Robust Expectation-Maximization Algorithm for Multiple Wide-Band Acoustic Source Localization in the Presence of Non-Uniform Noise Variances

Transcription

1 Robut Exectation-Maximization Algorithm for Multile Wide-B Acoutic Source Localization in the reence of on-uniform oie Variance Lu Lu, Hiao-Chun Wu, Senior Member, IEEE, Kun Yan, S S Iyengar, Fellow, IEEE Abtract Wideb ource localization uing acoutic enor network ha been drawing a lot of reearch interet recently The maximum-likelihood i the redominant objective which lead to a variety of ource localization aroache However, the robut efficient otimization algorithm are till being uruit by reearcher ince different aect about the effectivene of uch algorithm have to be addreed on different circumtance In thi aer, we would like to combat the ource localization baed on the realitic aumtion the ource are corruted by the noie with non-uniform variance We focu on the two oular ource localization method for olving thi roblem, namely the SC-ML (tewie-concentrated maximum likelihood AC-ML (aroximately-concentrated maximum likelihood algorithm We exlore the reective limitation of thee two method deign a new exectation maximization ( algorithm Furthermore, we rovide the Cramer-Rao lower bound (CRLB for all thee three method Through Monte Carlo imulation, we demontrate that our rooed algorithm outerform the SC-ML AC-ML method in term of the localization accuracy, the root-mean-quare (RMS error of our algorithm i cloer to the derived CRLB than both SC- ML AC-ML method Index Term algorithm, ource localization, CRLB I ITRODUCTIO Source localization uing low-cot low-comlexity enor array ha been the active reearch area in the field of radar, onar, geohyic, wirele ytem acoutic tracking for year ]- ] Recently, the wide-b ource localization in the near field ha drawn a lot of reearch interet in the ignal roceing alication ] 6] Extenive tudie for the wide-b ource localization can be found in ], 4] Among them, the maximum-likelihood (ML aroach in ] ha been regarded a the otimal robut cheme for coherent ource ignal However, when the multile ource are reent, the ML aroach facilitate a nonlinear otimization roblem, which i imractical eecially for the energy-contrained enor network In addition, many of the exiting ML etimator are baed on the unrealitic atiallywhite noie aumtion acro different enor 5] 7], Thi work wa uorted by Information Technology Reearch Award for ational rioritie (SF-ECCS 46644, etworking Technology Sytem Award (SF-CS 9679 from ational Science Foundation, DoD-DESCoR Grant ( from Office of aval Reearch Lu Lu, Hiao-Chun Wu, Kun Yan are with the Deartment of Electrical Comuter Engineering, Louiiana State Univerity, Baton Rouge, LA 78 ( llu4@tigerluedu, wu@eceluedu, kyan@tigerluedu S S Iyengar i with the Deartment of Comuter Science, Louiiana State Univerity, Baton Rouge, LA 78 ( iyengar@ccluedu the noie roce at each enor i aumed to be atiallyuncorrelated-white-gauian with an identical variance It i hown that under thi aumtion, the ML etimate of the unknown arameter (ource waveform/ectra noie variance can be exreed a the reective function of the ource location the number of indeendent arameter to be etimated i greatly reduced Thu, thi aumtion, although unrealitic, ubtantially reduce the earch ace uually lead to more efficient localization algorithm Hence, variou wide-b ML ource location etimator were rooed in ] However, thi atially-white noie aumtion i unrealitic in many alication In everal ractical alication 7], the enor are arely laced o that the enor noie rocee are atially uncorrelated However, the noie variance of each enor can till be quite different due to either the variation of the manufacturing roce, the imerfection of the enor array calibration or the unquiet background A a reult, the atial noie covariance