International Journal of Antennas and Propagation

Volume 2018, Article ID 3505918, 12 pages

https://doi.org/10.1155/2018/3505918

## DOA Estimation for a Mixture of Uncorrelated and Coherent Sources Based on Hierarchical Sparse Bayesian Inference with a Gauss-Exp-Chi^{2} Prior

^{1}Department of Electronic and Information Engineering, Jinling Institute of Technology, Nanjing 211169, China^{2}Department of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China^{3}No. 8511 Research of Institute of CASIC, Nanjing 210007, China

Correspondence should be addressed to Pinjiao Zhao; nc.ude.uebrh@oaijnipoahz

Received 29 November 2017; Accepted 31 May 2018; Published 10 July 2018

Academic Editor: Sotirios K. Goudos

Copyright © 2018 Pinjiao Zhao 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

Direction of arrival (DOA) estimation algorithms based on sparse Bayesian inference (SBI) can effectively estimate coherent sources without recurring to extra decorrelation techniques, and their estimation performance is highly dependent on the selection of sparse prior. Specifically, the specified sparse prior is expected to concentrate its mass on the zero and distribute with heavy tails; otherwise, these algorithms may suffer from performance degradation. In this paper, we introduce a new sparse-encouraging prior, referred to as “Gauss-Exp-Chi^{2}” prior, and develop an efficient DOA estimation algorithm for a mixture of uncorrelated and coherent sources under a hierarchical SBI framework. The Gauss-Exp-Chi^{2} prior distribution exhibits a sharp peak at the origin and heavy tails, and this property makes it an appropriate prior to encourage sparse solutions. A three-layer hierarchical sparse Bayesian model is established. Then, by exploiting variational Bayesian approximation, the model parameters are estimated by alternately updating until Kullback-Leibler (KL) divergence between the true posterior and the variational approximation becomes zero. By constructing the source power spectra with the estimated model parameters, the number and locations of the highest peaks are extracted to obtain source number and DOA estimates. In addition, some implementation details for algorithm optimization are discussed and the Cramér-Rao bound (CRB) of DOA estimation is derived. Simulation results demonstrate the effectiveness and favorable performance of the proposed algorithm as compared with the state-of-the-art sparse Bayesian algorithms.

#### 1. Introduction

Direction of arrival (DOA) estimation has been a crucial issue in various application areas involving radar, wireless communication, and navigation [1–3]. Multiple signal classification (MUSIC) [4] and estimation of signal parameter via rotational invariance technique (ESPRIT) [5], known as the two most classical subspace-based DOA estimation algorithms, have been proposed to resolve uncorrelated sources. However, in multipath propagation environments, sources from an identical target may undergo reflection from various surfaces, and thus, the received sources may be a mixture of uncorrelated and coherent sources. In such environments, the aforementioned subspace-based algorithms would suffer from serious performance deterioration or even fail [6].

In order to solve the aforementioned problem, several preprocessing techniques are developed for decorrelation. In the related studies, these preprocessing techniques are mainly classified into two categories: spatial smoothing (SS) [7] and matrix reconstruction (MR) [8]. Theoretically, the SS technique is implemented by dividing the whole array into multiple subarrays to combat the rank deficiency, while the MR technique is performed by rearranging the rank-deficient matrix into Hankel matrix, Toeplitz matrix, or other matrices to restore the rank. By combing the SS or MR techniques with subspace-based algorithms, several DOA estimation algorithms have been developed to handle the scenarios where uncorrelated and coherent sources coexist [9–13]. More specifically, the oblique projection spatial smoothing (OPSS) [9] and moduli property spatial smoothing (MPSS) [10] are the typical SS-based algorithms, and the oblique projection Toeplitz matrix reconstruction (OPT-MR) [11], symmetric non-Toeplitz matrix reconstruction (SNT-MR) [12], and real-valued Hankel matrix reconstruction (RVH-MR) [13] are the popular MR-based algorithms.

