International Journal of Antennas and Propagation

Volume 2019, Article ID 6941963, 11 pages

https://doi.org/10.1155/2019/6941963

## A 2D Nested Array Based DOA Estimator for Incoherently Distributed Sources via Sparse Representation Utilizing L_{1}-Norm

^{1}Equipment Management and UAV College, Air Force Engineering University, Xi’an 710051, Shaanxi, China^{2}Science and Technology on Combustion, Thermal-Structure and Internal Flow Laboratory, Northwestern Polytechnical University, Xi’an 710051 Shaanxi, China^{3}Center for Optical Imagery Analysis and Learning, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China^{4}School of Automation, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

Correspondence should be addressed to Tao Wu; moc.621@upwn_uwoat

Received 3 February 2019; Accepted 14 May 2019; Published 3 July 2019

Academic Editor: Ping Li

Copyright © 2019 Tao Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Nested arrays are sparse arrays composed of subarrays with nonuniform sensor spacing. Compared with traditional uniform arrays, nested arrays have more degree of freedoms (DOFs) and larger apertures. In this paper, a nested array has been proposed as well as a direction-of-arrival (DOA) estimation method for two-dimensional (2D) incoherently distributed (ID) sources. A virtual array is firstly obtained through vectorization of the cross-correlation matrix of subarrays. Sensor positions of the virtual array and the optimal configuration of the nested array are derived next. Then rotational invariance relationship for generalized steering matrix of the virtual array with respect to nominal azimuth is deduced. According to the rotational invariance relationship, sparse representation model under* l*_{1}-norm constraint is established, which is resolved by transferring the objective function to second-order cone constraints and combining a estimation residual error constraint for receive vector of the virtual array. Simulations are conducted to investigate the effectiveness of the proposed method in underdetermined situation and examine different experiment factors including* SNR*, snapshots, and angular spreads as well as sensor number of subarrays. Results show that the proposed method has better performance than uniform parallel arrays with the same number of sensors.

#### 1. Introduction

In the field of array signal processing, traditional DOA estimation of targets is based on point source models. Point source models suppose signal propagation from a target to receive array is one straight path; geometry of the target can be ignored, which can be considered as a point. In the surrounding of underwater acoustics detection, on one hand, there exist reflection paths from seabed and sea surface between a target and receive array; on the other hand, when the distance from the target to receive array is short, as many parts of the target reflect signal and spatial distribution and the geometry of the target cannot be ignored; signal propagations from the target to receive array cannot be supposed to a single path. Point source models cannot describe spatial properties of sources under such circumstance; estimations based on point source models are no longer effective, which should be replaced by distributed sources [1]. Distributed sources can be regarded as an assembly of point sources which can be called scatterers within a spatial distribution.

A distributed source may be coherently distributed (CD) source or incoherently distributed (ID) source according to the coherence of scatterers of a target. A coherently distributed (CD) source means that scatterers of the target are coherent. Spatial distribution of scatterers of a CD source is described by deterministic angular signal distribution function (ASDF). On the contrary, an incoherently distributed (ID) source means that scatterers of the target are uncorrelated. Spatial distribution of scatterers of an ID source is described by angular power density function (APDF). Both ASDF and APDF can be generally modeled as Gaussian or uniform distribution according to the geometry and spatial distribution of a target. In this paper, ID sources are considered.

Distributed sources may be one-dimensional (1D) or two-dimensional (2D) which depends on whether impinging signals of scatterers and arrays are in the same plane. As to a 1D source, parameters of ASDF or APDF include nominal angle and angular spread. Nominal angle can be called nominal DOA representing the center of a target while angular spread represents spatial extension of the target. As to a 2D source, parameters contain nominal azimuth, nominal elevation, azimuth spread, and elevation spread. Nominal azimuth and nominal elevation represent center of a target which also can be expressed as DOA; azimuth spread and elevation spread indicating 2D spatial extension of the target are collectively called angular spreads. As impinging signals of scatterers and arrays are not in the same plane practically, 2D sources are more general and reasonable.

As to CD sources, utilizing different array configurations representative estimators have been proposed in [2–10], which are mostly based on rotational invariance relationship derived by Taylor approximation of generalized steering vectors. For ID sources, several algorithms have been presented and can be roughly divided into four categories: subspace-based algorithms developed from multiple signal classification (MUSIC) such as distributed signal parameter estimator (DSPE) [1] and dispersed signal parametric estimation (DSPARE) [11], generalizations of Capon’s methods [12–14] involving high-order matrix inversions and spectral searches, the maximum likelihood (ML) approaches [15–17] requiring high computational complexity but with better accuracy, and covariance matching estimation techniques (COMET) [18–21] with lower computational complexity than ML approach but the same large sample behavior.

