propoed approach ue time-difference of arrival TDOA) location information [] related to peak in the cro-correlation function of multiple patially eparated receiver TDOA-baed localization ha wide application in wirele communication [ 5] and paive urveillance ytem [6 9] and i potentially more accurate compared to traditional triangulation approache that ue angular meaurement TDOA Location Information and Random Finite Set Fig how the hyperbolic TDOA location information for three non-cooperative ource provided by a ynchronized cooperative receiver pair Hyperbolic TDOA location information related to different receiver pair i expected to interect in the vicinity of true ource location Localization of multiple ource i complicated by data aociation uncertainty, ie, it i not known which TDOA meaurement i aociated to a pecific ource In addition, due to fading in the propagation path, the ignal of certain ource might be miing at ome receiver or fale TDOA meaurement which are not related to a ource might be erroneouly extracted Thee two effect are typically referred to a mied detection and fale alarm, repectively Thu, the number of extracted TDOA meaurement might be different among receiver pair Furthermore, the number of ource to be localized i unknown Traditional method for TDOA-baed Thi work wa upported by the NATO Supreme Allied Command Tranformation under project SAC6 and by the FWF under grant J886-N F Meyer i currently with Laboratory for Information and Deciion Sytem, Maachuett Intitute of Technology, Cambridge, MA, USA Fig The hyperbolic function related to TDOA meaurement of two receiver indicated by circle) for three ource indicated by croe) for different noie realization localization ee [,9,] and reference therein) typically aume a ingle ource and do not conider mied detection and fale alarm For the etimation of an unknown number of object in the preence of data aociation uncertainty, random finite et RFS) and finite et tatitic FISST) offer attractive olution [] Technique for tracking an unknown number of object baed on the FISST framework include the probability hypothei denity PHD) filter [ ], the cardinalized PHD filter [, 4, 5], labeled) multi-bernoulli filter [, 6, 7], and the recently propoed trackoriented marginal multi-bernoulli/poion TOMB/P) filter [8,9] The TOMB/P ue a hybrid Bernoulli/Poion RFS model and provide an accurate approximation of the full Bayeian RFS filter Contribution and Paper Organization In thi paper, we tate the Bayeian etimation problem for the TDOA-baed localization of an unknown number of tatic ource Baed on the hybrid Bernoulli/Poion model introduced in the TOMB/P filter [8, 9], we develop a particle-baed localization algorithm that overcome the challenge related to the highly nonlinear TDOA meaurement model, the data aociation uncertainty, and the uncertainty in the number of ource to be localized Simulation reult confirm that the number of ource can be determined correctly and accurate location etimate can be obtained when the number of fale alarm i low and the probability of detection i high Thi paper i organized a follow The baic of RFS are reviewed in Section The ytem model i decribed in Section In Section 4 the RFS baed etimation problem i formulated and the propoed method i preented Simulation reult are reported in Section /7/$ 7 IEEE 5 ICASSP 7

2 RFS BASICS A random quantity X whoe realization X = {x ),, x n) } are finite et of n x-dimenional vector x ),,x n) R nx i referred to a a RFS Both the number n = X, n N of vector in the et the cardinality of X ) and the vector x i) are choen randomly Thu, X conit of a random number n = X of random vector x ),,x n) For the cae of zero cardinality n = ), we have X = Note that the element x ),,x n) of a et X = {x ),,x n) } are unordered Adopting the FISST framework [], the tatitic of a RFS X can be decribed by the multiobject probability denity function pdf) f XX) that i briefly denoted fx) For any realization X = {x ),,x n) }, fx) can be evaluated a fx) = n!