Multi-Source DOA Estimation Using an Acoustic Vector Sensor Array Under a Spatial Sparse Representation Framework

Transcription

1 DOI /s Multi-Source DOA Estimation Using an Acoustic Vector Sensor Array Under a Spatial Sparse Representation Framework Yue-Xian Zou 1 Bo Li 1 Christian H Ritz 2 Received: 22 May 2014 / Revised: 5 June 2015 / Accepted: 6 June 2015 Springer Science+Business Media New York 2015 Abstract This paper investigates the estimation of the two-dimensional direction of arrival (2D-DOA) of sound sources using an acoustic vector sensor array (AVSA) within a spatial sparse representation (SSR) framework (AVS-SSR-DOA). SSR-DOA estimation methods rely on a pre-defined grid of possible source DOAs and essentially suffer from the grid-effect problem: Reducing the size of the grid spacing leads to increased computational complexity. In this paper, we propose a two-step approach to tackle the grid-effect problem. Specifically, omnidirectional sensor array-based SSR- DOA estimation firstly provides initial low-cost DOA estimates using a coarse grid spacing. Secondly, a closed-form solution is derived by exploring the unique subarray manifold matrix correlation and subarray signal correlation of the AVSA, which allows for DOA estimates between the pre-defined angles of the grid and potentially achieves higher DOA estimation accuracy. To further alleviate the estimation bias due to noise and sparse representation model errors, line-fitting (LF) techniques and subspace techniques (ST) are employed to develop two novel DOA estimation algorithms, referred to as AVS-SSR-LF and AVS-SSR-ST, respectively. Extensive simulations validate the effectiveness of the proposed algorithms when estimating the DOAs of B Yue-Xian Zou zouyx@pkusz.edu.cn Bo Li boli@sz.pku.edu.cn Christian H Ritz critz@uow.edu.au 1 Advanced Digital Signal Processing Laboratory, School of Electronic Computer Engineering, Peking University, Shenzhen , China 2 School of Electrical, Computer, and Telecommunications Engineering, University of Wollongong, Wollongong, Australia

2 multiple sound sources. The proposed AVS-SSR-ST algorithm achieves high DOA estimation accuracy and is robust to various noise levels and source separation angles. Keywords Direction-of-arrival estimation Acoustic vector sensor Sparse representation Grid-effect Line-fitting Subspace technique 1 Introduction Estimating the direction of arrival (DOA) of sound sources with an acoustic vector sensor array (AVSA) was first investigated by Nehorai and Paldi in the field of the underwater and aerial acoustic direction-finding applications [12]. In contrast to an M- element scalar sensor array (SSA), an M-element AVSA is composed of M identically oriented AVS units as shown in Fig. 1b. Generally, each AVS unit consists of an omnidirectional sensor spatially colocated with two or three orthogonally oriented directional sensors, where either particle velocity sensors or differential microphones are typically used as the directional sensors according to different applications [12]. The signals recorded by directional sensors represent the x, y, and z components of the soundfield and can be processed to provide an estimation of the source DOA. For such processing, many existing SSA-based DOA estimation methods, referred to as SSA-DOA estimation methods, have been extended to the single AVS or the AVSA [1,4,9,13,14,16 21], which are referred to as the AVS-DOA estimation methods in this paper. Nehorai et al. have studied beamforming and the Capon methods for the AVSA and the Cramer-Rao bound of the DOA estimation was derived in their initial work [4,12]. The group led by Wong has carried out significant research into DOA estimation with a single AVS and an AVSA [14,16 19]. They have explored the trigonometric interrelations among the sensors of each individual AVS to develop the non-spatial realization of ESPRIT with both a single AVS and an AVSA for DOA estimation of multiple uncorrelated sources. Furthermore, the AVS and AVSA have been proposed for the DOA estimation for different source types, alternative array structures [1,9,13, 20,21], and for source tracking [27,28]. It is noted that the methods discussed above can be viewed as the extensions of the traditional SSA-DOA estimation methods to AVS-DOA estimation methods. Recently, a new DOA estimation approach under the spatial sparse representation (SSR) framework with the scalar sensor array (SSA) has quickly gained intensive attention worldwide [2], which is referred to as the SSA-SSR-DOA estimation method in this paper. Research has shown that the SSA-SSR-DOA estimation method is able to provide higher DOA estimation accuracy compared to the SSA-DOA estimation counterpart [2,5,10,11,22 24,26]. Gorodnitsky and Rao firstly developed a SSA- SSR-DOA estimation algorithm by dividing the interested spatial region into a predefined discrete grid of potential DOAs. An overcomplete dictionary was formed using the steering vectors corresponding to the grid points which are related to the possible DOAs of the sources. As a result, the sparse representation model of the array measurement data was formulated and the DOA estimation was then transformed into an underdetermined inverse problem, which was solved by the iterative focal

3 underdetermined system solver (FOCUSS) [2]. Following the work in [2], Malioutov et al. explored the use of multiple measurement vectors (MMV) jointly with more signal information to formulate the sparse representation model of the measurement data [10,11]. Moreover, the cost function has been reformulated by imposing L 1 - norm penalties to enforce the sparsity, and the singular value decomposition (SVD) technique was further employed to reduce the adverse impact of the noise. According to their paper, the resultant SSA-SSR-DOA estimation method was referred to as the L 1 -SVD method, which is recognized as a popular SSA-SSR-DOA estimation method in this research area. Many methods based on the L 1 -SVD method were proposed to improve the DOA estimation accuracy or reduce the hardware cost [5,7,22 24,26]. Following careful evaluation of the existing SSA-SSR-DOA estimation methods discussed above, we have the following observations: (1) The SSA-SSR-DOA estimation methods have the capability to potentially achieve a higher DOA estimation accuracy with less sensitivity to noise than the traditional SSA-DOA estimation methods; and (2) The DOA estimation accuracy of SSA-SSR-DOA are determined by the grid spacing used to divide the spatial space. For example, if the grid spacing is set as 1, then the DOA estimation accuracy of the SSA-SSR-DOA estimation method is only guaranteed to be 0.5 in the worst case. The smaller the grid spacing used, the higher the DOA estimation accuracy achieved, while the higher the computational cost required. This phenomenon is referred to as the effect of the grid or grid-effect in SSA-SSR-DOA estimation techniques [8,10,11]. To tackle this problem, an iterative grid refinement strategy provides a solution to achieve the compromise between the computational cost and the DOA estimation accuracy [11]. Besides, the sparse total least- square (STLS) method was proposed to resolve the grid-effect problem [23]. Therefore, the research described in this paper is motivated by three concerns: (1) The AVSA has much more compact physical size compared with the SSA with the same sensors, which is desirable in many applications; (2) The SSR framework can be used for the AVSA-DOA estimation to provide high accuracy; and (3) The grideffect problem can be tackled by investigating the additional source DOA information embedded in the signals received by the AVS units. The main contributions of this study are twofold. Firstly, we derive the formulation of DOA estimation with an AVSA under the spatial sparse representation (SSR) framework, which is referred to as the AVS-SSR-DOA estimation method in Sect. 3. Clearly, the AVS-SSR-DOA estimation method is an extension of the SSA-SSR-DOA estimation method following the same principle, and the AVS-SSR-DOA estimation method can achieve much higher DOA estimation accuracy than traditional AVS-DOA estimation methods with more computation effort if the space is discretized densely using a small grid spacing. Secondly, a two-step DOA estimation method with the AVSA under the SSR framework is proposed to tackle the grid-effect problem. Specifically, the SSR-DOA estimation with the omnidirectional sensor array (o-array) firstly provides the initial DOA estimates at lower cost by using a coarse grid spacing. Secondly, a closed-form solution is derived by exploring the unique subarray manifold matrix correlation and subarray signal correlation of the AVSA. This allows for DOA estimates that fall between the pre-defined grid points with the aim of achieving higher DOA estimation accuracy. To further alleviate the estimation bias due to additive noise and the sparse representation modeling error, line- fitting (LF) and subspace techniques (ST) are employed to

