A Distributed Particle Filter for Acoustic Source Tracking Using an Acoustic Vector Sensor Network

Transcription

1 A Distributed Particle Filter for Acoustic Source Tracing Using an Acoustic Vector Sensor Networ Xionghu Zhong #1, Arash Mohammadi, A. B. Premumar, Amir Asif # School of Computer Engineering, College of Engineering, Nanyang Technological University, Singapore, xhzhong@ntu.edu.sg Engineering Department, Faculty of Engineering, University of Malaya, KualaLumpur, Malaysia benjamin.premumar@gmail.com Department of Computer Science and Engineering, Yor University, Toronto, Canada, MJ 1P. marash@cse.yoru.ca, asif@cse.yoru.ca Abstract This letter presents a distributed particle filtering (PF) approach for wideband acoustic source tracing using an acoustic vector sensor (AVS) networ. At each distributed AVS node of the AVS networ, the unscented information PF (UIPF) provides local estimates of the source location. A distributed consensus algorithm, based on the first and second order statistics of the localized UIPF estimated states, then obtains the global position estimate for the overall AVS networ. Since the UIPF inverts smaller order matrices, its computational complexity is significantly less as compared to its centralized counterpart. The performance of the proposed tracing algorithm is studied under both single source and multiple source tracing scenarios. Index Terms Acoustic vector sensor, sensor networ, particle filtering, multiple source tracing, unscented information filter. I. INTRODUCTION Acoustic vector sensor (AVS) employs a co-located sensor structure capable of providing -D (azimuth and elevation) direction of arrival (DOA) information. The AVS manifold is independent of source signal s frequency, which enhances its utility in wideband acoustic signal processing applications [1], including acoustic source detection [], localization [1] and tracing [], [] in underwater, battlefield, and room acoustic environments [5], []. More recent AVS applications include speech enhancement for hands free communication, sound source localization in air, and detection of seismic activities. This letter presents a distributed wideband acoustic source tracing algorithm for the distributed AVS networ. A diffusive estimation scheme, namely the distributed unscented information particle filter (), is developed. The proposed approach consists of three main steps: (i) At each AVS node, an unscented information filter (UIF) [7] is introduced to approximate the optimum importance function to draw the particles according to both current measurements and previous state estimates; (ii) Local particle filters are applied at each node to estimate the first order and the second order statistics for the localized state estimates, and; (iii) Each node diffuses its local statistics throughout the networ by communicating with nodes within its local neighborhood. A consensus filter is employed in step (iii) to fuse these local statistics to obtain the global estimates. With the proposed, the positions of multiple wideband acoustic sources can be estimated directly from the received AVS signals. There is no need to preprocess the received signals to extract the DOA measurements, nor to associate the DOA measurements to each source. Further, the information communicated within the AVS networ is limited to the first and second order statistics of the local estimates eeping the communication overhead manageable. In the past, different centralized approaches were developed for AVS based localization and tracing applications. In [1], for example, the Capon beamforming algorithm is used at each node to estimate the local DOAs associated with the target. The local nodes communicate the processed DOAs to the central processor (CP), where the position of the source is triangulated using the least squares (LS) approach. Such DOA based approaches, including [] developed for distributed microphone array based acoustic source tracing, are, however, not applicable to multiple sources and -D source tracing applications. Further, their performance is dependent on the accuracy of the DOA estimates. Among the multiple source tracing approaches, the centralized extended information based PF () [9] is promising. But its applicability to large distributed networs without a CP is somewhat limited. Our approach belongs to the distributed category of AVS tracers and has several differences from [1] and [9]. Compared to the reweighted LS (RLS) approach [1], the achieves superior performance in both singlesource and multiple-source tracing scenarios. Compared to the [9], both computation and communication cost are substantially reduced. The rest of this letter is organized as follows. In Section II, the AVS signal model is introduced. Section III presents the localized implementation of the at each AVS node. Consensus based information fusion used to integrate local state statistics to derive the global estimate is introduced in Section IV. Different simulations are organized in Section V with the letter concluded in Section VI. II. AVS SIGNAL MODEL To trac M acoustic sources located at x m, = [x m,, y m,, z m, ] T R 1, for (1 m M), at time instant, assume N AVSs at fixed locations x n = [x n, y n, z n ] T R 1, for (1 n N), are arbitrarily deployed. The DOA of

