Closed-form two-step weighted-leastsquares-based. localisation using invariance property of maximum likelihood estimator in multiplesample

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1 IET Commuiatios Researh Artile Closed-form two-step weighted-leastsquares-based time-of-arrival soure loalisatio usig ivariae property of maximum likelihood estimator i multiplesample eviromet Chee-Hyu Park, Joo-Hyuk Chag Shool of Eletroi Egieerig, Hayag Uiversity, Seoul , Korea jhag@hayag.a.kr ISSN Reeived o d Otober 015 Revised o d Jauary 016 Aepted o d February 016 doi: /iet-om Abstrat: I this study, the authors propose a losed-form time-of-arrival soure loalisatio method ad justify the employmet of the ivariae property of the maximum likelihood (ML) estimator i the soure loalisatio otext with multiple samples. The magitude of the bias of the proposed sample vetor futio (the statisti that osists of the multiple observatios set) usig the ivariae property of the ML estimator is smaller tha that based o the sample mea. Therefore, the mea squared error (MSE) of the weighted least squares estimate usig the proposed sample vetor futio is smaller tha that based o the sample mea whe the variaes of both sample vetor futios are the same. Furthermore, the authors ivestigate a situatio i whih sesors have erroeous positio iformatio. The simulatio results show that the averaged MSE performae of the proposed method is superior to that of the existig methods irrespetive of the umber of samples. 1 Itrodutio Soure loalisatio is a tehique that fids a geometrial poit of itersetio usig measuremets from eah reeiver suh as the time differee of arrival, the time of arrival (TOA), the reeived sigal stregth or the agle of arrival. Loalisig poit soures usig passive ad statioary sesors is of osiderable iterest ad has bee a repeated theme of researh i teleommuiatio areas, video ofereig ad global positioig systems (GPSs). Several loalisatio methods have existed. Oe of them uses the Gauss Newto algorithm based o the Taylor-series expasio [1]. The bias i the loatio estimate from the Gaussia measuremet oise ad sesor positio errors has bee aalysed []. The total least squares method was adopted to elimiate the bias i the ovetioal loatio estimator [3]. The problem of optimum sesor plaemet for soure loalisatio usig TOA measuremets was ivestigated [4]. A target loalisatio method for a multi-stati passive radar system was proposed usig TOA measuremets through ovex optimisatio ad adoptig semi-defiite relaxatio ad bisetio methods [5]. Loalisatio methods usig robust statistis have bee proposed for o-lie-of-sight eviromets [6, 7]. Also, a losed-form two-step weighted least squares (WLS) positioig algorithm that attais the the Cramer-Rao lower boud (CRLB) uder suffiietly small oise oditios was already developed [8 10]. However, a weakess of the two-step WLS method is that its performae is severely degraded whe the variae of oise is large. Reetly, a losed-form shrikage-based loalisatio method usig the miimum mea squared error (MSE) riterio was ivestigated [11]. This algorithm improved the loalisatio performae of the two-step WLS method i large oise variae oditios by modifyig the blid shrikage estimator [1] suh that it a be effiietly utilised i the TOA-based soure loalisatio otext. This shrikage method is appropriate for moderate ad large samples situatios, but its averaged MSE performae beomes iferior to that of the two-step WLS method with a small umber of samples beause the shrikage fator loses auray below a ertai sample size. To overome this disadvatage of the shrikage estimatio, we propose a loalisatio algorithm that adopts the maximum likelihood (ML) estimator based