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Passive source localization using compressively sampled vector sensor array N. Suresh Kumar∗ , Soniya Peter† , A Unnikrishnan‡ , C. Bhattacharya§ ∗ Naval
Physical and Oceanographic Laboratory, Kochi, India Email:
[email protected] † Rajagiri School of Engineering and Technology, Kochi, India ‡ Naval Physical and Oceanographic Laboratory, Kochi, India § Defence Institute of Advanced Technology, Pune, India
Abstract—Accurate estimation of direction of arrival (DoA) with minimum ocean bed array hardware is a challenging task. Any reduction in the data acquisition circuitry positioned in the ocean bed significantly improves the signal integrity and hence the localization performance. In this paper, a novel compressively sampled acoustic vector sensor (CS-AVS) array architecture is proposed to estimate the DoA of multiple acoustic sources. Bearing localization is effectively achieved by customizing multiple signal classification (MUSIC) and minimum variance distortionless response estimator (MVDR) to suit the CS-AVS array. This architecture guarantees a substantial reduction in the number of sensors, acquisition hardware, bandwidth, number of snapshots and the software complexity. Index Terms—Acoustic vector sensor array, DoA estimation, Compressive sampling, Compressive beamforming
I. I NTRODUCTION Multiple threat scenarios exist to high value assets on or near ports and its surveillance have become one of the most intricate and challenging problem faced by the Navy today. Traditionally, long ocean bed array, consisting of hundreds of uniformly distributed pressure sensors, are installed on the ocean bed, to detect and localize the subsurface target. The sensor output is conditioned and then digitized using multibit analog to digital converter (ADC). The digitized channel data are send to the base station through a radio frequency or satellite link. The deployment of long ocean bed array along with its signal conditioning hardware on the ocean bed is a challenging task involving huge cost and effort. The investments can be greatly reduced if any reduction in number of sensors, front-end circuitry or data rate can be achieved. It is well known that, the strategic performance of Waterside Security System(WSS) mainly depends upon the capability of the system to localize an underwater threat with minimum number of snapshots or time samples. In this paper, hardware efficient ocean bed array architecture is proposed which greatly reduces the complexity of the WSS with superior strategic performance. The proposed system utilizes a compressively sensed acoustic vector sensor (AVS) array along with high resolution compressive beamformer, suitable for localizing multiple acoustic sources. An AVS [1] completely characterizes the acoustic field at a point in space through simultaneous measurements of the three orthogonal components of the particle velocity along with the
acoustic pressure. Though each AVS provides the acoustic pressure and the tri-axial particle velocity components, in this work, the estimation of azimuth angle is focused and hence the acoustic pressure and the particle velocity component along the horizontal plane are used. Particle velocity component along the vertical axis is not considered. Throughout this paper, three measurements are used at each AVS output. In the proposed architecture, an array consists of linearly distributed AVS, is installed on the ocean bed for passive source localization. An AVS based ocean bed array requires 13 rd of the number of sensors [2] compared to the traditional pressure sensor ocean bed array to achieve the same localization performance. Though an AVS array utilizes lesser number of sensors, the major issues to be addressed in conjunction with AVS array deployment for a given localization performance are: (1) number of channels and the associated analog signal conditioning hardware remains same as that of the pressure sensor array. This is due to the fact that, reduction in the number of sensors in the AVS based array is compensated by increase in the number of measurements; (2) transmission data rate through RF link and the size of the spatial correlation matrix remains same as that of pressure sensor array. Thus, it is not worthy that, though an AVS array drastically reduces the number of sensors, the number of channels and the associated signal conditioning hardware, transmission data rate, and spatial correlation matrix dimension remains same as that of pressure sensor array for achieving a bearing performance. Compressive sampling is a new signal acquisition technique which requires fewer measurements or front end signal conditioning hardware chain to represent or reconstruct the signal, which are sparse in some basis vectors. In this present context, the array signal vector is sparse in angular spectral basis. The applications of compressive sampling was intitiated by Donoho [2] and Candes et al. [3] in the year 2006. A good review of compressive sensing is given in [4]. Compressive beamforming is a method to recover the sparse angular spectral vector from very few non-adaptive, linear measurements. The outline of the paper is as follows. In section II, we present the AVS array data model. The details of compressive sampling is discussed in section III. The DoA estimation techniques sparse beamformers are presented in section IV. Simulation results are presented in section V. Section VI
978-1-4673-5090-7/13/$31.00 ©2013 IEEE
261 where
concludes the paper.
