Closed-Form Formulae for Time-Difference-of-Arrival Estimation

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1 614 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE 008 Closed-Form Formulae or ime-dierence-o-arrival Estimation Hing Cheung So, Senior Member, IEEE, Yiu ong Chan, Senior Member, IEEE, and Frankie Kit Wing Chan Abstract For a positioning system with sensors, a maximum o ( 1) distinct time-dierence-o-arrival (DOA) measurements, which are reerred to as the ull DOA set, can be obtained In this paper, closed-orm expressions regarding optimum conversion o the ull DOA set to the nonredundant DOA set, which corresponds to ( 1) DOA measurements with respect to a common reerence receiver, in the case o white signal source and noise, are derived he most interesting inding is that optimum conversion can be achieved via the standard least squares estimation procedure Furthermore, the Cramér-Rao lower bound or DOA-based positioning is produced in closed-orm, which will be useul or optimum sensor array design Index erms Optimum processing, source localization, time delay estimation I INRODUCION Estimation o the time dierences o arrival (DOAs) between noisy versions o the same signal received at spatially separated sensors has been an important research topic [1] [] Classical applications or time delay estimation include source localization and speed sensing in sonar and radar systems Recent requirements or DOA-based positioning are speaker tracking by microphone arrays [4], third-generation mobile communication systems [5], wireless local area networks [6], as well as sensor networks [7] For two-dimensional (-D) positioning, each DOA measurement, which is proportional to the dierences in source-sensor range, deines a hyperbolic locus on which the source must lie and the position estimate can be obtained rom the intersection o two or more hyperbolas For a positioning system with L sensors, there are L(L01) distinct DOA measurements rom all possible sensor pairs, and we call them the ull DOA set Alternatively, the so-called spherical DOA set [8], renamed the nonredundant set here, is a subset o the ull set and contains only (L 0 1) DOAs measured rom sensor pairs with a common reerence sensor, can be utilized or localization In act, most o the DOA-based positioning algorithms in the literature, such as [4] and [9] [11], are based on the nonredundant set, while only a ew [1], [1] have considered the ull set he motivation o this paper is to answer the undamental question o whether the ull or nonredundant DOA sets can give optimum positioning perormance he results show that optimality can be attained by both sets i the nonredundant DOA measurements are properly converted rom the ull set his conclusion conlicts with [8], which did not consider optimum processing o the nonredundant set he contributions here are as ollows 1) Optimum conversion o the ull DOA set to the nonredundant set, assuming that each sensor receives a white source signal in Manuscript received October 1, 006; revised October 1, 007 he associate editor coordinating the review o this manuscript and approving it or publication was Dr Zhi ian he work described in this paper was supported by a grant rom the Research Grants Council o the Hong Kong Special Administrative Region, China (roject No CityU ) H C So and F K W Chan are with Department o Electronic Engineering, City University o Hong Kong, Kowloon, Hong Kong ( hcso@eecityu eduhk; kwchan@studentcityueduhk) Y Chan is with Department o Electrical and Computer Engineering, Royal Military College o Canada Kingston, ON K7K 7B4, Canada ( chan-yt@rmcca) Digital Object Identiier /S white noise Although the results