matrix (acro the enor can be modeled a a diagonal matrix the diagonal element in general are not identical ote that thi noie model i initely not a ecial cae of the ARMA model a wa exlained in 8] Furthermore, the ource location etimator derived from the atially-white noie (SW aumtion would often not rovide atifactory reult in the real environment ince the algorithm derived from the SW aumtion blindly treat all enor equally in the etimated likelihood Motivated by the argument above, in 7], two DOA calculation algorithm, namely tewie-concentrated maximum likelihood etimator (SC-ML aroximately-concentrated maximum likelihood algorithm (AC-ML, have been recently rooed for the multile wide-b ource Although both SC-ML AC-ML method can be extended for the ource localization, the robutne iue till remain challenging in thi reearch area Thi i the rimary reaon why we would like to dedicate thi aer to addreing thee two iue by deigning a new ource localization cheme Felder Weintein rooed the generic exectationmaximization ( algorithm in 9] to etimate the arameter aociated with the uerimoed ignal emloyed it for the array ignal roceing in ] -baed technique have alo been alied for the multi-enor ignal enhancement ] ] In addition, -baed narrow-b ource localization algorithm were rooed by 4], 5] In thi aer, we modify the algorithm to tackle with the general multile ource localization roblem when the wide-b

2 ource are reent in the near field, which evolve from the imle DOA etimation method for the narrow-b ource in the far field in 6] If the wide-b ource are conidered, the ource ignal ignature or characteritic i unavailable at the enor array the method in 4] 6] cannot be alied according to 7], 8] Therefore, imilar to 7], 8], we ue the dicrete-fourier tranform (DFT filter bank to decomoe the wide-b ignal collected by the enor then etimate the comlete et of arameter involving ource waveform (or ectra ource location ote that our reviou work in 7], 8] can only deal with the ource localization roblem under the unrealitic SW aumtion In thi aer, we reformulate the ource localization roblem for the realitic SW aumtion deign a new baed localization algorithm for multile wide-b ource it can be hown that our rooed algorithm i much more comutationally efficient robut than the exiting SC-ML AC-ML method (we have extended the original SC-ML AC-ML method in 7] which could only olve the DOA roblem in 8] to combat the ource localization roblem The ret of thi aer i organized a follow The roblem formulation the ignal model are introduced in Section II The maximum-likelihood ource-location etimator for both SW SW model are introduced in Section III The novel algorithm for wide-b ource localization in the near field under the SW aumtion i derived dicued in Section IV In addition, the Cramer-Rao lower bound (CRLB derivation will be manifeted in Section V Monte Carlo imulation reult for demontrating our rooed new method illutrating our newly derived robutne analyi will be rovided in Section VI Concluion will be drawn in Section VII omenclature: The et of all real comlex number are denoted by R C, reectively A vector i denoted by A a matrix i denoted by A The tatitical exectation oeration i exreed ae{ } Beide, A T, A, A H,det( A, A, trace( A t for the tranoe, conjugate, Hermitian adjoint, determinant, eudo-invere, trace of the matrix A, reectively In addition, t for the Hadamard matrix roduct oerator, t for the Euclidean norm II SIGAL MODEL According to ], we conider a romly ditributed array of enor to collect the data from M ource Since the ource are aumed to be in the near field, the ignal gain are different acro the enor Thu, the ignal collected by the th enor at a dicrete time intant n i given by x (n M m a ( n t +w (n, ( for n,,,l,,,, m,,m, a i the gain of the m th ource ignal arriving at the th