Research Article Semidefinite Relaxation Algorithm for Multisource Localization Using TDOA Measurements with Range Constraints

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1 Wreless Communcatons and Moble Computng Volume 2018, Artcle ID , 9 pages Research Artcle Semdefnte Relaxaton Algorthm for Multsource Localzaton Usng TDOA Measurements wth Range Constrants Changgu Ja, 1,2 Jexn Yn, 1,2 Dng Wang, 1,2 Yunlong Wang, 1,2 and L Zhang 1,2 1 Natonal Dgtal Swtchng System Engneerng and Technology Research Center, Zhengzhou , Chna 2 Zhengzhou Informaton Scence and Technology Insttute, Zhengzhou, Henan , Chna Correspondence should be addressed to Jexn Yn; Cndyn0807@163com Receved 18 Aprl 2018; Accepted 4 June 2018; Publshed 28 June 2018 Academc Edtor: Davd Plets Copyrght 2018 Changgu Ja et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted Multple sources localzaton based on tme dfference of arrval (TDOA) measurements s nvestgated n ths paper Dfferent from the tradtonal methods, a novel and practcal multsource localzaton algorthm s proposed by adoptng a pror nformaton of relatve dstance among emttng sources Snce the maxmum lkelhood (ML) cost functon for multsource estmaton s hghly nonconvex, the semdefnte relaxaton (SDR) s utlzed to reformulate the ML cost functon A robust estmator s obtaned, whch can be solved by semdefnte programmng (SDP) Moreover, the constraned Cramér-Rao bound s also derved as a benchmark by consderng the range constrants between sources Smulaton results verfy the superor performance of the proposed algorthm over the tradtonal methods 1 Introducton Multsource localzaton s an essental task n radar, sonar, navgaton, and other applcatons [1, 2 Accordng to [3, 4, tme dfference of arrval (TDOA) measurements can be utlzed to locate a source wth hgh accuracy, whch crcumvents the synchronzaton problem n the tme of arrval- (TOA-) based methods The postonng problem of usng TDOA measurements s a nontrval task due to ts hgh nonlnearty and nonconvexty In [3, the authors develop an teratve nonlnear leastsquares (NLS) algorthm based on Taylor-seres expanson, but t requres suffcently precse ntal estmates for the global soluton Otherwse, t may suffer from the problem of local convergence To allevate ths drawback, a twostep weghted least-squares (WLS) method s put forward to approxmate the maxmum lkelhood (ML) functon and a closed-form soluton s also obtaned [4 However, naccurate estmaton of the covarance matrx of measurement errors used n [4 can result n performance degradaton n practce In general, the least-squares (LS) methods can acheve the Cramér-Rao bound () when the measurement errors are suffcently small Snce the convex optmzaton has been appled to solve localzaton problems, many researchers prove that ths method s attractve to robustly acheve excellent estmate results and accuracy even at hgh nose levels [5 8 Ths s manly because of ts preferable property of provdng globally optmal soluton wthout needng any ntal estmates and effcent calculaton wth exstng software package Usually, there are two ways to apply the semdefnte relaxaton (SDR) technques The authors n [5 start from reformulatng the nonconvex ML problem to a convex optmzaton problem and then add a constrant based on a pror knowledge of the admssble source poston Another approach s presented n [6 8 where the robust LS cost functon s frstly establshed and then relaxed to a semdefnte programmng (SDP) problem It s worth pontng out that these meanngful trals lay the foundatons for our further exploraton of usng the SDR technques Despte the precedng progress, all of these works consder only sngle source Multsource localzaton s of great nterest for ts frequent emergence n practce Here we address the problem n some common scenaros where emttng sources le wthn a certan range or move n group, whch can be referred to as group targets The targets are typcally sensors n the wreless sensor