agree with [14], we are able to produce the corresponding closed-orm conversion expressions ) A proo that the Gauss Markov nonredundant set estimates can be obtained rom the ull set with the use o the standard least squares (LS) he asymptotic cases o high and low signal-to-noise ratio (SNR) conditions have also been investigated ) A veriication that or DOA-based positioning, the ull and optimized nonredundant sets give identical Cramér Rao lower bound (CRLB), which demonstrates their eiciency Hence, localization algorithms should always use the nonredundant set, since there is a less number o equations to solve In addition, the CRLB is in closed-orm which will be beneicial or the design o optimum -D sensor arrays he rest o the paper is organized as ollows In Section II, we consider DOA estimation or a white source signal in additive white noise Ater optimally estimating the ull DOA set, the Gauss Markov estimate o the nonredundant DOA set is derived in closed orm It is shown that Gauss Markov estimate is in act equal to the standard LS estimate he closed-orm expression o the CRLB or DOAbased positioning is given in Section III Section IV provides numerical examples to validate the theoretical indings Finally, conclusions are drawn in Section V II OIMUM CONVERSION OF FULL SE O NONREDUNDAN SE he discrete-time signal received at the ith sensor is z i (n) s(n0d i )+q i (n); i1; ; 111L; n 0; 1; 111N 01 (1) s(n) is the passive source signal, and q i (n) and D i are the additive noise and signal propagation delay, respectively, at the ith sensor Without loss o generality, the sampling requency is 1 Hz rior to sampling, the continuous-time signals are irst lowpass iltered with a cuto requency o 05 Hz hat is, the discrete-time signals are obtained at the Nyquist rate, and they are band-limited between 005 and 05 Hz Following standard practices, q i (n)g in (1) are zero-mean white Gaussian processes which are independent o s(n) he derivation below assumes that s(n) is also a zero-mean white Gaussian process his means that the continuous-time source signal has a lat spectrum over the lowpass ilter bandwidth hen ater sampling at the Nyquist rate, all samples in s(n) are independent and identically distributed Certainly not all sources have lat spectra, but many signals do possess approximately this property Examples are in the localization o an explosion underground, in the air or underwater In electronic warare, localizing a wideband jammer or a direct sequence spread spectrum transmitter will also have s(n) that are near white noise processes he task o DOA estimation is to ind the time dierence between D ig rom the L N measurements o z i(n)g Let D i;j D i 0 D j, i>j, be the DOA between z i (n) and z j (n), which is generally not an integral multiple o the sampling interval A ull set o D i;jg contains L(L 0 1) elements It is well known [15] that or continuous-time white Gaussian signal and noises, the maximum-likelihood estimate is given by the cross-correlator peak For the discrete-time signals o (1), the estimate o D i;j based on crosscorrelation, denoted by ^D i;j, is computed as ^D i;j arg max ~D r i;j( ~ D i;j) n0 r i;j ( ~ D i;j ) () z i(n)z j(n 0 ~ D i;j) () X/$ IEEE