enor; (n denote the m th ource ignal waveform; t i the roagation delay (in data amle incurred from the m th ource to the th enor; w (n rereent the zero-mean indeendently identically ditributed (iid noie roce Several crucial arameter are ecified a follow: t r F r v : the roagation delay from the m th ource to the th enor, r R : the m th ource location, r R : the th enor location, v: the ource ignal roagation eed in meter/econd, F : amling frequency Taking the -oint dicrete Fourier tranform (DFT of both ide in Eq ( reerving a half of them due to the ymmetry roerty, we have X(k D(kS(k+U(k, for k,,,, ( X(k X (k,,x (k] T C ( X (k i the k th DFT oint of x (n,,, The ymbol for the right-h ide of Eq ( are clarified a follow D(k d ( (k,,d (M (k] C M (4 conit of M teering vector, each given by d (k d (k,,d (k]t C, m,,m, (5 j ote that d (k a e jπkt, (6 S(k S ( (k,,s (M (k] T C M (7 conit of M individual ource ignal ectra, each given by S (k S (k i the k th DFT oint of (n, m,,m In reality, the ource ignal ectral vector S(k i unknown determinitic The noie ectral vector U(k C i a comlex-valued zero-mean atially-uncorrelated Gauian roce with the following covariance matrix: q Q E { U(kU(k H} q q C, k (8 In general, q,,,,, are not necearily identical to each other under the SW aumtion Hence, we need to deal with the realitic ource localization roblem in the reence of the non-uniform noie variance thereuon III MAXIMUM-LIKELIHOOD FOR SOURCE LOCALIZATIO rior to the etablihment of the log-likelihood for the ource localization in the reence of the non-uniform noie variance a tated by Eq (8, we tart from the conventional maximum-likelihood formulation for the identical noie variance acro the enor

3 A Conventional Maximum-Likelihood for Source Localization in the reence of Identical oie Variance (SW According to the ignal model given by Eq ( together with the noie variance contraint a Q σ I, σ i the noie variance I i a identity matrix, the maximum-likelihood ource localization formulation can be facilitated a ], ], 7] We highlight the relevant ivotal formulae here Let r, S, σ rereent all the unknown arameter in Eq ( neceary to be etimated, ( r r T,,r T (M,,r T] T R M, (9 S S( T,,S(/ T] T C ( M ( In addition, we denote the reidual vector a g(k g (k,,g (k] T X(k D(kS(k C ( Thu, the likelihood function i given by f(r, S,σ π / σ ex σ / g(k ( Taking the logarithm of Eq ( neglecting all the contant term, we can derive the log-likelihood function L(r, S,σ a follow: L(r, S,σ log( σ / σ g(k, ( the correonding maximum likelihood etimate are ( ˆr,ˆ S,ˆσ arg max (r, S,σ arg min (r, S,σ (r, S,σ { L (r, S,σ } / arg min / Thu, according to Eq (4, we can write g(k X(k D(kS(k ] H X(k D(kS(k ] (4 ˆS(k D(k X(k ( D(k H D(k D(k H X(k, (5 ˆr argmin r / X(k D(k X(k (6 B Maximum-Likelihood for Source Localization in the reence of on-uniform oie Variance (SW In thi ubection, we will introduce the nonuniform maximum-likelihood ource localization formulation according to the recent literature 7], 8] for a more realitic SW model Let r, S, q be the arameter to be etimated for thi cae, q q,,q ] T R i the vector coniting of the diagonal ( element in Q given by Eq (8 The likelihood function of r, S,q can be exreed a f ( r, S,q ( π det( Q / / ex g(k H Q g(k (7 ( Then we have the following log-likelihood function L r, S,q by taking the logarithm of Eq (7 neglecting all the contant term: ( L r, S,q g(k X(k D(k log(q / g(k, (8 Q / g(k X(k D(kS(k, (9 Q / X(k, ( Q / D(k ( Conequently, ( we may obtain the maximum-likelihood etimate for r, S,q a (ˆr,ˆ S,ˆq argmax (r, S,q ( L r, S,q ( Similar to the derivation in Section III-A, we can obtain the etimate of the th element in q a ˆq / g (k g ( g (k denote the th element of the reidual vector g(k ( ] T g g (,,g C / (4 Subtituting Eq (4, ( into Eq (8, we can convert the log-likelihood function to a new verion in term of r S then get the ML etimator for r S given by ( ( ˆr,ˆ S argmax logg, (5 (r, S ˆS(k D(k X(k (6

