# Utilizing Principal Singular Vectors for 2D DOA Estimation in Single Snapshot Case with Uniform Rectangular Array

Authors: Yuntao Wu; Xiaobing Pei; Hing Cheung So
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

The problem of azimuth and elevation directions of arrival (DOAs) estimation
using a uniform rectangular array (URA) in single snapshot case is addressed in this paper.
Using the principal singular vectors of the observed data matrix, an iterative procedure
based on the linear prediction property, and weighted least squares is proposed for finding
the DOAs with lower computational complexity. Furthermore, the azimuth and elevation
parameters are automatically paired. Computer simulations are included to demonstrate
the effectiveness of the proposed algorithm.


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## Body

## 1. Introduction

Two-dimensional (2D) DOA estimation of multiple narrowband signals is an active research topic in array signal processing due to its wide applications in radar, sonar, radio astronomy, and mobile communications and so forth. In recent years, many high-resolution subspace-based algorithms such as 2D MUSIC- [1] or 2D ESPRIT-based [2, 3] as well as the modified unitary ESPRIT algorithms [4] have been proposed to jointly estimate the 2D parameters with various array geometries. These algorithms are effective in certain conditions; however, high-dimension data stacking and eigen-decomposition of a constructed covariance of array output are needed, corresponding to high computational load. To reduce the computational complexity of eigendecomposition, fast algorithms have been reported [5, 6]. In general, a direction-finding algorithm in real application must provide a real-time solution that is computationally efficient and uses only a few array snapshots. In some real scenarios, such as fast time-variant channel in mobile communications and sonar signal processing due to physical constraints, only one or a few snapshots are available. Therefore, the problem of parameter estimation in the case of small number of samples has been addressed in the literature [7–9], where the advanced concept of compressed sensing (CS) is applied. In particular, in the worst case such as automotive radar systems [10], only a single snapshot is available for parameter estimation of multiple spatial sources. That is to say, the problem of single snapshot DOA estimation is important in certain application and corresponding DOA estimation algorithms in single-snapshot case have been recently proposed [11–17].In this paper, an algorithm of joint estimation of the azimuth and elevation angles using a uniform rectangular array (URA) is proposed for the case of single snapshot. The main idea is based on the principal-singular-vector utilization for model analysis (PUMA) [18, 19]. With the use of the PUMA technique, not only the estimation performance in single snapshot case is improved but the computational complexity of the proposed method is also largely reduced. Moreover, the 2D parameters are automatically paired. Simulation results show that the proposed method provides better performance but with largely reduced computational load compared with the existing 2D ESPRIT method.The rest of the paper is organized as follows. The problem formulation is given in Section1. In Section 2, the proposed 2D DOA estimator for single snapshot case with a URA array is devised. Simulation results are included in Section 3 to evaluate the performance of the proposed method by comparing with the 2D ESPRIT method as well as the corresponding Cramer-Rao lower bound (CRLB). Finally, conclusions are drawn in Section 4.


## 2. Problem Formulation

Figure1 shows the planar URA configuration for receiving the incoming signals. The array is composed of M×N omnidirectional antennas with interelement spacing d and we assume that P narrowband signal sources imping on this array from the 2D spatial directions ϕp,θp, p=1,2,…,P. The sensor located at a11(1,1) is the reference sensor.Figure 1
Uniform rectangular array geometry.The single-snapshot output of the sensor at(m,n) can be expressed as(1)xm,n=∑p=1Pspej2π/λn-1dsinϕpcos⁡θpej2π/λm-1dsinϕpsinθp+nm,nm=1,2,…,M,n=1,2,…,N,where sp denotes the wavefront of the pth signal source, λ denotes the wavelength of the impinging wavefront, and nm,n is the white Gaussian noise with power σn2 received at the (m,n)th sensor.Our task is to estimate the parameter pairsϕp,θp, p=1,…,P, from the received array output.