Unlike traditional subspace-based DOA estimation algorithms, the emerging sparse source reconstruction (SSR) algorithms [14–19], including matching pursuit (MP) algorithm [14], l*p*-norm optimization algorithms [15, 16], and sparse Bayesian inference (SBI) algorithms [17–19], provide a new perspective for DOA estimation. Since SSR-based algorithms realize DOA estimation via sparse source reconstruction, instead of calculating the covariance matrix, they can resolve the coherent sources directly without extra preprocessing techniques. Among these SSR-based algorithms, both the MP and the l*p*-norm optimization algorithms are restricted to solve an optimization problem and are relied on the point estimate, so that these algorithms ignore the uncertainty of the underlying source in the process of source reconstruction. By contrast, SBI-based algorithms specially consider the uncertainty of the underlying source and estimate the source via choosing an appropriate prior, which yield favorable reconstruction performance [20]. Moreover, they can provide good estimation performance in the case of low SNR or small number of snapshots [21]. In general, SBI-based algorithms first specify a sparsity-encouraging prior to the unknown source model and then the model parameters are estimated via Bayesian inference. Since the exact Bayesian inference is intractable, two mainstream approximation inference algorithms were presented to estimate the model parameters, in which one is evidence procedure [22] and the other is variational Bayesian approximation [23]. For the evidence procedure, some unknown hyperparameters with respect to the hierarchical prior model are estimated iteratively by maximizing the evidence. For the variational Bayesian approximation, the posterior distribution is approximated as the product of several tractable distributions, and the model parameters keep updating to minimize the Kullback-Leibler (KL) divergence between the true posterior and the variational approximation, which has attractive computational efficiency along with high estimation performance [24]. Note that these approximation inference algorithms operate under the premise that an appropriate sparse prior has been imposed on the source model for the purpose of encouraging sparse solutions. Many sparsity-encouraging prior models have been investigated in the SBI framework [25–27]. In [25], Bayesian compressed sensing (BCS) was proposed with a two-layer hierarchical Gaussian-inverse-gamma prior (or Student’s- prior), where the first layer is a Gaussian probability density function (pdf) and the second layer is a gamma pdf. Babacan et al. [26] proposed an equivalent Laplace prior that is generated by a Gaussian prior and an Exponential prior. In [27], a normal product (NP) prior was developed with two algorithms: NP-0 (using one-layer source model) and NP-1 (using two-layer source model). However, these priors are concentrated near the origin with relatively light tails, which may cause overshrinkage of the incident sources [28].

In this paper, we develop a new sparse-encouraging prior (called Gauss-Exp-Chi^{2} prior) whose pdf distribution exhibits a sharp peak at the origin and heavy tails. With the proposed prior, the DOA estimation for a mixture of uncorrelated and coherent sources is performed under the hierarchical SBI framework using a uniform linear array (ULA). A three-layer hierarchical Bayesian model is established based on the Gauss-Exp-Chi^{2} prior. Subsequently, according to the variational Bayesian approximation, the model parameters (including the mean and variance of sparse sources and hyperparameters) keep alternately updating until the KL divergence between the true posterior and the variational approximation tends to be zero. By exploiting the estimated model parameters, the source power spectra is constructed, from which the number and locations of the highest peaks are extracted to obtain source number and DOA estimates. Simulation results show that the proposed algorithm has superior estimation performance. Now we briefly summarize the contributions of this work as follows:
(i)To encourage sparse solutions, we develop a new sparse-encouraging prior, called Gauss-Exp-Chi^{2} prior, whose pdf distribution has a sharp peak at the origin and heavy tails.(ii)By constructing the source powers of all the potential directions in the angular space, both source number and DOA estimates are obtained.(iii)Variational approximations are adopted for the estimation of the hierarchical sparse Bayesian model parameters.(iv)Several implementation details for algorithm optimization including Woodbury matrix identity for dimension-reduction, pruning a basis function and the third kind Bessel function approximation are discussed, and the CRB of DOA estimation is derived.

The remainder of this paper is organized as follows. The DOA estimation model for mixed sources is formulated in Section 2. Section 3 presents the Gauss-Exp-Chi^{2} prior and DOA estimation algorithm for a mixture of uncorrelated and coherent sources within the hierarchical SBI framework. The algorithm optimization and CRB of DOA estimation are discussed in Section 4. Section 5 presents the simulation results of the proposed algorithm. Conclusions are drawn in Section 6.