2 2 Wreless Communcatons and Moble Computng network or formaton-flght arcraft and arcraft carrer fleet n the open space Such patterns usually ndcate that these group targets wll stay wthn a certan relatve dstance wth each other and keep the resemble velocty for most of the tmes For example, arcraft n formaton-flght usually keeps a dstance around several meters to tens of meters between each companon n the formaton [9, and t s also smlar for group targets wth unknown postons and veloctes n sensor network To date, only several papers have consdered the problem of localzng multple sources [1, 2, One of the challenges for multsource localzaton s data assocaton of multsensor and multsource When the sensors have no pror knowledge of ndvdual sgnal features of the sources, namely, the assocatons between the multple sgnal measurements at each sensor and the correspondng sources are unknown, a target acquston and localzaton algorthm (TALA) s developed n [10, whch utlzes hybrd angle-ofarrval(aoa)andtoameasurementsgeneratedwthna tme wndow by an array of sensors to detect and localze an unknown number of targets The authors n [11 apply TDOA measurements to localze multple acoustc sources by treatng the assocaton problem wthn a multple-hypothess framework Alternatvely, usng TOA measurements, [12, 13 proposes a three-stage algorthm to smultaneously estmate the source-measurement assocatons and the sources locatons, respectvely There are also other cases when the problem of matchng measurement data wth the orgnal emttng sources can be accomplshed through technques that make use of the dsjontness of dfferent sources [1, such as tme, frequency, or sgnal transents [2 Hence, the postonng parameters for multsource, eg, TDOA, can be obtaned separately Based on the perfect matchng, the multsource localzaton algorthms n [1, 14 explot the fact that the TDOAs from dfferent sources have the same sensor poston dsplacements to jontly estmate the sources postons and mprove the localzaton accuracy when the sensor postons are not known exactly However, the pror nformaton of relatve dstance among sources has not been consdered n these related works In ths paper, the number of sources to be located s assumed to be already known and the assocatons between the multple sgnal measurements at each sensor and the correspondng sources have been accomplshed Unlke the tradtonal approaches mentoned above, ths paper devses a novel method usng SDR technques to jontly locate the multple sources by ntroducng the pror nformaton constrants on relatve dstances between them Practcally, these relatve dstance nformaton could be obtaned through other observaton methods And t can mprove the localzaton accuracy wth such range constrants for group targets The contrbutons of ths paper are summarzed as follows (1) We formulate the TDOA-based ML problem for multsource localzaton as an SDP problem whch can be solved by the convex optmzaton tools The tghtness of our SDP s also enhanced by ntroducng the exstng methods, such as utlzng Cauchy-Schwartz nequalty (2) The pror nformaton of relatve dstance among sources s ntroduced as convex constrant for group targets and ts effectveness of mprovng localzaton accuracy s verfed by our smulatons (3) The constraned s derved n ths paper as a benchmark to evaluate the proposed method by ncorporatng the pror range constrants The rest of the paper s organzed as follows In Secton 2, the TDOA measurement model for multsource s gven Subsequently, the semdefnte relaxaton (SDR) method for solvng the localzaton problems s presented n Secton 3 and complexty analyss of the proposed algorthm s gven n Secton 4 Secton 5 derves the constraned wth nequalty constrants Smulaton results are llustrated n Secton 6 Fnally, conclusons are drawn n Secton 7 In the sequel of ths paper, boldface lowercase letters represent column vectors and boldface uppercase letters denote matrces a o s used to denote the nose-free value of a ( ) T denotes transpose operator and â represents the estmaton of a blkdag{a, B,,C} constructs block dagonal matrx from the matrces A, B,andCtr(A) means the trace of A,and A B means that A B s postve semdefnte 2 Problem Formulaton Consder that there are M sensors deployed at the known postons as s ( = 1,2,,M) to ntercept sgnals emtted by N sources at unknown postons denoted by u j (j = 1,2,,N)n the three-dmensonal (3D) scenaro Usually we have M 4and N 2 The unknown parameter vector to be determned s denoted