2 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE is the cross-correlation unction o z i(n) and z j (n) with lag D ~ i;j, which denotes the variable or D i;j, and thus its value is not necessarily an integer It is worthy to mention that since we do not assume ~D i;j, the results derived in this paper are not immediately applicable or typical discrete-time DOA estimation ^D i;j is restricted to be an integral multiple o the sampling interval Utilizing the 0j! ~D inverse discrete-time Fourier transorm or e, the time-shited signal z j(n 0 D ~ i;j) is generated as [16] z j (n 0 ~ D i;j ) i0 z j (n 0 i)sinc(i 0 ~ D i;j ) (4) sinc(v) sin(v)(v) is the sinc unction and, whose ideal value is ininity, should be chosen large enough to make the modeling error negligible Let s and q g be the variances o s(n) and q i (n)g, respectively, and their values are assumed known Consider that N is suiciently large and SNR s q is identical at each sensor with q q, i 1; ; 111;L When ^D i;j D i;j, the variance o ^D i;j at N!1, denoted by var( ^D i;j), is derived using a irst-order approximation as (see Appendix A) var( ^D (1 + SNR) i;j ) N SNR : (5) Because o the optimality o the cross-correlator or continuous-time white signals, there is no surprise that var( ^D i;j ) equals the CRLB or DOA estimation [17] between two discrete-time sensor outputs, denoted by CRLB It is noteworthy to point out that an asymptotic variance expression or cross-correlation-based time-delay estimate has been derived in [18] Apart rom dierent derivation approaches, the variance o [18] is larger than CRLB mainly because suboptimal parabolic interpolation is used while we have employed the optimum sinc interpolator he cross covariance o ^D i;j and ^D k;l at N! 1, denoted by cov( ^D i;j; ^D k;l ) is (see Appendix A) cov( ^D i;j; ^D k;l ) NSNR ; 0 NSNR ; 0; i 6 j 6 k 6 l: i k and j 6 l or i 6 k and j l i l and j 6 k or i 6 l and j k Note that in Appendix A, we have in act derived the general orms o var( ^D i;j) and cov( ^D i;j; ^D k;l ) when the SNR at each sensor is distinct Let D [D ;1 ;D ;1 ; 111;D L;1 ;D ; ; 111;D L;L01] L(L01)1, denotes transpose, be the ull set DOA vector and its estimate using cross correlation be ^D [ ^D ;1 ; ^D ;1 ; 111; ^D L;1 ; ^D ; ; 111; ^D L;L01] With the use o (5) and (6), it can be shown that the asymptotic covariance matrix o ^D, denoted by C L(L01)L(L01), is o the orm with C (6) N SNR JJ + 1 I (7) SNR J J 1 J 111 J L01 L J i 0 (L0i)(i01) 0 1 L0i I L0i (L0i)L and I i, 0 ij and 1 i represent the ii identity matrix, ij zero matrix and i 1 vector with all elements 1, respectively It is noteworthy that (7) agrees with the covariance matrix or the correlator scheme (8) measurement error vector given in [14] but we are able to produce it in closed orm In the noise-ree case, the nonredundant DOA set can generate the ull DOA set without errors, that is, the ull set contains redundant DOA measurements Without loss o generality, we take the irst sensor as the reerence sensor and denote the nonredundant DOA set as D s [D ;1;D ;1; 111;D L;1] (L01)1 he ull and nonredundant DOA sets are related by D HD s (9) H JK L(L01) L with K [0 (L01)1 I L01] L(L01) Let ^D s [^^D ;1; ^^D ;1; 111; ^^D L;1] be the Gauss Markov estimate o D s based on ^D Employing the inverse o C as the weighting matrix, ^D s is easily obtained as the weighted LS estimate ^D s H C 01 H 01 H C 01 ^D (10) 01 denotes the matrix inverse We have ound the closed-orm expression o C 01 as C 01 0 N SNR (1 + LSNR) JJ 0 1+LSNR SNR I : (11) With the use o (11), H C 01 can be simpliied to (see Appendix B) so that H C 01 N SNR (1 + LSNR) H (1) H C 01 H 01 H C 01 (H H) 01 H : (1) hus, the Gauss Markov estimate o D s is in act the standard LS estimate and computational complexity can be signiicantly reduced or optimum conversion o the ull DOA set to the nonredundant Furthermore, the closed-orm expression o (H H) 01 H is (see Appendix C) (H H) 01 H 1 L I L L011 L01 H (14) H is a partition o H which contains only its last (L01)(L0 ) columns For reerence sensors other than the irst, we only need to change H accordingly to obtain the corresponding estimate o the nonredundant