*Notations. *Vectors and matrices are denoted by lowercase and uppercase boldface letters, respectively. , , , and represent transpose, conjugate transpose, inverse, and the statistical expectation, respectively. is an identity matrix. denotes the Euclidean norm, and denotes a diagonal matrix. Additionally, represents the integral of from to .

#### 2. Problem Formulation

Consider a total of far-field narrowband sources impinging on the uniform linear array (ULA) consisting of omnidirectional sensors with the interspacing between adjacent sensors being a half of the carrier wavelength , that is, . Then, the array output vector at the th snapshot can be expressed as where is the number of snapshots, is the steering vector corresponding to the direction of the th impinging source , and denotes the noise vector. Without loss of generality, the impinging source is a mixture of uncorrelated and coherent sources. Note that there exists power fading among the coherent source, and fading coefficients are used to describe the degree of power fading [5]. Specifically, consider far-field narrowband sources impinging on the ULA, in which the first sources are uncorrelated, and the last sources are coherent. Then, the uncorrelated source set is denoted as and the coherent source set is denoted as with are the fading coefficients.

Divide the entire angular space into sampling grids , where represents the grid number and generally satisfies . Assume that is the nearest grid to true direction ; thus, we have . Thus, can be rewritten in a sparse form where and . Due to the fact that has non-zero elements in elements, it is a sparse vector. In the case of multiple snapshots, the sparse sources at all the snapshots share the same support, and the array output matrix of snapshots can be represented by where , , and . The goal of this paper is to provide the DOA estimation under the coexistence of uncorrelated and coherent sources from a sparse Bayesian perspective.

#### 3. DOA Estimation

In this section, a DOA estimation algorithm for a mixture of uncorrelated and coherent sources is proposed within the hierarchical SBI framework. A Gauss-Exp-Chi^{2} prior is developed to encourage sparse solutions, and then the parameters of three-layer hierarchical Bayesian model are estimated via variational Bayesian approximations. By constructing source power spectra, the source number and DOA estimation are obtained.

##### 3.1. Bayesian Model

In the Bayesian model, the pdf of a joint distribution with respect to all the unknown and observed quantities is required to be known. In this paper, the joint distribution can be expressed as where and are referred to as the hyperparameters; and are, respectively, referred to as the rate parameter and shape parameter. It is assumed that the components are the independently zero-mean stationary Gaussian noise with known variance . Thus, the pdf of the noise matrix is given by

Combining (3) and (5), the Gaussian likelihood model is obtained as follows:

From a Bayesian perspective, the pdf distribution of an assigned prior is appealing to exhibit a sharp peak at the origin and heavy tails, which favors strong shrinkage of noise sources and avoids overshrinkage of the interest sources. This property is generally considered as a desirable property for enforcing sparsity and variable selection [24]. Some typical sparse priors, such as Gaussian-inverse-gamma prior and Laplace prior, are widely used in the relevant research [25, 26], which, however, are concentrated near the origin with relatively light tails [28]. To alleviate this problem, we here develop a three-layer hierarchical prior, referred to as Gauss-Exp-Chi^{2} prior, for . In the first layer of prior, we adopt a zero-mean Gaussian prior
with being . In the second layer of prior, an exponential hyperprior is imposed on where denotes the exponential distribution. In the third layer of prior, a chi-square (Chi2) hyperprior is considered over , that is,
where denotes the gamma function.

The first two layers (7) and (8) of the proposed prior result in a generative Laplace (Gaussian-Exponential) distribution [26], and the last layer (i.e., a Chi2 distribution) is embedded to obtain the proposed three-layer Gauss-Exp-Chi^{2} prior. Thus, the proposed prior has more free parameters to control the degree of sparseness as compared with the Laplace prior [29].

Based on the above analysis, the directed graph of the sparse Bayesian model is shown in Figure 1, where arrows are used to point to the generative model. Note that the first five blocks from the left (corresponding to the variables , , , , and ) depict the three-layer hierarchical prior mentioned above.