by u =[u T 1, ut 2,,uT N T Note that the vector u s a necessty to construct our SDP n the followng sectons Wthout loss of generalty, the frst sensor s chosen as the reference sensor and lne-of-sght propagaton condton s consdered The TDOA measurement between a sensor par and 1 from source j s gven by τ 1,j = 1 c u j s 1 c u j s 1 +Δτ 1,j, =2,3,,M, j=1,2,,n, where c s the sgnal propagaton speed Δτ 1,j denotes the TDOA measurement nose that s assumed to be zero-mean Gaussan dstrbuted u j s represents the dstance between j th source and th sensor and s Eucld norm Multplyng c and TDOA measurement n (1), the range dfference of arrval (RDOA) s obtaned by r 1,j = u j s u j s 1 +Δr 1,j, =2,3,,M, j=1,2,,n, where Δr 1,j =cδτ 1,j Snce there are M sensors, the collecton of RDOA measurements for source j s r j =[r 21,j,r 31,j,, r M1,j T = r o j +Δr j,whereδr j =[Δr 21,j,Δr 31,j,,Δr M1,j T s the nose vector and r o j =[ u j s 2 u j s 1, u j s 3 (3) u j s 1,, u j s M u j s 1 T (1) (2)

3 Wreless Communcatons and Moble Computng 3 Then collect all measurements of N sources, and the N(M 1) 1 measurement vector s denoted by r =[r 1 T, r 2 T,,r N T T = r o +Δr, (4) where Δr = [Δr T 1,ΔrT 2,,ΔrT N T and r o = [r ot 1, r ot 2,, r o NT T Furthermore, Δr s assumed to be a zero-mean Gaussan random vector wth covarance matrx Q Δ = E[ΔrΔr T Usng smlar formaton method n [7, r o can be reformulated as where the NM 1 vector: r o = Gd, (5) d =[ u 1 s 1,, u 1 s M,, u N s 1,, u N s M T, and the N(M 1) NM matrx G = blkdag{[ 1 M 1 I M 1,,[ 1 M 1 I M 1 } 1 M 1 and I M 1 denote the (M 1) 1 all onecolumnvectorandthe(m 1) (M 1) dentty matrx, respectvely We can further rewrte (4) as (6) r = Gd +Δr (7) Then the problem of nterest s to estmate the unknown vector u gven the RDOA measurements r and the pror nformaton of relatve dstance among sources Here the pror range constrants for multple sources that accompany each other n group are ntroduced As nterpreted before, group targets usually mantan a certan formaton where the relatve dstances among each other can be bounded wth a certan value, and the range constrants are gven by u j u m δ, j,m=1,2,,n, (8) where δ s the upper range bound between two targets Smlar dea can also be found n [15, but t just consders the potental communcaton range of an anchor to confne an area that a source must le wthn Here we practcally consder a rough upper bound for all potental relatve dstances among sources snce they are not equal or may change slghtly n real stuatons Besdes, the loose upper rangeboundcanbetghtenedthroughmoreprecseobservatons Specfcally, the selecton of the upper range bound parameter δ n practce can be accomplshed by the followng ways: (1) If the group targets are n the open space, we can obtan a rough target poston by some preestmaton methods [3, 5, separately Then the relatve dstances between targets can be computed usng the preestmated target postons Next, based on the relatve dstances obtaned through the preestmaton process, we can select a large, but approprate, δ to establsh the range constrants (2) For a group of targets that exst n a formaton group [9, the relatve dstances between the targets can be obtaned accordng to the correspondng formaton rules and types, and then the value of δ canbeselectedwthproper consderaton 3 Semdefnte Relaxaton Method for Multsource Localzaton In ths secton, the SDR technques are employed to approxmate the ML problem for multple sources localzaton and ncorporate the range nequaltes as convex constrants Accordng to (7), the ML estmaton of u canbeformulated as û = arg mn (r Gd)T Q 1 u 1,,u N R 3 Δ (r Gd) (9) Then, (9) can be rewrtten as a constraned quadratc program as follows: mn u 1,,u N,d (r Gd) T Q 1 Δ (r Gd), (10a) st [d 2 M(j 1)+ = u j s 2, (10b) [d M(j 1)+ 0, =1,2,,M, j=1,2,,n (10c) It s proved n [16 that (10c) can be neglected snce they are always satsfed for any globally optmal soluton determned by (10a) and (10b) Addtonally, the objectve functon n (10a) s convex, and the quadratc equalty constrants n (10b) are nonconvex To transform the problem to standard convex optmzaton problem defned n [17, problem (10a) and (10b) can be equvalently reexpressed by decomposng the Eucld norms as mn u 1,,u N,d st [ d T 1 [ GT Q 1 Δ G GT Q 1 Δ r r T Q 1 Δ G rt Q 1 Δ r [ d, (11a) 1 T [d 2 M(j 1)+ =[u j 1 [ I s s T s T s [ u j 1, =1,2,,M, j=1,2,,n (11b) By applyng the basc property of x T Ax = tr{xx T A}, (11a) and (11b) can be reformulated as mn u,d,u,d st D d tr {[ F}, (12a) d T 1 [D M(j 1)+,M(j 1)+ = tr { J j UJ T j J j u [ { {[ (J j u) T [ I s 1 s T s T s } }, } D = dd T, U = uu T, =1,2,,M, j=1,2,,n, (12b) (12c) where F =[ GT Q 1 Δ G GT Q 1 Δ r, J r T Q 1 Δ G rt Q 1 Δ r j =[0 3 3(j 1) I (N j) s the 3 3Nmatrx, and t s easy to check that u j = J j u