o the ivariae property. The motivatio for this paper is give i the followig. This paper justifies the utilisatio of the ML estimator based o the ivariae property i the multiple-sample-based soure loalisatio. To the best of our kowledge, ay proof for the optimality of the ML estimator i the multiple-sample-based soure loalisatio otext has ot bee performed. As the MSE of a sample vetor futio (statisti that osists of the futio of the measuremets) beomes larger, the MSE of a WLS estimate usig the orrespodig sample vetor futio also beomes larger. Therefore, determiig a sample vetor futio with the smallest possible MSE is importat. We use the ML estimator to determie a more optimal sample vetor futio beause the ML estimator is a asymptotially effiiet estimator [13, 14]. Whe the variaes of two differet sample vetor futios are the same, as the magitude of the bias is larger, the MSE of the WLS estimate usig that sample vetor futio is also larger. The proposed WLS estimate is show to have less bias tha that attaied usig the existig WLS estimate. As a result, the MSE of the WLS estimate based o the proposed sample vetor futio is smaller tha that attaied from the existig sample vetor futio whe the variaes of the sample vetor futios are the same. Also, the estimated value outside the parameter spae is restrited. This ostraied estimate a be iterpreted as the ML estimate. Simulatio results show that the proposed method is superior to the ovetioal two-step WLS estimate based o the sample mea eve with a small umber of samples, whereas the shrikage estimator is iferior to the two-step WLS estimator i that oditio. Furthermore, the proposed method also outperforms the shrikage estimator whe the umber of measuremets is moderate or large. The orgaisatio of this paper is as follows. Setio explais the soure loalisatio problem to be solved i this paper. The proposed positioig method based o the two-step WLS algorithm usig the ivariae property of the ML estimator is addressed i Setio 3. The estimatio performaes 106 & The Istitutio of Egieerig ad Tehology 016

2 of the proposed method are evaluated ad ompared with the existig algorithms via simulatio results i Setio 4. Setio 5 presets our olusios ad suggestios for future works. Problem formulatio The TOA soure loalisatio method fids the positio of a soure by usig multiple irles whose etres represet the positios of sesors. I the otext of the lie-of-sight soure loalisatio, the measuremet equatio is represeted as follows: r i,j = d i + i,j = (x x o i ) + (y y o i ) + i,j, (1) i whih i, j follows N(0, s i ). Also, i =1,,, M, j =1,,, P with M ad P deotig the total umber of sesors ad the umber of samples i the ith sesor, r i, j is the measured distae betwee the soure ad the ith sesor at the jth samplig ad d i is the true distae betwee the soure ad the ith sesor. Also, [x y] T is the true soure positio ad [x o i y o i ]T is the aurate positio of the ith sesor without loatio error. Note that, throughout this paper, a lowerase boldfae letter deotes a vetor, a upperase boldfae letter idiates a matrix ad the supersript T sigifies the vetor/matrix traspose. Squarig (1) ad rearragig yields the followig equatio x o i x + y o i y 0.5R + m i,j = 0.5(x o i + y o i ri,j), i = 1,,..., M, j = 1,,..., P () where R = x + y, m i,j = d i i,j (1/) i,j, i = 1,..., M, j = 1,..., P. For oveiee, () a be simply represeted i matrix form as Note that Ax + q j = b j. (3) x o 1 y o A =..., x o M yo M 0.5 x = [x yr] T, q j = [m 1,j,..., m M,j ] T, x o 1 + y o 1 r b j = 1 1,j.. (4) x o M + y o M rm,j The aim of this paper is to fid the soure positio suh that the MSE of the positio estimate is miimised. 