4 Fig. 1 Illustration of an AVS unit and an M-element AVS array. a A single AVS unit; b M-element AVSA (M is the number of AVS unit used in the array) develop two novel DOA estimation algorithms referred to as AVS-SSR-LF and AVS- SSR-ST, respectively. Intensive experiments have been carried out to validate their performance under various settings, such as grid spacing, source separation, snapshot numbers, and SNR levels. The remainder of this paper is organized as follows. The AVS data models are presented in Sect. 2. The DOA estimation with an AVS array under the spatial sparse representation (SSR) framework is formulated in detail, and some discussions are presented in Sect. 3. Two novel AVS-SSR-DOA estimation algorithms are derived in Sect. 4. Simulation results are presented in Sect. 5. Section 6 gives the conclusion of our work. 2 Data Models An ideal AVS unit is composed of one omnidirectional sensor and three directional sensors (Fig.1a). Specifically, the omnidirectional sensor is denoted as the o-component, and the directional sensors are depicted as u-, v- and w-components, respectively. These directional sensors are orthogonally oriented along the x, y, and z axes, respectively. In addition, it is assumed that there are K (K < M) far-field spatial uncorrelated narrowband acoustic signals s k (t)(k = 1,...,K ) impinging upon the AVS array. The 4 1 direction vector of the AVS unit for the kth spatial source coming from (θ k,φ k ) can be expressed as [16]: v(θ k,φ k ) [u k,v k,w k, 1] T, (1) where [.] T denotes the vector/matrix transposition, θ k (0, 180 ) is the elevation angle, and φ k [0, 360 ) is the azimuth angle. Moreover, we have the following definitions

5 u k = sin θ k cos φ k, v k = sin θ k sin φ k and w k = cos θ k. (2) From the geometric point of view, u k, v k, and w k in (2) can be viewed as the projections of the kth spatial source onto the x, y, and z axes of the DOA, respectively. Hence, Eq. (2) isreferredtoasthedoadirection cosines, and correspondingly u k, v k and w k are specifically referred to as the x-axis direction cosine, y-axis direction cosine, and z-axis direction cosine, respectively. On the other hand, given the direction cosines, the 2D-DOA estimates of the spatial sources can be determined by φ k = tan 1 v k /u k, k = 1,...,K. (3) 2.1 Measurement Model for an M-Element AVS Array It is well known that an array with M scalar sensors generates the M 1 steering vector for the kth spatial source located at (θ k,φ k ) as q(θ k,φ k ) =[q 1 (θ k,φ k ),..., q M (θ k,φ k )] T, (4) where the mth component in the steering vector can be denoted as q m (θ k,φ k ) = exp[ j2π(x m u k + y m v k + z m w k )/λ], (5) in which λ and (x m, y m, z m ) denote the source wavelength and the location of the mth sensor in the Cartesian coordinates, respectively. In contrast to an M-element SSA, an M-element AVSA is composed of M identically oriented AVS units as shown in Fig. 1b. Its 4M 1 manifold vector for the kth spatial source impinging from (θ k,φ k ) can be denoted as [16] a(θ k,φ k ) v(θ k,φ k ) q(θ k,φ k ) (6) where denotes the Kronecker product operator, and v(θ k,φ k ) and q(θ k,φ k ) are given in (1) and (4), respectively. For all K spatial sources, the single snapshot measurement of an M-element AVSA at time t can be formed as follows where x(t) = K a(θ k,φ k )s k (t) + n(t) = As(t) + n(t), x(t) C 4M 1, (7) k=1 A [a(θ 1,φ 1 ),..., a(θ K,φ K )] C 4M K, (8) s(t) [s 1 (t),...,s K (t)] T R K 1, n(t) [n 1 (t),..., n 4M (t)] T R 4M 1, (9) and where x(t) and n(t) are the measurement data vector and the additive zero-mean white Gaussian noise vector, respectively. Matrix A in (8) is defined as the manifold matrix associated with K spatial sources. When L snapshots (t = t 1,...,t L ) are taken,