4 4 Wreless Communcatons and Moble Computng Usng the SDR prncple, we relax the constrants D = dd T as D dd T and U = uu T as U uu T,respectvelyBy utlzng the basc property of Schur-complements descrbed n [17, these relaxed constrants can be equvalently rewrtten as [ D d d T 1 0, [ U u u T 1 0, (13) where the matrces are of rank 1 and symmetrc postve semdefnte (PSD) Note that the PSD constrants n (13) are convex [17 Then we obtan a convex optmzaton problem whch can be seen as an approxmaton of the ML problem descrbed n (9) Itsnecessarytopontoutthat,asshownn[5,such relaxatons are not tght enough to yeld hghly precse solutons In order to remedy the SDR formulaton to enhance the tghtness and also make use of the pror nformaton of relatve dstance among sources, two types of addtonal constrants for D and U are devsed subsequently We consder usng the relatonshp between elements of D and dd T to generate the constrants for unknown parameters to be determned as the frst type Snce there are M recevers and by Cauchy-Schwartz nequalty, for j th source we have Then the pror nformaton of relatve dstance n (8) for group targets can be utlzed to form the second type of constrants Usng the above relatons, (8) can be represented as J ju J m u δ, j,m=1,2,,n, j>m (17) Squarng both sdes of (17) yelds whch s equvalent to J 2 ju J m u δ 2, (18) u T (J j J m ) T (J j J m ) u δ 2 (19) Note that the quadratc nequalty constrant n (19) s convex By applyng x T Ax = tr{xx T A} to (19), we have tr {uu T (J j J m ) T (J j J m )} δ 2 (20) Substtutng U = uu T nto (20) obtans tr {UJ jm } δ 2 j,m=1,2,,n, j>m, (21) where J jm = (J j J m ) T (J j J m ) In summary, addng the constrants of (13), (16), and (21) to (12a), (12b), and (12c), the proposed SDP s obtaned as follows: [D M(j 1)+,M(j 1)+n = [d M(j 1)+ [d M(j 1)+n = u j s u j s n (u j s ) T (u j s n ), (14) mn u,d,u,d st D d tr {[ d T 1 F}, [D M(j 1)+,M(j 1)+ where (u j s ) T (u j s n )=[ u T j 1 [ I s n s T s T s [ u j n 1 (15) = tr { J j UJ T j J j u [ { {[ (J j u) T [ I s n 1 s T s T s } } n } Therefore, t s straghtforward to obtan the followng nequaltes: = tr { J j UJ T j J j u [ { {[ (J j u) T [ I s 1 s T s T s } }, } tr {UJ jm } δ 2 [D M(j 1)+,M(j 1)+n j,m=1,2,,n, j>m, tr { J j UJ T j J j u [ { {[ (J j u) T [ I s n 1 s T s T s } } n } (22) [D M(j 1)+,M(j 1)+n,n=1,2,,M, n>, j=1,2,,n, tr { J j UJ T j J j u [ { {[ (J j u) T [ I s n 1 s T s T s } }, n },n=1,2,,m, n>, j=1,2,,n (16) Itsnecessarytopontoutthatthenequaltesabove mposeconstrantsontheproductofrangesforeachsource wth dfferent sensor pars That s, the correspondent elements that le n the upper trangular matrx around the prncple dagonal n D are restraned The effectveness of these nequalty constrants wll be verfed through our smulatons [ D d d T 1 0, [ U u u T 1 0 Ths canoncal convex optmzaton problem can be solved by CVX Toolbox n Matlab By solvng the SDP n (22), the estmaton of u s fnally obtaned Remark 1 The covarance matrx Q Δ n F can be estmated by takng the covarance of the TDOA measurements obtaned