2 the acoustic signal associated with the mth source at the nth AVS is ( φ n m, = tan 1 xm, x n ) y m, y n, ( ) ψm, n = tan 1 z m, z n, (1) (xm, x n ) + (y m, y n) where φ n m, ( π, π] and ψn m, [ π/, π/] represent the azimuth angle and the elevation angle respectively, and superscript T denotes the transpose operator. Let u n m, = [cos ψn m, cos φn m,, cos ψn m, sin φn m,, sin ψn m, ]T be the unit direction vector pointing out from the nth sensor towards the mth source. Assuming that at, T number of snapshots are considered, the collection of wideband source signals s m (), (1 m M), is given by S = [s 1 (),..., s M ()] T C M T. The received signal model for the nth AVS is as follows Y n = g n (X )S + ɛ n, () where X = [x T 1,,..., xt M, ]T is the source state, g n(x ) = [a 1 n(),..., a M n ()] with a m n () = [1, u n m, ]T is the steering vector, and ɛ n represent the channel noise including C T the pressure and velocity noise terms. Note that the particle velocity terms are normalized by multiplying by a constant term ρ c, where ρ and c represent the ambient density and the propagation speed of the acoustic wave in the medium respectively. The noise process ɛ n is a sequence of complexvalued i.i.d. circular Gaussian random variables with zero mean and covariance matrix Γ. Since dynamic sources are considered, the source state x m, is constructed by cascading the original position component x p m, with a velocity component xv m,. Constant velocity (CV) model [9] is employed here to model the source dynamics as X = FX 1 + Gv, () where v is the global uncertainties in the state process. The coefficient matrix F and G are defined respectively as [ ] [ I T I T F = I M ; G = I I M I ], () T I where I q denotes the qth order identity matrix, T represents the time period in seconds between the previous and current, and denotes the Kronecer product. Eqs. () and () present the state-space model for the AVS networ based tracing problem. The nodes of the networ are modeled as vertices of the graph G = (ν, E), namely as elements of the node set ν = {1,..., N}. The edge set E ν ν represents the networ s communication constraints, i.e., if node n can send information to node u then (n, u) E. For graph G, the maximum degree G = max n D (n), where D (n) is the number of neighboring nodes for node n. We present the proposed distributed acoustic source tracing algorithm for the AVS networ in the next two sections. III. ESTIMATION OF LOCAL STATISTICS The AVS networ uses a large number of snapshots at each which leads to the covariance matrix of large dimensions. Therefore, the unscented particle filter (UPF) [1], [11] and the [9], typically used for centralized nonlinear tracing, are computationally demanding and not suitable for distributed acoustic source tracing using the AVS networ. In this letter, a distributed unscented information particle filter () is developed, which reduces the size of the covariance matrix computed, and eeps the tracing algorithm computationally practical. In comparison with the [9], the does not require a CP and eliminates the need for computation of a large Hessian matrix. The proposed updates the Fisher information matrix Ĩ n 1 ( P n 1 ) 1 and the information state vector Z n 1 Ĩ n X 1 n 1 at each local AVS node instead of the source state X n 1 and the covariance matrix P n 1. Starting at the conclusion of iteration ( 1), the global source state estimate X n 1 and the global covariance estimate P n 1 (Step below) are available. The th iteration of the is explained next. Step 1: A set of sigma points is calculated at node n as follows { j X 1 = X n 1 ± (n x + κ) P 1} n, (5) where κ is a scaling parameter, n x is the dimension of X n 1, and term { (n x + κ) P 1} n j corresponds to the jth column of the square root of matrix (n x + κ) P n 1. The initial condition is given by X n, = X n 1. The associated weights for the sigma points are W j = 1/((n x + κ)) with the following initial condition W = κ/(n x + κ). Step : Sigma points are updated based on () as follows X 1 = FX 1 + Gv, for j =,..., n x. () The predicted measurements based on these sigma points are Y 1 = gn (X 1 )Sn, for j =,..., n x, (7) where S n is maximum lielihood estimate of the source signal, S n = ((g n (X 1 ))H g n (X 1 )) 1 (g n (X 1 ))Yn, () with superscript H denoting the Hermitian operator. The predicted source state, the error covariance matrix, and the cross-covariance are, respectively, obtained as where X and X n 1 = P n 1 = P n,x Y 1 = n x W j X 1, (9) j= n x j= n x j= W j X W j X ( X )T, (1) (Ỹ ) T, (11) = X 1 Xn 1 and Ỹ = Y 1 Yn.