η (t) = [η1 (t), η2 (t), . . . , ηJ (t)]T ∈ C J×1 ,
II. AVS ARRAY DATA MODEL Consider a uniform horizontal AVS array of N equispaced sensors positioned in a horizontal plane along the x- axis as shown in Fig.1. Consider J mutually uncorrelated narrowω band point sources of centre frequency 2π located at (rj , θj ) with respect to the first sensor of the array. The sources are positioned at azimuth angles θj , j = 1, 2, . . . , J are measured with respect to the axis of the array. The complex amplitude of acoustic pressure at the n-th sensor due to the j-th source, is given by pnj = ei(n−1)kd cos θj η j (t),
(1)
where k is the wavenumber, d is the inter-sensor spacing and η j (t) is the slowly varying complex envelope of the analytic signal from the j-th source. The amplitude η j (t) is a random process with mean zero and variance σj2 = E η 2j (t) .
(6)
is the source signal vector and A is the 3N ×J array manifold matrix defined as A = [a(θ1 ) a(θ2 ) . . . a(θJ )],
(7)
where a(θj ) is the steering vector corresponding to the source direction θj . a(θj ) = c(θj ) ⊗ d(θj ),
(8)
where ⊗ denotes Kronecker product and c(θj ) = [1, eikdcosθj , . . . , ei(N −1)kdcosθj ]T ,
(9)
2π where k= , λ being the wavelength of the received signal λ and √ √ (10) d(θj ) = [1, 2ρc cos(θj ), 2ρc sin(θj )]T . The array noise vector w(t) is given by w(t) = [w1 (t), . . . , w3N (t)]T ,
(11)
where w1 (t), . . . , w3N (t) are the i.i.d circular complex random variables with variance σ 2 and wn (t) = [wp,n (t), wvx ,n (t), wvy ,n (t)].
(12)
The SNR for the j-th source is defined as (SN R)j =
The relation between the acoustic pressure p and the particle velocity v at a point r = (x, y, z) and time t is governed by the law of conservation of momentum which is given by ∂vv (rr , t) + ∇p(rr , t) = 0. (2) ∂t Using equations (1) and (2) and invoking the plane wave and the far-field approximation, the complex amplitudes of x and y components, respectively, of the particle velocity at the n-th sensor due to the j-th source are given by
ρ
cos θj i(n−1)kd cos θj = η j (t), e ρck
(3)
sin θj i(n−1)kd cos θj η j (t), e ρck
(4)
vynj =
where ρ is the density of the water and c is the speed of sound in water. The received signal at the sensor array at time t can be expressed as x(t) = Aηη (t) + w(t) ∈ C 3N ×1 ,
(13)
The correlation matrix of the data vector y(t) is defined as
Fig. 1: Source array geometry.
vxnj
σj2 . σ2
(5)
R3N = E[y(t)y(t)H ].
(14)
In practical calculations, considering the received data is finite, the true correlation matrix can be estimated as the following sample correlation matrix ˆ 3N = 1 [y(t)y(t)H ], R L t=1 L
(15)
where L is the number of snapshots. III. C OMPRESSIVE S AMPLING ON AVS ARRAY PROCESSING
Compressive sampling is an emerging sampling technique which uses few linear measurements in comparison with the conventional Nyquist sampling theory together with a nonlinear recovery process. Compressive sampling enables sparse or compressible signals to be captured and stored at a rate much below the Nyquist rate. The reconstruction of the original signal from its random projections is possible by means of an optimization process as long as the measurements satisfy reasonable conditions such as incoherence and Restricted Isometry Property (RIP) [7].
262 Consider a signal vector x ∈ C N . If x is K-sparse in some orthonormal basis, then x can be represented as x = Ψ s,
(16)
where Ψ ∈ C N ×N is the sparsity basis matrix and s is a N × 1 vector with K N non-zero entries. The CS theory states that x can be recovered using H = KO(log N ) non-adaptive linear projection measurements on to a random matrix Φ ∈ C H×N ,which is typically called sensing matrix or measurement matrix. The compressed signal vector y can be written as y = Φ x = ΦΨ s = Ω s,
(17)
where Ω = ΦΨ ∈ C H×N . The choice of measurement matrix Φ is important since it decides the stability and reliability of the compressive sensing process. The measurements should satisfy incoherence with respect to the original basis and RIP for the recovery of the original signal from its randomized projections by means of an optimization process. Both the RIP and incoherence can be achieved with high probability simply by selecting Φ as a random matrix. Table.I summarises the hardware requirement of the proposed CS-AVS array architecture and compares it with that of the conventional scalar sensor array and the AVS array. It is clear that with greatly reduced complexity, the CS-AVS array can still achieve similar results in DoA estimation as a conventional large size AVS array [9]. IV. D OA E STIMATION TECHNIQUES AND S PARSE B EAMFORMERS We estimate the angle pseudo-spectrum by modifying MUSIC [10] for use with compressive sampled AVS array (Sparse MUSIC: SMUSIC). Let Enoise be a matrix whose columns are ˆ yc . If we perform a onethe H − J noise eigenvectors of R dimensional scanning of the function A(θ) along θ, where A(θ) is given by equation (7), we get J different A(θj ) (for j=1, . . . , J) which lie nearest to the signal sub-space. In compressive beamforming, we use V(θ) instead of A(θ), where V(θ) = Φ A(θ),
(18)
is the compressed array manifold matrix. The projection of V(θ) on the noise sub-space is given by proj. = V(θ)H Enoise EH noise V(θ).