3 Proposed losed-form two-step WLS-based TOA soure loalisatio method usig the ML estimator based o the ivariae property with multiple samples I this setio, we adopt the ML estimator based o the ivariae property to derive the estimate of the loatio i the multiple-sample eviromet, where the first ase is for the preisely kow sesor positios ad the seod ase is for the self-loalisatio seario. 3.1 Case 1: preisely kow sesor positios The existig TOA loalisatio algorithm for multiple samples is based o the followig likelihood futio [11] 1 p(b 1:P ; x) = (p) MP/ C q P/ { } (5) exp 1 P (b j Ax) T C 1 q (b j Ax) where C q deotes the ovariae matrix of q ad is represeted i the followig by egletig the seod-order oise term uder suffiietly small oise oditios { {C q } i,k = d i s i, if i = k; 0, if i = k, { (6) r i s i, if i = k; 0, if i = k, {C q } i,k represets the (i, k)th ompoet of C q ad d i is the true value of r i. Also, we adopt the sample vetor futio to extrat the loatio iformatio i the multiple-sample eviromet ad it is defied as the statisti whih osists of the ombiatio of the measuremet vetors. I this paper, two sample vetor futios are dealt with, that is, oe is (1/P) P b j usig the sample mea ad the other is b iv based o the ivariae property of the ML estimator. The first-step WLS estimator from (5) is determied as (A T C 1 q A) 1 A T C 1 q {(1/P) P b j}. The expeted value of the ith ompoet of the sample vetor futio based o the sample mea ({(1/P) P b j } i ) is {bo j } i (1/)s i,j ad b o j is the true value of b j. Here, {m} i is the ith ompoet of the vetor m. Whe the statioary state is assumed withi the estimatio iterval, the time idex j a be omitted ad the expeted value of the ith ompoet of the sample vetor futio based o the sample mea a be rewritte as {b o j } i (1/)s i = {Ax} i (1/)s i. The, the magitude of the bias for {(1/P) P b j } i is (1/)s i. We use the ML estimate based o the ivariae property to redue the bias of the sample vetor futio beause the ML estimator is a asymptotially ubiased estimator [14]. The ivariae property meas that if û is the ML estimator of θ ad g(θ) is ay trasformatio of θ, the the ML estimator for g(θ) is by defiitio g(û). I a TOA loalisatio otext, the ith elemet (1 i M) of the sample vetor futio that uses the ivariae property of the ML estimate, i.e. {b iv } i, is determied as (x o i + y o i ( r i ML ) )/ ad r i ML is P r i,j/p. The approximate Gaussiaity of the oise ompoet of b iv,defied as q iv, a be proved as follows. The ith ompoet of b iv is represeted as ( ) {b iv } i = 1 xo i + y o i 1 P r P i,j ( ) = 1 xo i + y o i d i + 1 P P i,j ( ) = 1 xo i + y o i di d P i P i,j 1 P P i,j { } 1 xo i + y o i di d P i P i,j. (7) I the derivatio of the last term of (7), the seod-order oise term was igored by assumig a suffiietly small oise oditio. The, & The Istitutio of Egieerig ad Tehology

3 the oise ompoet of {b iv } i is foud as {q iv } i d P i P i,j. (8) As a be see from (8), {q iv } i follows the Gaussia distributio uder a suffiietly small oise oditio, thus q iv follows the multivariate ormal distributio. Cosequetly, the improved likelihood futio is attaied as follows p(b iv 1 ; x) = (p) M/ exp 1/ C q iv { 1 } (9) (biv Ax) T C 1 q iv(biv Ax) where q iv = b iv Ax. Also, the oise ovariae matrix of q iv is determied i the followig {C q iv} i,k = E[{q iv } i {q iv } k ] 0, if i = k; = di s i, if i = k. P 0, if i = k; ( r i ML ) s i, if i = k. P (10) Differetiatig the logarithm of (9) with respet to x ad settig it to zero yields x iv = (A T C 1 q iva) 1 A T C 1 q ivbiv. (11) As E{b iv } i = {b o } i 1 P s i, the magitude of the bias for the sample vetor futio {b iv } i is (1/P)s i ad it is smaller tha that of {(1/P) P b j } i by