5 Wreless Communcatons and Moble Computng 5 from multple observatons n practce The correspondng estmaton methods are descrbed n detal n [18, 19 For clarty, the estmate of the element of Q Δ s gven by [Q Δ j ( 1),m (n 1) = where K 1 K 1 k=1 (r 1,j [k r 1,j )(r n1,m [k r n1,m ),,n=2,3,,m; j,m=1,2,,n, r 1,j = 1 K K r 1,j [k, k=1 r n1,m = 1 K K r n1,m [k, k=1 (23) (24) K s the number of observatons, and r 1,j [k s the TDOA measurement of the k th observaton Remark 2 Theoretcally, n terms of the effect of the range constrants ntroduced for group targets, the constrants may have lttle effect to mprove the estmaton performance for group targets when the measurement nose s small Thssmanlybecause,atsmallmeasurementnoselevel, the poston estmaton results wthout constrants may just slghtly devate from the true target postons; thus they wll probably fall nto the feasble doman confned by the range constrants and meet the constrants However, when the measurement nose grows larger, the poston estmaton results wthout constrants mght devate from the real target postons by a large amount wth a hgh probablty, whch are lkely to exceed the feasble doman In ths case, the range constrants wll defntely confne and reduce the devatons and sgnfcantly decrease the estmaton errors Therefore, the mprovement of the estmaton performance can be more sgnfcant by utlzng the range constrants at large measurement nose level Ths wll be llustrated by the smulaton results n Secton 6 4 Complexty Analyss Computatonal complexty s analyzed n ths secton Here, we apply the result of [6, 20 to analyze the computatonal complexty of the proposed multsource localzaton algorthm, denoted as, n them-dmensonal scenaro The localzaton algorthm for sngle target usng smlar SDR method n [7 except for the range constrants n (21), denoted as, and the classc two-step weghted leastsquare () method [4 are also dscussed In Table 1, algorthm complextes n terms of the number of teratons and operatons needed n each teraton are examned From the table, t can be seen that the complexty grows wth the number of targets to be estmated and our SDP algorthm s computatonally hgher than the classc Thus t needs further refnng [6 It s necessary to pont out that, Table 1: Complexty comparson of dfferent algorthms Algorthm Iteraton Number Operaton per Iteraton 1 O((2m + 1)M) O((2m) 1/2 ) O((mM) 2 ) O((2Nm) 1/2 ) O((mNM) 2 ) unlke sngle target postonng algorthm, whch locates the targets n a decoupled manner, our approach s to localze multple targets jontly Although the complexty of the proposed algorthm ncreases, the correspondng localzaton performance can be promoted Ths s verfed by the smulaton results n Secton 6 5 The Constraned The s usually seen as a benchmark aganst whch the statstcal effcency of any unbased estmators can be compared The s for sngle source and multsource localzaton usng TDOA measurements have been nvestgated n [5 and [1, respectvely, but they do not consder the stuaton where nequalty constrants are employed Inspred by the work of Gorman et al [21, we try to ncorporate the range constrant nequaltes and derve a constraned (C) The probablty densty functon (PDF) of jontly Gaussan dstrbuted measurement nose Δr s subsequently gven by p (r u) = exp { (1/2)(r Gd)T Q 1 Δ (r Gd)} (25) (2π) N(M 1) det (Q Δ ) The correspondng Fsher nformaton matrx (FIM) s obtaned as FIM = E [ 2 ln (p (r u)) u u T =[G d u T T Q 1 Δ d [G u T, (26) where d/ u T s the NM 3N Jacoban matrx defned as (u 1 s 1 ) T u 0 1 s d (u 1 s M ) T d u 0 1 s M u T = d (27) (u N s 1 ) T u N s 1 d [ (u N s M ) T [ u N s M

6 6 Wreless Communcatons and Moble Computng Table 2: Postons of sensors (m) Sensor no x y z The nequaltes n (8) can be wrtten n a functonal nequalty constrant of the form b (u) =[b 1,2 (u),, b 1,N (u),, b,j (u),, b N 1,N (u) T 0 N(N 1)/2 1, b,j (u) = u u j δ, =1,2,,N 1 j=2,3,,n (28) The constraned parameter space Θ C s defned by b(u), and t s composed of the equalty constrant b eq (u) =0 and pure nequalty constrant b neq (u) <0 Whenu s a regular pont that les n the nteror of Θ C,namely,u belongs to the set {u: b neq (u) <0} where the equalty constrant s nactve, t s demonstrated n [21 that the constraned s dentcal to the unconstraned However, due to the exstence of measurement errors, some ponts of the estmated parameter û are not regular Hence they wll not le n the nteror of Θ C Consequently, the equalty constrants b eq (u) = 0 wll take effect under such condton, thus leadng to the bound reducton For ths case, the classcal s nvald due to the actve equalty constrants Itsprovedn[21that,foranyunbasedestmatoru confned by nequalty constrants (28), the estmator error covarance matrx satsfes the matrx nequalty where Σ u B u, (29) B u =[ m u Q u (FIM) 1 [ m u T, (30) where m u s the 3N 