# Two 2-D DOA Estimation Methods with Full and Partial Generalized Virtual Aperture Extension Technology

Authors: Riheng Wu; Yangyang Dong; Zhenhai Zhang; Le Xu
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

We address the two-dimensional direction-of-arrival (2-D DOA) estimation problem for L-shaped uniform linear array (ULA) using two kinds of approaches represented by the subspace-like method and the sparse reconstruction method. Particular interest emphasizes on exploiting the generalized conjugate symmetry property of L-shaped ULA to maximize the virtual array aperture for two kinds of approaches. The subspace-like method develops the rotational invariance property of the full virtual received data model by introducing two azimuths and two elevation selection matrices. As a consequence, the problem to estimate azimuths represented by an eigenvalue matrix can be first solved by applying the eigenvalue decomposition (EVD) to a known nonsingular matrix, and the angles pairing is automatically implemented via the associate eigenvector. For the sparse reconstruction method, first, we give a lemma to verify that the received data model is equivalent to its dictionary-based sparse representation under certain mild conditions, and the uniqueness of solutions is guaranteed by assuming azimuth and elevation indices to lie on different rows and columns of sparse signal cross-correlation matrix; we then derive two kinds of data models to reconstruct sparse 2-D DOA via M-FOCUSS with and without compressive sensing (CS) involvements; finally, the numerical simulations validate the proposed approaches outperform the existing methods at a low or moderate complexity cost.


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## Body

## 1. Introduction

The 2-D DOA estimation has received considerable attention in the past decades and has met many applications in array signal processing society, such as, radar, sonar, and wireless communications [1]. Numerous array structures, such as L-shaped arrays, rectangular arrays, circular arrays, two parallel ULAs, etc., [2–9] are proposed and are applied in different scenario, among which, L-shaped ULA attracts wide concerns due to its simple structure and ease of implementation. Accordingly, researchers developed numerous methods for estimation of 2-D DOA based on L-shaped ULAs [10–24]. These methods are roughly categorized into two classes: one class is to decouple L-shaped ULA into two separate ULAs, where azimuths and elevations are estimated independently, and the additional 2-D DOA pairing procedure is needed. The other class developed the Vandermonde structure of steering matrix with automatic angles pairing, such as JSVD [18], 2-D MUSIC [20], 2-D PM, 2-D ESPRIT, and their variants [23, 25], of tensor techniques, such as PARAFAC [26], CS-PARAFAC [27], and the tensor-based PARAFAC [8]. Compared to other methods, JSVD and 2-D MUSIC can obtain better angle estimation performance at the expense of prohibitive complexity. Alternatively, the 2-D ESPRIT and 2-D PM show average performance with a moderate or low complexity, but the PARAFAC and CS-PARAFAC methods possess high complexity, thanks to numerous matrix calculations involved, and may converge slowly or even divergent under the case of low signal-to-noise ratio (SNR).


### 1.1. Our Contributions and Comparisons to Related Work

In order to achieve good estimation performance with automatic angles pairing at the cost of a low or moderate complexity, the conjugate symmetry property of L-shaped ULA is introduced to enlarge virtual array aperture [12]. Motivated by this work, we further generalize virtual aperture by developing not only the forward but also the backward potential rotational invariance property of samples’ cross-correlation matrix. As a result, a larger virtual received data matrix is constructed by concatenating the sub-cross-correlation matrices along the columns, and hence, two one-dimensional ESPRITs are constructed to obtain favorable azimuth estimate performance. The proposed methods demonstrate the performance gain as the number of snapshots and of subarray elements increase.Two kinds of approaches to estimating 2-D DOA are in this paper developed based on the full and partial generalized virtual received data model. One is called the subspace-like method, where the generalization version of PM-ESPRIT algorithm [12] is carried out at a low complexity. The other kind of approach is referred to as the sparse reconstruction method, in which two data models are used to reconstruct sparse 2-D DOA via the M-FOCUSS framework [28] with and without compressive sensing (CS) involvements. We provide a lemma to verify that the received data model is equivalent to its dictionary-based sparse representation; the uniqueness of solutions is guaranteed by assuming azimuth and elevation indices to lie on different rows and columns of sparse signal cross-correlation matrix. In order to demodulate the sparse 2-D DOA, the 2-D MMVs (two-dimensional multiple measurement vectors) are converted into one 1-D MMV and K 1-D SMV (one-dimensional single measurement vector) problems, respectively; the CS-based sparse reconstruction methods are also developed in this paper.The main contributions of our work are summarized as follows:(1)
We maximize the virtual array aperture of the received data model compared to other existing methods, which greatly benefits 2-D DOA estimation.(2)
We propose an efficient 2-D DOA estimation algorithm for L-shaped array, referred to as the subspace-like method, which outperforms existing methods with a low complexity.(3)
We propose the sparse reconstruction methods with and without CS involvement to estimate 2-D DOA. Particularly, it provides a general framework to estimate 2-D DOA for any array geometry.


### 1.2. Paper Organization and Notations

The rest of the paper is organized as follows. Section2 describes the signal model of L-shaped ULA. Section 3 presents two kinds of 2-D DOA estimation approaches based on the full and partial generalized data model. The simulation results and analysis are provided in Section 4. Finally, conclusions are drawn in Section 5.Throughout the paper, matrices and vectors are denoted by bold capitals and bold lowercases, respectively.⋅∗, ⋅T, ⋅−1, ⋅†, and ⋅H represent conjugation, transpose, inverse, pseudoinverse, and Hermitian transpose, respectively. E⋅ denotes the expectation operation. ⊙,⊗,Ip,Jp,andOm×n denote Hadamard product, Kronecker product, p×p identity matrix, p×p exchange matrix with ones on antidiagonal and zeros elsewhere, m×n zero matrix, respectively, phase⋅ and diag⋅ denote, respectively, the phase of its argument and the diagonalization on a vector. 1n denotes all one vector of size n×1, and rank⋅ denotes the rank of matrix.


## 1.1. Our Contributions and Comparisons to Related Work

In order to achieve good estimation performance with automatic angles pairing at the cost of a low or moderate complexity, the conjugate symmetry property of L-shaped ULA is introduced to enlarge virtual array aperture [12]. Motivated by this work, we further generalize virtual aperture by developing not only the forward but also the backward potential rotational invariance property of samples’ cross-correlation matrix. As a result, a larger virtual received data matrix is constructed by concatenating the sub-cross-correlation matrices along the columns, and hence, two one-dimensional ESPRITs are constructed to obtain favorable azimuth estimate performance. The proposed methods demonstrate the performance gain as the number of snapshots and of subarray elements increase.Two kinds of approaches to estimating 2-D DOA are in this paper developed based on the full and partial generalized virtual received data model. One is called the subspace-like method, where the generalization version of PM-ESPRIT algorithm [12] is carried out at a low complexity. The other kind of approach is referred to as the sparse reconstruction method, in which two data models are used to reconstruct sparse 2-D DOA via the M-FOCUSS framework [28] with and without compressive sensing (CS) involvements. We provide a lemma to verify that the received data model is equivalent to its dictionary-based sparse representation; the uniqueness of solutions is guaranteed by assuming azimuth and elevation indices to lie on different rows and columns of sparse signal cross-correlation matrix. In order to demodulate the sparse 2-D DOA, the 2-D MMVs (two-dimensional multiple measurement vectors) are converted into one 1-D MMV and K 1-D SMV (one-dimensional single measurement vector) problems, respectively; the CS-based sparse reconstruction methods are also developed in this paper.The main contributions of our work are summarized as follows:(1)
We maximize the virtual array aperture of the received data model compared to other existing methods, which greatly benefits 2-D DOA estimation.(2)
We propose an efficient 2-D DOA estimation algorithm for L-shaped array, referred to as the subspace-like method, which outperforms existing methods with a low complexity.(3)
We propose the sparse reconstruction methods with and without CS involvement to estimate 2-D DOA. Particularly, it provides a general framework to estimate 2-D DOA for any array geometry.


## 1.2. Paper Organization and Notations

The rest of the paper is organized as follows. Section2 describes the signal model of L-shaped ULA. Section 3 presents two kinds of 2-D DOA estimation approaches based on the full and partial generalized data model. The simulation results and analysis are provided in Section 4. Finally, conclusions are drawn in Section 5.Throughout the paper, matrices and vectors are denoted by bold capitals and bold lowercases, respectively.⋅∗, ⋅T, ⋅−1, ⋅†, and ⋅H represent conjugation, transpose, inverse, pseudoinverse, and Hermitian transpose, respectively. E⋅ denotes the expectation operation. ⊙,⊗,Ip,Jp,andOm×n denote Hadamard product, Kronecker product, p×p identity matrix, p×p exchange matrix with ones on antidiagonal and zeros elsewhere, m×n zero matrix, respectively, phase⋅ and diag⋅ denote, respectively, the phase of its argument and the diagonalization on a vector. 1n denotes all one vector of size n×1, and rank⋅ denotes the rank of matrix.


## 2. Signal Model

As shown in Figure1, suppose that there are K uncorrelated narrow-band sources (noncircular and/or circular) in far-field impinging upon the L-shaped ULA from directions θk,φk,k=1,2,…,K (note that here the azimuth definition is different from the traditional one, it is commonly defined with L-shaped array [12, 29]). The L-shaped ULA consists of two orthogonal X-subarray and Z-subarray with each one being M elements, and they share the reference element placed at the origin. The interelement spacing d=λ/2 for each subarray, where λ is the wavelength of the incident sources.Figure 1
L-shaped ULA geometry.Let theM×1 and M−1×1 data vectors received at the tth time of X-subarray and Z-subarray be xt=x1t,x2t,…,xMtT and zt=z2t,z3t,…,zMtT, respectively. The received data vectors from two subarrays can be written as(1)xt=Axθst+qxt,zt=Azφst+qzt,where Axθ and Axθ are steering matrices of X-subarray and Z-subarray, respectively.(2)Axθ=axθ1,axθ2,…,axθK,(3)axθk=1,μk,μk2,…,μkM−1T,k=1,…,K,(4)μk=expj2πdcosθkλ,(5)Azφ=azφ1,azφ2,…,azφK,(6)azφk=ηk,ηk2,…,ηkM−1T,k=1,…,K,(7)ηk=expj2πdcosφkλ,st=s1t,s2t,…,sKtT is the received signal vector; qxt=qx,1t,qx,2t,…,qx,MtT and qzt=qz,1t,qz,2t,…,qz,M−1tT are Gaussian noise vectors of two subarrays, respectively. The additive white Gaussian noise (AWGN) is temporally and spatially independently with zero mean and variance σn2 and is uncorrelated with the impinging sources. The highest order in (6) should be M−1 since both X-subarray and Z-subarray share the reference element placed at the origin.