ρn)f nx ),,x n) ), ) where, ρn) Pr{ X = n} i the probability ma function of the random cardinality n termed the cardinality ditribution and f nx ),,x n) ) i a pdf of n random vector x ),,x n) that i ymmetric, ie, invariant to argument permutation For the cae of zero cardinality, f ) = ρ) Next, we review three type of RFS that are relevant for the propoed etimation cheme The Poion RFS X ha a cardinality that follow a Poion ditribution with meanµ, ie,ρn) = e µ µ n /n!, n N Note that a Poion ditribution i fully characterized by it mean which i at the ame time it variance) For any n, the element of a Poion RFS x ),,x n) of X are independent and identically ditributed iid) according to it patial pdf fx), ie, f nx ),,x n) ) = n i= fxi) ) From ), the multiobject pdf can thu be obtained a fx) = e µ x Xµfx) The product µfx) i referred to a intenity function or PHD and fully characterize a Poion RFS A Bernoulli RFS X with probability of exitence r and patial pdf fx) i empty with probability r and contain one element x fx) with probability r Hence, the multiobject pdf i r, X =, fx) = rfx), X ={x},, otherwie A multi-bernoulli RFS X i the union of independent Bernoulli RFS X j), j {,,J} with exitence probabilitie r j) and patial pdf f j) x), ie, X = J j= Xj) Every Bernoulli RFS X j), j {,,J} i referred to a a component of X The multiobject pdf fx) of the multi-bernoulli RFS X i parametrized by r j) and f j) x), j {,,J} and can be obtained uing FISST convolution For an union of two RFS, ie, X = X X calculating the multiobject pdffx) by mean of FISST convolution i defined a [, 8] f XX) = Y Xf X Y)f X X\Y) ) SYSTEM MODEL Conider the localization of multiple paive ource uing n r receiver in D in known location q k) = [q x k) q y k) ] T The number of ource n t and their unknown location p j) = [p j) x p j) y ] T are random and can be denoted conitently uing a RFS P Receiver can exchange their received ignal and are perfectly ynchronized TDOA Meaurement A imple way to obtain location information of non-ynchronized ource with unknown waveform, i to compare ignal at the receiver pairwie More pecifically, for each receiver pair k, l) the ignal of receiver k and the ignal of receiver l are correlated and time delay, m =,,n related to peak in the reulting cro-correlation function are extracted Thee time delay are referred to a a TDOA meaurement [] Each TDOA meaurement i related to a poible ource locationp j) along a hyperbola For receiver pairk,l), the random TDOA that wa originated by ourcej i modeled a p j) q k) p j) q l) = v = h p j) )+z m), ) ) +z m) wherev i the propagation peed andz m) i the meaurement noie which i zero-mean Gauian with variance σz and tatitically independent acromandk,l) pair The dependence of a meaured TDOA on the location p j) of the generating ource j i decribed by the likelihood function f p j) ) that can be directly obtained from ) For receiver k,l located at q k) = [q x,] T and q l) = [ q x,] T, ) read = [ p j) ) ] p x q x +p j) j) ) y x +q x +p j) y +z m) v 4) Let be a meaured TDOA according to 4) and let u neglect it unknown noie realization, ie, z m) = After ome algebra, 4) can be written in the form of a hyperbola a p j) x pj) y = a b p j) x v) /4 j) py qx v) /4 = Fig how the related hyperbola for three ource with location [ ] T,[ ] T, and[ ] T, and different realization of z m) Multiobject Likelihood Function and Prior Ditribution Then TDOA meaurement related to receiver pairk,l) are modeled a a RFS R The probability that the ource j give rie to a meaurement i p d Therefore, the meaurement related to ource j i modeled a a Bernoulli RFS Θ ) p j) with exitence probability r = p d and patial pdf f ) m) = f r ) p j) Thi lead to the following RFS likelihood function related to ource j, f R p j)) p d, R =, = p df p j)), R { = },, otherwie We adopt the tandard aumption, that a meaurement cannot originate from more than one ource imultaneouly, and one ource can generate at mot one meaurement at each receiver pair [, ] Thu, all ource-originated meaurement form the multi- Bernoulli RFS n t ) j= Θ p j) Following [, ], it i aumed that the number of fale alarm at receiver pair k, l) i Poion ditributed with mean µ Furthermore, at each receiver pair k,l), the fale alarm denoted by K 5