4 4 Subtituting Eq (6 into Eq (5, we can obtain the maximum-likelihood etimate of r a ( (ˆr argmax logg (7 (r g i ined by Eq (4, g(k X(k D(k D(k X(k (8 IV WIDE-BAD SOURCE-LOCALIZATIO ALGORITHM FOR DISTICT OISE VARIACES A Individual Likelihood Formulation for Source Localization The algorithm i a well-known iterative algorithm for the maximum-likelihood etimation The comlicated nonlinear otimization roblem in Eq ( Eq (7 can be imlified uing the rocedure incororated with the augmented (comlete data correonding to the individual incident ource ignal Firt, we denote the received ignal ectrum a X (k,, m M, k from the m th ource { to the th enor Then we ine the } augmented data a X (k; m M, k ] T X (k X (k,,x (k C In addition, the relationhi between the oberved (incomlete data X(k the unoberved latent (comlete data i etablihed a M X(k X (k (9 m According to Eq (, (5, (7 (9, for a ingle ource ignal (the m th ource, we have X (k d (ks (k+u (k, for k,,,/, ( U (k C i the comlex-valued zero-mean uncorrelated Gauian noie in the ole reence of the m th ource According to Eq (, (7, (, we have (ˆr,ˆS (,ˆq arg max L r, (r,s,q S,q, m M, ( S S ( S (/ ] T ] C / q T q,,q C i the vector coniting of the diagonal element in Q { ( } H E U (k U (k C, k Let d (k ( Q / d (k, ( X ( Q / (k X (k ( According to Eq (4, we denote the th element of the articular reidual vectorg (k ag (k when only ource m i reent, g (k X (k d (k d (k X (k (4 Similar to the derivation in Section III-B, Eq ( yield ˆq g / ] g (k g, (5 g (,,g (/ ] T C / (6 Conequently, the maximum-likelihood etimate ˆr,i given by ( ( ( g ˆr argmax log (7 (r According to Eq (7, the ource localization roblem can be formulated a the indeendent maximization ub-roblem with reect to the individual likelihood function B ew Exectation-Maximization Algorithm for Source Localization In contrat to other exiting algorithm for the ource localization uing the enor ignal in the reence of noie with identical variance ], ], 7], 8], we reent a new algorithm here to olve the realitic ource localization roblem for enor ignal in the reence of noie with different variance, which ha been tackled by 7] recently everthele, our rooed algorithm can be demontrated to be more robut than the method rooed by 7] The detail of our rooed algorithm are introduced a follow (ince our rooed algorithm can be decouled acro different ource in each iteration, we only need to addre the te for the ource m it can be run for other ource a well in arallel: Initialization: Romly initialize ˆr ] ] Set the initial value for the entrie in ˆq ] ] ˆq] ] a ˆq ] ] M ]T R (8 ˆq] ] ] T R, (9 reectively Inut (Given arameter at Iteration i: ˆq ] i ], ˆr ] i ] Outut Variable at Iteration i: ˆq ] i], ˆr ] i] Given the inut arameter, the algorithm for the i th iteration i tated below Exectation Ste (E-Ste: Calculate { ˆ Q diag ˆq ] i ]}, (4

5 5 diag{ } convert the vector inide the aociated brace into a diagonal matrix containing the vector entrie a the diagonal element in the ame order Comute Calculate t α Q M m ˆ Q (4 ] trace(ˆ Q ] (4 trace(ˆ Q ˆr ] i ] r F v (4 According to Eq (4, (6, (5, (4 a, baed on 7], determine d (k D(k ext, follow Eq (, (, (6 to determine ˆS(k ˆS (k, k,,,/, ˆS (k i the m th element of ˆS(k Then determine { } ˆX (k E ˆX (k X(k d (kˆs (k+α(x(k D(kˆS(k, t k,,,/ (44 Maximization Ste (M-Ste: ow let r r F v, (45 r i the variable coordinate it ha to be etimated in thi te Then, follow Eq (45, (6, (5 to facilitate d (k, k,,,/, which involve the variable coordinate r Then according to d (k, contruct the following arameter d (k (ˆ Q ( / d (k, k,,,/, (46 which alo involve the variable coordinate r According to the reult from Eq (44, calculate ˆ X (ˆ Q ( / (k ˆX (k, k,,,/ Then, contruct ˆg (k ˆX (k d (k d (k ˆ X (k, (47 k,,,/, (48 which involve the variable coordinate r a well Denote the th element of ˆg (k a ˆg (k Facilitate ˆg ˆg (,,ˆg (/ ] T, (49 which involve the variable coordinate r R Carry out ˆr ] i] argmin log( ˆg (5 r Beide, calculate t ˆr F ] i] r v Let a, Enumerate the arameter given by Eq (44, (6, (5, (, (47, (48, (49 in thi equential order Then calculate Thu, obtain ˆq ] i] ˆq ] i] ˆg,,,, (5 ˆq ] i],,ˆq ] i]] T R (5 The above algorithm facilitate a recurive olution to multile wide-b ource localization V ROBUSTESS AALYSIS FOR SOURCE LOCALIZATIO ALGORITHMS CRLB i often ued to characterize the robutne of the etimation method In thi ection, by extending the CRLB reented in 7] for the imle DOA etimation roblem, we derive the CRLB for the ource localization roblem to benchmark our method the SC-ML/AC-ML cheme a CRLB G(k G(k D(k R (k R / ] { G(k H G(k D(k R (k T}}, (5 Q / G(k, (54 ] r ( d( (k,, r (M d(m (k, (55 I D(k D(k, (56 S(kS(k H (57 ote that Q, d (k, D(k, S(k are given by Eq (8, (5, (, (4, (7 We can rewrite Eq (55 a Ϝ a ( G(k D(k r T jf k π v Ϝ (58 e jπkt ( λ ( a ( e jπkt ( ( λ a ( e jπkt ( ( λ a ( e jπkt ( a (M a (M λ ( (M e jπkt (M λ e jπkt (M (M λ (59

6 6 λ d χ d y d χ ], d y χ r T T r r r R, χ ( (, χ χ + y y y ( ( (6 χ χ + y y y ote that r χ,y ] T r χ,y ] T VI SIMULATIO The comarion i made among our newly rooed baed multile wide-b ource localization cheme, the SC- ML method the AC-ML method here The amling frequency i khz The roagation eed i 45 meter/ec The data i imulated for a circularly-haed array of five enor uing the recorded acoutic data acquired from ] a hown in Figure (quare denote the enor location circle denote the actual ource location The amle ize i L the DFT ize i 56 Throughout the imulation, the minimization in our method characterized by Eq (5 i erformed by elder-mead direct earch ], while the otimization te in both SC-ML AC-ML method are erformed uing the AM algorithm, which would lead to better erformance than elder-mead direct earch in thee two cheme ], 7] Moreover, the additive noie in all exeriment are romly generated by a Gauian roce uing the comuter the ignal-to-noie ratio (SR i ined according to 7], 8] A A Localization Layout Examle Then we invetigate the erformance of the algorithm for etimating the two ource location in the reence of enor noie with non-uniform variance, comare with the SC-ML AC-ML algorithm The noie rocee acro different enor have the covariance matrix a Q σ diag{,,,5,9} A hundred Monte Carlo exeriment are carried out uing our method with romly initialized ource location for a articular ignal-to-noie ratio (SR db The localization reult from a certain exeriment i deicted in Figure the ultimate location are achieved after three iteration of algorithm We ault the number of iteration a in all Monte Carlo exeriment B Root-Mean-Square Error Comutational Comlexitie for Source Localization For each SR value ranging from to 4 db, we fix the initial ource location etimate a deicted in Figure carry out a hundred Monte Carlo exeriment to obtain the average localization accuracy in term of the root-mean-quare (RMS error in meter The three correonding RMS error curve to the three aforementioned cheme are deicted in Figure Then, we vary the initial location etimate around the circular area with a one-meter diameter with reect to the two initial ource-location etimate deicted in Figure redo a hundred Monte Carlo exeriment imilar to the et-u generating Figure The reult are deicted in Figure It i obviou that the accuracie of all three method degrade from Figure to Figure ince the initial condition change To further tudy thi effect, we read the initial location etimate over a broader area a deicted in Figure 4 redo a hundred Monte Carlo exeriment imilar to Figure