## 3. The Proposed Methods

### 3.1. Generalized Virtual Aperture Extension Technology

We note that bothAxθ and Azφ are Vandermonde structure-based steering matrices of centrosymmetric arrays and, therefore, satisfying [23, 30, 31].(8)JMAxθ∗=AxθΛx−M−1,(9)AxθTJM=ΛxM−1AxθH,(10)JM−1Azφ∗=AzφΛz−M,(11)AzφTJM−1=ΛzMAzφH,where Λx=diagμ1,…,μK and Λz=diagη1,…,ηK are two K×K diagonal matrices containing K azimuth angles θkk=1K and K elevation angles φkk=1K, respectively. A similar observation has been reported in [12], where only the Vandermonde structure of Azφ is exploited. We instead exploit the Vandermonde structures of both Axθ and Azφ and, hence, yields a larger data matrix by generalized virtual aperture extension technology. Our method is stated as follows:CollectJ snapshots for L-shaped ULA; the samples cross-correlation matrix R^x,z of xt and zt is defined by(12)R^x,z≈ExtztH=XZHJ=AxθR^sAzφH,where R^s=EststH=diagσ12,σ22,…,σK2 and σk2 is the received power of the kth source. The estimate of R^x,z is dependent on snapshots J. Defining two selection matrices ψxj=OM−1×j−1,IM−1,OM−1×2−j,j=1,2, and ψzj=OM−2×j−1,IM−2,OM−2×2−j,j=1,2, we divide R^x,z into four maximum overlapping submatrices with the size M−1×M−2, which are denoted by(13)Y1=ψx1R^x,zψz1T=Ax1θR^sAz1φH,(14)Y2=ψx1R^x,zψz2T=Ax1θR^sAz2φH,(15)Y3=ψx2R^x,zψz1T=Ax2θR^sAz1φH,(16)Y4=ψx2R^x,zψz2T=Ax2θR^sAz2φH,where Ax1θ are the first M−1 rows of Axθ; Az1φ are the first M−2 rows of Azφ; Ax2θ are the last M−1 rows of Axθ; and Az2φ are the last M−2 rows of Azφ; consequently,(17)Ax2θ=Ax1θΛx,Az2φ=Az1φΛz.As a result,Y2,Y3,andY4 in (14)–(16) can be, respectively, rewritten as(18)Y2=Ax1θΛz−1R^sAz1φH,Y3=Ax1θΛxR^sAz1φH,Y4=Ax1θΛxΛz−1R^sAz1φH.Using the conjugate symmetric properties in (8)–(11) and R^s=R^s∗, the additional four augmented virtual matrices are yielded(19)Y5=JM−1Y2∗JM−2=Ax1θΛx−M−2ΛzMR^sAz1φH,Y6=JM−1Y1∗JM−2=Ax1θΛx−M−2ΛzM−1R^sAz1φH,Y7=JM−1Y4∗JM−2=Ax1θΛx−M−1ΛzMR^sAz1φH,Y8=JM−1Y3∗JM−2=Ax1θΛx−M−1ΛzM−1R^sAz1φH.Consider a generalized virtual received data modelY consisting of concatenating Y1∼Y8 along the columns, which is denoted by(20)Y=Y1T,Y2T,Y3T,Y4T,Y5T,Y6T,Y7T,Y8TT=Axz−augθ,φR^sAz1φH,where(21)Axz−augθ,φ=Ax1θAx1θΛz−1Ax1θΛxAx1θΛxΛz−1Ax1θΛx−M−2ΛzMAx1θΛx−M−2ΛzM−1Ax1θΛx−M−1ΛzMAx1θΛx−M−1Λz−M−1,is the generalized steering matrix of size 8M−1×K; in other words, the effective array aperture is extended significantly, which will substantially improve the azimuth estimation performance, while the elevation estimation will not benefit from the generalized virtual aperture extensive technology. However, we can compromise on the performance between azimuth and elevation flexibly by concatenating Y1∼Y8 in other ways.To estimate 2-D DOA using the generalized virtual received data modelY in (20), we propose two kinds of methods, i.e., the subspace-like method and the sparse reconstruction method, which are formulated in a way that no angles pairing issue arises.


### 3.2. The Subspace-Like Method

One effective method to obtainθk,φkk=1K is to apply generalization of PM-ESPRIT algorithm [12] (Section 3) to the generalized steering matrix (21). We can find propagator matrix by using PM, which partitions Axz−augθ,φ into two submatrices Axz−1θ,φ and Axz−2θ,φ of size K×K and 8M−1−K×K, respectively.(22)Axz−augθ,φ=Axz−1θ,φTAxz−2θ,φTT.Under the hypothesis thatAxz−1θ,φ is a nonsingular matrix (which is sufficient for subspace-like methods while M−1≥K), the propagator matrix P is a unique linear operator, which satisfies(23)PHAxz−1θ,φ=Axz−2θ,φ.By solving a LS problem,P is achieved by(24)P^=Y˜1Y˜1H−1Y˜1Y˜2H,where Y=Y˜1T,Y˜2TT and Y˜1∈CK×M−2, Y˜2∈C8M−1−K×M−2. Let Q=IKP^H denotes an extended propagator matrix; we have QAxz−1θ,φ=Axz−augθ,φ. Define two azimuth angle selection matrices as follows(25)ϒxj=I8⊗OM−2×j−1,IM−2,OM−2×2−j,j=1,2.We observe that the rotational invariance property ofAxz−augθ,φ still holds, and hence,(26)ϒx1QAxz−1θ,φΛx=ϒx2QAxz−1θ,φ.By carrying out simple operations to (26), a new K×K nonsingular matrix Τx is achieved(27)Tx=ϒx1Q†ϒx2Q=Axz−1θ,φΛxAxz−1θ,φ−1.Applying eigenvalue decomposition (EVD) toΤx, the estimation of Λx can be written as Λ^x=ΔΛxΔ, where Δ is the K×K permutation matrix satisfying ΔT=Δ−1. Let A^xz−1θ,φ denotes the estimation of Axz−1θ,φ; we have A^xz−1θ,φ=Axz−1θ,φΔ. Note that the scaling ambiguity is not considered here since Λx is unitary matrix and its diagonal elements locate on the unit circle.In order to achieve 2-D DOA automatic pairing, letϒzj,j=1,2 be two elevation angle selection matrices, which is defined as(28)ϒzj=I4⊗OM−1×j−1M−1,IM−1,OM−1×2−jM−1.Again by utilizing the rotational invariance property ofAxz−augθ,φ used in (26) and substitute Axz−1θ,φ with A^xz−1θ,φ, we have(29)ϒz1QA^xz−1θ,φΔΛz−1=ϒz2QA^xz−1θ,φΔ.Left-multiply and right-multiply both sides of (29) by ϒz2QA^xz−1θ,φ† and ΛzΔ, respectively; the estimation Λ^z is denoted as(30)Λ^z=ΔΛzΔ=ϒz2QA^xz−1θ,φ†ϒz1QA^xz−1θ,φ.The angle pairing is achieved automatically viaA^xz−1θ,φ, which associates Λ^x with Λ^z without permutation ambiguity. Therefore, the estimates of θk,φkk=1K can be obtained by(31)θ^kk=1K=arccosphaseΛ^x⋅λ2πd,φ^kk=1K=arccosphaseΛ^z⋅λ2πd.Note: the azimuth angle estimation utilizes the rotational invariance property from two 8(M − 1) element data, while the elevation estimation only use the rotational invariance property from two 4(M − 1) element data; in such case, the azimuth estimation performance is better than that of elevation.The main steps of the proposed subspace-like method are summarized as follows:(1)
Calculate the extended propagator matrix via the generalized steering matrix.(2)
Develop the rotational invariance property of the generalized steering matrixAxz−augθ,φ to obtain a nonsingular matrix Τx via two azimuth angle selection matrices.(3)
Apply eigenvalue decomposition toΤx, and the estimations of eigenvalue matrix Λx and eigenvector matrix Axz−1θ,φ are obtained; the azimuths are estimated from Λ^x.(4)
Develop the rotational invariance property of the generalized steering matrixAxz−augθ,φ but replace Axz−1θ,φ with A^xz−1θ,φ estimated from step 3, the autopaired elevation estimation is achieved via two elevation angle selection matrices.


