Multi-Source DOA Estimation Using an Acoustic Vector ...web.pkusz.edu.cn/adsp/files/2015/10/Multi-Source-DOA-Estimation-U… · Keywords Direction-of-arrival estimation · Acoustic

Circuits Syst Signal ProcessDOI 10.1007/s00034-015-0102-9

Multi-Source DOA Estimation Using an Acoustic VectorSensor Array Under a Spatial Sparse RepresentationFramework

Yue-Xian Zou1 · Bo Li1 · Christian H Ritz2

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 ofarrival (2D-DOA) of sound sources using an acoustic vector sensor array (AVSA)within a spatial sparse representation (SSR) framework (AVS-SSR-DOA). SSR-DOAestimation methods rely on a pre-defined grid of possible source DOAs and essen-tially suffer from the grid-effect problem: Reducing the size of the grid spacing leads toincreased computational complexity. In this paper, we propose a two-step approach totackle the grid-effect problem. Specifically, omnidirectional sensor array-based SSR-DOA estimation firstly provides initial low-cost DOA estimates using a coarse gridspacing. Secondly, a closed-form solution is derived by exploring the unique subar-ray manifold matrix correlation and subarray signal correlation of the AVSA, whichallows for DOA estimates between the pre-defined angles of the grid and potentiallyachieves higher DOA estimation accuracy. To further alleviate the estimation bias dueto noise and sparse representation model errors, line-fitting (LF) techniques and sub-space techniques (ST) are employed to develop two novel DOA estimation algorithms,referred to as AVS-SSR-LF and AVS-SSR-ST, respectively. Extensive simulationsvalidate the effectiveness of the proposed algorithms when estimating the DOAs of

B Yue-Xian [email protected]

Bo [email protected]

Christian H [email protected]

1 Advanced Digital Signal Processing Laboratory, School of Electronic Computer Engineering,Peking University, Shenzhen 518055, China

2 School of Electrical, Computer, and Telecommunications Engineering, University ofWollongong,Wollongong, Australia

http://crossmark.crossref.org/dialog/?doi=10.1007/s00034-015-0102-9&domain=pdf

Circuits Syst Signal Process

multiple sound sources. The proposed AVS-SSR-ST algorithm achieves high DOAestimation accuracy and is robust to various noise levels and source separation angles.

Keywords Direction-of-arrival estimation · Acoustic vector sensor · Sparserepresentation · Grid-effect · Line-fitting · Subspace technique

1 Introduction

Estimating the direction of arrival (DOA) of sound sources with an acoustic vectorsensor array (AVSA) was first investigated by Nehorai and Paldi in the field of theunderwater and aerial acoustic direction-finding applications [12]. In contrast to anM-element scalar sensor array (SSA), anM-elementAVSA is composed ofM identicallyoriented AVS units as shown in Fig. 1b. Generally, each AVS unit consists of anomnidirectional sensor spatially colocated with two or three orthogonally orienteddirectional sensors, where either particle velocity sensors or differential microphonesare typically used as the directional sensors according to different applications [12].The signals recorded by directional sensors represent the x , y, and z components ofthe soundfield and can be processed to provide an estimation of the source DOA. Forsuch processing, many existing SSA-based DOA estimation methods, referred to asSSA-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 inthis paper.

Nehorai et al. have studied beamforming and the Capon methods for the AVSA andthe 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 estimationwith a single AVS and an AVSA [14,16–19]. They have explored the trigonometricinterrelations among the sensors of each individual AVS to develop the non-spatialrealization of ESPRIT with both a single AVS and an AVSA for DOA estimation ofmultiple uncorrelated sources. Furthermore, the AVS and AVSA have been proposedfor 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 abovecan be viewed as the extensions of the traditional SSA-DOA estimation methods toAVS-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 intensiveattention worldwide [2], which is referred to as the SSA-SSR-DOA estimationmethodin this paper. Research has shown that the SSA-SSR-DOA estimation method is ableto provide higher DOA estimation accuracy compared to the SSA-DOA estimationcounterpart [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 pre-defined discrete grid of potential DOAs. An overcomplete dictionary was formedusing the steering vectors corresponding to the grid points which are related to thepossible DOAs of the sources. As a result, the sparse representation model of the arraymeasurement data was formulated and the DOA estimation was then transformedinto an underdetermined inverse problem, which was solved by the iterative focal


underdetermined system solver (FOCUSS) [2]. Following the work in [2], Malioutovet al. explored the use of multiple measurement vectors (MMV) jointly with moresignal information to formulate the sparse representation model of the measurementdata [10,11]. Moreover, the cost function has been reformulated by imposing L1-norm penalties to enforce the sparsity, and the singular value decomposition (SVD)technique was further employed to reduce the adverse impact of the noise. Accordingto their paper, the resultant SSA-SSR-DOA estimation method was referred to as theL1-SVDmethod, which is recognized as a popular SSA-SSR-DOA estimationmethodin this research area. Many methods based on the L1-SVD method were proposed toimprove the DOA estimation accuracy or reduce the hardware cost [5,7,22–24,26].

Following careful evaluation of the existing SSA-SSR-DOA estimation methodsdiscussed above, we have the following observations: (1) The SSA-SSR-DOA esti-mation methods have the capability to potentially achieve a higher DOA estimationaccuracy with less sensitivity to noise than the traditional SSA-DOA estimation meth-ods; and (2) The DOA estimation accuracy of SSA-SSR-DOA are determined by thegrid spacing used to divide the spatial space. For example, if the grid spacing is setas 1◦, then the DOA estimation accuracy of the SSA-SSR-DOA estimation methodis 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 computationalcost required. This phenomenon is referred to as the effect of the grid or grid-effect inSSA-SSR-DOA estimation techniques [8,10,11]. To tackle this problem, an iterativegrid refinement strategy provides a solution to achieve the compromise between thecomputational cost and the DOA estimation accuracy [11]. Besides, the sparse totalleast- 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 thesame sensors, which is desirable in many applications; (2) The SSR framework canbe used for the AVSA-DOA estimation to provide high accuracy; and (3) The grid-effect problem can be tackled by investigating the additional source DOA informationembedded in the signals received by the AVS units. The main contributions of thisstudy are twofold. Firstly, we derive the formulation of DOA estimation with anAVSAunder the spatial sparse representation (SSR) framework, which is referred to as theAVS-SSR-DOA estimation method in Sect. 3. Clearly, the AVS-SSR-DOA estima-tion method is an extension of the SSA-SSR-DOA estimation method following thesame principle, and the AVS-SSR-DOA estimation method can achieve much higherDOA estimation accuracy than traditional AVS-DOA estimation methods with morecomputation effort if the space is discretized densely using a small grid spacing. Sec-ondly, a two-step DOA estimation method with the AVSA under the SSR frameworkis proposed to tackle the grid-effect problem. Specifically, the SSR-DOA estimationwith the omnidirectional sensor array (o-array) firstly provides the initial DOA esti-mates at lower cost by using a coarse grid spacing. Secondly, a closed-form solutionis derived by exploring the unique subarray manifold matrix correlation and subarraysignal correlation of the AVSA. This allows for DOA estimates that fall between thepre-defined grid points with the aim of achieving higher DOA estimation accuracy. Tofurther alleviate the estimation bias due to additive noise and the sparse representa-tion modeling error, line- fitting (LF) and subspace techniques (ST) are employed to


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 theirperformance under various settings, such as grid spacing, source separation, snapshotnumbers, and SNR levels.

The remainder of this paper is organized as follows. The AVS data models arepresented in Sect. 2. The DOA estimation with an AVS array under the spatial sparserepresentation (SSR) framework is formulated in detail, and some discussions arepresented in Sect. 3. Two novel AVS-SSR-DOA estimation algorithms are derived inSect. 4. Simulation results are presented in Sect. 5. Section 6 gives the conclusion ofour work.