set In act, it can also be deduced easily rom ^D s that the LS estimation procedure implies ^^D i;j + ^D j;k + ^^D k;i 0[1] hat is, ^^D i;j ^^D i;1 0 ^^D j;1 or i 6 j 6 1 Note that the zero residual o ^^D i;j g is also mentioned in [19] or consistency checks but no suggestion is provided or achieving it Furthermore, the covariance matrix o ^D s, denoted by C s (L01)(L01), is then C s H C 01 H 01 : (15) With the use o (1), the closed-orm expression or C s is (see Appendix C) (1 + LSNR) C s (H N SNR H) 01 (1 + LSNR) I LN SNR L L011 L01 (16) while its inverse is (see Appendix C) C 01 s N SNR LI L L011 L01 (1 + LSNR) : (17)

3 616 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE 008 Note that (17) will be employed or computing the CRLB or positioning using the estimate o the nonredundant DOA set in the next section Recall that the Fisher inormation matrix (FIM) or DOA estimates using the nonredundant set is [14] F N 0! S (!) 1+ L i1 S(!)Qi(!) tr Q 01 (!) Q 01 p (!) 0 Q 01 p (!)1 L011 L01Q 01 p (!) d! (18) S(!) and Q i (!) represent the power spectra o s(k) and q i (k), respectively, while Q(!) diag(q 1 (!);Q (!); 111;Q L (!)) and Q p(!) diag(q (!);Q (!); 111;Q L(!)) It is noteworthy that (18) is derived under the ollowing assumptions: 1) the continuous-time random signals are wide-sense stationary processes; ) instead o analyzing the signals in the time domain, they are analyzed in the requency domain by sampling their Fourier transorms with requency interval o is the observation interval; since we have assumed unity sampling interval in (1), we have N ; ) the observation interval-bandwidth product o the processes is suiciently large such that the Fourier samples are uncorrelated and thus the discrete summation can be changed to continuous integration For white signal and noises with variances s and q,wehaves(!) s, Q i(!) q, i 1; ; 111;L, Q(!) qi L and Q p(!) qi L01 Substituting these values into (18) with SNR s q,we can easily show that (17) and (18) are identical Since the CRLB or D s is given by the inverse o the FIM, it ollows that ^D s is the asymptotically optimum estimate o the nonredundant set his inding is again consistent with [14], but we are able to produce the corresponding closed orm expressions here On the other hand, although Schmidt [1] has also suggested to employ the LS technique or averaging or improving the ull set DOA estimates, there is no discussion regarding the optimality issue From the diagonal o (15), the CRLB or D i;1g, i ; ; 111;L, using L sensors, denoted by CRLB L,is CRLB L 6(1 + LSNR) LN SNR : (19) Clearly, (19) agrees with (5) when L he perormance improvement o using ^D s over the truncated set rom ^D, namely ^D ;1; ^D ;1; 111; ^D L;1g, can be investigated rom the ratio o CRLB L to CRLB as ollows: CRLB L CRLB (1 + LSNR) L(1 + SNR) : (0) For SNR 1, this ratio approaches 1, which indicates that there is no improvement and ^^D i;1 ^D i;1, i ; ; 111;L As a result, estimating only ^D i;1 g is suicient to achieve optimum processing while computations can be saved because estimation o the remaining ^D i;jg and the LS procedure or calculating the Gauss Markov estimate are not required It is noteworthy that this inding conorms with (7) as C will become an ill-conditioned matrix at SNR 1 due to the rank deiciency o JJ At SNR 1, CRLB LCRLB approaches L which means that the enhancement only increases linearly with respect to the sensor number Interestingly, we have also ound that or suiciently small SNR, the covariance matrix is C ( NSNR )I L(L01) With this value o C, (1) can be obtained much simpler III OSIIONING ERFORMANCE BOUND Let [x y] and [x i y i ], i 1; ; 111;L, be the unknown target position