P times. Usig the above property, the MSE of the first-step WLS estimate that uses the sample vetor futio based o the ivariae property is obtaied as follows (see Appedix for more details) MSE( x iv ) = 1 P tr{(at C 1 q A) 1 } + 1 tr[qs] (1) 4P where Q = C 1 q A(AT C 1 q A) A T C 1 q ad [S] i,k = s i s k. Meawhile, the MSE of the first-step WLS estimate that exploits the sample mea is represeted (after some algebra) as follows MSE( x e ) = 1 P tr{(at C 1 q A) 1 } + 1 tr[qs]. (13) 4 Sie MSE( x iv ) (1/P)tr{(A T C 1 q A) 1 } 0, tr[qs] is o-egative. Thus, the MSE of the first-step WLS estimate that uses the sample vetor futio based o the ivariae property is smaller tha that obtaied from the sample mea. Note that all pairs of estimators are futios of the omplete suffiiet statisti with regard to their respetive likelihood futio. The ompleteess of the suffiiet statisti of (5) ad (9) a be show as follows. The suffiiet statisti of (9) is A T C 1 q ivbiv. The probability desity futio of this suffiiet statisti is give by p(a T C 1 1 q ivbiv ; x) = (p) 3/ exp 1/ R [ 1 ] {AT C 1 q iv(biv Ax)} T R 1 {A T C 1 q iv(biv Ax)} where R = (1/P)(A T C 1 q A). By lettig AT C 1 q ivbiv = t, (14) the suffiiet statisti t is omplete whe E[f (t)] = 0 implies f (t) = 0, where f :R 3 R 1. I this regard, E[f (t)] = 0 a be rewritte as V f (t)p(t; x)dt = 0 (15) where Ω is the support for the suffiiet statisti t. Note that the expetatio of f (t) aot be zero uless f (t) = 0 everywhere from (15) so that t is the omplete suffiiet statisti. Furthermore, (11) beomes loser to the miimum variae ubiased estimator tha the WLS estimator derived from (5) beause the bias of the WLS estimator usig the ivariae property is smaller tha that of the outerpart equatio [13, 14]. The, the followig seod-step estimator is obtaied usig the relatioship betwee the first-step estimates where ẑ iv = (G T C 1 h ivg) 1 G T C 1 h ivhiv (16) h iv = [{ x iv } 1 { xiv } { xiv } 3 ] T, C h iv = 1 P diag(x, y, 1)(AT C 1 q A) 1 diag(x, y, 1), with x, y substituted as { x iv } 1,{ x iv } i the alulatio of C h iv. Also, it a be show that the MSE of the seod-step WLS estimate of the proposed method is smaller tha that based o the sample mea i the same maer. The fial loatio estimate is [ ] T x iv = sg({ x iv } 1 ) {ẑ iv } 1 sg({ x iv } ) {ẑ iv }. (17) 3. Case : self-loalisatio seario; sesors with erroeous oordiates exist with sesors that have preise positio iformatio I pratial appliatios, the sesor positios used for the positioig task might have errors [10]. For example, i the absolute self-loalisatio problem, the sesors are ategorised ito ahor odes ad pseudo-ahors. The positios of the ahor odes are kow with omparatively higher auray tha the other odes from the GPS ad the oordiates of the pseudo-ahors are estimated with the aid of the ahor odes. There are errors i the oordiates of the pseudo-ahors beause they are estimated values. Therefore, the effets of the erroeous positios of the pseudo-ahor odes should be take ito aout to loalise the eighbourig ukow sesors more preisely. I this setio, we develop a loalisatio algorithm that alleviates these adverse effets aused by sesor positio errors i a multiple-sample seario. We assume that the first L sesors are the ahor odes ad the remaiig M L sesors are pseudo-ahors. The measuremet equatios for the first L ahor odes are the same with (1) (4). The measuremet equatio for the sesors with erroeous positio iformatio is represeted by assumig a suffiietly high sigal-to-oise ratio oditio as follows 1 (x i + y i ri,j ) (x i x + y i y 0.5R) = (x o i x)w xi + (y o i y)w yi d i i,j, i = L + 1, L +,..., M, j = 1,,..., P. (18) where w xi ad w yi are the sesor positio errors with N(0, s s,i) ad s s,i is the variae of the ith sesor positio error [15 19]. 108 & The Istitutio of Egieerig ad Tehology 016