2 Data Models

An idealAVS unit is composed of one omnidirectional sensor and three directional sen-sors (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, respec-tively. In addition, it is assumed that there are K (K < M) far-field spatial uncorrelatednarrowband acoustic signals sk(t) (k = 1, . . ., K ) impinging upon the AVS array. The4× 1 direction vector of the AVS unit for the kth spatial source coming from (θk, φk)

can be expressed as [16]:

v(θk, φk) ≡ [uk, vk, wk, 1]T , (1)

where [.]T denotes the vector/matrix transposition, θk ∈ (0◦, 180◦) is the elevationangle, and φk ∈ [0◦, 360◦) is the azimuth angle. Moreover, we have the followingdefinitions


uk = sin θk cosφk, vk = sin θk sin φk and wk = cos θk . (2)

From the geometric point of view, uk , vk , and wk in (2) can be viewed as theprojections of the kth spatial source onto the x , y, and z axes of the DOA, respectively.Hence, Eq. (2) is referred to as the DOA direction cosines, and correspondingly uk ,vk and wk are specifically referred to as the x-axis direction cosine, y-axis directioncosine, and z-axis direction cosine, respectively. On the other hand, given the directioncosines, the 2D-DOA estimates of the spatial sources can be determined by

φk = tan−1 vk/uk, 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 steeringvector for the kth spatial source located at (θk, φk) as

q(θk, φk) = [q1(θk, φk) , . . . , qM (θk, φk)]T , (4)

where the mth component in the steering vector can be denoted as

qm(θk, φk) = exp[ j2π(xmuk + ymvk + zmwk)/λ], (5)

in which λ and (xm, ym, zm) denote the source wavelength and the location of themthsensor in the Cartesian coordinates, respectively.

In contrast to an M-element SSA, an M-element AVSA is composed of M identi-cally oriented AVS units as shown in Fig. 1b. Its 4M × 1 manifold vector for the kthspatial source impinging from (θk, φk) can be denoted as [16]

a(θk, φk) ≡ v(θk, φk) ⊗ q(θk, φk) (6)

where⊗ denotes theKronecker product operator, and v(θk, φk) and q(θk, φk) are givenin (1) and (4), respectively. For all K spatial sources, the single snapshot measurementof an M-element AVSA at time t can be formed as follows

x(t) =K∑

k=1

a(θk, φk)sk(t) + n(t) = As(t) + n(t), x(t) ∈ C4M×1, (7)

where

A ≡ [a(θ1, φ1), . . . , a(θK , φK )] ∈ C4M×K , (8)

s(t) ≡ [s1(t), . . . , sK (t)]T ∈ RK×1, n(t) ≡ [n1(t), . . . , n4M (t)]T ∈ R4M×1, (9)

and where x(t) and n(t) are the measurement data vector and the additive zero-meanwhite Gaussian noise vector, respectively. Matrix A in (8) is defined as the manifoldmatrix associatedwith K spatial sources.When L snapshots (t = t1, . . . , tL) are taken,


the so-called multiple measurement vectors (MMV) data model of an M-elementAVSA can be written into a matrix form as follows

X = AS + N,

X = [x(t1), . . . , x(tL)], S = [s(t1), . . . , s(tL)], N = [n(t1), . . . , n(tL)], (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 collocatedsubarrays 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 Mcomponents of the corresponding type. For example, the o-subarray can be consideredas one SSA formed by M omnidirectional components. In other words, the o-subarrayof 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(θ1, φ1) ⊗ q(θ1, φ1), . . . , v(θK , φK ) ⊗ q(θK , φK )]S + N. (11)

Substituting v(θk, φk) in (1) into (11), we expand the Kronecker product and obtain

X = AS + N =

⎡

⎢⎢⎣

u1q(θ1, φ1), . . . , uK q(θK , φK )

v1q(θ1, φ1), . . . , vK q(θK , φK )

w1q(θ1, φ1), . . . , wK q(θK , φK )

q(θ1, φ1), . . . , q(θK , φK )

⎤

⎥⎥⎦ S + N (12)

with the following definitions

⎧⎪⎪⎨

⎪⎪⎩

Au = [u1q(θ1, φ1), . . . , uK q(θK , φK )]Av = [v1q(θ1, φ1), . . . , vK q(θK , φK )]Aw = [w1q(θ1, φ1), . . . , wK q(θK , φK )]Ao = [q(θ1, φ1), . . . , q(θK , φK )]

⇒ A =[ATu AT

v ATw AT

o

]T(13)

In (13), q(θk, φk) is given in (4). Au,Av,Aw, and Ao are called the subarraymanifold matrices of the u-, v-, w-, o-subarray, respectively. Accordingly, X and Ncan also be partitioned similarly into four blocks. Equation (12) is identically rewrittenas

⎡

⎢⎢⎣

Xu

Xv

Xw

Xo

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

Au

Av

Aw

Ao

⎤

⎥⎥⎦ S +

⎡

⎢⎢⎣

Nu

Nv

Nw

No

⎤

⎥⎥⎦ , (14)


where S is given in (10),Xu , Xv , Xw and Xo represent the measurement data matri-ces of the u-, v-, w-, and o-subarray, respectively, and Nu,Nv,Nw, and No representthe additive noise data matrices of the u-, v-, w-, and o-subarray, respectively. Equa-tion (14) can be further expressed as follows

Xu = AuS + Nu, Xv = AvS + Nv, Xw = AwS + Nw, Xo = AoS + No. (15)

Moreover, from (13), the following correlations between the subarray manifoldmatrices and the direction cosines (called subarray manifold matrix correlation inshort) can be derived

Au = Ao�u, Av = Ao�

v, Aw = Ao�w, (16)

where

�u = diag(u1, . . . , uK ), �v = diag(v1, . . . , vK ), �w = diag(w1, . . . , wK ).

(17)

The three matrices in (17) are referred to as the K ×K - dimensional diagonal direc-tion cosines matrices since their elements on the diagonals are the direction cosinesdefined in (2). Obviously, from (16), it can be seen that Au is interconnected with Ao

by the diagonal direction cosinesmatrix (�u) defined in (17). Similar observations canbe found for the v-subarray and w-subarray. Furthermore, substituting (16) into (15)yields

Xu = Ao�uS + Nu = AoSu + Nu (18)

Xv = Ao�vS + Nv = AoSv + Nv (19)

Xw = Ao�wS + Nw = AoSw + Nw (20)

Xo = AoS + No = AoSo + No (21)

with the definitions as follows

Su = �uS, Sv = �vS, Sw = �wS, So = S (22)

Comparing the second equations in (18)–(21)with the ones in (15), it is straightforwardto see that Su , Sv , Sw, and So can be viewed as the new signal sources virtuallyimpinging on the o-subarray (with the subarray manifold matrix Ao), which carrythe DOA information of the original spatial sources. Hence, (22) can be viewed asthe correlations between the source signal and the direction cosines (referred to assubarray signal correlation in short) of the AVS array. Moreover, in (22), the diagonaldirection cosines matrices determine the linear magnitude relations between Su , Sv ,Sw and S (or So), respectively.


3 DOA Estimation with an AVS Array Under SSR Framework

Tomake the presentation complete, in this section,DOAestimationwith anM-elementAVS array within the SSR framework (AVS-SSR-DOA estimation in short) will be firstformulated in detail before providing some discussions about the algorithm advantagesand limitations.

