International Journal of Antennas and Propagation

Volume 2019, Article ID 4816183, 12 pages

https://doi.org/10.1155/2019/4816183

## DOA Estimation of Two-Dimensional Coherently Distributed Sources Based on Spatial Smoothing of Uniform Rectangular Arrays

^{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}School of Automation, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China^{4}School of Marine, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

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

Received 2 April 2019; Revised 29 August 2019; Accepted 7 September 2019; Published 7 October 2019

Academic Editor: Raffaele Solimene

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

Aiming at the direction-of-arrival (DOA) estimation of two-dimensional (2D) coherently distributed (CD) sources which are coherent with each other, we explore the propagator method based on spatial smoothing of a uniform rectangular array (URA). The rotational invariance relationships with respect to the nominal azimuth and nominal elevation are obtained under the small angular spreads assumption. A propagator operator is constructed through spatial smoothing of sample covariance matrices firstly. Then, combination of propagator and identical matrix is divided according to rotational operators, and the nominal angles can be obtained through eigendecomposition lastly. Realizing angle matching automatically, the proposed method can estimate multiple DOAs of 2D coherent CD sources without spectral peak searching and prior knowledge of deterministic angular signal distribution function. Simulations are conducted to verify the effectiveness of the proposed method.

#### 1. Introduction

In the field of array signal processing, traditional DOA estimation is based on point source models. In the real surroundings of radar and sonar systems, because of multipath propagation between receive arrays and targets, especially when the distances of targets and receive arrays are short, the spatial scattering of targets cannot be ignored, and the assumed condition of point source models is no longer valid. In such a condition, DOA estimation based on distributed source models has presented better accuracy [1]. A distributed source can be regarded as an assembly of point sources within a spatial distribution where point sources can be called scatterers.

According to coherence of scatterers, distributed sources can be classified as coherently and incoherently distributed sources [1]. Incoherently distributed (ID) sources are defined as scatterers within a source which are uncorrelated. On the contrary, coherently distributed (CD) sources are defined as scatterers within a source which are coherent. According to spatial distribution of sources, distributed sources can be classified as one-dimensional (1D) distributed sources and two-dimensional (2D) distributed sources. Under the assumption that scatterers of a target and receive array are not in the same plane, the 2D distributed sources should be more generally.

No matter CD or ID sources, most estimators suppose that signal from different sources is uncorrelated. In the field of underwater detection, the general process of detection is emitting narrowband pulse sound signal firstly, followed by receiving and analyzing the target’s backscatter signal. Thus, different CD sources are supposed to be coherent, which is more reasonable on account of the coherent multipath characteristics of underwater acoustic channel.

As to ID sources, utilizing different array configurations representative estimators have been proposed in [2–11]. Supposing that signals from different CD sources are coherent, sample covariance matrices are rank deficient. Thus, subspace-based algorithms [1, 2, 12–19] and ESPRIT class algorithms [20–23] which are based on eigendecomposition of sample covariance matrices cannot be applied for DOA estimation. PM class algorithms [24–27] based on linear operation of full rank sample covariance matrices are no longer applicable too. Considering signals from different CD sources are coherent, authors of [28] have proposed a DOA estimator by means of Toeplitz operation of sample covariance matrices. Authors of [29] have proposed a generalized MUSIC algorithm based on spatial smoothing. Both of the two algorithms above deal with 1D CD sources. Authors of [30] have proposed a general model of 2D CD sources coherent with each other which are defined as 2D coherent CD sources and proposed two spatial smoothing methods with double parallel linear arrays (DPLA).

In [1, 6–23], CD sources have been regarded as uncorrelated with each other; estimators cannot be applied for CD sources which are correlated. In [24], CD sources correlated with each other are discussed while sources are supposed to be 1D case. Literatures [10, 31] have considered 2D DOA estimators under uniform rectangular arrays, whereas they deal with ID sources and point sources, respectively. In [30], authors have considered 2D coherent CD sources and DOA estimators based on DPLA. In this paper, we are concerned on DOA estimation for 2D coherent CD sources under uniform rectangular array (URA). 2D coherent CD source and URA are introduced first. Then, the relationships of rotational invariance within and between subarrays have been derived under the assumption of small angular spreads. Afterwards, for decoherence of 2D coherent CD sources, spatial smoothing approach of URA is proposed, and a modified propagator is detailed for DOA estimation of 2D coherent CD sources. The method proposed need not angles matching and prior knowledge of deterministic angular signal distribution function. The simulation outcomes indicate that the proposed method is effective and advanced as for DOA estimation of 2D coherent CD sources.

#### 2. Array Configuration and Signal Model

Figure 1 shows the URA configuration placed in the *xoz* plane. Arrays consist of *M* × *K* sensors separated by *d* meters along the direction of the *x* and *z* axis. Suppose that there are *q* 2D CD sources with nominal angles (*θ*_{i}, *ϕ*_{i}) () impinging the arrays, where *θ*_{i} is nominal azimuth angle and *ϕ*_{i} is nominal elevation angle of *i*th source, *θ*_{i} ∈ (0, π), *ϕ*_{i} ∈ (0, π). The noise is assumed to be additive Gaussian white with zero mean and uncorrelated between sensors.