(19)
The estimated DoAs are given by θSMUSIC = arg min V(θ)H Enoise EH noise V(θ). θ
(20)
Therefore, the spatial pseudo-spectrum estimate of SMUSIC is given by PSMUSIC (θ) =
1 , V(θ)H Enoise EH noise V(θ)
(21)
where θ = θ1 , . . . , θNs (Ns is the total number of scanning angles).
The SMVDR pseudo-spectrum of the compressive sampling array can be derived [11] and is given by 1 PSMVDR (θ) = . (22) V(θ)H R−1 yc V(θ) V. RESULTS AND DISCUSSIONS In this section, the performance of sparse high resolution angular spectral estimators are compared with the conventional high resolution spatial filters by simulating multiple acoustic sources received through an AVS array. All the results have been obtained by averaging over 250 Monte Carlo simulations, each uses 200 snapshots (L=200) for the conventional AVS array and 20 for CS-AVS array, unless otherwise stated, to demonstrate the superior strategic performance of the proposed CS-AVS array processor. The compression matrix H has entries drawn from an i.i.d Gaussian random process with μ=0 1 and σ 2 = H . First, we present the pseudo-spectrum performance of the conventional MUSIC and SMUSIC algorithm, using a 36-element conventional AVS array and the CS-AVS array (N =36, H=10) architecture. We consider two sources at 40◦ and 43◦ with respect to the array end-fire direction with -3 dB SNR. Fig.2 shows a typical pseudo-spectrum response of MUSIC and SMUSIC angular spectral estimator. To demonstrate the superior bearing resolution performance of the CS-AVS array which uses 10 measurements, a 10-element conventional AVS array MUSIC response is also shown in the Fig.2. In this figure, it is observed that: (1) the performance of SMUSIC, which utilizes 10 measurements is not adversely affected in comparison with the conventional MUSIC which utilizes 36element AVS array, consisting of 108 measurements and (2) the conventional 10-element AVS array delivers a poor bearing resolution performance. This clearly indicates the superior performance of the CS-AVS architecture which utilizes only 10.8% of the front-end signal conditioning hardware, and requiring only 2% data rate compared to the conventional AVS array configuration. Fig.3 shows the pseudo spectrum response of MVDR and SMVDR using the same parameters used in the previous experiment. Fig 2 and 3 show that: (1) standard deviation or variance of the pseudo spectrum with respect to the source bearing, is large for sparse algorithm compared to the conventional algorithm; (2) source signal level to the mean pseudo spectrum level is almost same for both conventional and sparse algorithm and (3) 10-element conventional AVS array has poor bearing resolution capability. Higher fluctuations in the sparse algorithms are primarily due to the error in the estimation of the spatial correlation matrix using significantly reduced number of measurements and snapshots. It can be also viewed as a probabilistic reduction of signal entry into the reduced dimensional subspace. It is noteworthy that, AVS array processing algorithms which utilize the inverse of the spatial covariance matrix (say MVDR) to estimate the DoA requires a minimum of 3N snapshots to achieve a non-singular spatial covariance matrix. It is also known that, in the conventional AVS array processing higher
263 TABLE I: Comparison of the different array architectures Parameter No. of Channels No. of signal conditioning hardware Data rate (bits/sec) Spatial Correlation Matrix Minimum number of snapshots
Scalar Sensor Array N N N Fs K N ×N N
AVS Array 3N 3N 3N Fs K 3N × 3N 3N
CS-AVS Array J log (3N ) J log (3N ) J log (3N )Fs 2 J log (3N ) × J log (3N ) J log (3N )
Note: The hardware requirement for the CS-AVS array is greatly reduced for large values of N and K, K=2 bit precision for CS-AVS array
Fig. 2:
MUSIC and SMUSIC pseudo-spectrum of the conventional AVS array (N = 36, L = 200), conventional AVS array (N = 10, L = 200) and the CS-AVS array (N = 36, H = 10, L = 20). Two sources at 40◦ and 43◦ . SNR=-3 dB, f =78 Hz.