and known sensor positions, respectively, then their relationship with the DOAs is vd i;j (x 0 x i ) +(y 0 y i ) 0 (x 0 x j ) +(y 0 y j ) (1) v represents the signal propagation speed which is a known constant he minimum achievable mean square position error (MSE), denoted by E(^x 0 x) +(^y 0 y) g, ^x and ^y represent the unbiased estimates o x and y and E is the expectation operator, is developed as ollows he FIM or -D positioning with the parameter vector [x y] using the nonredundant set, denoted by F s,is given by [11] F s G scs 01 G s () v G s [g 1 g 1 111g L1] (L01) g i1 g i 0 g 1 g i g x;i g y;i p x0x (x0x ) +(y0y ) y0y p (x0x ) +(y0y ) In Appendix D, we have expressed G s C 01 s G s as G sc 01 s G s N SNR (1 + LSNR) G gx;1 gx; 111 gx;l g y;1 g y; 111 g y;l LG G 0 G 1 L1 LG () Substituting () into () and then taking the inverse, we can obtain the closed-orm CRLB or x and y rom its diagonal elements On the other hand, the FIM or -D positioning using the ull DOA set, denoted by F, is given by [0] F G C 01 G (4) v G [g 1 g g 1L g 111 g L 111 g L01L] : With the use o (15) and G G s H, it is easy to obtain G C 01 G G scs 01 G s Hence, localization using the ull or nonredundant sets will have the same CRLB his is not a surprising result in view o the development in Section II Hence, localization algorithms should always use the nonredundant set because there is a less number o equations to solve he indings in this section can also contribute to the research o optimum -D sensor array design Although Yang and Scheuing [8], [0] have perormed some pioneering study on the impact o the sensor array geometry on the DOA-based localization accuracy, their assumption o a scaled identity matrix or C is rather restrictive As seen in Section II, this is valid only or SNR 1 In contrast, () provides the basis or study o sensor array geometry eects on the CRLB IV SIMULAION SUDY Computer simulations have been perormed to veriy the theoretical development on DOA-based positioning A microphone array application scenario with eight sensors is considered and the speed o signal : :

4 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE Fig 1 Mean square position error when sensor outputs have same SNRs Fig Mean square position error when the irst sensor output has smaller SNR propagation is v 40 ms 01 he eight microphones are at (0,0)m, (0,50)m, (0,100)m, (50,100)m, (100,100)m, (100,50)m, (100,0)m, and (50,0)m, while the source position is at (10,0)m he MSE is employed as the perormance measure, and all results provided are averages o 1000 independent runs In the irst test, the SNRs o all sensor outputs are identical Based on the investigated geometry, the optimum ull set DOA estimates, namely, ^D, are generated with the covariance matrix o (7) according to a Gaussian distribution In doing so, we have implicitly assumed suiciently high SNR conditions such that there are no large errors in the DOA estimates It should be emphasized that the DOA estimation problem o () is nonlinear, and hence in practice, it will suer rom the threshold phenomenon when SNR is small enough he optimum nonredundant estimate o D s is then obtained rom ^D with the use o the LS procedure o (14) he perormance o the truncated set ^D ;1 ; ^D ;1 ; 111; ^D 8;1 g is also evaluated From each set o DOA measurements, the position estimate, namely, (^x; ^y), is determined by inding the maximum o the corresponding maximum likelihood cost unctions he Newton Raphson iterative procedure is employed or the maximum search and the true position is selected as the initial guess Again, we have ignored the threshold phenomenon Fig 1 shows the MSEs o the three sets o DOA estimates versus SNR It