Accurate three-step algorithm for joint source position and propagation speed estimation

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1 Signal Processing 8 (00) wwwelseviercom/locate/sigpro Accurate three-step algorithm for joint source position propagation speed estimation Jun Zheng, Kenneth WK Lui, HC So Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China Received 9 December 006; received in revised form May 00; accepted June 00 Available online 10 July 00 Abstract A popular strategy for source localization is to utilize the measured differences in arrival times of the source signal at multiple pairs of receivers Most of the time-difference-of-arrival (TDOA) based algorithms in the literature assume that the signal transmission speed is known which is valid for in-air propagation However, for in-solid scenarios such as seismic tangible acoustic interface applications, the signal propagation speed is unknown In this paper, we exploit the ideas in the two-step weighted least squares method [1] to design a three-step algorithm for joint source position propagation speed estimation Simulation results are included to contrast the proposed estimator with the linear least squares scheme as well as Crame r Rao lower bound r 00 Elsevier BV All rights reserved Keywords: Source localization; Propagation speed estimation; Time-difference-of-arrival; Weighted least squares 1 Introduction Passive source localization using time-differenceof-arrival (TDOA) information from an array of spatially separated sensors is an important problem in signal processing In the TDOA method, the differences in arrival times of the source signal at multiple pairs of sensors are measured Each TDOA measurement defines a hyperbolic locus on which the source must lie the position is given by the intersection of two or more hyperbolas for noise-free twodimensional localization Although there are numerous TDOA-based positioning algorithms in the literature, such as [1 6], most of them assume that Corresponding author Tel: ; fax: address: hcso@eecityueduhk (HC So) the signal transmission speed is known which is valid for in-air propagation However, for in-solid scenarios such as seismic [] tangible interface for human computer interaction [8] applications, the signal propagation speed is unknown, we need to find it together with the source position for accurate estimation In this paper, an efficient three-step algorithm for joint source position propagation speed estimation is derived by applying the ideas of [1], namely, employment of weighted least squares (WLS) exploitation of the relationship between parameter estimates Our major contributions are to address the positioning problem with unknown propagation speed exploit the nonlinear relationship between the source position speed parameters in the third step of the developed algorithm The rest of the paper is organized as follows The proposed three-step method is developed in Section /$ - see front matter r 00 Elsevier BV All rights reserved doi:101016/jsigpro000601

2 It is proved that the first step solution is in fact equal to the least squares (LS) algorithm of [6] Simulation results are included in Section to evaluate the estimation performance of the threestep algorithm by comparing with the LS method Cramer Rao lower bound (CRLB) Finally, conclusions are drawn in Section Joint source position propagation speed estimation algorithm In this section, we develop a three-step algorithm to jointly estimate source location propagation speed using TDOA measurements from M sensors The discrete-time signal received at the ith sensor can be expressed as r i ðkþ ¼sðk D i Þþq i ðkþ; i ¼ 1; ; ; M, (1) where sðkþ is the signal radiating from the source, D i q i ðkþ are the time-of-arrival additive noise, respectively, at the ith sensor Let ðx; yþ ðx i ; y i Þ, i ¼ 1; ; ; M, be the unknown source location known position of the ith sensor, respectively Denote d i;1 D i;1 ¼ D i D 1 as the range difference TDOA with respect to the first sensor, respectively, then we have the following relationship: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d i;1 ¼ cd i;1 ¼ ðx i xþ þðy i yþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx 1 xþ þðy 1 yþ, i ¼ ; ; ; M, ðþ where c is the unknown propagation speed Our task is to find ðx; yþ c with the use of fd i;1 g fðx i ; y i Þg Following the idea of [,], we rearrange square () to yield ðx i x 1 Þx þ ðy i y 1 Þy þ D i;1 u þ D i;1 v ¼ x i x 1 þ y i y 1 ; i ¼ ; ; ; M, ðþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where u ¼ c ðx 1 xþ þðy 1 yþ v ¼ c are introduced to make a linear representation In matrix form, we have Ah ¼ b, () where ðx x 1 Þ ðy y 1 Þ D ;1 D ;1 A ¼ 6, ðx M x 1 Þ ðy M y 1 Þ D M;1 D M;1 J Zheng et al / Signal Processing 8 (00) x y h ¼ 6 v, u x x 1 þ y y 1 b ¼ 6 x M x 1 þ y M y 1 In the presence of fq i ðkþg, the TDOA measurements are noisy which can be modelled as D i;1 ¼ D 0 i;1 þ n i;1; i ¼ ; ; ; M, () where fd 0 i;1g denote the noise-free TDOA s while each TDOA estimation error n i;1 is characterized by r i ðkþ r 1 ðkþ Based on (), the stard LS estimate of h, denoted by ^h 1, is simply ^h 1 ¼ðA T AÞ 1 A T b, (6) where T 1 denote the transpose operator matrix inverse, respectively It is noteworthy that (6) is in fact identical to the solution of [6], although we work on the hyperbolic equations from TDOA measurements while circular equations from timeof-arrival information are considered in the latter For more accurate estimation, we propose to utilize the ideas of [1], namely, employing WLS exploiting the relationship between x, y, u v, in the following two steps For sufficiently small noise conditions, the measured error vector in (), denoted by e, can be approximated as [1] e ¼ b Ah ½ðu þ vd 0 ;1 Þn ;1 ðu þ vd 0 ;1 Þn ;1 ðu þ vd 0 M;1 Þn M;1Š T ðþ The covariance matrix for e, denoted by U, is then U ¼ Efee T gb QB, (8) where E denotes expectation operator, B ¼ diagðu þ vd 0 ;1 ; u þ vd0 ;1 ; ; u þ vd0 M;1Þ Q is the covariance matrix for fn i;1 g which can be determined using the power spectra of sðkþ fq i ðkþg [1,9] For simplicity, we assume that the source signal noises in (1) are white processes the signal-to-noise ratios at all fr i ðkþg are