3 are aumed to be independent of the ource-originated meaurement and independent and identically ditributed iid) according to the patial pdf f c ) Auming that each meaurement originate either from a ource or i a fale alarm, the overall meaurement RFS at receiver pair k,l),r, i then obtained a n t R = Θ p j) ) K 5) j= The dependence of the overall meaurement RFS R on the overall ource location RFS P, i decribed by the multiobject likelihood function fr P), which can be obtain from 5) uing FISST convolution cf )) Let every receiver pair be one of n = n r ) enor with correponding ) receiver indexe k,l ) and let R :n Rk l,, R kn be an ordered lit of ingle enor meaurement The all-enor likelihood function then ln read n fr :n P) = fr kl P) = The prior pdf of P, fp), i modeled a a Poion RFS with mean parameterµ p and a patial pdff pp) that i choen uniform on the D region of interet 4 MULTIOBJECT ESTIMATION In a Bayeian RFS etting, multiobject etimation relie on the marginal poterior pdf fp R :n ) after making the meaurement R :n R k l,, R kn l n ) In our approach, thi pdf i calculated by performing an update tep for each enor recurively, ie, a ingle enor update tep i performed n time At recurion, fp R : ) i tranformed into fp R :) Note that fp R :) fp R ) andfp R :) fp)) We denote byp the RFS related to the pdf fp R :) Following [8] all updated multiobject tate RFS P are modeled a the union of independent RFS P u and P d, ie, P = P u P d, decribing the undetected and detected ource, repectively Let fp) u be the pdf of P u note that P u i independent of R : but not of ) and f d P R :) be the poterior pdf related to P d Thu, the poterior multiobject pdffp R :) i given by the FISST convolution cf )) Note that P u i a Poion RFS with mean parameter µ u, patial pdf fp), u and intenity function λ u p) = µ u fp) u Furthermore, P d i a multi-bernoulli RFS coniting of J Bernoulli component with exitence probabilitie r j) and patial pdff j) p),j {,,J }, where each Bernoulli component repreent a potential ource and it location For =, we have λ u = µ pf pp) and J = 4 Single Senor Update The intenity function λ u p) related to the Poion pdf f u P) at recurion, can be calculated directly from λ u p) a λ u p) = p D)λ u p) However, the update tep for the detected ource yield a weighted mixture of multi-bernoulli pdf with a number of component that grow exponentially in the number of enor n To obtain an approximation f d P R :) for f d P R :) that i till a multi- Bernoulli RFS and to avoid an exponential caling of the computational complexity in the number of enor n, an approximation preented in the context of multiobject tracking [8] wa adopted The approximated multi-bernoulli pdf f d P R :) i characterized by exitence probabilitie r j) and patial pdf f j) p) related to the Bernoulli component f j) P), j {,,J } Therefore, the update tep amount to calculating thee quantitie from λ u p), r j) and fj) p), j {,,J } Note that the number of Bernoulli component in f d P R :) i J = J + n kl, coniting of one legacy Bernoulli component j {,,J } for each component in f d P R : ) and one new component j {J +,,J + n kl } for each of the n kl meaurement, m {,,n kl } The calculation of updated Bernoulli component i baed on o-called marginal aociation probabilitie p j) m), j {,,J } [] For efficient calculation of marginal aociation probabilitie by mean of loopy belief propagation we refer the reader to [] The legacy component j {,,J }, can be obtained a f j) r j) ) rj) pd) n kl + r j) pd m= = p j) p) = fj) p) r j) + [ n kl m= pd) p j) ) rj) r j) pd p j) m) p j) m), f p ) ] m) f r p ) f j) p)dp 6) Furthermore, the new component j = J + m with m {,,n kl }, are given by r j) = pj) m) m)p D f r p ) λ u p)dp m)) m) λ c r +pd f r p ) 7) λ u p)dp, f j) p) = f p ) λ u p) m) f r p ) 8) λ u p)dp The expreion in 6), 7), and 8) can not be evaluated in cloed form Thu, we employ a equential Monte Carlo implementation for approximate evaluation of 6), 7), and 8) Firt, the integral f p ) f j) p)dp and f p ) λ u p)dp are evaluated uing Monte Carlo integration [] Next, for each j {,,J }, L particle and correponding weight { p j,l) w j,l) )} L l=, repreenting fj) p) are calculated by mean of importance ampling [] uing a propoal pdf f j) p) in 6) and f p) u in 8), repectively A particle-baed calculation of the marginal aociation probabilitie p j) m), j {,,J } i preented in [9] 4 Detection and Etimation The Bernoulli component j in f d P R :n ) i conidered to exit if r n j) > 5 For each component j that i conidered to exit, an etimate of the ource location p j) can then be calculated a ˆp j) L l= p j,l) n w n j,l) The entirety of all ˆn t detected location etimate ˆp j) form the etimated et ˆP = {ˆp ),,ˆpˆnt) } 5