4 where b,j = [b v j b T p,j ]T, A = [A T A T p ]T ad q,j = [q T j q T p,j ]T. b j, A ad q j are the same with those defied i Setio. The, the sample vetor futio (b iv ) foud usig the ML estimator based o the ivariae property is represeted as x o i + y o i ( r i ML ), i = 1,..., L, } i = x i + y i ( r i ML ), i = L + 1,..., M. {b iv () As a be see from (4) ad (0), the error vetor b iv A x, represeted as q iv, is the multivariate Gaussia distributio uder a suffiietly small oise oditios. The likelihood futio of the ombied model is represeted as Fig. 1 Deploymet of sesors Represetig (18) i a matrix form yields p(b iv 1 ; x) = (p) M/ exp 1/ C q iv { 1 } (biv A x) T C 1 (b iv A x) q iv (3) Note that A p x + q p,j = b p,j. (19) x L+1 y L A p =..., x M y M 0.5 q p,j = [v L+1,j,..., v M,j ] T, x b p,j = 1 L+1 + y L+1 rl+1,j., (0) x M + y M r M,j where i = L +1,, M, j =1,, P ad v i,j = (x o i x)w xi + (y o i y)w yi d i i,j. Combiig the measuremet equatios from (4) ad (0) leads to the followig: b,j = A x + q,j (1) where q iv = b iv A x. The first-step WLS estimator is the obtaied from (3) x iv = (A T C 1 A ) 1 A T C 1 q iv q iv b iv. (4) The ovariae matrix C q iv a be derived i the same maer with (8) (10) as follows: {C q iv} i,k = 0, if i = k; ( r i ML ) s i, if i = k ad i L; P ( r ML i ) s i + ( r ML i ) s s,i, if i = k ad i. L. P (5) The seod-step WLS solutio is obtaied usig the relatioship betwee the first-step WLS estimates ẑ iv = (G T C 1 G) 1 G T C 1 h iv h iv h iv (6) Fig. Compariso of averaged MSE of the proposed method with that of the existig methods (sample umber = 3) & The Istitutio of Egieerig ad Tehology

5 Fig. 3 Compariso of averaged MSE of the proposed method with that of the existig methods (sample umber = 10) where C h iv = diag(x, y, 1)(A T C 1 A ) 1 diag(x, y, 1), ad x, y are substituted as { x iv The fial loatio estimate is x iv,f = q iv } 1,{ x iv } i the alulatio of C h iv. [ ] T sg({ x iv } 1 ) {ẑ iv } 1 sg({ x iv } ) {ẑ iv }. (7) 4 Simulatio results I this setio, we ompare the performae of the proposed soure loalisatio algorithm with that of the ovetioal method. The first sesor at the oordiates (0, 0) was set as the referee sesor ad the soure was assumed to be withi a 0 m square ell to determie the performae over the etire regio. The umber of sesors was six. Next, 30 differet soure loatios were geerated with a uiform Fig. 4 Compariso of the averaged orm of the bias of the proposed method with that of the existig methods (sample umber = 10) distributio, sesors were fixed, as show i Fig. 1 ad 00 Mote-Carlo simulatios were ru for eah give variae of oise. The variae of the measuremet oise of all of the sesors was assumed to be idetial. Also the sigle- ad omi-diretioal soure was assumed to be i the statioary state. The averaged MSE was alulated as follows Averaged MSE i=1 k=1 = [( xk (i) x(i)) + (ŷ k (i) y(i)) ] (8) where x k (i), ŷ k (i) is the estimated positio of the soure i the ith positio set ad kth iteratio, x(i) ad y(i) idiate the ith true positio of the soure. The CRLBs whih are used i this simulatio as the behmark are give i [10, 0]. Fig. 1 ideed illustrates a arragemet of sesors, i whih the radius of the sesor etwork was set to 10 m. As desribed i the itrodutio, ay estimated positio outside the parameter spae is ostraied to the maximum or miimum value of the parameter spae. The auray ompariso of the loalisatio is show i Fig. betwee the two-step WLS method based o the ivariae property of the ML estimator ad the existig methods as a futio of the variae of the oise