6 the so-called multiple measurement vectors (MMV) data model of an M-element AVSA can be written into a matrix form as follows X = AS + N, X =[x(t 1 ),...,x(t L )], S =[s(t 1 ),..., s(t L )], N =[n(t 1 ),...,n(t L )], (10) where x(t), s(t) and n(t) are given in (7) and (9), respectively. 2.2 Measurement Models for the Subarrays in an AVS Array As shown in Fig. 1, an M-element AVSA can be viewed as four different collocated subarrays with scalar sensors, which can be specifically denoted as the u-subarray, v- subarray, w-subarray, and o-subarray, respectively. Each subarray is composed of M components of the corresponding type. For example, the o-subarray can be considered as one SSA formed by M omnidirectional components. In other words, the o-subarray of an M-element AVSA is identical to an M-element SSA discussed in [11]. The derivation of the measurement models for each subarray is shown as follows. Substituting (6)into(8), Eq. (10) can be written as X =[v(θ 1,φ 1 ) q(θ 1,φ 1 ),...,v(θ K,φ K ) q(θ K,φ K )]S + N. (11) Substituting v(θ k,φ k ) in (1) into(11), we expand the Kronecker product and obtain X = AS + N = u 1 q(θ 1,φ 1 ),...,u K q(θ K,φ K ) v 1 q(θ 1,φ 1 ),...,v K q(θ K,φ K ) w 1 q(θ 1,φ 1 ),...,w K q(θ K,φ K ) q(θ 1,φ 1 ),..., q(θ K,φ K ) S + N (12) with the following definitions A u =[u 1 q(θ 1,φ 1 ),...,u K q(θ K,φ K )] A v =[v 1 q(θ 1,φ 1 ),...,v K q(θ K,φ K )] A w =[w 1 q(θ 1,φ 1 ),...,w K q(θ K,φ K )] A o =[q(θ 1,φ 1 ),...,q(θ K,φ K )] A = [ A T u AT v AT w AT o ] T (13) In (13), q(θ k,φ k ) is given in (4). A u, A v, A w, and A o are called the subarray manifold matrices of the u-, v-, w-, o-subarray, respectively. Accordingly, X and N can also be partitioned similarly into four blocks. Equation (12) is identically rewritten as X u A u N u X v = A v S + N v, (14) X w X o A w A o N w N o

7 where S is given in (10), X u, X v, X w and X o represent the measurement data matrices of the u-, v-, w-, and o-subarray, respectively, and N u, N v, N w, and N o represent the additive noise data matrices of the u-, v-, w-, and o-subarray, respectively. Equation (14) can be further expressed as follows X u = A u S + N u, X v = A v S + N v, X w = A w S + N w, X o = A o S + N o. (15) Moreover, from (13), the following correlations between the subarray manifold matrices and the direction cosines (called subarray manifold matrix correlation in short) can be derived A u = A o u, A v = A o v, A w = A o w, (16) where u = diag(u 1,...,u K ), v = diag(v 1,...,v K ), w = diag(w 1,...,w K ). (17) The three matrices in (17) are referred to as the K K - dimensional diagonal direction cosines matrices since their elements on the diagonals are the direction cosines defined in (2). Obviously, from (16), it can be seen that A u is interconnected with A o by the diagonal direction cosines matrix ( u ) defined in (17). Similar observations can be found for the v-subarray and w-subarray. Furthermore, substituting (16) into(15) yields X u = A o u S + N u = A o S u + N u (18) X v = A o v S + N v = A o S v + N v (19) X w = A o w S + N w = A o S w + N w (20) X o = A o S + N o = A o S o + N o (21) with the definitions as follows S u = u S, S v = v S, S w = w S, S o = S (22) Comparing the second equations in (18) (21) with the ones in (15),it is straightforward to see that S u, S v, S w, and S o can be viewed as the new signal sources virtually impinging on the o-subarray (with the subarray manifold matrix A o ), which carry the DOA information of the original spatial sources. Hence, (22) can be viewed as the correlations between the source signal and the direction cosines (referred to as subarray signal correlation in short) of the AVS array. Moreover, in (22), the diagonal direction cosines matrices determine the linear magnitude relations between S u, S v, S w and S (or S o ), respectively.

8 3 DOA Estimation with an AVS Array Under SSR Framework To make the presentation complete, in this section, DOA estimation with an M-element AVS array within the SSR framework (AVS-SSR-DOA estimation in short) will be first formulated in detail before providing some discussions about the algorithm advantages and limitations. Essentially, the objective of DOA estimation with an M-element AVSA is to determine (θ k,φ k ), (k = 1,...,K ) from the 4M L MMV data matrix X given in (10). It is noted that the formulation of the AVS-SSR-DOA estimation problem follows the same principle and procedure as that of the SSA-SSR-DOA estimation except that they have different MMV data models and manifold matrices. The essential principle of SSA-SSR-DOA estimation is to recast the DOA estimation problem as a sparse signal reconstruction problem [11]. In order to make use of the spatial sparsity of the spatial sources, the whole spatial space is discretized into a grid of N points (N >> K ) to form a pre-defined grid ={( θ 1 φ 1 ),...,( θ N φ N )}. Theovercomplete manifold matrix associated with is given as follows: [ ( ) ( )] a θ 1 φ 1,...,a θ N φ N C 4M N, (23) where a is the manifold vector given in (6). Obviously, the larger the value of N, the smaller the grid spacing that is obtained, and it is more probable that the spatial sources would fall on a grid defined by, and a higher DOA estimation accuracy is then achieved. Therefore, replacing A by in (10) results in a sparse representation MMV data model of the AVSA in the following X = Z + N, (24) where Z is a matrix of size N L. It can be seen that the matrix Z should have only K nonzero rows, which correspond to the DOA estimates of the K spatial signal sources. Therefore, the DOA estimates of K sources from (24) can be achieved by locating the indices of nonzero rows in Z, which is the core concept of the SSR-DOA estimation methods. In (24), is pre-defined in (23); X is the measured data vector from the M-element AVSA, and Z is a row sparse matrix to be estimated. Practically, the estimation of Z can be recast as an optimization problem minimizing the following objective [11] X Z 2 f + λ 1 z (l 2 ) 1, (25) where 2 f is the Frobenius norm, λ 1 is the regularization parameter and 1 is the l 1 -norm. The second term in (25) enforces the row sparsity of Z, which is computed as follows [11] [ ] z (l λ 2 ) 1 1 = λ 1 z (l 2) 1,...,z (l 2) 1 N, (26)