Essentially, the objective of DOA estimation with an M-element AVSA is to deter-mine (θk, φk), (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 thesame principle and procedure as that of the SSA-SSR-DOA estimation except that theyhave different MMV data models and manifold matrices. The essential principle ofSSA-SSR-DOA estimation is to recast the DOA estimation problem as a sparse signalreconstruction problem [11]. In order to make use of the spatial sparsity of the spatialsources, the whole spatial space is discretized into a grid of N points (N >> K )

to form a pre-defined grid � = {(θ1 φ1), . . . , (θN φN )}. The overcomplete manifoldmatrix � associated with � is given as follows:

� ≡[a

(θ1 φ1

), . . . , a

(θN φN

)]∈ C4M×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 spatialsources would fall on a grid defined by �, and a higher DOA estimation accuracy isthen achieved. Therefore, replacing A by � in (10) results in a sparse representationMMV 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 haveonly K nonzero rows, which correspond to the DOA estimates of the K spatial signalsources. Therefore, the DOA estimates of K sources from (24) can be achieved bylocating the indices of nonzero rows in Z, which is the core concept of the SSR-DOAestimation methods. In (24), � is pre-defined in (23); X is the measured data vectorfrom the M-element AVSA, and Z is a row sparse matrix to be estimated. Practically,the estimation ofZ can be recast as an optimization problemminimizing the followingobjective [11]

‖X − �Z‖2f + λ1

∥∥∥z(�2)∥∥∥1, (25)

where ‖ · ‖2f is the Frobenius norm, λ1 is the regularization parameter and ‖ · ‖1 is the�1-norm. The second term in (25) enforces the row sparsity of Z, which is computedas follows [11]

λ1

∥∥∥z(�2)∥∥∥1

= λ1

∥∥∥[z(�2)1 , . . . , z(�2)N

]∥∥∥1, (26)


where we have z(�2)n = ‖[zn(t1), . . . , zn(tL)]‖2 , n = 1, . . . , N and zn(tl), l =1, . . . , L denotes the element of the nth row and lth column of Z. Research showsthat the reconstruction of Z using (25) is a convex optimization problem that can besolved by several techniques [2,11]. Similar to the L1-SVD algorithm [11], singularvalue decomposition (SVD) is adopted to reduce both the computational complexityand the sensitivity to noise. Taking the SVD of X, we have X = ULVH and define

XSV = XVDK , ZSV = ZVDK and NSV = NVDK , DK = [IK 0] , (27)

where IK is a K×K identitymatrix and 0 is a K×(L−K ) zerosmatrix.Applying (27),the sparse representation MMV data model of the AVSA in (24) can be rewritten withthe reduced dimension representation as follows

XSV = �ZSV + NSV . (28)

Similarly, the estimation of ZSV from (28) can be obtained by minimizing thefollowing unconstrained cost function [11]

‖XSV − �ZSV ‖2f + λ1

∥∥∥ z(�2)∥∥∥1

(29)

whereXSV andZSV are given in (27). The second term in (29) is computed as follows

λ1

∥∥∥ z(�2)∥∥∥1

= λ1

∥∥∥[z(�2)1 , . . . , z(�2)N

]∥∥∥1

(30)

in which we have z(�2)n = ||[zSVn (1), . . . , zSVn (K )]||2, n = 1, . . . , N , and zSVn (k)denotes the element at the nth row and kth columnof theZSV .Moreover, the estimationof ZSV by minimizing the cost function in (29) can be performed efficiently by asecond-order cone programming (SOCP) technique at the cost of O((4K N )3) [3].Assuming that the K dominant rows in the reconstructed ZSV are with indices Ik ∈{1, . . ., N }, k = 1, . . ., K , the DOA estimates can be denoted as

(θk, φk) = (θIk φIk ), k = 1, . . . , K , (31)

where (θIk φIk ) is the Ik-th angular grid point in the pre-defined grid �.In conclusion, the AVS-SSR-DOA estimation method has been formulated within

the SSR framework. Beforewemove to the derivation of the proposedDOAestimationalgorithms, we have some discussions: (1) The advantages of the AVS-SSR-DOA esti-mation method are the following: The performance of the AVS-SSR-DOA is expectedto outperform the SSA-SSR-DOA since additional data have been captured in anAVSA. 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 thearray configuration more flexible for 2D-DOA estimation applications with an AVSarray; (2) TheAVS-SSR-DOAestimationmethod for anM-elementAVSA formulatedabove has the obvious defect of requiring much higher computational cost compared


with that of the SSA-SSR-DOA estimation method with the M-element SSA if thesame grid spacing is used. One of the reasons is that the inverse problem definedin (28) requires high computational complexity. Besides, there is no fast algorithm tosolve the optimization problem in (29) due to the L1-norm being used in the cost func-tion; (3) From (31), it is clear to see that the AVS-SSR-DOA estimation method alsofaces the grid-effect problem, where the DOA estimates are confined to the grid pointsin the pre-defined grid �, and the DOA estimation accuracy of the AVS-SSR-DOAestimation is determined by the grid spacing. This disadvantage is inherited from theSSR-DOA estimation method using a finite discrete grid. A larger value of N leads tohigher estimation accuracy but higher computational cost. The compromise betweenthe computational complexity and the DOA estimation accuracy is an important issuefor 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 theindex information of the K dominating nonzero rows in the reconstructed ZSV toobtain the final DOA estimates. Other useful information derived from the AVSAsignals such as the subarray manifold matrix correlation in (16) and subarray signalcorrelation in (22) have not been taken into account. To develop efficient DOA estima-tion algorithms with a high DOA estimation accuracy but comparable computationalcost to that of the O-SSR-DOA algorithm, a two-step DOA estimation scheme with anAVSA under the SSR framework is proposed in this section. In the first step, the initialDOA 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 Ao is also estimated accordingly. Inthe second step, with the correlations in (16) and (22), a closed-form DOA estimationis derived, and higher DOA estimation accuracy can be achieved since the final DOAestimates are not confined to the grid points of the pre-defined grid. To alleviate theadverse impact of the noise and modeling error on the DOA estimates obtained fromthe AVSA recordings, we propose two alternative methods that are employed withinthe SSR framework. The first method is based on line-fitting (LF) and is referred toas the AVS-SSR-LF DOA estimation method. The second method is based on a sub-space technique (ST) and is referred to asAVS-SSR-STDOA estimation method. Thedetails 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 estimationmethod presented in Sect. 3,the MMV data model of the o-subarray can be denoted as

Xo = �oZ + No, (32)


where

�o ≡[q(θ1 φ1), . . . , q(θN1 φN1)

]. (33)

In (32),Z is the row sparsematrix and has only K nonzero rows corresponding to the Kspatial sources. N1 is the size of the pre-defined grid � = {(θ1 φ1), . . . , (θN1 φN1)}and satisfies the relation of N > N1 >> K . This means that the pre-defined gridspacing for � is bigger than that for � (referred as the coarse grid spacing). The termq(θk φk) represents the steering vector defined in (4). It is easy to understand that ifwe select N1 = N/g, g > 1, then the computational cost using � will be reducedto 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 tolocate the indices of nonzero rows of Z in (32), which can be achieved by using thepopular L1-SVD technique [11], as shown in Sect. 3. Let us denote the indices ofthe K dominant rows in ZSV as Ik (k = 1, . . . , K ), then the DOA estimates can bedetermined as follows

(θk, φk) = (θIk , φIk ), Ik ∈ {1, . . . , N1}, k = 1, . . . , K , (34)

where θIk φIk are the grid points in the predefined �. Moreover, the computationalcost by using the L1-SVD technique to estimate ZSV from (32) will be of O((K N1)

3).Obviously, the saving of the computational complexity is at the cost of degrading theDOA estimation accuracy, since the DOA estimates from (34) are confined on the gridpoints defined in �, which have larger estimation bias than those from (31) where theDOA estimates are confined on the grid points defined in �.

With the DOA estimates from (34), the corresponding o-subarray manifold matrixAo is estimated as

Ao = �o(:, I) ∈ CM×K , (35)

where I = {I1, . . . , IK } is estimated by (34). It is clear that Ao is an estimate of thetrue manifold matrix Ao with certain estimation bias.