is seen that the MSEs based on the ull set and optimized nonredundant set perorm equally and meet the CRLB or all SNR conditions, which agrees with the indings in Section III Furthermore, we observe that employing ^D ;1 ; ^D ;1 ; 111; ^D 8;1 g can provide optimum position estimates only at high SNRs, as predicted in Section II In the second test, we study the comparative positioning accuracy or dierent schemes o processing the DOA measurements when the sensor outputs have unequal SNRs We use the covariance matrix in Appendix A to generate the ull set DOA measurements For simplicity, we set q 0q and q q q 111 q with SNR s q, which implies that the truncated set ^D ;1; ^D ;1; 111; ^D 8;1g is now more noisy than other DOA estimates in the ull set he MSEs o dierent methods are plotted in Fig Note that or this scenario, (1) does not hold and thus the results or the Gauss Markov estimate o (10) are also included Again, we observe that the estimators using the ull set and optimized nonredundant set give optimum perormance or the whole SNR range Due to a smaller SNR condition at ^D ;1; ^D ;1; 111; ^D 8;1g, the MSE rom using the truncated set cannot meet the CRLB even at high SNRs On the other hand, it is interesting to see that the perormance o the simple LS procedure or DOA s attains optimality Fig Mean square position error when the second sensor output has smaller SNR or SNR [010; 0]dB his test is repeated with q 0q and q q q 111 q, and the results are plotted in Fig he observations are similar to those o Fig except that the truncated set can give optimum estimation perormance at higher SNRs because now ^D ;1 ; ^D ;1 ; 111; ^D 8;1 g is generally less noisy compared to the remaining DOA estimates in the ull set V CONCLUSION Optimal conversion o the ull time-dierence-o-arrival (DOA) set estimates to the nonredundant can be achieved using the standard least squares procedure and the corresponding closed orm expressions are derived or the case o white signal in additive white noise It is proved that both the ull and nonredundant sets o DOA estimates can give the same position estimation perormance Cramér Rao lower bound or DOA-based positioning is also derived in closed orm, and this should acilitate the design o optimum sensor placement or localization AENDIX A he DOA variance o the cross-correlator is irst derived as ollows For simplicity but without loss o generality, we only consider DOA estimation between z 1(n) and z (n) and let D 1 0and D D ;1 D be the time dierence he powers o the Gaussian processes s(k)

5 618 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE 008 and q i(k), i 1; ; 111;L, are denoted by s and q, respectively he corresponding cross-correlator unction is then r ;1( D) ~ z (n)z 1(n 0 D) ~ z 1 (n 0 ~ D) n0 i0 z 1 (n 0 i)sinc(i 0 ~ D): (A1) (A) We consider the asymptotic condition o N!1and is chosen suiciently large Denote ^D as the DOA estimate which is the ~ D which maximizes r ;1 ( ~ D) his implies r 0 ;1( ~ D) ~D ^D 0 (A) r 0 ;1( ~ D) is the irst derivative o r ;1 ( ~ D) with respect to ~ D When Er 0 ;1( ~ D)g is suiciently smooth around ~ D D, wehave the ollowing irst-order approximation [1]: de r 0 ;1( ~ D) d ~ D E r 00 ;1( ~ D) r;1( 0 D) ~ 0 ~D ^D r0 ;1( D) ~ (A4) ^D 0 D r;1( 00 D) ~ is the corresponding second derivative With the use o (A) and (A4), the mean value o ^D is E ^Dg E r;1( 0 D) ~ D 0 E r 00 ;1 ( D) ~ Employing (A1) and (A), r 0 ;1( ~ D)j is r;1( 0 D) ~ 0 n0 i0 : (A5) z (n)z 1 (n 0 i)sinc 0 (i 0 D): (A6) For uncorrelated q 1(n) and q (n), taking the expected value o (A6) yields E r 0 ;1( ~ D) 0 n0 i0 1 0 n0 i0 j01 E s(n 0 D)s(n 0 i)g sinc 0 (i 0 D) E s(n 0 i)s(n 0 j)g sinc 0 (i 0 D)sinc(j 0 D) 