The average RMS error curve are demontrated in Figure 5 ext, we would like to invetigate the erformance of the three aforementioned localization method for the enor noie with identical variance (SW Thu, we chooe the enor noie covariance matrix a Q σ diag{,,,,} now With thi new noie covariance matrix, we redo the Monte Carlo exeriment imilar to thoe generating Figure,, 5 The correonding reult are lotted in Figure 6, 7, 8, reectively According to thee two et of exeriment, our rooed algorithm greatly outerform both SC- ML AC-ML method in all condition In addition, the accuracie of all three method degrade due to the change in the initial condition for the SW cenario a well Beide, the erformance of all thee three cheme for the SW cae are not much different from thoe for the SW cae, ince the SW model i a articular cae of the SW model C Robutne Analyi of Source Localization We fix the initial ource location etimate a thoe generating Figure carry out a hundred Monte Carlo exeriment again The correonding CRLB for our method, the SC-ML (or AC-ML method are deicted in Figure 9 We alo deict the average RMS error curve in the ame figure According to Figure 9, we dicover that the RMS error reulted from our algorithm are much cloer to the CRLB than the SC-ML AC-ML method ote that all the three ource localization cheme in comarion are quite enitive to the initial condition Thi till remain a a very challenging roblem for the wide-b ource localization ote that our exerimental reult illutrated in thi aer can be generalized for other condition It mean that if we change the ource location ue all the three algorithm ubject to the ame initial condition, the exerimental reult under every different condition ecified in Section VI-A-VI-C will be very imilar to Figure -9 VII COCLUSIO In thi aer, we rooe a novel -baed multile wideb ource localization cheme in the reence of nonuniform noie variance For our method the conventional SC-ML AC-ML method, the erformance i rather enitive to the initial ource location etimate Our rooed algorithm can lead to an outting localization erformance given a reaonably good initial condition Moreover, our rooed algorithm can alway outerform the conventional SC-ML AC-ML method when the initial ource location etimate are romly choen The Monte Carlo

7 7 imulation reult demontrate the ueriority of our rooed method To rovide the robutne analyi for the ource localization algorithm, we reent the CRLB aociated with thee three cheme The CRLB analyi demontrate that our rooed algorithm i much cloer to the achievable minimum variance than the two other method in all ignal-tonoie ratio condition In addition, according to our comlexity analyi, the comlexity meaure for our rooed algorithm i ofo(m which i much le than thoe for the SC-ML AC-ML method (both with a comlexity meaure ofo(m y axi (meter Initial location etimate Actual ource location Ultimate location etimate Senor location REFERECES ] K Yan, H-C Wu, S S Iyengar, Robutne analyi of ource localization uing Gauianity meaure, in roceeding of IEEE Global Telecommunication Conference, ovember 8, 5 ] H Krim M Viberg, Two decade of array ignal roceing reearch: the arametric aroach, IEEE Signal roceing Magazine, vol, no 4, 67 94, July 996 ] J C Chen, R E Hudon, K Yao, Maximum-likelihood ource localization unknown enor location etimation for wideb ignal in the near-field, IEEE Tranaction on Signal roceing, vol 5, no 8, , Augut 4], Source localization of a wideb ource uing a romly ditributed beamforming enor array, in roceeding of International Society of Information Fuion,, TuC: 8 5] Stoica A ehorai, Muic, maximum likelihood Cramer-Rao bound, IEEE Tranaction on Acoutic, Seech Signal roceing, vol 7, no 5, 7 74, May 989 6], erformance tudy of conditional unconditional direction-ofarrival etimation, IEEE Tranaction on