For presentation clarity, the DOA estimation algorithm described in this subsectionis referred to the O-SSR-DOA algorithm since it is derived using the o-subarray ofAVSA only and is summarized in Table 1. The O-SSR-DOA estimates will serve asthe 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 inTable 1 is determined by the size of the grid spacing used. In the following sections, wewill explore the information provided by the subarray manifold matrix correlation andthe subarray signal correlation of theAVSA to develop a closed-formAVS-SSR-DOAestimation solution, which has the ability to partially address the grid-effect problemand obtain a higher DOA estimation accuracy.


Table 1 The O-SSR-DOA algorithm

Initial input: set parameter N1

Form the o-subarray MMV data Xo

Define the spatial grid � = {(θ1 φ1), . . . , (θN1 φN1 )}Start

(1) Construct �o ≡ [q(θ1 φ1), . . . , q(θN φN )] in (33) using (4)(2) Obtain ZSV from (32) by L1-SVDmethod, locate the nonzero rows in ZSV , get the indices of nonzerorows

(3) Calculate θk and φk using (34)

(4) Determine Ao from (35)

End

Output θk , φk

Let us revisit the correlations shown in (22). It is noted that if the direction cosinematrices �u , �v and �w are estimated properly, a closed-form DOA estimation solu-tion can be achieved by (3). This concept is the essential basis behind the developmentof the efficient DOA estimation algorithms for an AVS array proposed in this paper.

As discussed above, with the O-SSR-DOA algorithm, Ao can be estimated as Ao

in (35) where the DOA estimates are confined on the grid points defined in �. Replac-ing Ao with Ao, equations from (18) to (21) yield

Xu ≈ Ao�uS + Nu = AoSu + Nu, Xv ≈ Ao�

vS + Nv = AoSv + Nv,

Xw ≈ Ao�wS + Nw = AoSw + Nw, Xo ≈ AoS + No = AoSo + No, (36)

where Su , Sv , Sw and So are defined in (22). The approximation sign in (36) is addeddue to the replacement of Ao by its estimate Ao. The minimum mean square error(MMSE) estimation of Su , Sv , Sw and So from (36) can be obtained as

[Su, Sv, Sw, So] = A+o [Xu,Xv,Xw,Xo], (37)

where (·)+ denotes the pseudo-inverse operator. From (22), we can derive the follow-ing

Su = Su + �Su = �uSo + �SuSv = Sv + �Sv = �vSo + �Sv

Sw = Sw + �Sw = �wSo + �Sw

So = So + �So

⎫⎪⎪⎪⎬

⎪⎪⎪⎭⇒

Su = �u(So − �So

)+ �Su

Sv = �v(So − �So

)+ �Sv

Sw = �w(So − �So

)+ �Sw

So = So + �So

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

.(38)

With simple manipulation of (38), we have

Su = �u So + eu, Sv = �vSo + ev, Sw = �wSo + ew, (39)


Table 2 The closed-form AVS-SSR-DOA algorithm

(1) Use the O-SSR-DOA estimation method in Table 1 to obtain Ao

(2) Compute Su , Sv, Sw and So from (37)

(3) Compute �u , �v, �w from (41)

(4) Compute the DOA estimates from (3), Output θk , φk

where

eu = �Su − �u�So, ev = �Sv − �v�So and ew = �Sw − �w�So. (40)

The quantities eu, ev, ew in (40) can be treated as modeling the errors contributedby both the additive noise in the data model (32) and the estimation bias of Ao.According to the structure of theAVS unit, it is inferred that the variations of eu, ev, ew

in (39) and the adverse impact of the additive noise on the subarray signal correlationin (36) can be suppressed due to the directional projection of the AVS unit. Hence, thedesirable subarray signal correlation shown in (22) can be maintained approximatelyfor Su, Sv, Sw and So, which yields

Su ≈ �u So, Sv ≈ �vSo, Sw ≈ �wSo. (41)

According to (37) and (41), a closed-form DOA estimation method for an AVSAcan be achieved, which is summarized in Table 2 for presentation clarity.

However, it is clear that when the variations of eu, ev, ew in (39) cannot be ignoreddue to strong additive noise at each sensor and a smaller value of N1 (spatial gridspacing is too large), the subarray signal correlations shown in (41) will be affectedand the DOA estimation accuracy of the closed-form AVS-SSR-DOA algorithm inTable 2 will degrade. The following two sections propose the use of both a line- fittingand a subspace techniques to deal with the modeling errors in (39) introduced byadditive noise and estimation bias in Ao.

4.3 The Proposed AVS-SSR-LF DOA Estimation Method

Observing the subarray signal correlations under the noise condition given in (39), theunderlying linear relationship between Su, Sv, Sw and So can be extracted using line-fitting. Without loss of the generality, we take the first equation in (39) to demonstratethe derivation of the line-fitting-based algorithm. The kth row of the first equationin (39) can be formulated equivalently as follows

Su(k, t) = uk So(k, t) + eu(k, t) (t = t1, · · · , tL , k = 1, . . . , K ), (42)

where Su(k, t), So(k, t) and eu(k, t) denote the element at the kth row and the t thcolumn in matrix Su , So and e respectively. Equation (42) can be viewed as a homoge-neous linear function between So(k, t) and Su(k, t) with the slope coefficient uk and


the error term eu(k, t). Moreover, uk is the x axis direction cosine and thus real, whileSo(k, t) and Su(k, t) are complex. The linear relation in (42) holds for both the realand the imaginary parts as follows

(Su(k, t)

)= uk

(So(k, t)

)+ (eu(k, t)) ,

(Su(k, t)

)= uk

(So(k, t)

)+ (eu(k, t)) , (43)

in which (·) and (·) extract the real and imaginary parts of a complex number,respectively. From (43), the following data vectors can be formed

�Su ≡[(Su(k, t1)), . . . ,(Su(k, tL)), (Su(k, t1)), . . . ,(Su(k, tL))

]T,

�So ≡[(So(k, t1)), . . . ,(So(k, tL)), (So(k, t1)), . . . ,(So(k, tL))

]T. (44)

It is clear that if the error term is not presented in (42), the scatter plot of �Su versus�So will form a line with the slope of uk , which corresponds to the direction cosine ofthe kth spatial source. With the error term, the points in the scatter plot of �Su versus�So will cluster along that line. The least- squares line-fitting technique [15] is adoptedto get a good estimate of the slope of the line, i.e., the direction cosine, as follows

uk = 2K L�STu �So − 1T �Su1T �So2K L�STo �So − 1T �So1T �So

, (45)

where 1 denotes the vector of size 2K L × 1 with all elements equal to 1. Similarly,vk and wk can be estimated following the same procedure as that for estimating uk .

With the direction cosine estimates uk , vk and wk , the closed-form DOA estimatefor the kth spatial source can be determined by

θk = cos−1 wk, φk = tan−1 vk/uk . (46)

At this point, the closed-form 2D-DOA estimation method under the SSR frame-work for an AVS by employing the least-squares line-fitting technique has beenderived, 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 issummarized 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 esti-mates by the O-SSR-DOA algorithm (Table 1) are confined to the discrete anglesdefined in �. The grid-effect is essentially introduced by the DOA estimationmethod under the SSR framework. However, the DOA estimates resulting fromthe proposed AVS-SSR-LF algorithm (Table 3) are not confined to � anymore.With the models shown in (39), the direction cosines (uk , vk , wk) can be estimatedby the least-squares solutions and accordingly the closed-form DOA estimationresults are achieved in (46).