9 where we have z (l 2) n = [z n (t 1 ),...,z n (t L )] 2, n = 1,...,N and z n (t l ), l = 1,...,L denotes the element of the n th row and l th column of Z. Research shows that the reconstruction of Z using (25) is a convex optimization problem that can be solved by several techniques [2,11]. Similar to the L 1 -SVD algorithm [11], singular value decomposition (SVD) is adopted to reduce both the computational complexity and the sensitivity to noise. Taking the SVD of X, wehavex = ULV H and define X SV = XVD K, Z SV = ZVD K and N SV = NVD K, D K = [I K 0], (27) where I K is a K K identity matrix and 0 is a K (L K ) zeros matrix. Applying (27), the sparse representation MMV data model of the AVSA in (24) can be rewritten with the reduced dimension representation as follows X SV = Z SV + N SV. (28) Similarly, the estimation of Z SV from (28) can be obtained by minimizing the following unconstrained cost function [11] X SV Z SV 2 f + λ 1 z (l 2 ) 1 (29) where X SV and Z SV are given in (27). The second term in (29) is computed as follows [ ] λ 1 z (l 2 ) 1 = λ 1 z (l 2) 1,..., z (l 2) 1 N (30) in which we have z (l 2) n = [zn SV(1),...,zSV n (K )] 2, n = 1,...,N, and zn SV(k) denotes the element at the nth row and kth column of the Z SV. Moreover, the estimation of Z SV by minimizing the cost function in (29) can be performed efficiently by a second-order cone programming (SOCP) technique at the cost of O((4KN) 3 ) [3]. Assuming that the K dominant rows in the reconstructed Ẑ SV are with indices I k {1,...,N}, k = 1,...,K, the DOA estimates can be denoted as ( ˆθ k, ˆφ k ) = ( θ Ik φ Ik ), k = 1,...,K, (31) where ( θ Ik φ Ik ) is the I k -th angular grid point in the pre-defined grid. In conclusion, the AVS-SSR-DOA estimation method has been formulated within the SSR framework. Before we move to the derivation of the proposed DOA estimation algorithms, we have some discussions: (1) The advantages of the AVS-SSR-DOA estimation method are the following: The performance of the AVS-SSR-DOA is expected to outperform the SSA-SSR-DOA since additional data have been captured in an AVSA. In addition, for a specific array type, such as the uniform linear array (ULA), the AVSA has the ability to resolve the azimuth angle ambiguity [21]. This makes the array configuration more flexible for 2D-DOA estimation applications with an AVS array; (2) The AVS-SSR-DOA estimation method for an M-element AVSA formulated above has the obvious defect of requiring much higher computational cost compared

10 with that of the SSA-SSR-DOA estimation method with the M-element SSA if the same grid spacing is used. One of the reasons is that the inverse problem defined in (28) requires high computational complexity. Besides, there is no fast algorithm to solve the optimization problem in (29) due to the L 1 -norm being used in the cost function; (3) From (31), it is clear to see that the AVS-SSR-DOA estimation method also faces the grid-effect problem, where the DOA estimates are confined to the grid points in the pre-defined grid, and the DOA estimation accuracy of the AVS-SSR-DOA estimation is determined by the grid spacing. This disadvantage is inherited from the SSR-DOA estimation method using a finite discrete grid. A larger value of N leads to higher estimation accuracy but higher computational cost. The compromise between the computational complexity and the DOA estimation accuracy is an important issue for SSR-DOA estimation methods. 4 Proposed AVS-SSR-DOA Estimation Algorithms From (31), it is noted that the AVS-SSR-DOA estimation method utilizes only the index information of the K dominating nonzero rows in the reconstructed Ẑ SV to obtain the final DOA estimates. Other useful information derived from the AVSA signals such as the subarray manifold matrix correlation in (16) and subarray signal correlation in (22) have not been taken into account. To develop efficient DOA estimation algorithms with a high DOA estimation accuracy but comparable computational cost to that of the O-SSR-DOA algorithm, a two-step DOA estimation scheme with an AVSA under the SSR framework is proposed in this section. In the first step, the initial DOA estimates will be obtained by the SSA-SSR-DOA estimation using only the o- subarray with a coarse grid spacing, which leads to less computational cost. Besides, the subarray manifold matrix of the o-subarray A o is also estimated accordingly. In the second step, with the correlations in (16) and (22), a closed-form DOA estimation is derived, and higher DOA estimation accuracy can be achieved since the final DOA estimates are not confined to the grid points of the pre-defined grid. To alleviate the adverse impact of the noise and modeling error on the DOA estimates obtained from the AVSA recordings, we propose two alternative methods that are employed within the SSR framework. The first method is based on line-fitting (LF) and is referred to as the AVS-SSR-LF DOA estimation method. The second method is based on a subspace technique (ST) and is referred to as AVS-SSR-ST DOA estimation method. The details of each of these methods are provided in the following subsections. 4.1 The DOA Estimation Under SSR Framework Using the o-subarray Following the principle of the AVS-SSR-DOA estimation method presented in Sect. 3, the MMV data model of the o-subarray can be denoted as X o = o Z + N o, (32)

11 where o [ ] q( θ 1 φ 1 ),..., q( θ N1 φ N1 ). (33) In (32), Z is the row sparse matrix and has only K nonzero rows corresponding to the K spatial sources. N 1 is the size of the pre-defined grid ={( θ 1 φ 1 ),...,( θ N1 φ N1 )} and satisfies the relation of N > N 1 >> K. This means that the pre-defined grid spacing for is bigger than that for (referred as the coarse grid spacing). The term q( θ k φ k ) represents the steering vector defined in (4). It is easy to understand that if we select N 1 = N/g, g > 1, then the computational cost using will be reduced to 1/(4g) 3 compared with the computational cost using (23) for the AVS-SSR- DOA method. As discussed, to estimate the DOA of the spatial sources, we need to locate the indices of nonzero rows of Z in (32), which can be achieved by using the popular L 1 -SVD technique [11], as shown in Sect. 3. Let us denote the indices of the K dominant rows in Ẑ SV as I k (k = 1,...,K ), then the DOA estimates can be determined as follows ( ˆθ k, ˆφ k ) = ( θ Ik, φ Ik ), I k {1,...,N 1 }, k = 1,...,K, (34) where θ Ik φ Ik are the grid points in the predefined. Moreover, the computational cost by using the L 1 -SVD technique to estimate Ẑ SV from (32) will be of O((KN 1 ) 3 ). Obviously, the saving of the computational complexity is at the cost of degrading the DOA estimation accuracy, since the DOA estimates from (34) are confined on the grid points defined in, which have larger estimation bias than those from (31) where the DOA estimates are confined on the grid points defined in. With the DOA estimates from (34), the corresponding o-subarray manifold matrix A o is estimated as  o = o (:, I) C M K, (35) where I ={I 1,...,I K } is estimated by (34). It is clear that  o is an estimate of the true manifold matrix A o with certain estimation bias. For presentation clarity, the DOA estimation algorithm described in this subsection is referred to the O-SSR-DOA algorithm since it is derived using the o-subarray of AVSA only and is summarized in Table 1. The O-SSR-DOA estimates will serve as the first step of the proposed two-step method. 4.2 The Proposed Closed-Form AVS-SSR-DOA Estimation Method As discussed in Sect. 4.1, the estimation accuracy of the O-SSR-DOA algorithm in Table 1 is determined by the size of the grid spacing used. In the following sections, we will explore the information provided by the subarray manifold matrix correlation and the subarray signal correlation of the AVSA to develop a closed-form AVS-SSR-DOA estimation solution, which has the ability to partially address the grid-effect problem and obtain a higher DOA estimation accuracy.