Acoutic, Seech, Signal roceing, vol 8, no, , 99 7] C E Chen, F Lorenzelli, R E Hudon, K Yao, Maximum likelihood DOA etimation of multile wideb ource in the reence of nonuniform enor noie, EURASI Journal on Advance in Signal roceing, vol 8, 8 8] M eavento A B Gerhman, Maximum-likelihood direction-ofarrival etimation in the reence of unknown nonuniform noie, IEEE Tranaction on Signal roceing, vol 49, no 7, 4, 9] A Demter, M Laird, D B Rubin, Maximum likelihood from incomlete data via the algorithm, Journal of the Royal Statitical Society, vol 9, no, 8, 977 ] M Feder E Weintein, Multiath multile ource array roceing via the algorithm, roceeding of IEEE International Conference on Acoutic, Seech, Signal roceing, vol, 5 56, 986 ], arameter etimation of uerimoed ignal uing the algorithm, IEEE Tranaction on Acoutic, Seech, Signal roceing, vol 6, no 4, , 988 ] E Weintein, V Oenheim, M Feder, Iterative equential algorithm for multienor ignal enhancement, IEEE Tranaction on Signal roceing, vol 4, no 4, , 994 ] J Chung J F Bohme, Recurive SAGE algorithm, roceeding of IEEE Workho on Statitical Signal roceing, 54 54, 4] J Chung, J F Bohme, A O Hero, Tracking of multile moving ource uing recurive algorithm, EURASI Journal on Alied Signal roceing, vol, no, 5 6, 5 5] L Frenkel M Feder, Recurive exectation-maximization (em algorithm for time-varying arameter with alication to multile target tracking, IEEE Tranaction on Signal roceing, vol 47, no, 6, 999 6] M I Miller D R Fuhrmann, Maximum-likelihood narrow-b direction finding the algorithm, IEEE Tranaction on Acoutic, Seech, Signal roceing, vol 8, no 9, , 99 7] K K Mada H-C Wu, algorithm for multile wideb ource, in roceeding of IEEE Global Telecommunication Conference, 5, 6 8] K K Mada, H-C Wu, S S Iyengar, Efficient robut algorithm for multile wideb ource localization, IEEE Tranaction on Vehicular Technology, vol 58, no 6, 7 75, July x axi (meter Fig The localization of two wide-b (acoutic ource in the near field corruted by the noie with non-uniform variance (ignal-to-noie ratio i db The initial location etimate the ultimate location etimate reulted from the algorithm ( iteration are taken are alo demontrated Average RMS Error (meter SR (db Fig Average RMS localization error veru SR for the ource corruted by the noie with non-uniform variance The initial location etimate are lotted in Figure Average RMS Error (meter SR (db Fig Average RMS localization error veru SR for the ource corruted by the noie with non-uniform variance The initial ource location etimate here are romly choen within the area which are one meter around the initial location etimate ued in Figure

8 8 y axi (meter Initial location etimate for Source Initial location etimate for Source x axi (meter Average RMS Error (meter SR (db Fig 4 The eighteen different initial ource location etimate Fig 7 Average RMS localization error veru SR for the ource corruted by the noie with identical variance The initial ource location etimate are romly drawn from the area which are one meter around the initial ource location etimate in Figure Average RMS Error (meter SR (db Fig 5 Average RMS localization error veru SR for the ource corruted by the noie with non-uniform variance The initial ource location etimate are lotted in Figure 4 Average RMS Error (meter SR (db Fig 8 Average RMS localization error veru SR for the ource corruted by the noie with identical variance The initial ource location etimate are lotted in Figure 4 Average RMS Error (meter SR (db Fig 6 Average RMS localization error veru SR for the ource corruted by the noie with identical variance The initial ource location etimate are lotted in Figure Average RMS Error (meter CRLB 4 SR (db Fig 9 Cramer-Rao lower bound imulated RMS localization error (actual variance veru different SR value for the three cheme in comarion