The aforementioned estimations of ID sources suppose the sources and receive arrays are in the same plane, which are special cases modeled as 1D ID sources. Involving more parameters, there have been relatively few studies on estimation of 2D ID sources. An extension of COMET has been proposed in [22] which employs alternating projection principle separating source powers and noise variances from objective functions and then estimates parameters by a four-dimensional nonlinear optimization. Utilizing double parallel uniform linear arrays, a TLS-ESPRIT-like approach has been proposed for DOA estimation in [23], where nominal elevations are firstly estimated via rotational invariance relationship derived by the first-order Taylor series expansions of the generalized steering vectors; then nominal azimuths are obtained through 1D searching. An estimator based on uniform rectangular arrays has been proposed in [24], where DOAs are estimated through an ESPRIT like algorithm by virtue of rotational invariance relationship of receive vectors which are described by generalized signal vector and generalized steering vector composed of nominal steering vector and its first-order partial derivatives.

Most distributed sources estimation techniques mentioned above are based on uniform arrays such as uniform linear arrays, uniform rectangular arrays, or circular arrays, which require the distances between sensors less than or equal to half of impinging signal wave. For achieving better accuracy and resolved more sources, estimation should be performed with larger aperture, which needs to increase number of sensors. Consequently, this would lead to an increase in complexity and cost of sonar system. The recent proposed nested arrays [25] possess larger apertures and more degree of freedoms (DOFs) compared with uniform arrays with the same number of sensors. DOA estimations for point sources under different kinds of nested arrays have been proposed in [26–30]. As for distributed sources, based on a linear nested array, a spatial smoothing with spectral search technique has been proposed in [8] for 1D distributed sources supposing a priori knowledge of the angular spreads is known, which is impractical in practice. Using the same array configuration, the authors of [21] have proposed annihilating filter technique combined with structured low rank approximation for 1D distributed sources without a priori knowledge of the angular spreads, where the aperture of virtual array has not been fully utilized as half of the sensors for estimation of DOA and the other half for angular spreads.

In this paper, DOA estimation for 2D ID sources is proposed based on a 2D nested array composed of double parallel uniform linear subarrays. According to the concept of difference coarray, the cross-correlation matrix of subarrays is firstly vectorized by Khatri-Rao product to obtain a hole-free virtual array; and the closed form expression for sensor positions of the virtual array is presented. Then rotational invariance relationship with regard to nominal azimuth is derived under the assumption of small angular spreads. Sparse representation with* l*_{1}-norm constraint for DOA estimation of 2D ID sources is established next. Objection function is then converted to second-order cone constraints to avoid determining the regularization parameters. Based on estimation residual error for receive vector of virtual array, a robust constraint is derived. Followed by the fact that nominal azimuths are resolved by the sparse representation framework, paired nominal elevations are obtained by least-square techniques. The proposed method can detect 2D ID sources more than sensors without angles matching and without information of APDF of sources. Simulations are conducted to investigate the effectiveness and superior of the proposed method lastly.

#### 2. The Nested Array and Source Model

Generally, nested arrays are composed of several levels of uniform linear subarrays; the distances between sensors within the first level subarray are less than or equal to half of impinging signal wave; distances between sensors within the rest subarrays are far greater than the first level subarray. As shown in Figure 1, the proposed nested array is composed of double parallel uniform subarrays on* xoz* plane X1 and X2. Subarray X1 is parallel to* x* axis and consists of the middle sensor placed on* z* axis and* M*_{1} sensors located on both sides. The sensor number of subarray X1 is* N*_{1}=2*M*_{1}+1; the distance between sensors of subarray X1 is* d*. Subarray X2 is on* x* axis with* M*_{2} sensors located on both positive and negative semiaxes. The sensor number of subarray X2 is* N*_{2}=2*M*_{2}; the distance between sensors of subarray X2 is* D*=*N*_{1}*d*. The distance between subarrays X1 and X2 is also* d. *Suppose that there are* q* 2D ID narrowband sources with nominal angles (*θ*_{i}, *φ*_{i}) (*i*=1,2,⋯,*q*) impinging into the nested array. *θ*_{i} denotes a spatial nominal azimuth between* x* axis and propagation path of the* i*th source; *φ*_{i} is nominal elevation of the* i*th source. *θ*_{i}, *φ*_{i}. *λ* is the wavelength of the impinging signal.