12 Table 1 The O-SSR-DOA algorithm Initial input: set parameter N 1 Form the o-subarray MMV data X o Define the spatial grid ={( θ 1 φ 1 ),...,( θ N1 φ N1 )} Start (1) Construct o [q( θ 1 φ 1 ),...,q( θ N φ N )] in (33)using(4) (2) Obtain Ẑ SV from (32)byL 1 -SVD method, locate the nonzero rows in Ẑ SV, get the indices of nonzero rows (3) Calculate ˆθ k and ˆφ k using (34) (4) Determine  o from (35) End Output ˆθ k, ˆφ k Let us revisit the correlations shown in (22). It is noted that if the direction cosine matrices u, v and w are estimated properly, a closed-form DOA estimation solution can be achieved by (3). This concept is the essential basis behind the development of the efficient DOA estimation algorithms for an AVS array proposed in this paper. As discussed above, with the O-SSR-DOA algorithm, A o can be estimated as  o in (35) where the DOA estimates are confined on the grid points defined in. Replacing A o with  o, equations from (18) to(21) yield X u  o u S + N u =  o S u + N u, X v  o v S + N v =  o S v + N v, X w  o w S + N w =  o S w + N w, X o  o S + N o =  o S o + N o, (36) where S u, S v, S w and S o are defined in (22). The approximation sign in (36) is added due to the replacement of A o by its estimate  o. The minimum mean square error (MMSE) estimation of S u, S v, S w and S o from (36) can be obtained as [Ŝ u, Ŝ v, Ŝ w, Ŝ o ]= + o [X u, X v, X w, X o ], (37) where ( ) + denotes the pseudo-inverse operator. From (22), we can derive the following ( ) Ŝ u = S u + S u = u Ŝ S o + S u = u u Ŝ v = S v + S v = v ( Ŝ o S o + S u ) S o + S v Ŝ v = v Ŝ o S o + S Ŝ w = S w + S w = w S o + S ( v ).(38) w Ŝ o = S o + S Ŝ w = w Ŝ o S o + S w o Ŝ o = S o + S o With simple manipulation of (38), we have Ŝ u = u Ŝ o + e u, Ŝ v = v Ŝ o + e v, Ŝ w = w Ŝ o + e w, (39)

13 Table 2 The closed-form AVS-SSR-DOA algorithm (1) Use the O-SSR-DOA estimation method in Table 1 to obtain  o (2) Compute Ŝ u, Ŝ v, Ŝ w and Ŝ o from (37) (3) Compute u, v, w from (41) (4) Compute the DOA estimates from (3), Output ˆθ k, ˆφ k where e u = S u u S o, e v = S v v S o and e w = S w w S o. (40) The quantities e u, e v, e w in (40) can be treated as modeling the errors contributed by both the additive noise in the data model (32) and the estimation bias of  o. According to the structure of the AVS unit, it is inferred that the variations of e u, e v, e w in (39) and the adverse impact of the additive noise on the subarray signal correlation in (36) can be suppressed due to the directional projection of the AVS unit. Hence, the desirable subarray signal correlation shown in (22) can be maintained approximately for Ŝ u, Ŝ v, Ŝ w and Ŝ o, which yields Ŝ u u Ŝ o, Ŝ v v Ŝ o, Ŝ w w Ŝ o. (41) According to (37) and (41), a closed-form DOA estimation method for an AVSA can be achieved, which is summarized in Table 2 for presentation clarity. However, it is clear that when the variations of e u, e v, e w in (39) cannot be ignored due to strong additive noise at each sensor and a smaller value of N 1 (spatial grid spacing is too large), the subarray signal correlations shown in (41) will be affected and the DOA estimation accuracy of the closed-form AVS-SSR-DOA algorithm in Table 2 will degrade. The following two sections propose the use of both a line- fitting and a subspace techniques to deal with the modeling errors in (39) introduced by additive noise and estimation bias in  o. 4.3 The Proposed AVS-SSR-LF DOA Estimation Method Observing the subarray signal correlations under the noise condition given in (39), the underlying linear relationship between Ŝ u, Ŝ v, Ŝ w and Ŝ o can be extracted using linefitting. Without loss of the generality, we take the first equation in (39) to demonstrate the derivation of the line-fitting-based algorithm. The kth row of the first equation in (39) can be formulated equivalently as follows Ŝ u (k, t) = u k Ŝ o (k, t) + e u (k, t) (t = t 1,, t L, k = 1,...,K ), (42) where Ŝ u (k, t), Ŝ o (k, t) and e u (k, t) denote the element at the kth row and the tth column in matrix Ŝ u, Ŝ o and e respectively. Equation (42) can be viewed as a homogeneous linear function between Ŝ o (k, t) and Ŝ u (k, t) with the slope coefficient u k and

14 the error term e u (k, t). Moreover, u k is the x axis direction cosine and thus real, while Ŝ o (k, t) and Ŝ u (k, t) are complex. The linear relation in (42) holds for both the real and the imaginary parts as follows ) ) R (Ŝu (k, t) = u k R (Ŝo (k, t) +R(e u (k, t)), ) ) I (Ŝu (k, t) = u k I (Ŝo (k, t) +I(e u (k, t)), (43) in which R( ) and I( ) extract the real and imaginary parts of a complex number, respectively. From (43), the following data vectors can be formed [ T S u R(Ŝ u (k, t 1 )),..., R(Ŝ u (k, t L )), I(Ŝ u (k, t 1 )),..., I(Ŝ u (k, t L ))], [ T S o R(Ŝ o (k, t 1 )),..., R(Ŝ o (k, t L )), I(Ŝ o (k, t 1 )),...,I(Ŝ o (k, t L ))]. (44) It is clear that if the error term is not presented in (42), the scatter plot of S u versus S o will form a line with the slope of u k, which corresponds to the direction cosine of the kth spatial source. With the error term, the points in the scatter plot of S u versus S o will cluster along that line. The least- squares line-fitting technique [15] is adopted to get a good estimate of the slope of the line, i.e., the direction cosine, as follows û k = 2KL S T u S o 1 T S u 1 T S o 2KL S T o S o 1 T S o 1 T S o, (45) where 1 denotes the vector of size 2KL 1 with all elements equal to 1. Similarly, ˆv k and ŵ k can be estimated following the same procedure as that for estimating u k. With the direction cosine estimates û k, ˆv k and ŵ k, the closed-form DOA estimate for the kth spatial source can be determined by ˆθ k = cos 1 ŵ k, ˆφ k = tan 1 ˆv k /û k. (46) At this point, the closed-form 2D-DOA estimation method under the SSR framework for an AVS by employing the least-squares line-fitting technique has been derived, which is referred to as the AVS-SSR-LF DOA estimation algorithm (AVS- SSR-LF for short). For presentation clarity, the proposed AVS-SSR-LF algorithm is summarized in Table 3. From the derivation above, we have the following observations: (1) Reduction in the grid-effect: As discussed before, it is noted that the DOA estimates by the O-SSR-DOA algorithm (Table 1) are confined to the discrete angles defined in. Thegrid-effect is essentially introduced by the DOA estimation method under the SSR framework. However, the DOA estimates resulting from the proposed AVS-SSR-LF algorithm (Table 3) are not confined to anymore. With the models shown in (39), the direction cosines (û k, ˆv k, ŵ k ) can be estimated by the least-squares solutions and accordingly the closed-form DOA estimation results are achieved in (46).