3 098 J Zheng et al / Signal Processing 8 (00) identical In doing so, Q will be proportional to we will substitute this matrix for Q in our study With the use of (8), the WLS estimate of h, denoted by ^h,is[10] ^h ¼ðA T U 1 AÞ 1 A T U 1 b (9) Note that the technique of WLS has already been utilized in localization [1,], although A U in (9) are of different forms because we have the unknown parameter of speed as well To compute U in practice, fd 0 i;1 g in B are replaced by fd i;1 g in () v u are approximated by ^v ¼ DT d D T (10) D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^u ¼ ^vðð½^h 1 Š 1 x 1 Þ þð½^h 1 Š y 1 Þ Þ, (11) where ½^h 1 Š 1 ½^h 1 Š represent the first second elements of ^h 1, which are the LS estimates of x y, respectively The D ¼½D ;1 ; D ;1 ; ; D M;1 Š T d ¼½d ;1 ; d ;1 ; ; d M;1 Š T are the TDOA vector range difference vector, respectively, where d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i;1 is computed as d i;1 ¼ ðx i ½^h 1 Š 1 Þ þðy i ½^h 1 Š Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx 1 ½^h 1 Š 1 Þ þðy 1 ½^h 1 Š Þ It is noteworthy that choosing the above initial estimates of u v is based on the fact that the location estimate in (6) is more accurate than the speed estimate [] The covariance matrix for the WLS estimate in (9) is [10] covð^h Þ¼ðA T U 1 AÞ 1 (1) The relationship between x, y, u v has not been exploited so far In the third step, we utilize their relationship u ¼ vððx x 1 Þ þðy y 1 Þ Þ (1) We first define the matrices ðx x Þ H ¼ ;! ¼ 6 ðy y 1 Þ v ½^h Š ½^h Š 0 ð½^h Š 1 x 1 Þ ð½^h Š y 1 Þ p ¼ 6 ½^h Š ½^h Š The ^h can be written in terms of h as ½^h Š i ¼½hŠ i þ e i ; i ¼ 1; ; ;, (1) where fe i g are the estimation errors in (9) Now we have H! p (1) Similar to (), the error vector for (1) can be approximated as n ¼ p H! ½ðx x 1 Þe 1 ; ðy y 1 Þe ; e ; ue re Š T, ð16þ where r ¼ðx x 1 Þ þðy y 1 Þ Following [1], the covariance matrix for n, denoted by U,is U ¼ Efnn T gb covð^h ÞB þ½0 GŠ, (1) where B ¼ diagððx x 1 Þ; ðy y 1 Þ; 1; uþ, 0 is a zero matrix G ¼½ rðx x 1 ÞEfe 1 e g; rðy y 1 ÞEfe e g; refe g; r Efe g urefe e gš T with Efe i e j g corresponding to the ði; jþ entry of covð^h Þ in (1) In practice, the values of x, y u are approximated by the estimates of (9) The WLS estimate of!, denoted by ^!, isthen ^! ¼ðH T U 1 HÞ 1 H T U 1 p (18) The final estimates of the location speed are then computed as ^x ¼ ½^!Š 1 þ x 1, ^y ¼ ½^!Š þ y 1, ^c ¼ ½ ^!Š ð19þ The values of ^x ^y which are closest to the corresponding estimates in ^h are chosen as the position estimate Note that when the square root is a complex number, the imaginary component will be set to zero, this happens when ðx 1 ; y 1 Þ is very close to ðx; yþ The proposed three-step algorithm for joint source position propagation speed estimation