### 3.3. The Sparsity Reconstruction Method

Assume that 2-D DOA search ranges are divided intoNθ and Nφ angle grids denoted by θp,p=1,2,…,Nθ, and φq,q=1,2,…,Nφ, respectively. Define in the spatial domain two basis matrices Ωθ of size M×Nθ,Nθ≫M and Ωφ of size M−1×Nφ,Nφ≫M−1 as(32)Ωθ=ηθθ1,ηθθ2,…,ηθθNθ,ηθθp=1,ej2πdcosθp/λ,…,ej2πM−1dcosθp/λTM,Ωφ=ηφφ1,ηφφ2,…,ηφφNφ,ηφφq=ej2πdcosθq/λ,…,ej2πM−1dcosθq/λTM−1,where p=1,…,Nθ, q=1,…,Nφ; ηθθp and ηφφq are the array steering basis vectors of θp and φq, respectively. Suppose the pairing θk,φkk=1K of K sources just fall on the 2-D DOA angle grids and indexed by pk,qkk=1K, respectively. Taking the samples cross-correlation matrix (12) into account, we can recast (12) into the dictionary-based sparse representation(33)R^x,z=AxθR^sAzφH=ΩθΘsΩφH,where Θs is a Nθ×Nφ sparse matrix containing only K nonzero coefficients, i.e., Θspk,qk=σk2; the similar observation is achieved in [32], wherein it is only studied numerically without theoretical analysis. The following lemma proves that equation (33) holds subject to certain mild constraints on the basis matrices and the sources.Lemma 1.
Considering (33) with the assumptions that any M columns of Ωθ and any M−1 columns of Ωφ are linearly independent, the received signals are uncorrelated and independently across the array which ensures rankΩθ=M and rankΩφ=M−1; the columns of basis matrices Ωθ and Ωφ are not nearly collinear to the columns that form the basis for the Θs, which guarantee the uniqueness of solutions, but to what extent depends on the signal-to-noise ratio (SNR) and angular interval between the columns of Ωθ and Ωφ, if rankΘs=K≤M−1, i.e., pi≠pj,qi≠qj,i≠j, then rankY1=K, and (33) is satisfied.Proof.
From the assumptions above, we haverankΩθ=M, rankΘs=K, and rankΩφ=M−1. Therefore, rankΘsΩφH≤minrankΘs,rankΩφ=K. Let Z=ΩθΘsΩφH, rankΖ=rankΩθΘsΩφH≤minrankΩθ,rankΘsΩφH=K. Without loss of generality, we assume that the first K rows and the first K columns of Θs (the case where those linear independent columns are dispersed randomly across the matrix Θs can be dealt in a similar manner) are linearly independently, yielding a new matrix Ts, which is equivalent to R^s, i.e., Γs=ΞR^sΞ. Accordingly, the generating matrices Ω˜θ and Ω˜φH, consisting of the first K columns of Ωθ and the first K rows of ΩφH, are intrinsically equivalent matrices of Ax1θ and Az1φH, respectively; that is to say Ω˜θ=Ax1θΞ and Ω˜φ=Az1φΞ; Ξ is the K×K permutation matrix satisfying ΞT=Ξ−1. As a result, ΩθΘsΩφH=Ω˜θΓsΩ˜φH, Ω˜θΓsΩ˜φH=Ax1θΞΞR^sΞΞAz1φH=Ax1θR^sAz1φH; therefore, Y1=Ax1θR^sAz1φH=ΩθΘsΩφH, rankY1=K, we complete the proof. Lemma 1 associates the parameter-based data models with its sparse representations.
The multiple measurement vector (MMV) sparse reconstruction is first addressed in [28], which is used to compute the sparse solutions given MMV that shares a common sparsity structure, wherein it exploits the sparsity in angle domain. Later, [32] extends 1-D MMV to 2-D MMV. In this context, the 2-D DOA sparse reconstruction is derived, and the sparse solutions are reconstructed via the M-FOCUSS [28] with and without the involvements of CS based on two received data models.


#### 3.3.1. On the Data Model (33) with and without CS Involvements

First, we make use of (33) as the data model. In order to estimate Θs, we set up the cost function as follows:(34)Ζθ,φ∗=argminΖθ,φΖθ,φ2,s,s.t.R^x,z=ΩθΖθ,φ,where Ζθ,φ=ΘsΩφH, Ζθ,φ2,s≜∑r=1Nθzr2s, s∈0,1zr=zr1,…,zrM−1 is the rth row of Zθ,ϕ, and the row norm is 2-norm defined by zr2=∑l=1M−1zrl21/2; M−1 denotes the number of MMVs; Nθ denotes rows vectors of sparse solutions; s is the sparsity measure that balances the performance and the complexity. Equation (34) converts equation (33) 1-D MMV problem, where Ζθ,φ contains only K nonzero rows vector index supports corresponding to azimuth angles θkk=1K. By letting certain termination condition be satisfied, we take out K nonzero rows with the maximum amplitudes from Nθ rows vectors to form Ζθ,φ∗, i.e., Ζθ,φ∗=Ωθ,0†R^x,z, Ωθ,0=ηθθn1,ηθθn2,…,ηθθnK, IK=n1,n2,…,nK is the rows sparse index supports. In order to estimate elevations φkk=1K, we need to solve additional K 1-D SMV problems of(35)θs,k∗=argminθs,kθs,kH2,s,s.t.zθ,φ,k∗H=Ωφθs,kH,where zθ,φ,k∗ is the kth row vector of Ζθ,φ∗; θs,k∗ is the kth row vector of Θs; the kth nonzero coefficient is found from θs,k∗; the angles pairing is achieved automatically. Note: K azimuth angles θkk=1K are first found from K nonzero rows vector index supports at a single time slot, and then K elevation angles φkk=1K are found from K different row vectors by K time slots, which implies a fact that small azimuth angle separation will lead to poor azimuth estimation performance, but elevation estimation performance will not be affected even if the elevation angle separation is small.The proposed methods, inspired by [32], extend 2-D sparse reconstruction methods to CS-based sparse reconstruction in such a way that no angles pairing issue arises. This method allows for recovery of 2-D DOA assuming they stay on the predefined 2-D angle grids.Define two spatial domain compression matricesΦx of size Ml×M,Ml≤M and Φz of size Mk×M−1,Mk≤M−1, respectively, whose elements are drawn i.i.d from a Gaussian distribution with zero mean and unit variance. Left-multiply Φx and right-multiply ΦzH by R^x,z, the CS-based samples cross-correlation matrix R^x,zCS becomes(36)R^x,zCS=ΦxR^x,zΦzH=ΦxΩθΘsΩφHΦzH=Ω˜θΘsΩ˜φH,where Ω˜θ=ΦxΩθ and Ω˜φ=ΦzΩφ are the composite azimuth and elevation angle basis matrices, respectively; the incorporation of compression matrices into the received data matrix will benefit lower storage request. The CS-based sparse reconstruction method is implemented using the same procedures as shown in (34) and (35).


#### 3.3.2. On the Data Model (20) with and without CS Involvements

The new insight into the 2-D sparse reconstruction method can also be obtained by developing the common sparse structure of the partial generalized data model (20), which will improve the azimuth estimation performance even when the azimuth separations are close. From Lemma 1, we know the basis matrix Ωθ and Ωφ must be row full rank and rankΘs=K; hence, we select from Y in (20) the group Y1,Y3 to construct a column full rank data matrix Y13=Y1T,Y⌢3TT of size M×M−2, where Y⌢3=eM−1TY3, eM−1=0,…,0,1T is the unit vector of size M−1. Taking the Hermitian transpose over Y13, Y13H can be represented as(37)Y13H=Az1φR^sHBx1θH,where Bx1θH=Ax1θHΛxHAx1θHeM−1, from Lemma 1, we recast (37) as(38)Y13H=A˜z1φR˜sHB˜x1θH,where A˜z1φ, R˜s, and B˜x1θ are M−2×Nφ, Nθ×Nφ, and M×Nθ basis matrices of Az1φ, R^s, and Bx1θ, respectively. B˜x1θH=A˜x1θHΛ˜xHA˜x1θHeM−1, Λ˜x=diagμ1,μ2,…,μNθ is Nθ×Nθ diagonal matrix; A˜x1θ is M−1×Nθ basis matrix of Ax1θ. Note: one can also define the groups Y2,Y4,Y5,Y7,Y6,Y8 to make new data matrices which provide the similar performance as Y13, but Y=Y1,…,Y8 may not provide better performance since the generalized steering basis matrix is rank deficit and does not suffice Lemma 1. Similar to (34), we establish the cost function(39)Z˜θ,φ∗=argminZ˜θ,φZ˜θ,φ2,s,s.t.Y13H=A˜z1φZ˜θ,φ,where Z˜θ,φ=R˜sHB˜x1θH, by enforcing some termination condition to (39), we select out K nonzero rows with the maximum amplitudes to form Z˜θ,φ∗, i.e., Z˜θ,φ∗=A˜z1,0φ†Y13H, A˜z1,0φ=A˜z1φm1,A˜z1φm2,…,A˜z1φmK, IK=m1,m2,…,mK is the row sparse index supports. Again additional K 1-D SMV problems of(40)r˜s,k∗=argminr˜s,kr˜s,kH2,s,s.t.z˜θ,φ,k∗H=B˜x1θr˜s,kH,have to be solved to obtain θkk=1K; the angles pairing is achieved automatically. Note: K elevation angles θkk=1K are first found from K nonzero rows vector index supports of Ζ˜θ,φ at a single time slot, and then K azimuth angles θkk=1K are found from K different row vectors at K time slots, which suggests a fact that small elevation separation will yield poor elevation estimation performance, but azimuth estimation performance will not be affected even if the azimuth separations are small.Define two spatial domain compression matricesΦx13 of size Ml×M,Ml≤M, and Φz13 of size Mk×M−2,Mk≤M−2, whose elements are drawn i.i.d from a Gaussian distribution with zero mean and unit variance, respectively. Left-multiply Φz13 and right-multiply Φx13H by Y13H, the CS-based data matrices Y13CSH become(41)Y13CSH=Φz13Y13HΦx13H=Φx13A˜z1φR˜sHB˜x1θHΦx13H=A˜˜z1φR˜sHB˜˜x1θH,where A˜˜z1φ=Φz13A˜z1φ and B˜˜x1θ=Φx13B˜x1θ are the composite 2-D DOA basis matrices. The CS-based sparse reconstruction procedure is implemented using the same methods as those in (39) and (40).The major steps of the sparse reconstruction methods are summarized as follows:(1)
Establish the sparse representation models based on the conventional data model and the partial generalized data model.(2)
Establish cost functions to minimize the errors based on M-FOCUSS framework.(3)
Transform the 2-D DOA estimation into two associated 1-D MMV problems, and the autopaired angles are achieved by solving the two optimization problems successively.Remark 1.
From (26) and (29), it is obvious that the array aperture, of θkk=1K and φkk=1K, is enhanced greatly from M to 8M−1 and from M to 4M−1, respectively.Remark 2.
The subspace-like method provides super-resolution when the received data matrix is modeled accurately for the L-shaped array. By contrast, the sparse reconstruction methods do not impose any restricts on the array structure, either nonlinear or nonuniform. On the other hand, the number of the maximum identifiable sparse solutionsK [33] should satisfy K<M−1/2 for two kinds of approaches.