3 Step : The predicted information matrix and state vector are expressed as I n 1 = (P n 1) 1, Z n 1 = I n 1X n 1. The information contribution equations are given by K n = (H n ) T (R n ) 1 ( Ξ n + H n X n 1), (1) G n = (H n ) T (R n ) 1 H n, (1) where (H n ) T = ( P n 1 ) 1P n,x Y 1, () Ξ n = Y n g n (X n 1)S n. (15) Step : The Fisher information matrix and the information state vector are updated as follows Ĩ n = K n +I n 1, Zn = G n +Z n 1. () The incorporates the global state statistics from the previous iteration ( X n 1 and P n 1 ) into Step 1 to compute Ĩ n 1 and Z n 1 (Step ), which are then used below in Step 5. Step 5: The computation savings come at this step. Local PF at node n, uses Ĩn 1 and Z n 1 (computed at Step ) to approximate the optimal importance function q(x n Xn 1, Yn ), i.e., the particles are generated as follows X n,(l) q(x n X n 1, Y n ) = N( X n, P n ), (17) 1 where Pn = (Ĩn ), and Xn = P n Z n. Note that the size of the matrix to be inverted in the is of O(M M), while for the UPF, the size of the matrix to be inverted is of O(NT NT ) where NT M. Step : The importance weights at node n are calculated as = 1 p(y n Xn,(l) where transition density p(x n,(l) p(y n Xn,(l) )p(x n,(l) 1 ) q(x n,(l) 1, Yn ), (1) 1 ) and lielihood ) are obtained from () and () respectively. Step 7: After resampling, the minimum mean square error (MMSE) estimate of the local state vector X n and its corresponding error covariance P n is computed as X n = P n = L l=1 L l=1 X n,(l) ; (19) (X n,(l) X n )(X n,(l) X n ) T. () To summarize, Steps 1 to of the compute the local Fisher information matrix Ĩn and information vector Z n at node n using the previous global statistics ( X n 1 and P n 1 ) to be used as the proposal distribution at local PFs (Steps 5 to 7). The next step of the, Step introduced below, is to fuse local statistics to obtain the global state estimates. IV. DISTRIBUTED INFORMATION FUSION Step : The local particles and their associated weights are computed based on the local measurements Y n, (Steps 5 to 7). The local state estimates E(X n Yn ) are, therefore, localized and different across the AVS networ. In order to provide consistency among the local estimates, an information fusion step is introduced to combine local statistics X n,p n,z n 1, and I n 1 into a common set of global statistics X n and P n across the networ. Based on the Chong-Mori-Chang tracfusion theorem [1], the following fusion rules are derived [ Pn ] 1=I n 1 + N n=1 ] 1 [ X n = [ Pn Z n 1+ {[ ] P n 1 } n I 1 ; (1) } {{ } P c( ) N {[ n=1 P n ] 1X n Z n 1 } {{ } x c( ) }].() Using average consensus algorithms [11], {x c ( ), P c ( )} are obtained by iterating the following consensus rules X n c (t + 1) = X n c (t) + ɛ j ℵ (n) (X j c(t) X n c (t)), () P n c (t + 1) = P n c (t) + ɛ j ℵ (n) (P j c(t) P n c (t)), () where ɛ (, 1/ G ) with the following initial conditions [ ] P n c (t = ) = P n 1 n I 1, (5) [ ] 1X X n c (t = ) = P n n Z n 1. () Eqs. () and () converge to {X c ( ), P c ( )} after convergence of the consensus algorithm. The consensus rules in Eqs. () and () are distributed where communication is limited to local neighborhood. After a burn-in period, the global estimation of source state X n and corresponding covariance P n can be obtained. V. SIMULATIONS Both single source tracing and multiple source tracing scenarios are organized to study the performance of the proposed approach. Two different source trajectories are simulated: one is from ( 1,, )m to ( 1,, 1)m, and the other from (11,, )m to (1,, 1)m with s. For single source tracing, only the first source is considered. For multiple source tracing, both sources are simultaneously active. Such motions result in a velocity of ±.5m/s roughly along each coordinate axis. 15 AVSs are deployed to formulate a distributed AVS networ. The sensor locations are randomly drawn in the -D space. The wideband source signals are uncorrelated from each other and from one snapshot to the next so that each signal has a flat spectral density and its bandwidth equals the sampling frequency. The bacground noise (evaluated by the signal to

4 5 1 x position (m) 5 1 x position (m) y position (m) 5 y position (m) 5 z position (m) ground truth z position (m) ground truth Fig. 1. Tracing results of x (up), y (middle), and z (bottom) coordinate under SNR = db and T = for (a) single source and (b) two sources. Fig.. Tracing results of x (up), y (middle), and z (bottom) coordinate under SNR = db and T = for (a) single source and (b) two sources. noise ratio (SNR)) is simulated by adding the complex circular white Gaussian noise to the received signal. The tracing performance of the proposed approach is compared with that using localization algorithm [1], tracing algorithm [] and tracing algorithm [9]. The tracing algorithm is not available in a straightforward manner for the AVS based tracing problem here. We have replaced the pseudo lielihood in [] by the Capon beamforming response [] for single source case and concentrated lielihood [9] for multiple source case. It