0Ns sinc 0 (i 0 D)sinc(i 0 D) i0 0Ns sinc 0 (0) 0 (A7) because sinc 0 (0) 0 In a similar manner, the term Er 00 ;1( ~ D)gj is evaluated as E r 00 ;1( ~ D) n0 i0 E s(n 0 D)s(n 0 i)g sinc 00 (i 0 D) N s sinc 00 (0) 0 N s (A8) with the use o sinc 00 (0) 0 [] Substituting (A7) and (A8) into (A5) yields E ^Dg D (A9) which shows the approximate unbiasedness o the cross-correlator Squaring both sides o (A4) with the use o (A) and (A9), the variance o ^D based on the irst-order approximation, is var( ^D) E ^D 0 E ^Dg E ( ^D 0 D) E r 0 ;1( ~ D) E r 00 ;1 ( ~ D) : (A10) It is noteworthy that (A10) can also be derived by expanding r 0 ;1( ~ D) in a aylor series about D and retaining only linear terms [] By squaring (A6), the numerator o (A10) is E r 0 ;1(D) n0 m0 i0 j0 E z 1 (n 0 i)z 1 (m 0 j)z (n)z (m)g sinc 0 (i 0 D)sinc 0 (j 0 D): (A11) Using the properties that the signal and noise are uncorrelated and white Gaussian processes as well as the sinc unction, namely, 1 i01 sinc0 (v) or any real value o v [] and sinc 0 (0) 0, we have simpliied (A11) as E r 0 ;1(D) N s q + s q + q q : (A1) Substituting (A8) and (A1) into (A10) with q q q and SNR s q yields (5) Following the variance development or (A10), the covariance between ^D i;j and ^D k;l is then cov( ^D i;j; ^D k;l ) E ( ^D i;j 0D i;j)( ^D k;l 0D k;l ) E r 0 i;j( ~ D i;j )r 0 k;l ( ~ D k;l ) E r 00 i;j ( ~ D i;j ) E r 00 k;l ( ~ D k;l ) With the use o (A8), it is easy to see that E r 00 i;j( ~ D i;j ) ~D D E r00 k;l( ~ D k;l ) ~D D ; ~ D D : (A1) s ~D D 0N : (A14) In a similar manner, the numerator o (A1) can be derived as E r 0 i;j( ~ D i;j )r 0 k;l( ~ D k;l ) N ~D D ; ~ D D ; i k and j 6 l or i 6 k and j l 0 N ; i l and j 6 k or i 6 l and j k 0; i 6 j 6 k 6 l: utting (A14), (A15) into (A1) with q q q yields (6) (A15)

6 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE AENDIX B In this Appendix, (1) will be proved, namely, H C 01 is proportional to H First o all, J JJ is evaluated as ollows Recall (8) J J 1 J 111 J L01 J i [0(L0i)(i01) 0 1 L0i I L0i], wehave Hence J i J i 0(i01)(i01) 0(i01)1 0(i01)(L0i) 01(i01) L 0 i 01 L0i 0(L0i)(i01) 01 L0i I L0i L01 J J J i J i i From (B), we get L L L : (B1) LI L 0 1 L 1 L : (B) J JJ LI L 0 1 L 1 L J LJ (B) since 1 L 1LJ 0 L(L(L01)) Using (11) and (B), H C 01 is then evaluated as H C 01 K J C 01 0 N SNR which is (1) (1+LSNR) N SNR (1+LSNR) H LK J 0K J 1+LSNR SNR (B4) AENDIX C In this Appendix, (14), (16), and (17) will be proved Using (B), (H H) 01 becomes (H H) 01 (K J JK) 01 K LI L 0 1 L 1 L K 01 LI L L011 L01 01 : (C1) With the use o (C1), taking the inverse o (16) yields (17) Using the matrix inversion lemma, (C1) can also be expressed as LI L L011 L LI L L011 L01 (LI L01) 01 + (0LI L01) 01 1 L011 L01(0LI L01) L01 (0LI L01) 01 1 L01 1 L I L L011 L01 : (C) From (1) and (C), we easily obtain (16) Using (C), (H H) 01 H is evaluated as ollows: (H H) 01 H 1 L H + 1 L011 L01 01 K J 1 K J K J 111 K J L01 1 L H + 1 L011 L01 I L01 K J K J 111 K J L01 1 L H + 1 L01 1 L01 0 (L0)1 0 (L0) L I L L011 L01 AENDIX D H : (C) In this Appendix, we will prove () We irst write G s as By using (B1), we have Moreover, by the property o J G s G J 1 : J 1 J1 L L01 01 L01 I L01 J 1 1 L01 ) J 1 1 L011 L01J1 0L +1 1 L01 (1 0 L) (1 0 L)1L01 (1 0 L)1 L01 1 L011L01 : (D1) (D)