15 Table 3 The proposed AVS-SSR-LF algorithm Initial input: Form the subarray measured data X o, X u, X v, X w Pre-define the spatial grid ={( θ 1 φ 1 ),...,( θ N1 φ N1 )} Start (1) Construct o [q( θ 1 φ 1 ),...,q( θ N φ N )] using (4) (2) Obtain Ẑ from (32)byL 1 -SVD method, locate the nonzero rows in Ẑ, get the indices of nonzero rows (3) Determine  o from (35) (4) Calculate Ŝ u, Ŝ v, Ŝ w and Ŝ o using (37) (5) Form data vectors S u and S o using (44), and estimate û k using (45) (6) Form data vectors S v and S o using the formula as (44), and estimate ˆv k using the formula as (45), where S u is replaced by S v (7) Form data vectors S w and S o using the formula as (44), and estimate ŵ k using the formula as (45), where S u is replaced by S w (8) Calculate ˆθ k and ˆφ k using (46) End Output ˆθ k, ˆφ k (2) Robustness to additive noise: There is no doubt that additive noise will adversely degrade the performance of the AVS-SSR-LF algorithm. The AVS-SSR-LF algorithm using the line-fitting technique to estimate the direction cosine matrices is based on the least-squares solutions obtained in (37) and the data model introduced in (39). More specifically, the least-squares estimation of Ŝ u, Ŝ v, Ŝ w and Ŝ o in (37) can be considered as the optimal solution when the additive noise has a stationary Gaussian distribution. For DOA estimation using the AVS-SSR-LF algorithm with non-stationary and non-gaussian noise, a robust estimator must be employed to replace the estimation in (37). Moreover, when the additive noise level goes higher, the error terms e u, e v, e w in (40) play more dominating roles in the subarray signal correlation and the least-squares estimation of the direction cosines using the line-fitting technique is biased due to the contribution of the error terms. As a result, the DOA estimation accuracy of the AVS-SSR-LF algorithm degrades with the increase in the additive noise level. To ensure the DOA estimation method is robust to noise, a more sophisticated algorithm should be considered, which motivates us to adopt the subspace technique described in the next subsection. (3) Computational complexity: From the AVS-SSR-LF algorithm shown in Table 3, the computational cost for estimating  o from (35) iso((kn 1 ) 3 and the cost for estimating direction cosines are about O(4KL), where K, L, and N 1 are the number of the spatial sources, the snapshots and the pre-defined grid, respectively. One example of the comparisons of the computational cost is shown in Table 5.It can be seen that the proposed AVS-SSR-LF algorithm provides a good trade-off between the DOA estimation accuracy and the computational cost.

16 4.4 The Proposed AVS-SSR-ST DOA Estimation Method Aiming at further reducing the adverse impact of strong additive noise on the AVS- SSR-LF algorithm (Table 3), the subspace technique is introduced to reformulate the estimation of the direction cosine matrices. Accordingly, a closed-form DOA estimation method in the signal subspace is derived. Let us denote the 4M 4M autocorrelation matrix of the received data x(t) as R xx, which is practically approximated by XX T /L with X being the MMV given in (10) and L being the number of snapshots used. The eigendecomposition of R xx results in the 4M K dimensional signal subspace matrix E s, which is composed of K signal eigenvectors of R xx. It is noted that the signal subspace matrix E s holds the following relation with the manifold matrix A [16]: E s AM T, (47) where M T is a non-singular K K transform matrix and A is given in (8). Substituting the row partitions of A defined in (13) into(47), we have [ ] T [ ] T Esu T, ET sv, ET sw, ET so Au T, AT v, AT w, AT o MT. (48) Let us take the u-subarray and o-subarray to illustrate the algorithm derivation. Substituting the subarray manifold matrix correlation between the u-subarray and o- subarray in (16)into(48), the subarray signal subspace matrices satisfy the following [16]: E su A u M T = A o u M T, (49) E so A o M T. (50) Replacing A o from (50) by its estimate  o given in (35), it gives E so =  o M T. (51) In (51), E so can be obtained by the eigendecomposition of R xx. Hence, the transform matrix M T can be estimated as ˆM T =  + o E so. (52) Therefore, the direction cosine matrix u in (49) can be estimated as Substituting (52) into(53), we have ˆ u = argmin u Esu  o u ˆM T 2 f. (53) ˆ u = argmin u Esu  o u  + o E so 2. (54) f

17 Table 4 The proposed AVS-SSR-ST algorithm Initial input: subarray measured data X o, X u, X v, X w Pre-defined the spatial grid ={( θ 1 φ 1 ),...,( θ N1 φ N1 )} Start (1) Construct o [q( θ 1 φ 1 ),...,q( θ N φ N )] using (4) (2) Obtain Ẑ from (32) bythel 1 -SVD method, locate the nonzero rows in Ẑ and obtain the indices of nonzero rows (3) Determine  o from (35) (4) Form the MMV in (10) and calculate R XX = XX T /L (5) Complete the eigendecomposition of R XX to generate E s =[Esu T, ET sv, ET sw, ET so ]T (6) Estimate u, v and w from (55)and(56) (7) Obtain û k, ˆv k and ŵ k using (57) (8) Calculate ˆθ k and ˆφ k using (46) End Output ˆθ k, ˆφ k Employing the least-squares estimation technique, the solution of (54) is given as ˆ u =  + o E sue + soâo. (55) We note that the approach starting from (49) (55) can be directly extended to analyze the relations between E sv, E sw and E so, respectively. Then, the least-squares estimates of v and w can be expressed as follows ˆ v =  + o E sve soâo, + ˆ w =  + o E swe soâo. + (56) From the definitions of u, v and w in (17), the estimates for u k, v k and w k (k = 1,...,K ) can be obtained accordingly as û k =[ˆ u ] kk, ˆv k =[ˆ v ] kk, ŵ k =[ˆ w ] kk, (57) where [.] kk denotes the kth diagonal element of a matrix. Following the definition of u k, v k, and w k, the closed-form DOA estimate is given by (46). The derived closed-form DOA estimation algorithm based on the subspace technique (ST) is referred to as the AVS-SSR-ST DOA estimation algorithm (AVS-SSR-ST for short). For presentation clarity, the proposed AVS-SSR-ST algorithm is summarized in Table 4. In summary, we have the following observations and discussions: (1) Reduction in the grid-effect: Comparing Tables 1, 3 and 4, we can see that the proposed AVS-SSR-LF and AVS-SSR-ST algorithms bear the advantages of the