whe the umber of samples is three. The dbm i Fig. deotes the square meter (m ) represeted with the deibel (db) uits. The proposed algorithm is learly superior to the existig estimators with large variaes of oise, whereas the averaged MSE performae of the existig shrikage estimator is iferior to that of the two-step WLS estimator beause the shrikage fator aot be estimated with suffiiet preisio with a small umber of samples. I additio, Fig. 3 shows the averaged MSE ompariso of the proposed ad existig methods whe the umber of measuremets is te. Agai, the averaged MSE performae of the proposed algorithm is lower tha that of the existig methods, but the shrikage method outperforms the two-step WLS method beause it has a suffiietly large umber of samples. That is, the averaged MSE of the proposed two-step WLS algorithm based o the ivariae property is smaller tha that of two-step WLS estimator regardless of the umber of samples, whereas the shrikage estimator is iferior to the two-step WLS method whe the umber of measuremets is small. Furthermore, the averaged MSE of the proposed method is superior to that of the shrikage estimator eve whe the umber of samples is suffiiet, as a be see i 110 & The Istitutio of Egieerig ad Tehology 016

6 Fig. 5 Compariso of averaged MSE of the proposed method with that of the existig methods (sample umber = 3, umber of ahor odes = 3) Fig. 3. Also, Fig. 4 shows that the orm of bias of the proposed method was muh smaller tha that of the other methods whe the sample umber was te. The orm of bias is obtaied i the followig Norm of Bias ( 1 ) ( ) 00 = x k x ŷ k y. 00 k=1 k=1 (9) irrespetive of the umber of measuremets, whereas that of the shrikage estimator is worse tha the two-step WLS estimator whe the umber of samples is small. Also, the averaged MSE of the proposed method was still smaller tha that of the shrikage estimator whe the umber of measuremets is te. Figs. 7 ad 8 show the averaged MSE whe the variae of oise is 10 dbm as a futio of the umber of sesors ad the radius of the sesor etwork, respetively. The averaged MSE beomes smaller as the umber of sesors ad the radius of the sesor etwork are larger. The bias dereases as the umber of samples ireases beause the bias of the proposed method is iversely proportioal to the umber of measuremets. The averaged MSE show i Figs. 5 ad 6 aouts for three sesors with erroeous positio iformatio. The variae of the sesor loatio oise was set to 0.01 m. Agai, the averaged MSE performae of the proposed method is superior to that of the two-step WLS method 5 Colusios We proposed a losed-form TOA soure loalisatio method ad justified the adoptio of the ivariae property of the ML estimator i the multiple-sample-based soure loalisatio otext. We have show that the MSE of the WLS estimator that uses a sample vetor futio based o the ivariae property of the ML estimator was smaller tha that usig the sample mea. The Fig. 6 Compariso of averaged MSE of the proposed method with that of the existig methods (sample umber = 10, umber of ahor odes = 3) & The Istitutio of Egieerig ad Tehology

7 Fig. 7 Compariso of averaged MSE of the proposed method with that of the existig methods as a futio of the umber of sesors (variae of oise = 10 dbm ) Fig. 8 Compariso of averaged MSE of the proposed method with that of the existig methods as a futio of the radius of the sesor etwork (variae of oise = 10 dbm ) averaged MSE of the proposed estimator was superior to that of the two-step WLS estimator i the large oise variaes regimes regardless of the umber of samples, ulike the shrikage estimator. Also, the proposed estimator outperformed the shrikage estimator i large oise variaes oditio whe the umber of measuremets was moderate. I future works, we will ivestigate a loalisatio method i the distae-depedet oise model. 