## 3.1. Generalized Virtual Aperture Extension Technology

We note that bothAxθ and Azφ are Vandermonde structure-based steering matrices of centrosymmetric arrays and, therefore, satisfying [23, 30, 31].(8)JMAxθ∗=AxθΛx−M−1,(9)AxθTJM=ΛxM−1AxθH,(10)JM−1Azφ∗=AzφΛz−M,(11)AzφTJM−1=ΛzMAzφH,where Λx=diagμ1,…,μK and Λz=diagη1,…,ηK are two K×K diagonal matrices containing K azimuth angles θkk=1K and K elevation angles φkk=1K, respectively. A similar observation has been reported in [12], where only the Vandermonde structure of Azφ is exploited. We instead exploit the Vandermonde structures of both Axθ and Azφ and, hence, yields a larger data matrix by generalized virtual aperture extension technology. Our method is stated as follows:CollectJ snapshots for L-shaped ULA; the samples cross-correlation matrix R^x,z of xt and zt is defined by(12)R^x,z≈ExtztH=XZHJ=AxθR^sAzφH,where R^s=EststH=diagσ12,σ22,…,σK2 and σk2 is the received power of the kth source. The estimate of R^x,z is dependent on snapshots J. Defining two selection matrices ψxj=OM−1×j−1,IM−1,OM−1×2−j,j=1,2, and ψzj=OM−2×j−1,IM−2,OM−2×2−j,j=1,2, we divide R^x,z into four maximum overlapping submatrices with the size M−1×M−2, which are denoted by(13)Y1=ψx1R^x,zψz1T=Ax1θR^sAz1φH,(14)Y2=ψx1R^x,zψz2T=Ax1θR^sAz2φH,(15)Y3=ψx2R^x,zψz1T=Ax2θR^sAz1φH,(16)Y4=ψx2R^x,zψz2T=Ax2θR^sAz2φH,where Ax1θ are the first M−1 rows of Axθ; Az1φ are the first M−2 rows of Azφ; Ax2θ are the last M−1 rows of Axθ; and Az2φ are the last M−2 rows of Azφ; consequently,(17)Ax2θ=Ax1θΛx,Az2φ=Az1φΛz.As a result,Y2,Y3,andY4 in (14)–(16) can be, respectively, rewritten as(18)Y2=Ax1θΛz−1R^sAz1φH,Y3=Ax1θΛxR^sAz1φH,Y4=Ax1θΛxΛz−1R^sAz1φH.Using the conjugate symmetric properties in (8)–(11) and R^s=R^s∗, the additional four augmented virtual matrices are yielded(19)Y5=JM−1Y2∗JM−2=Ax1θΛx−M−2ΛzMR^sAz1φH,Y6=JM−1Y1∗JM−2=Ax1θΛx−M−2ΛzM−1R^sAz1φH,Y7=JM−1Y4∗JM−2=Ax1θΛx−M−1ΛzMR^sAz1φH,Y8=JM−1Y3∗JM−2=Ax1θΛx−M−1ΛzM−1R^sAz1φH.Consider a generalized virtual received data modelY consisting of concatenating Y1∼Y8 along the columns, which is denoted by(20)Y=Y1T,Y2T,Y3T,Y4T,Y5T,Y6T,Y7T,Y8TT=Axz−augθ,φR^sAz1φH,where(21)Axz−augθ,φ=Ax1θAx1θΛz−1Ax1θΛxAx1θΛxΛz−1Ax1θΛx−M−2ΛzMAx1θΛx−M−2ΛzM−1Ax1θΛx−M−1ΛzMAx1θΛx−M−1Λz−M−1,is the generalized steering matrix of size 8M−1×K; in other words, the effective array aperture is extended significantly, which will substantially improve the azimuth estimation performance, while the elevation estimation will not benefit from the generalized virtual aperture extensive technology. However, we can compromise on the performance between azimuth and elevation flexibly by concatenating Y1∼Y8 in other ways.To estimate 2-D DOA using the generalized virtual received data modelY in (20), we propose two kinds of methods, i.e., the subspace-like method and the sparse reconstruction method, which are formulated in a way that no angles pairing issue arises.


## 3.2. The Subspace-Like Method

One effective method to obtainθk,φkk=1K is to apply generalization of PM-ESPRIT algorithm [12] (Section 3) to the generalized steering matrix (21). We can find propagator matrix by using PM, which partitions Axz−augθ,φ into two submatrices Axz−1θ,φ and Axz−2θ,φ of size K×K and 8M−1−K×K, respectively.(22)Axz−augθ,φ=Axz−1θ,φTAxz−2θ,φTT.Under the hypothesis thatAxz−1θ,φ is a nonsingular matrix (which is sufficient for subspace-like methods while M−1≥K), the propagator matrix P is a unique linear operator, which satisfies(23)PHAxz−1θ,φ=Axz−2θ,φ.By solving a LS problem,P is achieved by(24)P^=Y˜1Y˜1H−1Y˜1Y˜2H,where Y=Y˜1T,Y˜2TT and Y˜1∈CK×M−2, Y˜2∈C8M−1−K×M−2. Let Q=IKP^H denotes an extended propagator matrix; we have QAxz−1θ,φ=Axz−augθ,φ. Define two azimuth angle selection matrices as follows(25)ϒxj=I8⊗OM−2×j−1,IM−2,OM−2×2−j,j=1,2.We observe that the rotational invariance property ofAxz−augθ,φ still holds, and hence,(26)ϒx1QAxz−1θ,φΛx=ϒx2QAxz−1θ,φ.By carrying out simple operations to (26), a new K×K nonsingular matrix Τx is achieved(27)Tx=ϒx1Q†ϒx2Q=Axz−1θ,φΛxAxz−1θ,φ−1.Applying eigenvalue decomposition (EVD) toΤx, the estimation of Λx can be written as Λ^x=ΔΛxΔ, where Δ is the K×K permutation matrix satisfying ΔT=Δ−1. Let A^xz−1θ,φ denotes the estimation of Axz−1θ,φ; we have A^xz−1θ,φ=Axz−1θ,φΔ. Note that the scaling ambiguity is not considered here since Λx is unitary matrix and its diagonal elements locate on the unit circle.In order to achieve 2-D DOA automatic pairing, letϒzj,j=1,2 be two elevation angle selection matrices, which is defined as(28)ϒzj=I4⊗OM−1×j−1M−1,IM−1,OM−1×2−jM−1.Again by utilizing the rotational invariance property ofAxz−augθ,φ used in (26) and substitute Axz−1θ,φ with A^xz−1θ,φ, we have(29)ϒz1QA^xz−1θ,φΔΛz−1=ϒz2QA^xz−1θ,φΔ.Left-multiply and right-multiply both sides of (29) by ϒz2QA^xz−1θ,φ† and ΛzΔ, respectively; the estimation Λ^z is denoted as(30)Λ^z=ΔΛzΔ=ϒz2QA^xz−1θ,φ†ϒz1QA^xz−1θ,φ.The angle pairing is achieved automatically viaA^xz−1θ,φ, which associates Λ^x with Λ^z without permutation ambiguity. Therefore, the estimates of θk,φkk=1K can be obtained by(31)θ^kk=1K=arccosphaseΛ^x⋅λ2πd,φ^kk=1K=arccosphaseΛ^z⋅λ2πd.Note: the azimuth angle estimation utilizes the rotational invariance property from two 8(M − 1) element data, while the elevation estimation only use the rotational invariance property from two 4(M − 1) element data; in such case, the azimuth estimation performance is better than that of elevation.The main steps of the proposed subspace-like method are summarized as follows:(1)
Calculate the extended propagator matrix via the generalized steering matrix.(2)
Develop the rotational invariance property of the generalized steering matrixAxz−augθ,φ to obtain a nonsingular matrix Τx via two azimuth angle selection matrices.(3)
Apply eigenvalue decomposition toΤx, and the estimations of eigenvalue matrix Λx and eigenvector matrix Axz−1θ,φ are obtained; the azimuths are estimated from Λ^x.(4)
Develop the rotational invariance property of the generalized steering matrixAxz−augθ,φ but replace Axz−1θ,φ with A^xz−1θ,φ estimated from step 3, the autopaired elevation estimation is achieved via two elevation angle selection matrices.


## 3.3. The Sparsity Reconstruction Method

Assume that 2-D DOA search ranges are divided intoNθ and Nφ angle grids denoted by θp,p=1,2,…,Nθ, and φq,q=1,2,…,Nφ, respectively. Define in the spatial domain two basis matrices Ωθ of size M×Nθ,Nθ≫M and Ωφ of size M−1×Nφ,Nφ≫M−1 as(32)Ωθ=ηθθ1,ηθθ2,…,ηθθNθ,ηθθp=1,ej2πdcosθp/λ,…,ej2πM−1dcosθp/λTM,Ωφ=ηφφ1,ηφφ2,…,ηφφNφ,ηφφq=ej2πdcosθq/λ,…,ej2πM−1dcosθq/λTM−1,where p=1,…,Nθ, q=1,…,Nφ; ηθθp and ηφφq are the array steering basis vectors of θp and φq, respectively. Suppose the pairing θk,φkk=1K of K sources just fall on the 2-D DOA angle grids and indexed by pk,qkk=1K, respectively. Taking the samples cross-correlation matrix (12) into account, we can recast (12) into the dictionary-based sparse representation(33)R^x,z=AxθR^sAzφH=ΩθΘsΩφH,where Θs is a Nθ×Nφ sparse matrix containing only K nonzero coefficients, i.e., Θspk,qk=σk2; the similar observation is achieved in [32], wherein it is only studied numerically without theoretical analysis. The following lemma proves that equation (33) holds subject to certain mild constraints on the basis matrices and the sources.Lemma 1.
Considering (33) with the assumptions that any M columns of Ωθ and any M−1 columns of Ωφ are linearly independent, the received signals are uncorrelated and independently across the array which ensures rankΩθ=M and rankΩφ=M−1; the columns of basis matrices Ωθ and Ωφ are not nearly collinear to the columns that form the basis for the Θs, which guarantee the uniqueness of solutions, but to what extent depends on the signal-to-noise ratio (SNR) and angular interval between the columns of Ωθ and Ωφ, if rankΘs=K≤M−1, i.e., pi≠pj,qi≠qj,i≠j, then rankY1=K, and (33) is satisfied.Proof.
From the assumptions above, we haverankΩθ=M, rankΘs=K, and rankΩφ=M−1. Therefore, rankΘsΩφH≤minrankΘs,rankΩφ=K. Let Z=ΩθΘsΩφH, rankΖ=rankΩθΘsΩφH≤minrankΩθ,rankΘsΩφH=K. Without loss of generality, we assume that the first K rows and the first K columns of Θs (the case where those linear independent columns are dispersed randomly across the matrix Θs can be dealt in a similar manner) are linearly independently, yielding a new matrix Ts, which is equivalent to R^s, i.e., Γs=ΞR^sΞ. Accordingly, the generating matrices Ω˜θ and Ω˜φH, consisting of the first K columns of Ωθ and the first K rows of ΩφH, are intrinsically equivalent matrices of Ax1θ and Az1φH, respectively; that is to say Ω˜θ=Ax1θΞ and Ω˜φ=Az1φΞ; Ξ is the K×K permutation matrix satisfying ΞT=Ξ−1. As a result, ΩθΘsΩφH=Ω˜θΓsΩ˜φH, Ω˜θΓsΩ˜φH=Ax1θΞΞR^sΞΞAz1φH=Ax1θR^sAz1φH; therefore, Y1=Ax1θR^sAz1φH=ΩθΘsΩφH, rankY1=K, we complete the proof. Lemma 1 associates the parameter-based data models with its sparse representations.
The multiple measurement vector (MMV) sparse reconstruction is first addressed in [28], which is used to compute the sparse solutions given MMV that shares a common sparsity structure, wherein it exploits the sparsity in angle domain. Later, [32] extends 1-D MMV to 2-D MMV. In this context, the 2-D DOA sparse reconstruction is derived, and the sparse solutions are reconstructed via the M-FOCUSS [28] with and without the involvements of CS based on two received data models.