3N dentty matrx for unbased estmator [21, and Q u s the 3N 3Nmatrx whch s defned as follows: Q u = I (FIM) 1 [ b (u) T {[ b (u) (FIM) 1 [ b (u) T } + [ b (u), (31) b (u) = b (u) u T =[( b T 1,2 (u) u T ) ( b T 1,N (u) u T ) ( b T,j (u) u T ) ( b T T N 1,N (u) u T ), (32) b,j (u) (u u T = [ u j ) T 0 1 3( 1) [ u u j 0 1 3(j 2) (u u j ) T u u j 0 1 3(N j+1), =1,2,,N 1 j=2,3,,n, (33) where the ( ) + denotes pseudo-nverse ItcanbeseenthatQ u has already contaned the exact pror nformaton of relatve dstance among the sources, whch leads to a reducton n Hence, we have the followng constraned (C) matrx of the lowest varance for the effcent constraned estmator û as C C = B u (34) The dagonal elements of C C gve the lowest estmate varance of the source coordnates For example, n the mdmensonal scenaro, the C for u j n ths case s the sum of the j th m dagonal elements of C C after takng square root operaton 6 Smulaton Results In ths secton, smulaton has been conducted to evaluate the TDOA-based localzaton performance of the proposed SDR algorthm The localzaton algorthm developed n ths paper for multple sources, denoted as, s compared wth the sngle target SDR algorthm whch s smlar to [7 usng only TDOA measurements, namely, the, and the classc two-step weghted least-square () method [4 The for sngle target case [5 and C derved n ths paper are also ncluded In the smulaton, sx statonary sensors are employed to locate more than two targets n the sensor network The postons of the sensors are lsted n Table 2; they are the same as the settng n [1 The postonng performance s evaluated by the root mean square errors (RMSE), defned by RMSE =[ ( L =1 û 1/2 u 2 ), (35) L where û stheestmateofthetruesourcepostonu and the RMSE of j th source poston u j (j = 1,2,N) can be drectly computed n the same way when the estmate û j s

7 Wreless Communcatons and Moble Computng 7 Poston RMSE of target 1 Poston RMSE of target 2 Poston RMSE (m) Poston RMSE C =300m =40m (a) RMSE of target 1 C =300m =40m (b) RMSE of target 2 Fgure 1: Comparson of RMSE and measurement nose σ 2 for two targets wth dfferent δ obtanedthermsesobtanedthroughl = 500 ndependent Monte-Carlo runs n our smulaton The RDOA measurement nose s assumed to be a correlated Gaussan process [5, and ts covarance matrx s of the form Q Δ = σ 2 blkdag{σ, Σ,,Σ} wth Σ =2 [ d, (36) [ where Σ s the (M 1) (M 1) matrx, and Q Δ s the N(M 1) N(M 1)matrx In smulaton, the value of the range dfference measurement error varance σ 2 s modfed to acheve dfferent nose condtons, and the measurement noses of dfferent sources are assumed to be ndependent TheSDPnthspaperssolvedusngtheMatlabtoolboxCVX [22, where the solver s SDPT3 [23 In the frst test, the mpact of δ for localzaton performance s frst nvestgated here Wthout loss of generalty, the group targets are assumed to consst of two emttng sources The sources are located at 652, 805, and 710 m and 650, 814, and 708 m Thus the relatve dstance between them s about 10m, and we examne two stuatons where the upper range bound δ s set as 300m and 40m, respectvely The estmated poston RMSEs of the two targets aganst the measurement error varance σ 2 are shown n Fgure 1 At δ = 300m and 40m, Fgure 1 shows that when the upper range bound s more precse, the proposed method can acheve better result than that of large δ and eventually attan the correspondng C It can also be notced that even when δ s large, for example, δ = 300m, our method can slghtly mprove the localzaton accuracy compared wth the localzaton algorthms for sngle source Addtonally, t can be observed from Fgure 1 that the localzaton RMSE oftheproposedmethodcanapproachthecasthe measurement error gradually grows large Ths ndcates that the range constrants between sources are more effectve as the measurement nose ncreases The result corresponds to the analyss of Remark 2 n Secton 3 In the second test, the number of targets s ncreased to three Ther postons are 654, 803, and 708 m, 652, 808, and 710 m, and 650, 814, and 705 m Thus the true ranges between the targets are approxmately up to 12m Here the observed upper range bound nformaton δ s set as 60m The smulatons results are plotted n Fgure 2, from whch we can see that