4 J Zheng et al / Signal Processing 8 (00) is summarized as follows: (i) Compute the LS solution using (6) (ii) Compute the second step solution of (9) with the use of (10) (11) (iii) Compute the third step solution using (18) (19) with the use of ^h Simulation results Computer simulations are conducted to evaluate the proposed three-step algorithm for source localization speed estimation by comparing it with the LS solution of (6) or [6], aswellascrlb[] We consider a tangible acoustic interface application of interactive displays [11] Five sensors are placed on a 1m 1 m pane of glass with coordinates (0, 0) m, (0, 0) m, (1, 0) m, (1, 1) m, (0, 1) m while the unknown source position is located at (0, 01) m Note that the acoustic propagation speed in solid is dependent on the material of medium as well as the type of tactile interaction Here we assume a 1-cm thick pane with a knuckle tap, the propagation speed is set to 100 ms 1 [11] For this relatively high speed, the TDOA s become much smaller than the location coordinates As a result, the values in the last two columns of A are significantly less than those of the first two columns, which will make A abadlyscaled matrix result in inaccuracy of the solution To avoid this problem, we multiply both sides of () by 10 This operation scales up x; y; x i ; y i D i;1 by 10 In doing so, the elements in the first, second fourth columns will be multiplied by 10 while those of the third column will be multiplied by 10 6 thus the condition number of A will be decreased The solution of ðx; yþ in (19) is scaled down accordingly to obtain our solution The same scaling operation is also applied to the LS method in [6] The noise-free TDOA s are added by the correlated Gaussian noises with covariance matrix given by Q with diagonal elements equal s all other elements equal s All simulation results are averages of 000 independent runs Figs 1 compare the positioning speed estimation accuracy, respectively, of the proposed method with the LS approach CRLB for different values of s The mean square error (MSE) is used as the performance measure The MSE of the position estimate is defined as E ðx ^xþ þðy ^yþ where ^x ^y denote the estimates of x y, respectively On the other h, the MSE of the speed estimate is mean square position error (db) mean square speed error (db) defined as E ðc ^cþ where ^c is the estimate of c From Figs 1, it is observed that the MSE s of the proposed method are very close to CRLB, are smaller than those of the LS method, for both location speed estimation Conclusion LS proposed CRLB σ (db) Fig 1 Mean square position errors LS proposed CRLB σ (db) Fig Mean square speed errors A three-step algorithm for joint source localization propagation speed estimation is developed with the use of TDOA measurements Two intermediate variables are introduced to make a linear representation of the nonlinear TDOA equations The LS solution in the first step provides the initial estimates The second step refines the estimation by employing WLS while the final step further

5 100 J Zheng et al / Signal Processing 8 (00) improves the estimates by another WLS via utilizing the relationship between the source position, speed intermediate variables For sufficiently small noise conditions, it is shown that the accuracy of the proposed method approaches CRLB outperforms the LS method of [6] Acknowledgement The work described in this paper was supported by a grant from CityU (Project no 001) References [1] YT Chan, KC Ho, A simple efficient estimator for hyperbolic location, IEEE Trans Signal Process (August 199) [] KW Cheung, HC So, YT Chan, W-K Ma, A constrained least squares approach to mobile positioning: algorithms optimality, EURASIP J Appl Signal Processing 006 (006) 1, Article ID 088 [] JO Smith, JS Abel, Closed-form least-squares source location estimation from range-difference measurements, IEEE Trans Acoust Speech Signal Process (December 198) [] B Friedler, A passive localization algorithm its accuracy analysis, IEEE J Ocean Eng 1 (January 198) [] CW Reed, R Hudson, K Yao, Direct joint source localization propagation speed estimation, in: Proceedings of the ICASSP 99, vol, March 1999, pp [6] A Mahajan, M Walworth, -D position sensing using the differences in the time-of-flights from a wave source to various receivers, IEEE Trans Robotics Automation 1 (February 001) 91 9 [] JC Chen, K Yao, RE Hudson, Source localization beamforming, IEEE Signal Process Magazine 19 () (March 00) 0 9 [8] G De Sanctis, D Rovetta, A Sarti, G Scarparo S Tubaro, Localization of tactile interactions through TDOA analysis: geometric vs inversion-based method, in: Proceeding of 006 European Signal Processing Conference, Florence, Italy, September 006 [9] WR Hahn, SA Tretter, Optimum processing for delayvector estimation in passive signal arrays, IEEE Trans Inform Theory 19 (September 19) [10] SM Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall, Englewood Cliffs, NJ, 199 [11] JA Paradiso, et al, Passive acoustic sensing for tracking knocks atop large interactive displays, in: Proceedings of IEEE International Conference on Sensors, vol1, Orlo, Florida, June 00, pp 1