## 3. Proposed Method

In this section, the PUMA algorithm [18] is exploited to estimate the 2D DOA parameters of multiple narrowband sources with URA in the case of single snapshot. Define the spatial frequencies up,vp as up=2π/λdsin(ϕp)cos⁡(θp) and vp=2π/λdsin(ϕp)sin(θp); we first express (1) in matrix form as(2)X=S+N,where [X]m,n=xm,n, [S]m,n=sm,n, and [N]m,n=nm,n.Thesm,n has the form of(3)sm,n=∑p=1Pspejm-1up+n-1vp.It is easy to find that S can be factorized as(4)S=GΣHT,where(5)G=g1g2⋯gP,(6)Σ=diag⁡s1s2⋯sP,(7)H=h1h2⋯hP,(8)gp=upup2⋯upMT,(9)hp=vpvp2⋯vpNT.Here, (·)T denotes the transpose operator and diag(x) is a diagonal matrix with vector x as its main diagonal. It is seen from (8)-(9) that the elements of gp and hp satisfy the linear prediction (LP) relations of [gp]k+1=up[gp]k and [hp]k+1=vp[hp]k, where [·]k stands for the kth element of [·].On the other hand,X can be decomposed using singular value decomposition (SVD) as(10)X=UΛVH,where U and V contain the corresponding left and right singular vectors, respectively, Λ is the diagonal matrix of singular values sorted in descending order, and (·)H stands for the conjugate transpose.From the decomposition in (4)–(10), we have S, rank(S)=P, and thus the best rank-P approximation of S according to (10), denoted by S^, is(11)S^=UsΛsVsH,where(12)Us=u1u2⋯uP,Σ=diag⁡λ1λ2⋯λP,Vs=v1v2⋯vPare the corresponding signal subspace components.SinceG and Us have the same subspace, we have(13)Us=GT1,where T1 is an unknown nonsingular matrix with dimension P×P.For eachu^p, we have the following LP property:(14)∑l=1Pαlu^pm-l=0,α0=1,p=1,2,…,P,m=P+1,…,M,where αl are the LP coefficients. The spatial frequencies up are related to the following polynomial:(15)∑l=1PαlzP-l=0whose roots are z=ejup, p=1,2,…,P. According to (14), the LP error vector e can be constructed by(16)e=Db-f,where(17)D=D1TD2T⋯DPTT,f=f1Tf2T⋯fPTT,b=α1α2⋯αPT,Dp=upPupP-1⋯up1upP+1upP⋯up2⋮⋮⋱⋮upM-1upM-2⋯upM-P,fp=-upP+1upP+2⋯upMT,p=1,2,…,P.Let W be the symmetric weighting matrix. The weighted least squares (WLS) estimate of b, denoted by b^, is(18)b^=arg⁡min⁡b⁡eHWe=DHWfDHWD.Defining a Toeplitz matrix C=Toeplitz(bP01×M-P-1T, bPbP-1⋯b1101×(M-P-1)), where 01×(M-P-1) is a zero matrix with dimension 1×(M-P-1) and us=u1Tu2T⋯uPT, the optimal W is derived as [19](19)W=σ2EeeH-1=σ2ECususHCH-1≈diag⁡λ12,λ22,…,λP2⊗CCH-1,where ⊗ is the Kronecker product. As (19) depends on the unknown b, we follow [18] to estimate the b in an iterative manner and the estimation procedure is as follows:(1)
SetW=diag([λ12,λ22,…,λP2])⊗IN-P, where IN-P is the identity matrix with dimension N-P.(2)
Calculateb^ using (18).(3)
Compute an updated version ofW using (19) with b=b^.(4)
Repeat Steps (2)-(3) until a stopping criterion is reached.Substituting b^=b in (15) and solving for the roots, denoted by r^p, p=1,2…,P, we have the spatial frequency estimate up:(20)u^p=∠r^p,where ∠(·) denotes the phase angle of (·). In order to obtain the 2D parameter pairing in an automatic manner, another estimation method for the parameters vp, p=1,2,…,P, is presented as follows.From (3)-(4), we have(21)X≈G^QT,where G^ is the estimate of G which is constructed using rp=r^p and(22)Q^T=ΣHT=q1q2⋯qPT,qp=sphp.From (21), the least squares estimate of Q is(23)Q^=XTG^†T,where (·)† denotes the pseudoinverse. Noting that the elements of qp satisfy the same LP property as in hp, we extract q^p from Q to construct the equations:(24)q^p,lcp≈q^p,f,where q^p,l=q^p(1:N-1) and q^p,f=q^p(2:N), respectively. Following [18], the WLS estimate of cp is computed as(25)c^p=min⁡cp⁡q^p,lcp-q^p,fHΦq^p,lcp-q^p,f=q^p,lHΦq^p,fq^p,lHΦq^p,l,p=1,2,…,P,where the optimum weighting matrix Φ has the form(26)Φ=Eq^p,lcp-q^p,fq^p,lcp-q^p,fH-1=BpBpH-1,where(27)Bp=Toeplitz-cp01×N-2T,-cp101×N-2.Similar to the iterative estimation of b in finding up, we begin with Φ=IN-1 in the iterations between (25) and (26) to obtain c^p, p=1,2,…,P. Finally, the spatial frequencies vp, p=1,2,…,P, are estimated as(28)v^p=∠c^p,p=1,2,…,P.Note that u^p and v^p are automatically paired in the whole procedure.Finally, the estimated azimuth angleϕp and elevation angle θp are computed as(29)ϕ^p=λ2πdsin-1⁡u^p2+v^p2,θ^p=tan-1⁡v^pu^p,p=1,2,…,P.


## 4. Simulation Results

Computer simulations have been conducted to evaluate the 2D DOA estimation performance of the proposed scheme in the presence of white Gaussian noise by comparing with the 2D ESPRIT method [3] and CRLB. The number of iterations in the proposed algorithm is I=3 and the initial parameter estimates are provided by the ESPRIT algorithm [3]. Note that the larger values for I have been tried but no significant improvement is observed. The number of sensors in the URA is 180 where M=15 and N=12. Assume that two narrowband signals imping on this received array from the directional angles [ϕ1,ϕ2]=[20°,40°] and [θ1,θ2]=[10°,30°], respectively, while the additive noise is white Gaussian process. All results provided are averages of 200 independent runs.Figures2–5 plot the root mean square error (RMSE) performance of the 2D DOA estimates versus SNR. It is seen that the proposed method is superior to the 2D ESPRIT-based method at all signal-to-noise ratio (SNR) conditions. It is because the latter approach is based on the splitting the measurement space into signal subspace and noise subspace and generally gives a higher threshold SNR value. Nevertheless, all the RMSEs of the proposed method are 2 dB above the corresponding CRLBs at all SNRs. The average computational times of the proposed method and the 2D ESPRIT method are 0.0399 s and 0.2071 s, respectively, indicating that the former is much more computationally efficient.Figure 2
Root mean square error forϕ1 versus SNR.Figure 3
Root mean square error forθ1 versus SNR.Figure 4
Root mean square error forϕ2 versus SNR.Figure 5
Root mean square error forθ2 versus SNR.


## 5. Conclusion

A 2D DOA estimation algorithm with a URA in single snapshot case is proposed using the PUMA method. Compared with the 2D ESPRIT method, the proposed method can obtain higher estimation accuracy and its performance is close to the CRLBs at higher SNR. Moreover, the proposed method shows the advantage of lower computational complexity over the 2D ESPRIT method, and the 2D DOA parameters are automatically paired.



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*Source: 681251-2015-05-17.xml*