6 Akowledgmet This work was supported by the Natioal Researh Foudatio of Korea (NRF) grat fuded by the Korea govermet (MSIP) (No. 014R1AA1A ). 7 Referees 1 Torrieri, D.J.: Statistial theory of passive loatio systems, IEEE Tras. Aerosp. Eletro. Syst., 1983, 0, (), pp Rui, L., Ho, K.C.: Bias aalysis of maximum likelihood target loatio estimator, IEEE Tras. Aerosp. Eletro. Syst., 013, 50, (4), pp Barto, R.J., Rao, D.: Performae apabilities of log-rage UWB-IR TDOA loalizatio systems, EURASIP J. Adv. Sigal Proess., 008, Artile ID: Su, M., Ho, K.C.: Optimum sesor plaemet for fully ad partially otrollable sesor etworks: a uified approah, Sigal Proess., 014, 10, pp Chalise, B.K., Zhag, Y., Ami, M.G., et al.: Target loalizatio i a multi-stati passive radar system through ovex optimizatio, Sigal Proess., 014, 10, pp Qiao, T., Liu, H.: Improved least media of squares loalizatio for o-lie-of-sight mitigatio, IEEE Commu. Lett., 014, 18, (8), pp & The Istitutio of Egieerig ad Tehology 016

8 7 Yi, F., Fritshe, C., Gustafsso, F., et al.: EM- ad JMAP-ML based joit estimatio algorithms for robust wireless geoloatio i mixed LOS/NLOS eviromets, IEEE Tras. Sigal Proess., 014, 6, (1), pp Cha, Y.T., Ho, K.C.: A simple ad effiiet estimator for hyperboli loatio, IEEE Tras. Sigal Proess., 1994, 4, (8), pp So, H.C., Li, L.: Liear least squares approah for aurate reeived sigal stregth based soure loalizatio, IEEE Tras. Sigal Proess., 011, 59, (8), pp Su, M., Yag, L., Ho, K.C.: Aurate sequetial self-loalizatio of sesor odes i losed-form, Sigal Proess., 01, 9, (1), pp Park, C.-H., Chag, J.-H.: Shrikage estimatio-based soure loalizatio with miimum mea squared error riterio ad miimum bias riterio, Digit. Sigal Proess., 014, 9, pp Be-Haim, Z., Eldar, Y.C.: Blid miimax estimatio, IEEE Tras. Sigal Proess., 007, 53, (9), pp Casella, G., Berger, R.L.: Statistial iferee. Cegage Learig, Kay, S.M.: Fudametals of statistial sigal proessig vol. I: estimatio theory (Pretie-Hall, 1993) 15 Ho, K.C., Lu, X., Kovavisaruh, L.: Soure loalizatio usig TDOA ad FDOA measuremets i the presee of reeiver loatio errors: aalysis ad solutio, IEEE Tras. Sigal Proess., 007, 55, (), pp Yag, L., Ho, K.C.: A approximately effiiet TDOA loalizatio algorithm i losed-form for loatig multiple disjoit soures with erroeous sesor positios, IEEE Tras. Sigal Proess., 009, 57, (1), pp Ma, Z., Ho, K.C.: A study o the effets of sesor positio error ad the plaemet of alibratio emitter for soure loalizatio, IEEE Tras. Wirel. Commu., 014, 13, (6), pp Wag, Y., Ho, K.C.: A asymptotially effiiet estimator i losed-form for 3D AOA loalizatio usig a sesor etwork, IEEE Tras. Wirel. Commu., 015, 14, (1), pp Rokah, Y., Shultheiss, P.: Array shape alibratio usig soures i ukow loatios. Part II: ear-field soures ad estimator implemetatio, IEEE Tras. Aoust. Speeh Sigal Proess., 1987, 35, (6), pp Cheug, K.W., So, H.C., Cha, Y.T.: Least squares algorithms for time-of-arrival-based mobile loatio, IEEE Tras. Sigal Proess., 004, 5, (4), pp Appedix 8 1 Derivatio of (1) The MSE of the first-step WLS estimate based o the ivariae property is derived as follows: MSE( x iv ) = tr{var[ x iv ]} + Bias( x iv ) T Bias( x iv ) where Q = C 1 q 1 P tr{(at C 1 q A) 1 } + (E[b iv ] b o ) T C 1 q A T C 1 q (E[b iv ] b o ) A(AT C 1 q A) = 1 P tr{(at C 1 q A) 1 } + tr[q{e[b iv ] b o } {E[b iv ] b o } T ] { = 1 [ P tr{(at C 1 q A) 1 } + tr Q 1 P s 1 1 ] T P s M [ 1 P s 1 1 ]} P s M = 1 P tr{(at C 1 q A) 1 } + 1 4P tr[qs] (30) A(AT C 1 q A) A T C 1 q ad [S] i,k = s i s k. & The Istitutio of Egieerig ad Tehology