Table 3 The proposed AVS-SSR-LF algorithm

Initial input:

Form the subarray measured data Xo, Xu , Xv , Xw

Pre-define the spatial grid � = {(θ1 φ1), . . . , (θN1 φN1 )}Start

(1) Construct �o ≡ [q(θ1 φ1), . . . , q(θN φN )] using (4)(2) Obtain Z from (32) by L1-SVDmethod, locate the nonzero rows in Z, get the indices of nonzero rows


(4) Calculate Su , Sv, Sw and So using (37)

(5) Form data vectors �Su and �So using (44), and estimate uk using (45)

(6) Form data vectors �Sv and �So using the formula as (44), and estimate vk using the formula as (45),where �Su is replaced by �Sv

(7) Form data vectors �Sw and �So using the formula as (44), and estimate wk using the formula as (45),where �Su is replaced by �Sw


End

Output θk , φk

(2) Robustness to additive noise: There is no doubt that additive noise will adverselydegrade the performance of the AVS-SSR-LF algorithm. The AVS-SSR-LF algo-rithm using the line-fitting technique to estimate the direction cosine matrices isbased on the least-squares solutions obtained in (37) and the data model intro-duced in (39). More specifically, the least-squares estimation of Su, Sv, Sw andSo in (37) can be considered as the optimal solution when the additive noise hasa stationary Gaussian distribution. For DOA estimation using the AVS-SSR-LFalgorithm with non-stationary and non-Gaussian noise, a robust estimator mustbe employed to replace the estimation in (37). Moreover, when the additive noiselevel goes higher, the error terms eu, ev, ew in (40) play more dominating roles inthe subarray signal correlation and the least-squares estimation of the directioncosines using the line-fitting technique is biased due to the contribution of theerror terms. As a result, the DOA estimation accuracy of the AVS-SSR-LF algo-rithm degrades with the increase in the additive noise level. To ensure the DOAestimation method is robust to noise, a more sophisticated algorithm should beconsidered, which motivates us to adopt the subspace technique described in thenext subsection.

(3) Computational complexity: From the AVS-SSR-LF algorithm shown in Table 3,the computational cost for estimating Ao from (35) is O((K N1)

3 and the costfor estimating direction cosines are about O(4K L), where K , L , and N1 are thenumber 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. Itcan be seen that the proposed AVS-SSR-LF algorithm provides a good trade-offbetween the DOA estimation accuracy and the computational cost.


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 reformulatethe estimation of the direction cosine matrices. Accordingly, a closed-form DOAestimation method in the signal subspace is derived.

Let us denote the 4M×4M autocorrelation matrix of the received data x(t) asRxx,which is practically approximated by XXT/L with X being the MMV given in (10)and L being the number of snapshots used. The eigendecomposition of Rxx results inthe 4M × K dimensional signal subspace matrix Es , which is composed of K signaleigenvectors of Rxx. It is noted that the signal subspace matrix Es holds the followingrelation with the manifold matrix A [16]:

Es ≈ AMT , (47)

whereMT is a non-singular K ×K transformmatrix andA is given in (8). Substitutingthe row partitions of A defined in (13) into (47), we have

[ETsu, E

Tsv, E

Tsw, ET

so

]T ≈[ATu , AT

v , ATw, AT

o

]TMT . (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]:

Esu ≈ AuMT = Ao�uMT , (49)

Eso ≈ AoMT . (50)

Replacing Ao from (50) by its estimate Ao given in (35), it gives

Eso = AoMT . (51)

In (51),Eso can be obtained by the eigendecomposition ofRxx. Hence, the transformmatrixMT can be estimated as

MT = A+o Eso. (52)

Therefore, the direction cosine matrix �u in (49) can be estimated as

�u = argmin

�u

∥∥∥Esu − Ao�uMT

∥∥∥2

f. (53)

Substituting (52) into (53), we have

�u = argmin

�u

∥∥∥Esu − Ao�uA+

o Eso

∥∥∥2

f. (54)


Table 4 The proposed AVS-SSR-ST algorithm

Initial input: subarray measured data Xo, Xu , Xv , Xw

Pre-defined the spatial grid � = {(θ1 φ1), . . . , (θN1 φN1 )}Start

(1) Construct �o ≡ [q(θ1 φ1), . . . , q(θN φN )] using (4)(2) Obtain Z from (32) by the L1-SVD method, locate the nonzero rows in Z and obtain the indices ofnonzero rows


(4) Form the MMV in (10) and calculate RXX = XXT /L

(5) Complete the eigendecomposition of RXX to generate Es = [ETsu , ET

sv, ETsw, ET

so]T(6) Estimate �u , �v and �w from (55) and (56)

(7) Obtain uk , vk and wk using (57)


End

Output θk , φk

Employing the least-squares estimation technique, the solution of (54) is given as

�u = A+

o EsuE+soAo. (55)

We note that the approach starting from (49)–(55) can be directly extended toanalyze the relations between Esv , Esw and Eso, respectively. Then, the least-squaresestimates of �v and �w can be expressed as follows

�v = A+

o EsvE+soAo,

�w = A+

o EswE+soAo. (56)

From the definitions of�u ,�v and�w in (17), the estimates for uk , vk andwk (k =1, . . ., K ) can be obtained accordingly as

uk = [�u]kk, vk = [�v]kk, wk = [�w]kk, (57)

where [.]kk denotes the kth diagonal element of a matrix. Following the definition ofuk , vk , and wk , the closed-form DOA estimate is given by (46).

The derived closed-form DOA estimation algorithm based on the subspace tech-nique (ST) is referred to as theAVS-SSR-STDOAestimation algorithm (AVS-SSR-STfor short). For presentation clarity, the proposed AVS-SSR-ST algorithm is summa-rized 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 theproposed AVS-SSR-LF and AVS-SSR-ST algorithms bear the advantages of the


Table 5 Computational costs of different SSR-DOA algorithms

Algorithm Computationalcomplexity

One exampleparameter setting:

Operationsinvolved

L1-SVD [11] O((K N )3) K = 2; N = 180; N1 = 90 I =3; J = 4; M = 8; L = 100

46.7 × 106

L1-SVD-GR [11] O((K N1)3) × I 17.5 × 106

AVS-SSR-LF O((K N1)3) + O(J K L) 5.8 × 106

AVS-SSR-ST O((K N1)3) + O(JM3) 5.8 × 106

The last two columns give a concrete example with explicit numerical valuesItems in bold are the two proposed algorithms

O-SSR-DOAestimation algorithm.This is because three algorithms are developedunder the SSR framework. However, the closed-form DOA estimates from theAVS-SSR-LF and AVS-SSR-ST algorithms are not confined to the pre-definedgrid, which addresses the grid-effect problem of the O-SSR-DOA algorithm in aneffort 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 cosinematrices utilizing the subarraymanifoldmatrix correlation in (16) and the subarraysignal correlations in (22). Obviously, from Tables 3 and 4, it can be seen thatthe AVS-SSR-LF algorithm has less computational complexity than that of theAVS-SSR-ST algorithm, but is less robust to the additive noise. The increasedrobustness of the AVS-SSR-ST algorithm results from estimating the DOA in thesignal subspace of (47) obtained from the eigendecomposition of the received datasignals to remove the noise subspace.

(3) Computational complexity: The AVS-SSR-ST algorithm provides the optimalestimation 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 withthat of the O-SSR-DOA algorithm, the increase in complexity mainly lies inthe eigendecomposition of Rxx and the pseudo-inversion of Ao, both of whichare about O(4M3) [6]. To compare the computational complexity, we providea simple example listing the computational costs of L1-SVD, AVS-SSR-LF andAVS-SSR-ST in Table 5.

4.5 Validation of the Subarray Signal Correlation

It is noted that the derivation of the AVS-SSR-LF and AVS-SSR-ST DOA estimationalgorithms assume the estimated Su, Sv, Sw and So in (37) satisfy or approximatelysatisfy the subarray signal correlation in (22). In this subsection, the validity of thisassumption is discussed by analyzing the proposed closed-form method applied tosome 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


Ao = q(θ1, φ1), and A+o Ao = co, (58)

in which co is a scalar constant.Case 1: noiseless condition

With some manipulations, from (37), (58) and (18) to (21), the following equationsare derived

Su = A+o Xu = A+

o (Ao�uS) = (A+

o Ao)�uS = �uA+

o (AoS)=�u(A+o Xo)=�u So.

