International Journal of Antennas and Propagation

Volume 2018, Article ID 5247919, 16 pages

https://doi.org/10.1155/2018/5247919

## Estimation for Two-Dimensional Nonsymmetric Coherently Distributed Source in L-Shaped Arrays

^{1}School of Automation, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China^{2}School of Marine, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

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

Received 28 December 2017; Revised 11 April 2018; Accepted 13 June 2018; Published 3 September 2018

Academic Editor: Shiwen Yang

Copyright © 2018 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

We explore the estimation of a two-dimensional (2D) nonsymmetric coherently distributed (CD) source using L-shaped arrays. Compared with a symmetric source, the modeling and estimation of a nonsymmetric source are more practical. A nonsymmetric CD source is established through modeling the deterministic angular signal distribution function as a summation of Gaussian probability density functions. Parameter estimation of the nonsymmetric distributed source is proposed under an expectation maximization (EM) framework. The proposed EM iterative calculation contains three steps in each cycle. Firstly, the nominal azimuth angles and nominal elevation angles of Gaussian components in the nonsymmetric source are obtained from the relationship of rotational invariance matrices. Then, angular spreads can be solved through one-dimensional (1D) searching based on nominal angles. Finally, the powers of Gaussian components are obtained by solving least-squares estimators. Simulations are conducted to verify the effectiveness of the nonsymmetric CD model and estimation technique.

#### 1. Introduction

Although point source models are commonly used in applications such as wireless communications, radar and sonar systems, distributed source models which have considered multipath propagation and the surface features of targets tend to perform better. Position localization estimators based on distributed source models have proved to be more precise in multipath scenarios compared with point source models. With information of surface features, the distributed source models have potential application for high-resolution underwater acoustical imaging.

Sources may be classified into coherently and incoherently distributed sources [1]. A coherently distributed (CD) source is defined as the signal components arriving from different angles within the range of extension that are coherent. For an incoherently distributed (ID) source, these components are uncorrelated. In this paper, the modeling and estimation of a CD source are considered.

Some classical point source estimation techniques have been extended to CD sources. DSPE [1], DISPARE [2], and vec-MUSIC [3] are developed from MUSIC for distributed sources, where parameters are estimated by two-dimensional (2D) spectral searching. In [4], ESPRIT is extended for distributed sources, where the TLS-ESPRIT algorithm is used to estimate nominal angles of sources firstly, and then angular spreads are obtained by one-dimensional (1D) spectral searching. The authors of [5] have developed an efficient DSPE algorithm and proposed generalized beamforming estimators in [6]. Generally, the parameters of CD sources are approximate solutions under the assumption of small angular spreads; some comparative studies on different methods have been proposed. The performance of two beamforming methods is analyzed in [7]. In the presence of model errors, the performance of DSPE algorithm is analyzed in [8], and the performance of MUSIC is analyzed in [9]. Considering mismodeling of the spatial distribution of distributed sources, a robust estimator is proposed in [10] by means of exploiting some properties of the generalized steering vector in the case of CD sources and the covariance matrix in the case of ID sources. All these methods are based on the 1D distributed source models, which assume that signal sources are in the same plane, where distributed sources are characterized by the nominal direction of arrival (DOA) and angular spread. However, impinging signals are not generally in the same plane but in a three-dimensional space practically. The 2D distributed source models are usually described by the nominal azimuth DOA, nominal elevation DOA, azimuth angular spread, and elevation angular spread. Involving more parameters, 2D CD source estimation problem is more complicated.

Based on DSPE, several spectral searching methods for 2D CD sources have been proposed. In [11–15], algorithms for exponential CD sources are presented under L-shaped arrays, uniform circular arrays, and nested arrays, which transform a four-dimensional parameter searching problem into a 2D parameter searching problem. The authors of [16] have proposed an estimator for Gaussian CD sources under L-shaped arrays.

Using an array with sensors oriented along three axes of the Cartesian coordinates, the authors of [17] have extended the matrix pencil method for 2D Gaussian and Laplacian CD sources. The method needs not spectral searching, but its array structure is not beneficial to engineering realization and its concern is DOAs without angular spreads.

In [18], a sequential one-dimensional searching (SOS) algorithm is proposed for Gaussian CD sources using a pair of uniform circular arrays, which first estimates the nominal elevation by the TLS-ESPRIT method, followed by sequential one-dimensional searching to seek the nominal azimuth. As the method presented in [17], the SOS algorithm only deals with the DOAs.

Several low-complexity algorithms have been presented in [19–24] utilizing two closely spaced parallel ULAs, treble parallel ULAs, and conformal arrays. It is a common characteristic that though preliminary Taylor approximation to generalized steering vectors, the nominal elevation and azimuth are solved under ESPRIT or modified propagator. The authors of [25] have proposed a method using a centrosymmetric crossed array, where DOAs are obtained through the symmetric properties of the special array. These methods need not any spectral searching and it can deal with unknown angular distribution functions. The disadvantage of these algorithms is that angular spreads are not within consideration.

Utilizing two closely spaced parallel ULAs and L-shaped arrays, two estimators for CD noncircular signals are proposed [26, 27] which exhibit better accuracy and resolution compared with circular signals.

Estimation techniques for distributed sources mentioned above are performed based on the assumption that the spatial distributions of sources are symmetric. Nevertheless, scatters are distributed irregularly or nonuniformly around targets; the distributions of scatters in practice are generally nonsymmetric. Considering complexity, people have paid less attention to nonsymmetric distributed sources.

Based on the principle that a nonsymmetric probability distribution can be composed of several symmetric distributions, some methods for 1D nonsymmetric ID sources have been presented. In [28], a nonsymmetric distribution is modeled by two Gaussian distributions. The shape of the nonsymmetric distribution would be figured via the variation of the ratio between two Gaussian distributions. In [29], the Gaussian mixture model (GMM) is employed to characterize the 1D nonsymmetric ID sources, and the expectation maximization (EM) algorithm is applied to solve the problem.

To the best of our knowledge, there are no algorithms for a 2D nonsymmetric distributed source. In this paper, we are concerned on the estimation of a 2D nonsymmetric CD source. Compared with a symmetric source, the modeling and estimation of a nonsymmetric source are more complex. Through modeling the nonsymmetric deterministic angular signal distribution function by Gaussian mixture, we have presented a parameter estimation method under an EM framework based on L-shaped arrays. In general EM frameworks, the maximization step is to maximize the likelihood function to get the best parameters which are hidden in expectation step results. In our method, the DOA parameters of Gaussian components of the 2D CD source are obtained through two approximate rotational invariance relations. By solving least-squares estimators, powers of Gaussian components are obtained.

#### 2. Signal Model

Figure 1 shows the L-shaped array configuration, which uses the *xoy* plane. Each linear array consists of sensor elements separated by meters, and the two linear arrays share an origin sensor. Suppose that there is a CD source with a nominal azimuth angle and a nominal elevation angle *ϕ*, and . is the wavelength of the impinging signal. The CD source is considered as stationary narrowband stochastic process.