Fig. 3: MVDR and SMVDR pseudo-spectrum of the conventional AVS array
(N = 36, L = 200), conventional AVS array (N = 10, L = 200) and the CS-AVS array (N = 36, H = 10, L = 20). Two sources at 40◦ and 43◦ . SNR=-3 dB, f =78 Hz.
bearing resolution is obtained by increasing the array aperture or increasing the number of sensors. It adversely affects the strategic performance of the system and also increases the size of the spatial correlation matrix. However, the CSAVS processing ensures the bearing resolution with minimum number of measurements and hence achieves a high degree of strategic performance. Fig 4 and 5 show, respectively, the plots of bias vs. SNR
Fig. 4: Bias vs. SNR of MVDR, MUSIC, SMUSIC and SMVDR on the conventional AVS array (N = 36, L = 200) and the CS-AVS array (N = 36, H = 10, L = 20), sources at 85◦ and 25◦ , f =78 Hz.
Fig. 5: RMSE vs. SNR of MVDR, MUSIC, SMUSIC and SMVDR on the conventional AVS array (N = 36, L = 200) and the CS-AVS array (N = 36, H = 10, L = 20), sources at 85◦ and 25◦ , f =78 Hz.
and root mean square error (RMSE) vs. SNR for a 36-element conventional AVS array along with that of the CS-AVS array (N=36, H=10) architecture. We have plotted for sources at 85◦ and 25◦ angles. It is seen that, the bias and the RMSE of the bearing estimate is slightly higher with CS-AVS, especially near the end-fire direction. We also compute the CRB [1] for the given signal, noise and array model. It is seen that the CRB varies from 3.8∗10−4 (SNR = -15 dB) to 4.8∗10−6 (SNR = 0 dB) for source bearing 25◦ and 2.6 ∗ 10−4 to 3.2 ∗ 10−6 for source bearing 85◦ .
264 VI. CONCLUSIONS In this work, the high resolution spectral estimator algorithms MUSIC and MVDR were customized to suit the CSAVS array. Simulation results confirmed that, localization and bearing resolution performance obtained using the CS-AVS array is similar to that obtained using a conventional AVS array that utilizes larger number of measurements. The superior strategic performance of the new architecture is demonstrated by localizing multiple acoustic sources with lesser number of snapshots (L=20) in comparison with the conventional AVS array processor. Using compressive measurements, size of the array data vector and the spatial covariance matrix is reduced which leads to minimum number of snapshots to realize nonsingular sparse spatial covariance matrix. ACKNOWLEDGEMENTS The authors express their gratitude to S. Ananthanarayanan of NPOL, Kochi and Dr. Abraham Thomas, of Rajagiri School of Engineering and Technology, Kochi for their time and effort in reviewing the initial manuscript and providing valuable comments and suggestions. R EFERENCES [1] A. Nehorai, E. Paldi, Acoustic vector-sensor array processing, IEEE Transactions on Signal Processing 42 (9) (September 1994) 2481-2491. [2] David L. Donoho, Compressed Sensing, IEEE Transactions on Information Theory 52 (4) (April 2006) 1289-1306. [3] E. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete Fourier information. IEEE Transactions on Information Theory 52 (2) (February 2006) 489-509. [4] K P Soman, R Ramanathan. Digital Signal And Image Processing-The Sparse way, ISA. [5] M.A. Iwen, Combinatorial Sublinear-Time Fourier Algorithms, Foundations of Computational Mathematics 10 (3) (June 2010) 303338. [6] Avinash P, Gandhiraj R and Soman K P. Spectrum Sensing using Compressed Sensing Techniques for Sparse Multiband Signals, In International Journal of Scientific & Engineering Research, 3(5), May 2012. [7] E.J. Candes., The restricted isometry property and its implications for compressed sensing, C. R. Acad. Sci. Sr, 346:589592, 2008. [8] Ying Wang, Geert Leus and Ashish Pandharipande, Direction Estimation Using Compressive Sampling Array Processing, Proc. IEEE/SP 15th Workshop on Statistical Signal Processing (August 2009) 626-629 . [9] Dinesh Ramasamy, Sriram Venkateswaran, Upamanyu Madhow, Compressive Adaptation of Large Steerable Arrays, Proc. Information Theory and Applications Workshop (February 2012) 234-239. [10] R. O. Schmidt, Multiple emitter location and signal parameter estimation, in: Proceedings of RADC Spectrum Estimation Workshop, Rome Air Development Center, Rome, pp.243-258, 1979. [11] Harry L Van Trees. Optimum array processing- Part 4 of Detection, Estimation and Modulation theory, Wiley publications. [12] I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Communications on Pure and Applied Mathematics 57 (11) (November 2004) 1413-1457.