18 Table 5 Computational costs of different SSR-DOA algorithms Algorithm Computational complexity One example parameter setting: Operations involved L 1 -SVD [11] O((KN) 3 ) K = 2; N = 180; N 1 = 90 I = 3; J = 4; M = 8; L = L 1 -SVD-GR [11] O((KN 1 ) 3 ) I AVS-SSR-LF O((KN 1 ) 3 ) + O(JKL) AVS-SSR-ST O((KN 1 ) 3 ) + O(JM 3 ) The last two columns give a concrete example with explicit numerical values Items in bold are the two proposed algorithms O-SSR-DOA estimation algorithm. This is because three algorithms are developed under the SSR framework. However, the closed-form DOA estimates from the AVS-SSR-LF and AVS-SSR-ST algorithms are not confined to the pre-defined grid, which addresses the grid-effect problem of the O-SSR-DOA algorithm in an effort to improve the DOA estimation accuracy. (2) Robustness to additive noise: The basic principle for developing the AVS-SSR- LF and AVS-SSR-ST algorithms lies in efficiently estimating the direction cosine matrices utilizing the subarray manifold matrix correlation in (16) and the subarray signal correlations in (22). Obviously, from Tables 3 and 4, it can be seen that the AVS-SSR-LF algorithm has less computational complexity than that of the AVS-SSR-ST algorithm, but is less robust to the additive noise. The increased robustness of the AVS-SSR-ST algorithm results from estimating the DOA in the signal subspace of (47) obtained from the eigendecomposition of the received data signals to remove the noise subspace. (3) Computational complexity: The AVS-SSR-ST algorithm provides the optimal estimation under the Gaussian noise distribution assumption since the Eqs. (55) and (56) are the least-squares solutions of the optimization problem in (53). Regarding the computational cost of the AVS-SSR-ST algorithm compared with that of the O-SSR-DOA algorithm, the increase in complexity mainly lies in the eigendecomposition of R xx and the pseudo-inversion of  o, both of which are about O(4M 3 ) [6]. To compare the computational complexity, we provide a simple example listing the computational costs of L 1 -SVD, AVS-SSR-LF and AVS-SSR-ST in Table Validation of the Subarray Signal Correlation It is noted that the derivation of the AVS-SSR-LF and AVS-SSR-ST DOA estimation algorithms assume the estimated Ŝ u, Ŝ v, Ŝ w and Ŝ o in (37) satisfy or approximately satisfy the subarray signal correlation in (22). In this subsection, the validity of this assumption is discussed by analyzing the proposed closed-form method applied to some simple DOA estimation problems. To make the problem easier to understand, let us consider the case of one spatial source (K = 1) as an example. Specifically, from (13), we have

19 A o = q(θ 1,φ 1 ), and  + o A o = c o, (58) in which c o is a scalar constant. Case 1: noiseless condition With some manipulations, from (37), (58) and (18) to(21), the following equations are derived Ŝ u =  + o X u =  + o (A o u S) = ( + o A o) u S = u  + o (A os)= u ( + o X o)= u Ŝ o. (59) A similar derivation can be carried out for Ŝ v and Ŝ w. As a result, we have Ŝ u = u Ŝ o, Ŝ v = v Ŝ o, Ŝ w = w Ŝ o. (60) Compared with (22), (60) indicates that Ŝ u, Ŝ v, Ŝ w and Ŝ o satisfies the desired subarray signal correlation under the noiseless condition. Moreover, under the noiseless condition, we note that the derivation in (59) and the subarray signal correlation in (60) ) T also hold for multiple spatial sources with  + o A o = (Â+ o A o. To verify our analysis above, two simulation examples are conducted to evaluate the DOA estimation performance of the closed-form SSR-DOA algorithm in Table 2. Example 1 A single spatial source at (θ 1,φ 1 ) is considered under the noiseless condition. An eight-element AVSA is considered. The elevation angle θ 1 varies from 20 to 90 witha1 increment. Meanwhile the azimuth angle φ 1 is fixed to 0. The grid spacing is set to 4 (coarse grid spacing). The absolute error between the true elevation angle and its estimate (referred to as E abs ) is shown in Fig. 2a. The black dash line and the green line show the results of the O-SSR-DOA algorithm (in Table 1) and the closed-form AVS-SSR-DOA algorithm (in Table 2), respectively. When the true DOA corresponds to a grid point, then E abs will be 0. If the true DOA does not correspond to a grid point, then the maximum value of E abs will be 2 (half of the grid spacing). From Fig. 2a, it can be seen that E abs of the O-SSR-DOA method ranges from 0 to 2, which clearly reflects the grid-effect problem since the grid spacing is set to be 4. However, for the closed-form AVS-SSR-DOA algorithm, E abs is near zero. This clearly indicates the grid-effect reduction capability of the proposed closed-form AVS-SSR-DOA algorithm (in Table 2). We can infer that the subarray signal correlations in (60) are maintained well when there is a single spatial source for the noiseless condition. Example 2 Two spatial sources at (θ 1,φ 1 ) and (θ 2 = 135 o,φ 2 = 0 o ) are considered with no additive noise. The setup of (θ 1,φ 1 ) and the simulation parameters are the same as those used in Example 1. The simulation results are shown in Fig. 2b. Obviously, E abs of the O-SSR-DOA algorithm ranges from 1 to 2.2, which indicates that the grid-effect problem is more serious than for the single source case. For the closedform AVS-SSR-DOA algorithm, E abs ranges from 0 to 0.25, which is much smaller than that of the O-SSR-DOA algorithm. These results verify that the closed-form AVS-SSR-DOA algorithm is able to efficiently reduce the grid-effect compared with