### 3.3.1. On the Data Model (33) with and without CS Involvements

First, we make use of (33) as the data model. In order to estimate Θs, we set up the cost function as follows:(34)Ζθ,φ∗=argminΖθ,φΖθ,φ2,s,s.t.R^x,z=ΩθΖθ,φ,where Ζθ,φ=ΘsΩφH, Ζθ,φ2,s≜∑r=1Nθzr2s, s∈0,1zr=zr1,…,zrM−1 is the rth row of Zθ,ϕ, and the row norm is 2-norm defined by zr2=∑l=1M−1zrl21/2; M−1 denotes the number of MMVs; Nθ denotes rows vectors of sparse solutions; s is the sparsity measure that balances the performance and the complexity. Equation (34) converts equation (33) 1-D MMV problem, where Ζθ,φ contains only K nonzero rows vector index supports corresponding to azimuth angles θkk=1K. By letting certain termination condition be satisfied, we take out K nonzero rows with the maximum amplitudes from Nθ rows vectors to form Ζθ,φ∗, i.e., Ζθ,φ∗=Ωθ,0†R^x,z, Ωθ,0=ηθθn1,ηθθn2,…,ηθθnK, IK=n1,n2,…,nK is the rows sparse index supports. In order to estimate elevations φkk=1K, we need to solve additional K 1-D SMV problems of(35)θs,k∗=argminθs,kθs,kH2,s,s.t.zθ,φ,k∗H=Ωφθs,kH,where zθ,φ,k∗ is the kth row vector of Ζθ,φ∗; θs,k∗ is the kth row vector of Θs; the kth nonzero coefficient is found from θs,k∗; the angles pairing is achieved automatically. Note: K azimuth angles θkk=1K are first found from K nonzero rows vector index supports at a single time slot, and then K elevation angles φkk=1K are found from K different row vectors by K time slots, which implies a fact that small azimuth angle separation will lead to poor azimuth estimation performance, but elevation estimation performance will not be affected even if the elevation angle separation is small.The proposed methods, inspired by [32], extend 2-D sparse reconstruction methods to CS-based sparse reconstruction in such a way that no angles pairing issue arises. This method allows for recovery of 2-D DOA assuming they stay on the predefined 2-D angle grids.Define two spatial domain compression matricesΦx of size Ml×M,Ml≤M and Φz of size Mk×M−1,Mk≤M−1, respectively, whose elements are drawn i.i.d from a Gaussian distribution with zero mean and unit variance. Left-multiply Φx and right-multiply ΦzH by R^x,z, the CS-based samples cross-correlation matrix R^x,zCS becomes(36)R^x,zCS=ΦxR^x,zΦzH=ΦxΩθΘsΩφHΦzH=Ω˜θΘsΩ˜φH,where Ω˜θ=ΦxΩθ and Ω˜φ=ΦzΩφ are the composite azimuth and elevation angle basis matrices, respectively; the incorporation of compression matrices into the received data matrix will benefit lower storage request. The CS-based sparse reconstruction method is implemented using the same procedures as shown in (34) and (35).


### 3.3.2. On the Data Model (20) with and without CS Involvements

The new insight into the 2-D sparse reconstruction method can also be obtained by developing the common sparse structure of the partial generalized data model (20), which will improve the azimuth estimation performance even when the azimuth separations are close. From Lemma 1, we know the basis matrix Ωθ and Ωφ must be row full rank and rankΘs=K; hence, we select from Y in (20) the group Y1,Y3 to construct a column full rank data matrix Y13=Y1T,Y⌢3TT of size M×M−2, where Y⌢3=eM−1TY3, eM−1=0,…,0,1T is the unit vector of size M−1. Taking the Hermitian transpose over Y13, Y13H can be represented as(37)Y13H=Az1φR^sHBx1θH,where Bx1θH=Ax1θHΛxHAx1θHeM−1, from Lemma 1, we recast (37) as(38)Y13H=A˜z1φR˜sHB˜x1θH,where A˜z1φ, R˜s, and B˜x1θ are M−2×Nφ, Nθ×Nφ, and M×Nθ basis matrices of Az1φ, R^s, and Bx1θ, respectively. B˜x1θH=A˜x1θHΛ˜xHA˜x1θHeM−1, Λ˜x=diagμ1,μ2,…,μNθ is Nθ×Nθ diagonal matrix; A˜x1θ is M−1×Nθ basis matrix of Ax1θ. Note: one can also define the groups Y2,Y4,Y5,Y7,Y6,Y8 to make new data matrices which provide the similar performance as Y13, but Y=Y1,…,Y8 may not provide better performance since the generalized steering basis matrix is rank deficit and does not suffice Lemma 1. Similar to (34), we establish the cost function(39)Z˜θ,φ∗=argminZ˜θ,φZ˜θ,φ2,s,s.t.Y13H=A˜z1φZ˜θ,φ,where Z˜θ,φ=R˜sHB˜x1θH, by enforcing some termination condition to (39), we select out K nonzero rows with the maximum amplitudes to form Z˜θ,φ∗, i.e., Z˜θ,φ∗=A˜z1,0φ†Y13H, A˜z1,0φ=A˜z1φm1,A˜z1φm2,…,A˜z1φmK, IK=m1,m2,…,mK is the row sparse index supports. Again additional K 1-D SMV problems of(40)r˜s,k∗=argminr˜s,kr˜s,kH2,s,s.t.z˜θ,φ,k∗H=B˜x1θr˜s,kH,have to be solved to obtain θkk=1K; the angles pairing is achieved automatically. Note: K elevation angles θkk=1K are first found from K nonzero rows vector index supports of Ζ˜θ,φ at a single time slot, and then K azimuth angles θkk=1K are found from K different row vectors at K time slots, which suggests a fact that small elevation separation will yield poor elevation estimation performance, but azimuth estimation performance will not be affected even if the azimuth separations are small.Define two spatial domain compression matricesΦx13 of size Ml×M,Ml≤M, and Φz13 of size Mk×M−2,Mk≤M−2, whose elements are drawn i.i.d from a Gaussian distribution with zero mean and unit variance, respectively. Left-multiply Φz13 and right-multiply Φx13H by Y13H, the CS-based data matrices Y13CSH become(41)Y13CSH=Φz13Y13HΦx13H=Φx13A˜z1φR˜sHB˜x1θHΦx13H=A˜˜z1φR˜sHB˜˜x1θH,where A˜˜z1φ=Φz13A˜z1φ and B˜˜x1θ=Φx13B˜x1θ are the composite 2-D DOA basis matrices. The CS-based sparse reconstruction procedure is implemented using the same methods as those in (39) and (40).The major steps of the sparse reconstruction methods are summarized as follows:(1)
Establish the sparse representation models based on the conventional data model and the partial generalized data model.(2)
Establish cost functions to minimize the errors based on M-FOCUSS framework.(3)
Transform the 2-D DOA estimation into two associated 1-D MMV problems, and the autopaired angles are achieved by solving the two optimization problems successively.Remark 1.
From (26) and (29), it is obvious that the array aperture, of θkk=1K and φkk=1K, is enhanced greatly from M to 8M−1 and from M to 4M−1, respectively.Remark 2.
The subspace-like method provides super-resolution when the received data matrix is modeled accurately for the L-shaped array. By contrast, the sparse reconstruction methods do not impose any restricts on the array structure, either nonlinear or nonuniform. On the other hand, the number of the maximum identifiable sparse solutionsK [33] should satisfy K<M−1/2 for two kinds of approaches.


## 3.3.1. On the Data Model (33) with and without CS Involvements

First, we make use of (33) as the data model. In order to estimate Θs, we set up the cost function as follows:(34)Ζθ,φ∗=argminΖθ,φΖθ,φ2,s,s.t.R^x,z=ΩθΖθ,φ,where Ζθ,φ=ΘsΩφH, Ζθ,φ2,s≜∑r=1Nθzr2s, s∈0,1zr=zr1,…,zrM−1 is the rth row of Zθ,ϕ, and the row norm is 2-norm defined by zr2=∑l=1M−1zrl21/2; M−1 denotes the number of MMVs; Nθ denotes rows vectors of sparse solutions; s is the sparsity measure that balances the performance and the complexity. Equation (34) converts equation (33) 1-D MMV problem, where Ζθ,φ contains only K nonzero rows vector index supports corresponding to azimuth angles θkk=1K. By letting certain termination condition be satisfied, we take out K nonzero rows with the maximum amplitudes from Nθ rows vectors to form Ζθ,φ∗, i.e., Ζθ,φ∗=Ωθ,0†R^x,z, Ωθ,0=ηθθn1,ηθθn2,…,ηθθnK, IK=n1,n2,…,nK is the rows sparse index supports. In order to estimate elevations φkk=1K, we need to solve additional K 1-D SMV problems of(35)θs,k∗=argminθs,kθs,kH2,s,s.t.zθ,φ,k∗H=Ωφθs,kH,where zθ,φ,k∗ is the kth row vector of Ζθ,φ∗; θs,k∗ is the kth row vector of Θs; the kth nonzero coefficient is found from θs,k∗; the angles pairing is achieved automatically. Note: K azimuth angles θkk=1K are first found from K nonzero rows vector index supports at a single time slot, and then K elevation angles φkk=1K are found from K different row vectors by K time slots, which implies a fact that small azimuth angle separation will lead to poor azimuth estimation performance, but elevation estimation performance will not be affected even if the elevation angle separation is small.The proposed methods, inspired by [32], extend 2-D sparse reconstruction methods to CS-based sparse reconstruction in such a way that no angles pairing issue arises. This method allows for recovery of 2-D DOA assuming they stay on the predefined 2-D angle grids.Define two spatial domain compression matricesΦx of size Ml×M,Ml≤M and Φz of size Mk×M−1,Mk≤M−1, respectively, whose elements are drawn i.i.d from a Gaussian distribution with zero mean and unit variance. Left-multiply Φx and right-multiply ΦzH by R^x,z, the CS-based samples cross-correlation matrix R^x,zCS becomes(36)R^x,zCS=ΦxR^x,zΦzH=ΦxΩθΘsΩφHΦzH=Ω˜θΘsΩ˜φH,where Ω˜θ=ΦxΩθ and Ω˜φ=ΦzΩφ are the composite azimuth and elevation angle basis matrices, respectively; the incorporation of compression matrices into the received data matrix will benefit lower storage request. The CS-based sparse reconstruction method is implemented using the same procedures as shown in (34) and (35).