is worth mentioning that the approach cannot be employed for multiple source localization scenario since it is difficult to associate the DOA with the correct source. In our implementation, we assume perfect DOA association is available for the algorithm, i.e., the measured DOAs are associated with ground truth DOA of each source. The approach is then applied to estimate the source positions. The parameters for proposed are set to: L = 5, Σ v =.1I M, κ =.5 and Γ = 5I T. This parameter setup is found adequate for the following simulations. The source velocities are initialized around the ground truth. The initial positions are coarsely estimated by using a maximum lielihood estimator. Figure 1 and Fig. show the -D position estimation results from a single implementation under SNR = db and T =. Under both single source and two sources scenario, the is able to trac the trajectories of sources accurately. The tracing result of the is significantly better than that of the and approaches. It is also noted that the tracing result of the is similar to that of its centralized counterpart (). To evaluate the average performance, the root mean square error (RMSE) over 5 Monte Carlo (MC) runs under different SNRs and different numbers of snapshots for single source and multiple sources are presented in Fig. and Fig., respectively. The number of snapshots is fixed to for studying the performance under different SNRs and the SNR is fixed to db for studying the performance under different numbers of snapshots. The results show that the proposed is able to provide good tracing accuracy for multiple sources as well as single source. It performs much better than the and approaches, and its performance is favorably close to the. When the SNR is relatively high and the number of snapshots is relatively large, the performance of the proposed is virtually overlaps with that of the. However, due to distributed implementation, both communication and computation cost of the are lower than the. VI. CONCLUSIONS A distributed tracing approach for -D source tracing using an AVS networ is proposed in this paper. The pro-

5 SNR (db) 1 SNR (db) Fig.. SNRs num. of snapshots (b) RMSE over 5 MC runs for single source tracing versus different num. of snapshots (b) Fig.. RMSE over 5 MC runs for multiple source tracing versus different numbers of snapshots. posed is able to directly fuse the information from the received signals and trac multiple wideband acoustic sources. It outperforms the localization approach [1] and centralized PF approach []. Also, the performance of the is favorably comparable with that of [9] when SNR is relatively high and the number of snapshots is relatively large (e.g., SNR > db and T >1). The performance bound and applications of the proposed approach in real acoustic environments will be studied in future wors. REFERENCES [1] M. Hawes and A. Nehorai, Wideband source localization using a distributed acoustic vector-sensor array, IEEE Trans. Signal Process., vol. 51, no., pp ,. [] V. N. Hari, G. V. Anand, A. B. Premumar, and A. S. Madhuumar, Design and performance analysis of a signal detector based on suprathreshold stochastic resonance, Signal Process., vol. 9, no. 7, pp , 1. [] X. Zhong, A. B. Premumar, and A. S. Madhuumar, Particle filtering and posterior Cramér-Rao bound for -D direction of arrival tracing using an acoustic vector sensor, IEEE Sensors J., vol. 1, no., pp. 77, 1. [] X. Zhong and A. B. Premumar, Particle filtering approaches for multiple acoustic source detection and -D direction of arrival estimation using a single acoustic vector sensor, IEEE Trans. Signal Process., vol., no. 9, pp , 1. [5] D. Levin, E. A. P. Habets, and S. Gannot, On the angular error of intensity vector based direction of arrival estimation in reverberant sound fields, J. Acoust. Soc. Amer., vol. 1, no., pp , 1. [] A. Song, A. Abdi, M. Badiey, and P. Hursy, Experimental demonstration of underwater acoustic communication by vector sensors, IEEE J. Ocean. Eng., vol., no., pp. 5 1, 11. [7] Deo-Jin Lee, Nonlinear estimation and multiple sensor fusion using unscented information filtering, IEEE Signal Process. Lett., vol. 15, pp. 1,. [] D.B. Ward, E.A. Lehmann, and R.C. Williamson, Particle filtering algorithms for tracing an acoustic source in a reverberant environment, IEEE Trans. Speech Audio Process., vol. 11, no., pp.,. [9] X. Zhong, A. B. Premumar, and C.-T. Lau, An extended information pf for wideband acoustic source tracing using a distributed AVS array, in Proc. th Conf. European Signal Process., pp , 1. [1] A. Mohammadi and A. Asif, A constraint sufficient statistics based distributed particle filter for bearing only tracing, in Proc. IEEE Int. Conf. Commun., pp. 7 75, 1. [11] A. Mohammadi and A. Asif, Distributed particle filter implementation with intermittent/irregular consensus convergence, IEEE Trans. Signal Process., In press 1. [1] C.Y. Chong, S. Mori, and K.C. Chang, Multitarget-Multisensor Tracing: Advanced Applications, chapter Distributed Multitarget Multisensor Tracing, Artech House, 199.