(59)

A similar derivation can be carried out for Sv and Sw. As a result, we have

Su = �u So, Sv = �vSo, Sw = �wSo. (60)

Compared with (22), (60) indicates that Su, Sv, Sw and So satisfies the desired sub-array signal correlation under the noiseless condition. Moreover, under the noiselesscondition, we note that the derivation in (59) and the subarray signal correlation in (60)

also hold for multiple spatial sources with A+o Ao =

(A+o Ao

)T.

To verify our analysis above, two simulation examples are conducted to evaluatethe DOA estimation performance of the closed-form SSR-DOA algorithm in Table 2.

Example 1 A single spatial source at (θ1, φ1) is considered under the noiseless con-dition. An eight-element AVSA is considered. The elevation angle θ1 varies from 20◦to 90◦ with a 1◦ increment. Meanwhile the azimuth angle φ1 is fixed to 0◦. The gridspacing is set to 4◦ (coarse grid spacing). The absolute error between the true elevationangle and its estimate (referred to as Eabs) is shown in Fig. 2a. The black dash lineand the green line show the results of the O-SSR-DOA algorithm (in Table 1) and theclosed-form AVS-SSR-DOA algorithm (in Table 2), respectively. When the true DOAcorresponds to a grid point, then Eabs will be 0◦. If the true DOA does not correspondto a grid point, then the maximum value of Eabs will be 2◦ (half of the grid spacing).From Fig. 2a, it can be seen that Eabs of the O-SSR-DOA method ranges from 0◦to 2◦, which clearly reflects the grid-effect problem since the grid spacing is set tobe 4◦. However, for the closed-form AVS-SSR-DOA algorithm, Eabs is near zero.This clearly indicates the grid-effect reduction capability of the proposed closed-formAVS-SSR-DOA algorithm (in Table 2). We can infer that the subarray signal correla-tions in (60) are maintained well when there is a single spatial source for the noiselesscondition.

Example 2 Two spatial sources at (θ1, φ1) and (θ2 = 135o, φ2 = 0o) are consideredwith no additive noise. The setup of (θ1, φ1) and the simulation parameters are the sameas those used in Example 1. The simulation results are shown in Fig. 2b. Obviously,Eabs of the O-SSR-DOA algorithm ranges from 1◦ to 2.2◦, which indicates that thegrid-effect problem is more serious than for the single source case. For the closed-form AVS-SSR-DOA algorithm, Eabs ranges from 0◦ to 0.25◦, which is much smallerthan that of the O-SSR-DOA algorithm. These results verify that the closed-formAVS-SSR-DOA algorithm is able to efficiently reduce the grid-effect compared with


(a)

(b)

20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Elevation angle (degrees)

RM

SE

of θ

est

imat

ion

(deg

rees

)

L1-SVDClosed-form method

10 20 30 40 50 60 70 80 90 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

True elevation angle of the source(degree)

Abs

olut

e er

ror o

f ele

vatio

n es

timat

ion

(deg

rees

) L1-SVDClosed-form method

Fig. 2 RMSE performance comparison between the O-SSR-DOA method (labeled as L1-SVD) given inTable 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). bPerformance 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 noiseconsidered.

Case 2: additive noise conditionIt is a difficult task to theoretically verify that the Su, Sv, Sw and So estimated

by (37) satisfy the desired subarray signal correlations defined in (22). However, fromthe discussions and derivations in Sect. 4.2, we are inferring that the Su, Sv, Sw and Soestimated by (37) will approximately satisfy the desired subarray signal correlations


defined in (22) since the variations of eu, ev, ew in (39) and the adverse impact of theadditive noise on the subarray signal correlations in (36) can be suppressed to a certainextent due to the directional projection of the AVSA. A more detailed performanceevaluation and discussion will be given in Sect. 5.

5 Simulation Results

To evaluate the performance of the SSR-DOA estimation algorithms, several simula-tions are performed. The comparisons are made between our proposed AVS-SSR-LFand AVS-SSR-ST algorithms as well as the L1-SVD and L1-SVD-GR algorithms [11]for far-field narrowband source DOA estimation in the presence of white Gaussiannoise 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 abilityof the proposed SSR-DOA estimation algorithms. The simulation settings are as fol-lows: (1) There are seven narrowband spatial sources with their true DOA angles listedin the second column of Table 6. The narrowband sources (sk(t), k = 1, . . ., 7) arerandomly generated with fixed spatial locations; (2) Gaussian noise is added and theSNR is set to be 30dB; (3) A 13-element AVS array (M = 13) is used with the AVSunits located at the Cartesian coordinates (λ/2) × {(0 0 0), (±1 0 0), (±2 0 0), (0 ±1 0), (0 ± 2 0), (0 0 ± 1), (0 0 ± 2)}, respectively; (4) Three DOA estimation algo-rithmsdevelopedunder theSSR framework are considered including theAVS-SSR-LF,the AVS-SSR-ST and L1-SVD algorithms. It is noted that the L1-SVD algorithm isperformed using the data from the o-subarray [11]; (5) The space is sampled withthe angle set {(5◦, 5◦), . . ., (5◦, 360◦), (10◦, 5◦), . . ..(180◦, 360◦)}, where the gridspacing is purposely set as 5◦ for three algorithms to make all spatial sources off thegrid points.

The root-mean-squared errors (RMSE) of the DOA estimation are used as theperformance metric, which is defined as

RMSE =√

1

Nest

∑Nest

n=1(θtrue − θn)2, (61)

where Nest, θtrue and θn denote the number of estimation trials, the true DOA and theestimated DOA in the nth estimate, respectively.

The DOA estimation results of three algorithms are listed in Table 6, where thelast row lists the RMSE. As shown in the third column of Table 6, it can be seen thatthe DOA estimates of the L1-SVD algorithm for these seven spatial sources are allgiven as the closest discrete grid points. Under this experimental setting, among threealgorithms, 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 abouta tenth of that given by the L1-SVD algorithm (1◦). From this experiment, it can be


Table 6 DOA estimation results for multiple sources

Spatial source True DOA (◦) L1-SVD (◦) AVS-SSR-LF (◦) AVS-SSR-ST (◦)

1 (31, 291) (30, 290) (30.71, 290.59) (31.10, 291.08)

2 (61, 111) (60, 110) (60.96, 110.90) (61.12, 110.95)

3 (61, 311) (60, 310) (61.26, 310.63) (61.02, 310.95)

4 (81, 201) (80, 200) (81.16, 201.07) (81.01, 200.99)

5 (101,291) (100, 290) (101.03, 291.32) (100.96, 290.81)

6 (121, 71) (120, 70) (120.98, 70.99) (120.93, 71.09)

7 (151, 251) (150, 250) (151.28, 250.18) (150.99, 251.24)

RMSE NA 1 0.31 0.10

Here seven sources are presentedThe 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 multiplesources 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 theother two algorithms.