## 3.3.2. On the Data Model (20) with and without CS Involvements

The new insight into the 2-D sparse reconstruction method can also be obtained by developing the common sparse structure of the partial generalized data model (20), which will improve the azimuth estimation performance even when the azimuth separations are close. From Lemma 1, we know the basis matrix Ωθ and Ωφ must be row full rank and rankΘs=K; hence, we select from Y in (20) the group Y1,Y3 to construct a column full rank data matrix Y13=Y1T,Y⌢3TT of size M×M−2, where Y⌢3=eM−1TY3, eM−1=0,…,0,1T is the unit vector of size M−1. Taking the Hermitian transpose over Y13, Y13H can be represented as(37)Y13H=Az1φR^sHBx1θH,where Bx1θH=Ax1θHΛxHAx1θHeM−1, from Lemma 1, we recast (37) as(38)Y13H=A˜z1φR˜sHB˜x1θH,where A˜z1φ, R˜s, and B˜x1θ are M−2×Nφ, Nθ×Nφ, and M×Nθ basis matrices of Az1φ, R^s, and Bx1θ, respectively. B˜x1θH=A˜x1θHΛ˜xHA˜x1θHeM−1, Λ˜x=diagμ1,μ2,…,μNθ is Nθ×Nθ diagonal matrix; A˜x1θ is M−1×Nθ basis matrix of Ax1θ. Note: one can also define the groups Y2,Y4,Y5,Y7,Y6,Y8 to make new data matrices which provide the similar performance as Y13, but Y=Y1,…,Y8 may not provide better performance since the generalized steering basis matrix is rank deficit and does not suffice Lemma 1. Similar to (34), we establish the cost function(39)Z˜θ,φ∗=argminZ˜θ,φZ˜θ,φ2,s,s.t.Y13H=A˜z1φZ˜θ,φ,where Z˜θ,φ=R˜sHB˜x1θH, by enforcing some termination condition to (39), we select out K nonzero rows with the maximum amplitudes to form Z˜θ,φ∗, i.e., Z˜θ,φ∗=A˜z1,0φ†Y13H, A˜z1,0φ=A˜z1φm1,A˜z1φm2,…,A˜z1φmK, IK=m1,m2,…,mK is the row sparse index supports. Again additional K 1-D SMV problems of(40)r˜s,k∗=argminr˜s,kr˜s,kH2,s,s.t.z˜θ,φ,k∗H=B˜x1θr˜s,kH,have to be solved to obtain θkk=1K; the angles pairing is achieved automatically. Note: K elevation angles θkk=1K are first found from K nonzero rows vector index supports of Ζ˜θ,φ at a single time slot, and then K azimuth angles θkk=1K are found from K different row vectors at K time slots, which suggests a fact that small elevation separation will yield poor elevation estimation performance, but azimuth estimation performance will not be affected even if the azimuth separations are small.Define two spatial domain compression matricesΦx13 of size Ml×M,Ml≤M, and Φz13 of size Mk×M−2,Mk≤M−2, whose elements are drawn i.i.d from a Gaussian distribution with zero mean and unit variance, respectively. Left-multiply Φz13 and right-multiply Φx13H by Y13H, the CS-based data matrices Y13CSH become(41)Y13CSH=Φz13Y13HΦx13H=Φx13A˜z1φR˜sHB˜x1θHΦx13H=A˜˜z1φR˜sHB˜˜x1θH,where A˜˜z1φ=Φz13A˜z1φ and B˜˜x1θ=Φx13B˜x1θ are the composite 2-D DOA basis matrices. The CS-based sparse reconstruction procedure is implemented using the same methods as those in (39) and (40).The major steps of the sparse reconstruction methods are summarized as follows:(1)
Establish the sparse representation models based on the conventional data model and the partial generalized data model.(2)
Establish cost functions to minimize the errors based on M-FOCUSS framework.(3)
Transform the 2-D DOA estimation into two associated 1-D MMV problems, and the autopaired angles are achieved by solving the two optimization problems successively.Remark 1.
From (26) and (29), it is obvious that the array aperture, of θkk=1K and φkk=1K, is enhanced greatly from M to 8M−1 and from M to 4M−1, respectively.Remark 2.
The subspace-like method provides super-resolution when the received data matrix is modeled accurately for the L-shaped array. By contrast, the sparse reconstruction methods do not impose any restricts on the array structure, either nonlinear or nonuniform. On the other hand, the number of the maximum identifiable sparse solutionsK [33] should satisfy K<M−1/2 for two kinds of approaches.


## 4. Simulation Results and Analysis

### 4.1. Computational Complexity Analysis

In this subsection, we present the complexity analysis for the competitive methods, the complexity of EAET [12] is OM2N+8MK2+4M4M−KK+8M−1K2, of JSVD [18] is OM2N+2MM−12+2MK2+K3+NsM2K, of PARAFAC [26] is O4M−12N+NitM−12K2+13M−12K+8M−1K2+8M−1K, of CSAP [29] is OM2N+16M3+Ns16M2+2+4M22M−K, of AAEA [10] is OM2N+9M3+6M2K+2MK3+18MK2+3K3, and of the proposed subspace-like method is OM2N+2K2M−2+KM−28M−1−K+16M−2K2+4K3, where Ns, Nit denote the total search times within the search range and the number of iteration, respectively. As Ns≫M,Ns≫K, and M>K, the subspace-like method is more computationally efficient than the JSVD and CSAP and is close to AAEA method, the complexity of sparse reconstruction methods depends on the implementation of M-FOCUSS [28]. Meanwhile, it is quite difficult to compare the proposed methods with the PARAFAC method, since the number of iterations depends on heavily the received data and possibly varies dramatically. Therefore, the other complexity benchmark is to numerically compare the CPU runtime of the competitive methods as used, for example, in [12, 29]. In this paper, all simulations are carried out on a laptop configured as follows: MATLAB R2016b, Intel ® Core™ i5-6200U CPU@ 2.30 GHz and 8 GB RAM.


### 4.2. Simulation and Parameter Settings

Simulations are conducted to evaluate azimuth estimation performances of the proposed approaches in this section, since the array aperture is significantly extended in azimuths but is slightly extended in elevations. The search range of azimuths and elevations is0∘,180∘ with step 0.1∘ for JSVD [18] and CSAP [29]. The performance of the proposed subspace-like method is compared with EAET [12], JSVD [18], PARAFAC [26], CSAP [29], AAEA [10], and CRB (Cramer-Rao Bound). The K sources are drawn from the standard normal distribution skt∼N0,1,t=1,…,J. I=500 is the number of Monte Carlo trials used throughout all simulations, and SNR is defined as 10log10σs2/σw2, where σs2,σw2 denote the power of signals and noise, respectively.


#### 4.2.1. 2-D DOA Estimation Performance

Example 1.
The 2-D DOA estimation performance of the proposed subspace-like method is compared to EAET [12] in this example. There are in the far-field K=3 uncorrelated narrow-band sources from the directions 60.25°,45.39°, 30.88°,80.19°, 133.05°,160.60° impinging on the L-shaped ULA. Both M=7 and J=500 are fixed. We set SNR=20dB and run I=500 Monte Carlo trials to observe scatters of 2-D DOA estimations.
As shown in Figure2, we observe that the estimation variance derived by the subspace-like method is smaller than EAET [12] because the proposed method develops generalized virtual aperture.Figure 2
The scatters of 2-D DOA estimation with (a) the subspace-like method and (b) EAET.
(a)
(b)


#### 4.2.2. RMSE vs. SNR for the Subspace-Like Method

Example 2.
The performance of the subspace-like method, against SNR varying from −5 dB to 20 dB with interval 2.5 dB, is studied in this example. The sources angles are99°,91°, 100°,65°, and 101°,76°; the other parameters are the same as those in Example 1. The RMSE of azimuth estimation vs. SNR is shown in Figure 3. As observed in the figure, the azimuth estimation performance for the proposed method is better than other methods because the subspace-like method utilizes the rotational invariance property from two 8M−1 elements array data while EAET [12] only use the rotational invariance property from two 4M−1 elements array data. One of the most significant advantages is the proposed subspace-like method possesses the super-resolution capability.Figure 3
The RMSE of azimuth estimation versus SNR withK=3, M=7, and J=500.


#### 4.2.3. RMSE vs. Snapshots for the Subspace-Like Method

Example 3.
LetSNR=10dB be fixed and the number of snapshots varies from 100 to 1000 with step 100; the other parameters are the same as those in Example 2. The RMSE of azimuth estimation versus snapshots is illustrated in Figure 4. Again, we observe that the estimation performance of the proposed method improves as snapshots increase and is the best of all competitive methods, from which we have similar observation and analysis.Figure 4
The RMSE of azimuth estimation vs. snapshots withK=3, M=7, and SNR=10dB.