20 (a) L1-SVD Closed-form method RMSE of θ estimation (degrees) (b) Absolute error of elevation estimation (degrees) Elevation angle (degrees) True elevation angle of the source(degree) L1-SVD Closed-form method Fig. 2 RMSE performance comparison between the O-SSR-DOA method (labeled as L 1 -SVD) given in Table 1 and the closed-form AVS-SSR-DOA algorithm (labeled as Closed-form method) given in Table 2. a Performance comparison: RMSE of estimated DOA at different angle (single source and no noise). b Performance comparison: RMSE of estimated DOA at different angle (two sources and no noise) the O-SSR-DOA algorithm when there are two spatial sources and there is no noise considered. Case 2: additive noise condition It is a difficult task to theoretically verify that the Ŝ u, Ŝ v, Ŝ w and Ŝ o estimated by (37) satisfy the desired subarray signal correlations defined in (22). However, from the discussions and derivations in Sect. 4.2, we are inferring that the Ŝ u, Ŝ v, Ŝ w and Ŝ o estimated by (37) will approximately satisfy the desired subarray signal correlations

21 defined in (22) since the variations of e u, e v, e w in (39) and the adverse impact of the additive noise on the subarray signal correlations in (36) can be suppressed to a certain extent due to the directional projection of the AVSA. A more detailed performance evaluation and discussion will be given in Sect Simulation Results To evaluate the performance of the SSR-DOA estimation algorithms, several simulations are performed. The comparisons are made between our proposed AVS-SSR-LF and AVS-SSR-ST algorithms as well as the L 1 -SVD and L 1 -SVD-GR algorithms [11] for far-field narrowband source DOA estimation in the presence of white Gaussian noise since all of these methods are developed under the SSR framework. 5.1 DOA Estimation for Multiple Sources This experiment is conducted to evaluate the multi-source 2D-DOA estimation ability of the proposed SSR-DOA estimation algorithms. The simulation settings are as follows: (1) There are seven narrowband spatial sources with their true DOA angles listed in the second column of Table 6. The narrowband sources (s k (t), k = 1,...,7) are randomly generated with fixed spatial locations; (2) Gaussian noise is added and the SNR is set to be 30 db; (3) A 13-element AVS array (M = 13) is used with the AVS units located at the Cartesian coordinates (λ/2) {(0 0 0), (±1 0 0), (±2 0 0), (0 ± 10), (0 ± 20), (00 ± 1), (00 ± 2)}, respectively; (4) Three DOA estimation algorithms developed under the SSR framework are considered including the AVS-SSR-LF, the AVS-SSR-ST and L 1 -SVD algorithms. It is noted that the L 1 -SVD algorithm is performed using the data from the o-subarray [11]; (5) The space is sampled with the angle set {(5, 5 ),...,(5, 360 ), (10, 5 ),...(180, 360 )}, where the grid spacing is purposely set as 5 for three algorithms to make all spatial sources off the grid points. The root-mean-squared errors (RMSE) of the DOA estimation are used as the performance metric, which is defined as 1 Nest RMSE = N (θ true ˆθ n ) 2, (61) est n=1 where N est, θ true and ˆθ n denote the number of estimation trials, the true DOA and the estimated DOA in the nth estimate, respectively. The DOA estimation results of three algorithms are listed in Table 6, where the last row lists the RMSE. As shown in the third column of Table 6, it can be seen that the DOA estimates of the L 1 -SVD algorithm for these seven spatial sources are all given as the closest discrete grid points. Under this experimental setting, among three algorithms, the AVS-SSR-ST algorithm achieves the smallest RMSE value (0.10 ) which is about a third of that given by the AVS-SSR-LF algorithm (0.31 ), and about a tenth of that given by the L 1 -SVD algorithm (1 ). From this experiment, it can be

22 Table 6 DOA estimation results for multiple sources Spatial source True DOA ( ) L 1 -SVD ( ) AVS-SSR-LF ( ) AVS-SSR-ST ( ) 1 (31, 291) (30, 290) (30.71, ) (31.10, ) 2 (61, 111) (60, 110) (60.96, ) (61.12, ) 3 (61, 311) (60, 310) (61.26, ) (61.02, ) 4 (81, 201) (80, 200) (81.16, ) (81.01, ) 5 (101,291) (100, 290) (101.03, ) (100.96, ) 6 (121, 71) (120, 70) (120.98, 70.99) (120.93, 71.09) 7 (151, 251) (150, 250) (151.28, ) (150.99, ) RMSE NA Here seven sources are presented The lowest RMSE results are indicated in bold and correspond in all cases to the proposed methods concluded that the three algorithms are able to estimate the DOAs of the multiple sources with a good accuracy when the grid spacing is set as 5. Moreover, the AVS- SSR-ST algorithm is able to provide much higher DOA estimation accuracy than the other two algorithms. 5.2 DOA Estimation Performance Under Different Noise Levels To further evaluate the performance of the proposed algorithms in the presence of various noise levels, we use the following experimental setup: (1) Two spatial sources are considered. Without loss of the generality, the elevation angles θ 1, θ 2 are fixed as 85.5 and 95.5, respectively, while the azimuth angles are randomly chosen from [0, 360 ) in each trial; (2) The whole elevation angular space is sampled uniformly with the grid spacing of θ = 3 ; (3) For the AVS-SSR-LF and AVS-SSR-ST algorithms, the simulation parameters shown in Table 5 are set as follows: M = 8, J = 4, L = 100, K = 2, N 1 = 180/ θ. The distance between the adjacent AVS units is half of the source wavelength; (4) For the L 1 -SVD algorithm and L 1 -SVD-GR algorithm, the simulation parameters shown in Table 5 are selected according to those used in [11]: M = 8, shrinking rate is 3 (γ = 3), and 3 refinement levels (I = 3) are used. The number of snapshots is set to be 100 (L = 100). K = 2 and N = 180/ θ; (5) Gaussian noise is added and the SNR varies from 20 to 40 db. For each noise level, 200 trials are carried out. The simulation results are shown in Fig. 3. The Cramer Rao lower bound (CRLB) for the performance of an AVS for DOA estimation was previously derived in [27] and we use the same derivation here to plot the CRLB as a function of SNR in Fig. 3 for comparison. The method in [16] estimates the direction cosines separately using the ESPRIT method. One challenge of this method is pairing the direction cosines that correspond to each individual source target. We observe that the pairing scheme in [16] is severely defected by strong noise and small direction cosines. For comparison, we also implement the method in [16] with perfect pairing using the true direction cosines, which achieves much more accurate DOA estimates. The performance of the L 1 -SVD algorithm improves as the SNR increases from 20 to 5 db. It performs consistently