7 60 IEEE RANSACIONS ON SIGNAL ROCESSING, VOL 56, NO 6, JUNE 008 G sc 01 s G s is derived with the use o (C1), (D1) and (D) as ollows, which is (): G sc 01 s G s N SNR (1 + LSNR) G J 1 LI L L011 L01 J 1 G N SNR (1 + LSNR) G L L01 01 L01 LI L L011 L01 N SNR (1 + LSNR) G LI L 0 1 L1 L G: (D) ACKNOWLEDGMEN he authors would like to thank the anonymous reviewers or their careul reading and constructive comments, which improved the clarity o this paper REFERENCES [1] G C Carter, IEEE rans Acoust, Speech, Signal rocess (Special Issue on ime Delay Estimation), vol 9, no, Jun 1981 [] G C Carter, Coherence and ime Delay Estimation: An Applied utorial or Research, Development, est, and Evaluation Engineers New York: IEEE ress, 199 [] J Chen, Y Huang, and J Benesty, ime delay estimation, in Audio Signal rocessing or Next-Generation Multimedia Communication Systems, Y Huang and J Benesty, Eds Norwell, MA: Kluwer, 004, ch 8 [4] Y Huang, J Benesty, G W Elko, and R M Mersereau, Real-time passive source localization: A practical linear-correction least-squares approach, IEEE rans Speech, Audio rocess, vol 9, pp , Nov 001 [5] Y Zhao, Standardization o mobile phone positioning or G systems, IEEE Commun Mag, vol 40, no 7, pp , Jul 00 [6] R Yamasaki, A Ogino, amaki, Uta, N Matsuzawa, and Kato, DOA location system or IEEE 8011b WLAN, in roc WCNC 005, 005, pp 8 4 [7] N atwari, J N Ash, S Kyperountas, A O Hero, III, R L Moses, and N S Correal, Locating the nodes: Cooperative localization in wireless sensor networks, IEEE Signal rocess Mag, vol, no 4, pp 54 69, Jul 005 G [8] B Yang and J Scheuing, A theoretical analysis o -D sensor arrays or DOA based localization, in roc Int Con Acoustics, Speech, Signal rocess (ICASS), oulouse, France, May 006, vol IV, pp [9] B Friedlander, A passive localization algorithm and its accuracy analysis, IEEE J Ocean Eng, vol 1, no 1, pp 4 45, Jan 1987 [10] J O Smith and J S Abel, Closed-orm least-squares source location estimation rom range-dierence measurements, IEEE rans Acoust, Speech, Signal rocess, vol 5, pp , Dec 1987 [11] Y Chan and K C Ho, A simple and eicient estimator or hyperbolic location, IEEE rans Signal rocess, vol 4, no 8, pp , Aug 1994 [1] R O Schmidt, A new approach to geometry o range dierence location, IEEE rans Aerosp Electron Syst, vol AES-8, pp 81 85, Nov 197 [1] R O Schmidt, Least squares range dierence location, IEEE rans Aerosp Electron Syst, vol, no 1, pp 4 4, Jan 1996 [14] W R Hahn and S A retter, Optimum processing or delay-vector estimation in passive signal arrays, IEEE rans In heory, vol 19, pp , Sep 197 [15] C H Knapp and G C Carter, he generalized correlation method or estimation o time delay, IEEE rans Acoust, Speech, Signal rocess, vol ASS-4, no 4, pp 0 7, Aug 1976 [16] Y Chan, J M F Riley, and J B lant, Modeling o time-delay and its application to estimation o nonstationary delays, IEEE rans Acoust, Speech, Signal rocess, vol ASS-9, no, pp , Jun 1981 [17] A H Quazi, An overview on the time delay estimate in active and passive system or target localization, IEEE rans Acoust, Speech, Signal rocess, vol ASS-9, no, pp 57 5, Jun 1981 [18] G Jacovitti and G Scarano, Discrete time techniques or time delay estimation, IEEE rans Signal rocess, vol 41, no, pp 55 5, Feb 199 [19] G C Carter, ime delay estimation or passive sonar signal processing, IEEE rans Acoust, Speech, Signal rocess, vol 9, no, pp , Jun 1981 [0] B Yang and J Scheuing, Cramér-Rao bound and optimum sensor array or source localization rom time dierences o arrival, in roc Int Con Acoustics, Speech, Signal rocessing (ICASS), hiladelphia, A, Mar 005, vol IV, pp [1] V H MacDonald and M Schultheiss, Optimum passive bearing estimation in a spatially incoherent noise environment, J Acoust Soc Amer, vol 46, no 1, pt 1, pp 7 4, Jul 1969 [] H C So, C Ching, and Y Chan, A new algorithm or explicit adaptation o time delay, IEEE rans Signal rocess, vol 4, no 7, pp , Jul 1994 [] J Ianniello, Large and small error perormance limits or multipath time delay estimation, IEEE rans Acoust, Speech, Signal rocess, vol 4, no, pp 45 51, Apr 1986