#### 4.2.4. RMSE vs. Azimuth Separations for the Subspace-Like Method

Example 4.
From the previous examples, we note that the performance of the proposed subspace method is a little higher than that of EAET [12]. In this example, we hope to disclose the discrimination in azimuth separations for two algorithms, where the initial angles are 60.25∘,45.39∘; the azimuth angle incremental interval is 0.1∘,4.9∘ with step 0.2∘, and the elevation angle incremental interval is 2∘,50∘ with step 2∘, as shown in Figure 5. We observe that the proposed subspace-like method has the similar angle discrimination performance as EAET [12].Figure 5
The RMSE of azimuth estimation vs. azimuth separations.


#### 4.2.5. Complexity vs. Number of Subarray Elements

Example 5.
The complexity with respect to the number of subarray elements is presented in Figure6, where SNR=10dB, J=500, K=3, and the range of M is set from 4 to 100 with step 4. We observe that the subspace-like method has relative low running time among the compared methods. It is worth noting that the subspace-like method possesses the best estimation performance when azimuth separation is small, which is particularly suitable for the scenarios where sources are close in spatial azimuths. On the other hand, we observe that the subspace-like method has lower complexity than the PARAFAC method when M < 100.Figure 6
Runtime vs. the number of subarray elements withK=3, J=500, and SNR=10dB.


#### 4.2.6. RMSE vs. SNR for the Sparse Reconstruction Methods

Example 6.
The performance of the sparse reconstruction methods, with respect to SNR varying from −5 dB to 20 dB with the interval 2.5 dB, is investigated in this example. The angle grids of two basis matrices span1∘,180∘ with step 1∘, i.e., Nθ=180 and Nφ=180. Three sources angles are the same as those in Example 2, and other parameters are set the same values as those in Example 1 except SNR. To compare all methods fairly, we take only the azimuth estimation into account. The RMSE of azimuth estimation vs. SNR for the sparse reconstruction methods with and without CS involvements are shown in Figure 7. It is observed that the azimuth estimation improves as SNR increases. The (38) performs significantly the best among the methods across the SNR range, which is expected as explained in Section 3.3.2 (see the note), since the sparse reconstruction method using (38) is developed regardless of whether azimuths are apart or close. The performance of (33), as stated in Section 3.3.1 (see the note), is however significantly degraded because azimuth angles are close. On the other hand, the CS versions (41) and (36) perform slightly worse than their counterparts (38) and (33), respectively, and their performance is unstable to varying basis matrices during the Monte Carlo trials. Comparing Figure 6 with Figure 3, the RMSE of the sparse reconstruction method using (38) is better than that of JSVD when SNR<10dB. Meanwhile, it is superior to PARAFAC.Figure 7
The RMSE of azimuth estimation vs. SNR for sparse reconstruction methods with and without CS involvements.


#### 4.2.7. Complexity vs. Number of Subarray Elements

Example 7.
The complexity with respect to the number of subarray elements for the sparse reconstruction methods is presented in Figure8, where SNR=10dB, J=500, K=3, and the range of M is set from 4 to 100 with step 4. We observe that the sparse reconstruction (38) has the lowest running time among the compared methods; it is worth noting that (38) still holds the best azimuth estimation performance. The incorporation of CS into conventional data models leads to higher complexity. By comparing Figures 7 with Figure 5, it is found that the runtime of (38) is less than that of PARAFAC; this can be explained by the fact that the search ranges are set to 0∘,180∘ with step 0.1∘ for PARAFAC, but the same search ranges with step 1∘ for sparse reconstruction methods, since 0.1∘ angle separation will destroy coherence of basis matrix and lead to performance degradation. The data models (33), (36), (38), and (41) are computationally comparable with PARAFAC, but more efficient than CSAP [29] and JSVD [18]. The other observation is that there is no significant increase in the complexity for the sparse reconstruction methods with the number of subarray elements, which shows advantages over the competitive methods. Thus, developing applications with large array size is an interesting topic.Figure 8
Runtime vs. the number of subarray elements for sparse reconstruction methods with and without CS involvements.


### 4.3. Analysis and Discussion

From all the examples above, it is clearly concluded that the performance benefiting from the generalized virtual received data model can outperform those from the conventional data models. We propose two kinds of methods to address the 2-D DOA estimation. One is the generalization of PM-ESPRIT, referred to as the subspace-like method, which fully utilize generalized virtual aperture to enhance the performance with a low computational complexity. The proposed sparse reconstruction methods provide the general theoretical framework with rigorous proof, which is suitable for any array geometry. Following this, we carry out the derivation of 2-D DOA estimation using partial generalized virtual aperture based on M-FOCUSS and obtain satisfactory results with a moderate computational complexity. One drawback of the proposed methods is that they only deal with the underdetermined signals using doubled elements. This problem to handle overdetermined sources has been addressed in the other paper.


## 4.1. Computational Complexity Analysis

In this subsection, we present the complexity analysis for the competitive methods, the complexity of EAET [12] is OM2N+8MK2+4M4M−KK+8M−1K2, of JSVD [18] is OM2N+2MM−12+2MK2+K3+NsM2K, of PARAFAC [26] is O4M−12N+NitM−12K2+13M−12K+8M−1K2+8M−1K, of CSAP [29] is OM2N+16M3+Ns16M2+2+4M22M−K, of AAEA [10] is OM2N+9M3+6M2K+2MK3+18MK2+3K3, and of the proposed subspace-like method is OM2N+2K2M−2+KM−28M−1−K+16M−2K2+4K3, where Ns, Nit denote the total search times within the search range and the number of iteration, respectively. As Ns≫M,Ns≫K, and M>K, the subspace-like method is more computationally efficient than the JSVD and CSAP and is close to AAEA method, the complexity of sparse reconstruction methods depends on the implementation of M-FOCUSS [28]. Meanwhile, it is quite difficult to compare the proposed methods with the PARAFAC method, since the number of iterations depends on heavily the received data and possibly varies dramatically. Therefore, the other complexity benchmark is to numerically compare the CPU runtime of the competitive methods as used, for example, in [12, 29]. In this paper, all simulations are carried out on a laptop configured as follows: MATLAB R2016b, Intel ® Core™ i5-6200U CPU@ 2.30 GHz and 8 GB RAM.


## 4.2. Simulation and Parameter Settings

Simulations are conducted to evaluate azimuth estimation performances of the proposed approaches in this section, since the array aperture is significantly extended in azimuths but is slightly extended in elevations. The search range of azimuths and elevations is0∘,180∘ with step 0.1∘ for JSVD [18] and CSAP [29]. The performance of the proposed subspace-like method is compared with EAET [12], JSVD [18], PARAFAC [26], CSAP [29], AAEA [10], and CRB (Cramer-Rao Bound). The K sources are drawn from the standard normal distribution skt∼N0,1,t=1,…,J. I=500 is the number of Monte Carlo trials used throughout all simulations, and SNR is defined as 10log10σs2/σw2, where σs2,σw2 denote the power of signals and noise, respectively.


### 4.2.1. 2-D DOA Estimation Performance

Example 1.
The 2-D DOA estimation performance of the proposed subspace-like method is compared to EAET [12] in this example. There are in the far-field K=3 uncorrelated narrow-band sources from the directions 60.25°,45.39°, 30.88°,80.19°, 133.05°,160.60° impinging on the L-shaped ULA. Both M=7 and J=500 are fixed. We set SNR=20dB and run I=500 Monte Carlo trials to observe scatters of 2-D DOA estimations.
As shown in Figure2, we observe that the estimation variance derived by the subspace-like method is smaller than EAET [12] because the proposed method develops generalized virtual aperture.Figure 2
The scatters of 2-D DOA estimation with (a) the subspace-like method and (b) EAET.
(a)
(b)


### 4.2.2. RMSE vs. SNR for the Subspace-Like Method

Example 2.
The performance of the subspace-like method, against SNR varying from −5 dB to 20 dB with interval 2.5 dB, is studied in this example. The sources angles are99°,91°, 100°,65°, and 101°,76°; the other parameters are the same as those in Example 1. The RMSE of azimuth estimation vs. SNR is shown in Figure 3. As observed in the figure, the azimuth estimation performance for the proposed method is better than other methods because the subspace-like method utilizes the rotational invariance property from two 8M−1 elements array data while EAET [12] only use the rotational invariance property from two 4M−1 elements array data. One of the most significant advantages is the proposed subspace-like method possesses the super-resolution capability.Figure 3
The RMSE of azimuth estimation versus SNR withK=3, M=7, and J=500.


### 4.2.3. RMSE vs. Snapshots for the Subspace-Like Method

Example 3.
LetSNR=10dB be fixed and the number of snapshots varies from 100 to 1000 with step 100; the other parameters are the same as those in Example 2. The RMSE of azimuth estimation versus snapshots is illustrated in Figure 4. Again, we observe that the estimation performance of the proposed method improves as snapshots increase and is the best of all competitive methods, from which we have similar observation and analysis.Figure 4
The RMSE of azimuth estimation vs. snapshots withK=3, M=7, and SNR=10dB.


### 4.2.4. RMSE vs. Azimuth Separations for the Subspace-Like Method

Example 4.
From the previous examples, we note that the performance of the proposed subspace method is a little higher than that of EAET [12]. In this example, we hope to disclose the discrimination in azimuth separations for two algorithms, where the initial angles are 60.25∘,45.39∘; the azimuth angle incremental interval is 0.1∘,4.9∘ with step 0.2∘, and the elevation angle incremental interval is 2∘,50∘ with step 2∘, as shown in Figure 5. We observe that the proposed subspace-like method has the similar angle discrimination performance as EAET [12].Figure 5
The RMSE of azimuth estimation vs. azimuth separations.


### 4.2.5. Complexity vs. Number of Subarray Elements

Example 5.
The complexity with respect to the number of subarray elements is presented in Figure6, where SNR=10dB, J=500, K=3, and the range of M is set from 4 to 100 with step 4. We observe that the subspace-like method has relative low running time among the compared methods. It is worth noting that the subspace-like method possesses the best estimation performance when azimuth separation is small, which is particularly suitable for the scenarios where sources are close in spatial azimuths. On the other hand, we observe that the subspace-like method has lower complexity than the PARAFAC method when M < 100.Figure 6
Runtime vs. the number of subarray elements withK=3, J=500, and SNR=10dB.