4 a) b) c) d) Fig Particle-baed repreentation of the patial pdf related to the three Bernoulli component with the highet exitence probabilitie in f d P R: ): a) =, b) =, c) =, and d) = 6 = n The location of receiver involved in the current update tep are indicated by dot, the location of other receiver are indicated by circle 5 SIMULATION RESULTS pd, µ) In our imulation the ource are randomly placed on a region of m) interet of [, ] [, ] The ource-originated TDOA r at receiver pair k, l) are ditributed according to ) with a tandard m) deviation of σz = /v for the noie z The fale alarm pdf m) ffa r at receiver pair k, l) i uniform on kqk) ql) k/v The receiver are located evenly on a circle of radiu The mean parameter of the prior pdf fp P) wa choen a µp = 5 The number of particle wa et to L = 5 For a ingle realization, Fig how imulation reult for a cenario with nr = 4 n = 6 ), pd = 9, and µ = More pecifically, Fig how particle repreenting the patial pdf of the three Bernoulli component with the highet exitence probabilitie of f d P R ) in Fig a), of f d P R: ) in Fig b), of f d P R: ) in Fig c), and of f d P R:6 ) in Fig d), repectively A few remark are in order: The firt receiver pair = ) ha one mied detection and no fale alarm meaurement nk l = ); f d P R ) con) it only J = component The patial pdf f p) and ) f p) meet the true location of two ource The econd receiver pair = ) ha no mied detection and one fale alarm meaurement nk l = 4); f d P R: ) conit of J = J + nk l = 6 component The two Bernoulli component with the highet exitence probabilitie are the legacy component that were generate from the meaurement of the firt receiver pair Their patial pdf ) ) f p) and f p) have mode which correpond to the interection point of the hyperbola related to previou meaurement and current meaurement The Bernoulli compo) nent f p) with the third-highet exitence probability i related to a new meaurement that doe not interect with an exitent Bernoulli component The third receiver pair = ) ha no mied detection and no fale alarm meaurement nk l = ); f d P R: ) conit of J = J + nk l = 9 component The patial pdf ) ) ) f p), f p), and f p) are till multimodal but more concentrated around the true ource location compared to the = reult After the update tep ha been performed for all n = 6 receiver pair, f d P R:n ) conit of Jn = component and the patial pdf related to the three component with the highet exitence probabilitie have a ingle mode that i well localized around the true location of the three ource 54 99, ) 9, ) 8, ) 99, ) 9, ) 8, ) 99, ) 9, ) 8, ) nr = 4 e L nr = 5 e L pc pc 6 Table Simulated mean localization error e L and probability of cardinality error pc for different ytem parameter To invetigate how the performance of our method depend on ytem parameter, we imulated cenario with three different pd {99, 9, 8}, three different µ {,, } at each receiver pair, and two different nr {4, 5} The performance of the propoed method in term of the probability of a cardinality error pc and the mean localization error e L i evaluated by imulation For each realization, the localization error el i evaluated in term of the OSPA metric [] with a cut-off parameter of and with the Euclidean ditance ued a inner metric el only contribute to the mean localization error if the cardinality wa etimated correctly Table how pc and e L for different value of nr, pd, and µ For each poible combination of µ, pd, and nr, we performed 5 imulation run It can be een that for low value of µ and high value of pd all ource can be accurately localized Furthermore, by comparing the reult for nr = 4 and nr = 5 it i demontrated how an increaed number of receiver can improve both robutne of the cardinality etimate and localization accuracy in ituation with an increaed number of fale alarm and a reduced probability of detection 6 CONCLUSION We preented a Bayeian algorithm for localizing an unknown number of tatic ource from TDOA meaurement The propoed algorithm i baed on a recently introduced hybrid Bernoulli/Poion filter and i implemented uing equential Monte Carlo technique Numerical reult howed that for a low number of fale alarm and a high probability of detection, all ource can be accurately localized A poible direction for further reearch include an improved approximation for the update tep where data aociation i performed acro multiple receiver pair [4]

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