5.2 DOA Estimation Performance Under Different Noise Levels

To further evaluate the performance of the proposed algorithms in the presence ofvarious noise levels, we use the following experimental setup: (1) Two spatial sourcesare considered. Without loss of the generality, the elevation angles θ1, θ2 are fixed as85.5◦ 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 withthe 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, N1 = 180/�θ . The distance between the adjacent AVS units is halfof the source wavelength; (4) For the L1-SVD algorithm and L1-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 40dB. 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. 3for comparison. The method in [16] estimates the direction cosines separately usingthe ESPRITmethod. One challenge of this method is pairing the direction cosines thatcorrespond 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, wealso implement themethod in [16] with perfect pairing using the true direction cosines,which achieves much more accurate DOA estimates. The performance of the L1-SVDalgorithm improves as the SNR increases from−20 to−5dB. It performs consistently


Fig. 3 RMSE of 2D-DOAestimation versus SNR (twosources)

2

when the SNR is larger than −5dB and results in a DOA RMSE of around 1◦. Theperformance limit is imposed by the grid-effect, i.e., off-grid targets are approximatedby the closest grid points. The L1-SVD-GR algorithm reduces the grid-effect but theminimum RMSE is only 0.33◦. If the grid spacing is reduced too much, the coherenceof the overcomplete manifold matrix will increase greatly, which in turn will affectthe performance of sparse recovery. Therefore, the L1-SVD-GR algorithm cannoteliminate the grid-effect completely but only to a certain extent. From Fig. 3, we cansee that the AVS-SSR-ST algorithm outperforms other compared algorithms for SNRlevels higher than 15dB and is comparable to all other sparsity-based methods withSNR levels smaller than 10dB. The AVS-SSR-LF algorithm performs inferior to thatof the AVS-SSR-ST algorithm, but superior to that of the L1-SVD algorithm whenthe SNR is higher than approximately 2dB. The computational time for L1-SVD,L1-SVD-GR, AVS-SS-LF and AVS-SS-ST, are respectively, 0.33, 1.3, 0.33 and 0.4 s.Although the L1-SVD-GR algorithm performs the best when the SNR is less than10dB, the advantage is reduced by the fact that at least three times more computationsare required.

5.3 DOA Estimation Performance for Different Numbers of Snapshots andSensors

In this subsection, we present in Fig. 4 the results for different numbers of snapshotsand AVS units. Also included for comparison is the CRLB derived in the same wayas for Fig. 3. Figure 4a shows the RMSE of the elevation angle estimates when thenumber of snapshots L changes from 20 to 100. The simulation setup is the same asthat in Sect. 5.2 except that the SNR is fixed as 20dB. The method in [16] performsworse than all other methods and the RMSE remains almost constant for all numbersof snapshots evaluated. This is because the performance of the method of [16] islimited due to the use of a matching method. As we can see from the dashed blackcurve, when perfect matching is assumed for the method in [16], the performanceis greatly improved, especially with a larger number of snapshots. The L1-SVD andL1-SVD-GR algorithms are also very slightly impacted by L . The reason is that theperformance limit of L1-SVD is restricted by the grid-effect and that L1-SVD-GR


(a) (b)

Fig. 4 a RMSE of 2D-DOA estimation versus number of snapshots; b RMSE of 2D-DOA estimationversus the number of AVS units

can only reduce the grid-effect to a certain extent. The performance of the proposedAVSA-SS-LF and AVSA-SS-ST methods also improves as the number of snapshotsincreases.

Figure 4b shows the RMSE of elevation angle estimates when the number of AVSunits increases from 6 to 16 (M = 6 to 16). The simulation setting is the same asbefore except that the number of snapshots is fixed at 100. Similar observations canbe made for the proposed methods. Note that the L1-SVD-GR method produces asmaller estimation error when a larger number of AVS units are used. This is dueto the fact that the coherence of the overcomplete manifold matrix is reduced witha larger number of AVS units. Therefore, the L1-SVD-GR algorithm can reduce thegrid-effect to a larger extent for a larger number of AVS units.

5.4 Impact of the Grid Spacing on DOA Estimation Accuracy

This experiment evaluates the performance of the SSR-DOAestimationmethodswhenusing different pre-defined grid spacings. The experimental setup is as follows: (1) Theexperimental setup is the same as that in Sect. 5.2, except that different grid spacingsare considered and the SNR is set to be 20dB; (2) The whole elevation angular rangeis sampled uniformly with a grid spacing of �θ . For each grid spacing, 200 trials arecarried out; (3) We consider two spatial sources with elevation angles θ1 randomlychosen from [30◦, 150◦] and θ2 = θ1 + 16◦, while the azimuth angles are randomlychosen from [0◦, 360◦) in each trial. The DOA RMSE and the computational timeversus the grid spacing are computed as the performance metrics, where �θ variesfrom 0.2◦ to 6◦. It is easy to find that with the simulation setup shown above, theelevation angles of the sources are off-grid with high probability. The simulationresults are plotted in Fig. 5. From Fig. 5a, it is clear to see that the DOA RMSEof the L1-SVD algorithm linearly increases with the size of the grid spacing. Thisresult directly reflects the grid-effect of the L1-SVD algorithm, where a larger gridspacing leads to less computational cost but poorer DOA estimation accuracy. It isnoted that our proposed AVS-SSR-LF and AVS-SSR-ST algorithms result in higher


Grid Space (degree)

RM

SE o

f e

stim

atio

n (d

egre

e)

Grid Space (degree)

θ

(a) (b)

Fig. 5 a RMSE of the DOA estimation versus grid spacing; b the computational cost versus grid spacing(SNR = 20dB, N1 = N )

DOA estimation accuracy than that of the L1-SVD-GR and L1-SVD algorithms asshown in Fig. 5a. From the RMSE curve of the L1-SVD-GR algorithm, we can see thatthe grid refinement method effectively reduces the grid-effect and performs constantlyfor all grid spacings and the DOARMSE remains relatively constant at approximately0.4◦. Moreover, from Fig. 5b, we can clearly see that the L1-SVD, AVS-SSR-LFand AVS-SSR-ST algorithms have similar computational cost under the simulationsetup, but the L1-SVD-GR algorithm requires a much higher computational cost. Inconclusion, in this experimental condition, the L1-SVD-GR and our proposed AVS-SSR-LF andAVS-SSR-ST algorithms are all able to suppress the grid-effect. However,our proposed methods are able to provide higher DOA estimation accuracy with lesscomputational cost than that of the L1-SVD -GR algorithm.

5.5 Performance Under Different Source Separations

This experiment aims to evaluate the performance of the SSR-DOA estimation meth-ods under different spatial source separations. The experimental setup is as follows:(1) All experimental parameters are the same as that in Sect. 5.3, except that differentspatial source separations are considered and the grid spacing is set to be 3◦; (2) Theelevation angle of the first source is fixed to 42.5◦, while the elevation angle of thesecond source varies from 46.5◦ to 77.5◦.

The DOA RMSE of the estimated elevation angles versus the source separationare plotted in Fig. 6. It is obvious that the proposed AVS-SSR-LF and AVS-SSR-STalgorithms demonstrate similar performance and perform much better than the L1-SVD algorithm for all source separations. In addition, the AVS-SSR-ST algorithmperforms slightly better than the AVS-SSR-LF algorithm, especially when the sourceseparation is small. Furthermore, the proposed algorithms offer better DOA estimationresults than that of the L1-SVD and the L1-SVD-GR algorithms when the sourceseparation is smaller than 15◦. Specifically, when the source separation is 4◦, theDOA RMSE is 0.12◦, 0.23◦, 1.12◦ and 1.14◦ for the AVS-SSR-ST, the AVS-SSR-LF,the L1-SVD and the L1-SVD-GR algorithms, respectively. It can be concluded that


10 15 20 25 30

0.2

0.4

0.6

0.8

1

1.2

Source Separation (degree)

RM

SE

of

θ es

timat

ion

(deg

ree)

L1-SVD

L1-SVD-GR

AVS-SS-LF

AVS-SS-ST

Fig. 6 RMSE of the DOA estimation versus source separation (the first source is located at (42.5◦, 0◦),SNR = 30 dB, Grid spacing = 2◦)

the capability to provide high DOA estimation accuracy at small source separationsis a favorable advantage of our proposed DOA estimation algorithms. Finally, we cansee that the performance of the L1-SVD-GR is much better than that of the L1-SVDalgorithm when the source separation is larger than 10◦, and the DOA RMSE of theL1-SVD algorithm is fixed to about 0.5◦ when the source separation is larger than15◦.