### 4.2.6. RMSE vs. SNR for the Sparse Reconstruction Methods

Example 6.
The performance of the sparse reconstruction methods, with respect to SNR varying from −5 dB to 20 dB with the interval 2.5 dB, is investigated in this example. The angle grids of two basis matrices span1∘,180∘ with step 1∘, i.e., Nθ=180 and Nφ=180. Three sources angles are the same as those in Example 2, and other parameters are set the same values as those in Example 1 except SNR. To compare all methods fairly, we take only the azimuth estimation into account. The RMSE of azimuth estimation vs. SNR for the sparse reconstruction methods with and without CS involvements are shown in Figure 7. It is observed that the azimuth estimation improves as SNR increases. The (38) performs significantly the best among the methods across the SNR range, which is expected as explained in Section 3.3.2 (see the note), since the sparse reconstruction method using (38) is developed regardless of whether azimuths are apart or close. The performance of (33), as stated in Section 3.3.1 (see the note), is however significantly degraded because azimuth angles are close. On the other hand, the CS versions (41) and (36) perform slightly worse than their counterparts (38) and (33), respectively, and their performance is unstable to varying basis matrices during the Monte Carlo trials. Comparing Figure 6 with Figure 3, the RMSE of the sparse reconstruction method using (38) is better than that of JSVD when SNR<10dB. Meanwhile, it is superior to PARAFAC.Figure 7
The RMSE of azimuth estimation vs. SNR for sparse reconstruction methods with and without CS involvements.


### 4.2.7. Complexity vs. Number of Subarray Elements

Example 7.
The complexity with respect to the number of subarray elements for the sparse reconstruction methods is presented in Figure8, where SNR=10dB, J=500, K=3, and the range of M is set from 4 to 100 with step 4. We observe that the sparse reconstruction (38) has the lowest running time among the compared methods; it is worth noting that (38) still holds the best azimuth estimation performance. The incorporation of CS into conventional data models leads to higher complexity. By comparing Figures 7 with Figure 5, it is found that the runtime of (38) is less than that of PARAFAC; this can be explained by the fact that the search ranges are set to 0∘,180∘ with step 0.1∘ for PARAFAC, but the same search ranges with step 1∘ for sparse reconstruction methods, since 0.1∘ angle separation will destroy coherence of basis matrix and lead to performance degradation. The data models (33), (36), (38), and (41) are computationally comparable with PARAFAC, but more efficient than CSAP [29] and JSVD [18]. The other observation is that there is no significant increase in the complexity for the sparse reconstruction methods with the number of subarray elements, which shows advantages over the competitive methods. Thus, developing applications with large array size is an interesting topic.Figure 8
Runtime vs. the number of subarray elements for sparse reconstruction methods with and without CS involvements.


## 4.2.1. 2-D DOA Estimation Performance

Example 1.
The 2-D DOA estimation performance of the proposed subspace-like method is compared to EAET [12] in this example. There are in the far-field K=3 uncorrelated narrow-band sources from the directions 60.25°,45.39°, 30.88°,80.19°, 133.05°,160.60° impinging on the L-shaped ULA. Both M=7 and J=500 are fixed. We set SNR=20dB and run I=500 Monte Carlo trials to observe scatters of 2-D DOA estimations.
As shown in Figure2, we observe that the estimation variance derived by the subspace-like method is smaller than EAET [12] because the proposed method develops generalized virtual aperture.Figure 2
The scatters of 2-D DOA estimation with (a) the subspace-like method and (b) EAET.
(a)
(b)


## 4.2.2. RMSE vs. SNR for the Subspace-Like Method

Example 2.
The performance of the subspace-like method, against SNR varying from −5 dB to 20 dB with interval 2.5 dB, is studied in this example. The sources angles are99°,91°, 100°,65°, and 101°,76°; the other parameters are the same as those in Example 1. The RMSE of azimuth estimation vs. SNR is shown in Figure 3. As observed in the figure, the azimuth estimation performance for the proposed method is better than other methods because the subspace-like method utilizes the rotational invariance property from two 8M−1 elements array data while EAET [12] only use the rotational invariance property from two 4M−1 elements array data. One of the most significant advantages is the proposed subspace-like method possesses the super-resolution capability.Figure 3
The RMSE of azimuth estimation versus SNR withK=3, M=7, and J=500.


## 4.2.3. RMSE vs. Snapshots for the Subspace-Like Method

Example 3.
LetSNR=10dB be fixed and the number of snapshots varies from 100 to 1000 with step 100; the other parameters are the same as those in Example 2. The RMSE of azimuth estimation versus snapshots is illustrated in Figure 4. Again, we observe that the estimation performance of the proposed method improves as snapshots increase and is the best of all competitive methods, from which we have similar observation and analysis.Figure 4
The RMSE of azimuth estimation vs. snapshots withK=3, M=7, and SNR=10dB.


## 4.2.4. RMSE vs. Azimuth Separations for the Subspace-Like Method

Example 4.
From the previous examples, we note that the performance of the proposed subspace method is a little higher than that of EAET [12]. In this example, we hope to disclose the discrimination in azimuth separations for two algorithms, where the initial angles are 60.25∘,45.39∘; the azimuth angle incremental interval is 0.1∘,4.9∘ with step 0.2∘, and the elevation angle incremental interval is 2∘,50∘ with step 2∘, as shown in Figure 5. We observe that the proposed subspace-like method has the similar angle discrimination performance as EAET [12].Figure 5
The RMSE of azimuth estimation vs. azimuth separations.


## 4.2.5. Complexity vs. Number of Subarray Elements

Example 5.
The complexity with respect to the number of subarray elements is presented in Figure6, where SNR=10dB, J=500, K=3, and the range of M is set from 4 to 100 with step 4. We observe that the subspace-like method has relative low running time among the compared methods. It is worth noting that the subspace-like method possesses the best estimation performance when azimuth separation is small, which is particularly suitable for the scenarios where sources are close in spatial azimuths. On the other hand, we observe that the subspace-like method has lower complexity than the PARAFAC method when M < 100.Figure 6
Runtime vs. the number of subarray elements withK=3, J=500, and SNR=10dB.


## 4.2.6. RMSE vs. SNR for the Sparse Reconstruction Methods

Example 6.
The performance of the sparse reconstruction methods, with respect to SNR varying from −5 dB to 20 dB with the interval 2.5 dB, is investigated in this example. The angle grids of two basis matrices span1∘,180∘ with step 1∘, i.e., Nθ=180 and Nφ=180. Three sources angles are the same as those in Example 2, and other parameters are set the same values as those in Example 1 except SNR. To compare all methods fairly, we take only the azimuth estimation into account. The RMSE of azimuth estimation vs. SNR for the sparse reconstruction methods with and without CS involvements are shown in Figure 7. It is observed that the azimuth estimation improves as SNR increases. The (38) performs significantly the best among the methods across the SNR range, which is expected as explained in Section 3.3.2 (see the note), since the sparse reconstruction method using (38) is developed regardless of whether azimuths are apart or close. The performance of (33), as stated in Section 3.3.1 (see the note), is however significantly degraded because azimuth angles are close. On the other hand, the CS versions (41) and (36) perform slightly worse than their counterparts (38) and (33), respectively, and their performance is unstable to varying basis matrices during the Monte Carlo trials. Comparing Figure 6 with Figure 3, the RMSE of the sparse reconstruction method using (38) is better than that of JSVD when SNR<10dB. Meanwhile, it is superior to PARAFAC.Figure 7
The RMSE of azimuth estimation vs. SNR for sparse reconstruction methods with and without CS involvements.


## 4.2.7. Complexity vs. Number of Subarray Elements

Example 7.
The complexity with respect to the number of subarray elements for the sparse reconstruction methods is presented in Figure8, where SNR=10dB, J=500, K=3, and the range of M is set from 4 to 100 with step 4. We observe that the sparse reconstruction (38) has the lowest running time among the compared methods; it is worth noting that (38) still holds the best azimuth estimation performance. The incorporation of CS into conventional data models leads to higher complexity. By comparing Figures 7 with Figure 5, it is found that the runtime of (38) is less than that of PARAFAC; this can be explained by the fact that the search ranges are set to 0∘,180∘ with step 0.1∘ for PARAFAC, but the same search ranges with step 1∘ for sparse reconstruction methods, since 0.1∘ angle separation will destroy coherence of basis matrix and lead to performance degradation. The data models (33), (36), (38), and (41) are computationally comparable with PARAFAC, but more efficient than CSAP [29] and JSVD [18]. The other observation is that there is no significant increase in the complexity for the sparse reconstruction methods with the number of subarray elements, which shows advantages over the competitive methods. Thus, developing applications with large array size is an interesting topic.Figure 8
Runtime vs. the number of subarray elements for sparse reconstruction methods with and without CS involvements.


## 4.3. Analysis and Discussion

From all the examples above, it is clearly concluded that the performance benefiting from the generalized virtual received data model can outperform those from the conventional data models. We propose two kinds of methods to address the 2-D DOA estimation. One is the generalization of PM-ESPRIT, referred to as the subspace-like method, which fully utilize generalized virtual aperture to enhance the performance with a low computational complexity. The proposed sparse reconstruction methods provide the general theoretical framework with rigorous proof, which is suitable for any array geometry. Following this, we carry out the derivation of 2-D DOA estimation using partial generalized virtual aperture based on M-FOCUSS and obtain satisfactory results with a moderate computational complexity. One drawback of the proposed methods is that they only deal with the underdetermined signals using doubled elements. This problem to handle overdetermined sources has been addressed in the other paper.


## 5. Conclusions

In this paper, we exploit both the forward and backward conjugate symmetric properties of the sample cross-correlation matrix of X-subarray and Z-subarray for L-shaped ULA. A generalized virtual received data model is therefore constructed, which improves substantially the azimuth estimation performance. We propose two kinds of approaches to estimate the 2-D DOA with automatic angles pairing. Compared to existing methods, the subspace-like method is efficient when the data model matches the physical model well. On the other hand, the sparse reconstruction methods are able to reduce complexity by converting 2-D search procedures to two successive 1-D search procedures. Numerical examples validate the effectiveness of the theoretical analysis and confirm the superb performance with low complexity for the subspace-like method and with moderate complexity for the sparse reconstruction methods.



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*Source: 3924569-2019-12-24.xml*