the performance of proposed collaboratve localzaton method outperforms other methods And t can be also notced that the localzaton accuracy can be mproved wth the ad of the range constrants, as the measurement error gradually becomes ntensve Moreover, based on the abovetwotests,tcanbefoundthatourmethodcanyelda promoton of localzaton accuracy by utlzng a rough upper range bound when the range nformaton between sources s not known precsely In the thrd test, the group targets are assumed to consst of three emttng sources that are farther from the sensors than that n the second test Ther postons are 1260, 1440, and1193m,1256,1450,and1195m,and1251,1456,and1188m Thus the maxmum relatve dstance among them s around 20mandtheupperrangeboundδ ssetas90mtheestmated poston RMSEs of the three targets aganst the measurement error varance σ 2 are shown n Fgure 3 when δ=90m We can observe that the RMSE of the proposed poston estmates usng the SDP method s much lower than the results of tradtonal sngle target localzaton algorthms, and t can attan the correspondng C as the measurement error ncreases Comparng the results of the thrd test wth those of the second test, the localzaton performance of all the three algorthms n the thrd test s worse than that n the second test because of the poor geometry Nevertheless, our method can yeld a promoton of localzaton accuracy by utlzng the range constrants compared wth other algorthms n the both tests 7 Concluson In ths paper, we propose to jontly localze multple emttng sources by utlzng the pror relatve dstance nformaton

8 8 Wreless Communcatons and Moble Computng Poston RMSE of target 1 Poston RMSE of target 2 Poston RMSE (m) Poston RMSE (m) C C (a) RMSE of target 1 (b) RMSE of target 2 Poston RMSE of target 3 Poston RMSE (m) C (c) RMSE of target 3 Fgure 2: Comparson of RMSE and measurement nose σ 2 forthreetargetswthδ=60m Poston RMSE of target 1 Poston RMSE of target 2 Poston RMSE (m) Poston RMSE (m) C C (a) RMSE of target 1 (b) RMSE of target 2 Poston RMSE of target 3 Poston RMSE (m) C (c) RMSE of target 3 Fgure 3: Comparson of RMSE and measurement nose σ 2 forthreetargetswthδ=90m

9 Wreless Communcatons and Moble Computng 9 between them based on TDOA measurements The SDR technque s appled to reformulate the orgnal nonconvex MLproblemformultsourcetoobtananSDPMoreover, the constraned, whch ncorporates the nequalty constrants of pror relatve dstance nformaton, s also derved n ths paper Smulaton results show that the proposed method sgnfcantly mproves the localzaton accuracy and can acheve the correspondng C when the range nformaton s more precse In addton, the proposed method performs well even when the measurement error s ntensve Data Avalablty The authors clam that the data used n ths artcle are provded by ther smulatons accordng to some real localzaton scenaros, and ths artcle s developed wthout usng any data n a publshed artcle to support ther results Conflcts of Interest The authors declare that they have no conflcts of nterest Acknowledgments The authors acknowledge support from the Natonal Natural Scence Foundaton of Chna (Grants no , no , and no ), Chna Postdoctoral Scence Foundaton (Grant no 2016M592989), the Self-Topc Foundaton of Informaton Engneerng Unversty (Grant no ), and the Outstandng Youth Foundaton of Informaton Engneerng Unversty (Grant no ) References [1 L Yang and K C Ho, An approxmately effcent TDOA localzaton algorthm n closed-form for locatng multple dsjont sources wth erroneous sensor postons, IEEE Transactons on Sgnal Processng,vol57,no12,pp ,2009 [2 D Carevc, Automatc estmaton of multple target postons andveloctesusngpassvetdoameasurementsoftransents, IEEE Transactons on Sgnal Processng, vol 55, no 2, pp , 2007 [3 C Mensng and S Plass, Postonng Algorthms for Cellular Networks Usng TDOA, n Proceedngs of the 2006 IEEE Internatonal Conference on Acoustcs Speed and Sgnal Processng, vol4,pp ,Toulouse,France,2006 [4 K C Ho and W Xu, An accurate algebrac soluton for movng source locaton usng TDOA and FDOA measurements, IEEE Transactons on Sgnal Processng,vol52,no9,pp , 2004 [5 K W K Lu, F K W Chan, and H C So, Semdefnte 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