6 Conclusions

In this paper, two efficient multi-source 2D-DOA estimation algorithms proposed foran AVS array have been developed within a spatial sparse representation frameworkand incorporating line- fitting and subspace techniques, respectively. The motivationfor the development lies in the potential high DOA estimation accuracy of the SSR-DOA estimation methods. With a reasonable choice of the angular grid spacing, weare able to obtain good initial DOA estimates using the o-subarray manifold matrixat a lower computational cost. Utilizing the unique subarray manifold matrix correla-tion and subarray signal correlation of the AVS array, a closed-form DOA estimationsolution is achieved, and the line-fitting technique and the subspace technique haveboth been employed to further suppress the adverse impact of the additive noise andthe estimation bias of the o-subarray manifold matrix. Experiments show that our pro-posed algorithms have the ability to estimate the DOA of multiple sources with highaccuracy. The performance under various parameter settings and different SNR levelshas been evaluated. It is encouraging to see that the proposed AVS-SSR-ST algorithmoutperforms other compared algorithms under all SNR levels and when the sourceseparation is smaller than 15o.


Acknowledgments This work is supported by National Natural Science Foundation of China (No:61271309) and theShenzhenScience andTechnologyFundamentalResearchProgram(No: JCY201110006).

References

1. H. Chen, J. Zhao, Coherent signal-subspace processing of acoustic vector sensor array for DOA esti-mation of wideband sources. Signal Process. 85(4), 837–847 (2005)

2. I. Gorodnitsky, B. Rao, Sparse signal reconstruction from limited data using FOCUSS: a re-weightedminimum norm algorithm. IEEE Trans. Signal Process. 45(3), 600–616 (1997)

3. M. Grant, S. Boyd, CVX: matlab software for disciplined convex programming. http://cvxr.com/cvx/(2011)

4. M. Hawkes, A. Nehorai, Acoustic vector-sensor beamforming and Capon direction estimation. IIEEETrans. Signal Process. 46(9), 2291–2304 (1998)

5. M.M. Hyder, K.Mahata, Direction-of-arrival estimation using a mixed l2,0 norm approximation. IEEETrans. Signal Process. 58(9), 4646–4655 (2010)

6. H. Krim, M. Viberg, Two decades of array signal processing research: the parametric approach. IEEESignal Process. Mag. 13(4), 67–94 (1996)

7. B. Li, Y.X. Zou, Y.S. Zhu, Direction estimation under compressive sensing framework: a review andexperimental results, inProceedings of IEEE International Conference on Information and Automation( ICIA) (2011), pp. 63-68

8. B. Li, Y.X. Zou, Improved DOA estimation with acoustic vector sensor arrays using spatial sparsityand subarray manifold, in Proceedings of the ICASSP (2012), pp. 2557–2560

9. Z. Liu, X. Ruan, J. He, Efficient 2-D DOA estimation for coherent sources with a sparse acousticvector-sensor array. Multidimens. Syst. Sign. Process. 24(1), 105–120 (2013)

10. D.M. Malioutov, A sparse signal reconstruction perspective for source localization with sensor arrays.http://ssg.mit.edu/~dmm/publications/malioutov_MS_thesis.pdf (2003)

11. D.M.Malioutov,M.Cetin,A.Willsky,A sparse signal reconstructionperspective for source localizationwith sensor arrays. IEEE Trans. Signal Proces. 53(8), 3010–3022 (2005)

12. A. Nehorai, E. Paldi, Acoustic vector-sensor array processing. IEEE Trans. Signal Process. 42(9),2481–2491 (1994)

13. P. Palanisamy, N. Kalyanasundaram, P.M. Swetha, Two-dimensional DOA estimation of coherentsignals using acoustic vector sensor array. Signal Process. 92(1), 19–28 (2012)

14. P. Tichavský, K.T. Wong, M.D. Zoltowski, Near-field/far-field azimuth and elevation angle estimationusing a single vector hydrophone. IEEE Trans. Signal Process. 49(11), 2498–2510 (2001)

15. E.W.Weisstein, Least squares fitting. http://mathworld.wolfram.com/LeastSquaresFitting.html (2012)16. K.T.Wong,M.D. Zoltowski, Closed-formunderwater acoustic direction-findingwith arbitrarily spaced

vector hydrophones at unknown locations. IEEE J. Ocean. Eng. 22(3), 649–658 (1997)17. K.T. Wong, M.D. Zoltowski, Extended-aperture underwater acoustic multisource azimuth/elevation

direction-finding using uniformly but sparsely spaced vector hydrophones. IEEE J. Ocean. Eng. 22(4),659–672 (1997)

18. K.T. Wong, M.D. Zoltowski, Root-MUSIC-based azimuth-elevation angle-of-arrival estimation withuniformly spaced but arbitrarily oriented velocity hydrophones. IEEE Trans. Signal Process. 47(12),3250–3260 (1999)

19. K.T. Wong, M.D. Zoltowski, Self-initiating MUSIC-based direction finding in underwater acousticparticle velocity-field beamspace. IEEE J. Ocean. Eng. 25(2), 262–273 (2000)

20. Y.I. Wu, K.T.Wong, Acoustic near-field source-localization by two passive anchor-nodes. IEEE Trans.Aerosp. Electron. Syst. 48(1), 159–169 (2012)

21. Y. Xu, Z. Liu, J. Cao, Noncircularity-exploitation in direction estimation of noncircular signals withan acoustic vector-sensor. Digit. Signal Process. 18(5), 777–796 (2008)

22. X. Xu, X. Wei, Z. Ye, DOA estimation based on sparse signal recovery utilizing weighted l1-normpenalty. IEEE Signal Process. Lett. 19(3), 155–158 (2012)

23. J. Yin, T. Chen, Direction-of-Arrival estimation using a sparse representation of array covariancevectors. IEEE Trans. Signal Process. 59(9), 4489–4493 (2011)

24. J. Zheng, M. Kaveh, H. Tsuji, Sparse spectral fitting for Direction of Arrival and power estimation, inIEEE/SP 15th Workshop on Statistical Signal Processing (2009), pp. 429–432

http://cvxr.com/cvx/

http://ssg.mit.edu/~dmm/publications/malioutov_MS_thesis.pdf

http://mathworld.wolfram.com/LeastSquaresFitting.html


25. J. Zheng, M. Kaveh, Direction-of-arrival estimation using a sparse spatial spectrum model with uncer-tainty, in Proceedings of the ICASSP (2011), pp. 2848–2851

26. C. Zheng, G. Li, H. Zhang, X. Wang, An approach of DOA estimation using noise subspace weightedl1 minimization, in Proceedings of the ICASSP (2011), pp. 2856–2859

27. X. Zhong, A.B. Premkumar, A.S. Madhukumar, Particle filtering and posterior Cramér-Rao bound for2-D direction of arrival tracking using an acoustic vector sensor. IEEE Sens. J. 12(2), 363–377 (2012)

28. X. Zhong, A.B. Premkumar, Particle filtering approaches for multiple acoustic source detection and2-D direction of arrival estimation using a single acoustic vector sensor. IEEE Trans. Signal Proces.60(9), 4719–4733 (2012)

29. H. Zhu, G. Leus, G.B. Giannakis, Sparsity-cognizant total least-squares for perturbed compressivesampling. IEEE Trans. Signal Process. 59(5), 2002–2016 (2011)

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Multi-Source DOA Estimation Using an Acoustic Vector ...web.pkusz.edu.cn/adsp/files/2015/10/Multi-Source-DOA-Estimation-U… · Keywords Direction-of-arrival estimation · Acoustic