Off-grid DOA estimation under nonuniform noise via variational sparse Bayesian learning

Highlights

• An off-grid DOA estimation method under nonuniform noise is proposed.

• The proposed method alleviates the nonuniformity of senor noise.

• A weighted partial virtual array output is exploited.

• A hierarchical Bayesian model is built with an almost Jeffrey’s prior incorporated.

• The proposed method can work without the knowledge of the number of sources.

Abstract

In this paper, the problem of direction-of-arrival (DOA) estimation in the presence of nonuniform noise is investigated, where the inherent off-grid effects in traditional sparsity-inducing algorithms are also considered. By formulating a sparse signal recovery problem for weighted partial virtual array (PVA) response, we develop a sparse Bayesian learning based method by exploiting joint sparsity between the power distribution of incident signals and the off-grid difference. In our proposed algorithm, a weighted partial covariance vector is obtained through the deliberate projection and decorrelation operations, which facilitates a sparse representation free from the nonuniform noise variances. Meanwhile, a variational Bayesian inference is implemented upon a hierarchical Bayesian learning model with an almost Jeffrey’s prior adopted, which strongly induces the sparsity and involves adaptively tuning sparseness-controlling parameters. Moreover, the proposed method works without the knowledge of the number of sources. Simulation results demonstrate it provides superiority in estimation precision and robustness against nonuniform noise.

Introduction

Direction-of-arrival (DOA) estimation using sensor arrays [1] frequently arises in a variety of applications, such as radar, sonar, wireless communications, etc. A large number of DOA estimation methods have been provided to exploit the array output or the covariance matrix which contains the directional information of the incident signals and well concentrates the signal energy distributed among all the snapshots. Commonly, it is explicitly or implicitly assumed in the DOA estimation methods that the sensor noises are spatially uncorrelated Gaussian white. In this case, the noise covariance matrix is diagonal with unknown identical entries and the conventional uniform maximum-likelihood (ML) methods [2], [3] can be expected to give good results. However, it has been shown that in some practical applications, such as, where exists the nonuniform sensor response, the non-ideality of the receiving channel and media inhomogeneity [4], [5], the sensor noises are spatially white but their variances are not identical. Such a noise model also becomes true in certain situations with the measurements obtained from sparse arrays under the prevalent external noise, e.g., reverberation noise in sonar or external seismic noise [5], [6]. In some cases, the sensor noise can be essentially uncorrelated and the noise covariance matrix can be modeled as a diagonal matrix with each diagonal element being different. In presence of this nonuniform uncorrelated noise, it should be noted that the DOA estimators derived from either the uniform white or the general colored noise assumption [7], [8] may not give satisfactory results, since the former assumption blindly treats all sensors equally and the latter one neglects the fact that the sensor noises are uncorrelated [4], [9]. As a consequence, much attention has been paid for the problem of DOA estimation in nonuniform noise environments.

To deal with such a problem, many ML DOA estimators for the nonuniform noise have been proposed in [4], [10], [11], [12]. ML DOA estimators operate by stepwise concentrating the objective function with respect to the signal and noise nuisance parameters. Although these ML methods achieve attractive estimation precision, most of them require time-consuming multi-dimensional search, moderately high input signal-to-noise ratio (SNR) and the prior information of the incident signal number, which significantly restrict their applications. To relax these requirements and enhance immunity to the lack of prior information on source number, a series of sparsity-inducing methods have been designed for DOA estimation, by exploiting the spatial sparsity of the DOAs of sources and solving a sparse signal reconstruction (SSR) problem. The algorithms based on ℓ₁-optimization [13] and its varieties [14] have well addressed the SSR problem with a directional overcomplete dictionary based on array response. To make use of the second-order statistics of the array output, and even the virtual array formed by Khatri-Rao (KR) product approach [15], some methods [16], [17], [18], [19] have been proposed for DOA estimation by using covariance fitting criterion. Among them, in [18], a partial manifold of virtual array by a deliberate selection has been exploited and the DOA estimator applies to the nonuniform noise case. Sparse Bayesian learning (SBL) [20] is another popular technique to solve the SSR problem, because it induces less structural error and convergence error. The SBL based method [21] performs well in capturing local signal properties to facilitate the refined DOA estimation process, which can achieve better results than ℓ₁-optimization based method.

It is worth noting, however, that the sparsity-inducing algorithms aforementioned involve division of the potential angular space of the incident signals into a mass of grids, and they are based on the on-grid assumption that all DOAs should fall right on the predefined computation grid so that reliable estimations can be guaranteed [13], [22]. Unfortunately, the on-grid scenario can rarely be met in reality, where there is a gap between the true DOA and its nearest computation grid which the estimated DOAs are constrained on. The employment of a coarse grid set often leads to unsatisfactory estimation accuracy because of the off-grid error. It should be noted that ML methods are also based on this assumption by default, otherwise they will find solutions by coarse grid search followed by denser grid search which leads to much more computation burden. It is evident that the most natural way to deal with the grid error is to refine the grid, which also can be adopted by the traditional sparsity-inducing algorithm [13]. Nevertheless, a too coarse grid set cannot achieve satisfactory estimation accuracy because of off-grid error. On the other hand, a dense grid set may result in unacceptable degradation of computational efficiency and the large reconstruction error because of the correlation between two adjacent steering vectors. And thus there are still some methods have been recently proposed for off-grid DOA estimation, where the unknown DOAs are not limited onto the grids. In [23], a sparse and parametric approach (SPA) is proposed by solely processing the array covariance matrix without grid division, where the covariance matrix fitting problem is cast as semidefinite programming (SDP) and Vandermonde decomposition lemma for positive semidefinite Toeplitz matrices can be exploited. Although SPA naturally takes nonuniform noise into account, it may suffer from low input SNR. Another popular way to achieve off-grid DOA estimation under a coarse grid is by exploiting an approximation of array response [24], [25], [26]. In [25] and [26], the SBL based methods presented can alleviate the sensitiveness to the low input SNR, but they mismodel the nonuniform noise, which deteriorates the estimation precision and computational efficiency of the learning process. Therefore, we focus on the case of the coexistence of the off-grid error and the nonuniform noise which can be viewed as a practically important generalization of the case of only involving either of them.

In this paper, we propose a SBL based method of off-grid DOA estimation under nonuniform noise, where both nonuniformity of the noise variance and off-grid error are addressed. In our proposed algorithm, we first build the partial covariance vector from the partial sample covariance matrix with the unknown noise variances removed. Thus, we can get rid of estimating the noise variance so that the Bayesian learning process may not suffer from mismodeling the nonuniform noise variances. Then, a weighted partial virtual array (PVA) manifold is approximated so that the dictionary becomes a linear function of the off-grid difference, and a SSR problem is formulated by exploiting the underlying joint sparsity between the power distribution of the sources and the off-grid difference. To tackle this problem within SBL framework where a variational Bayesian (VB) technique is employed, a hierarchical probabilistic model featuring a heavy-tailed prior is imposed on the spatial power of the sources to facilitate convenient inference. Thus, the estimator named as OGVSBL-PVA is proposed to estimate off-grid DOAs in presence of nonuniform noise. A major characteristic of heavy-tailed priors is their sparsity inducing nature. Instead of the hierarchical Laplace prior [25], [26], [27] for the sparse vector, an almost Jeffrey’s prior whose limiting case is parameter-free Jeffrey’s prior [28] is considered to induce the sparsity. The almost Jeffrey’s prior is derived from generalized inverse Gaussian (GIG) distribution [29] with the sparsity regularizing parameters adaptively inferred from the data within a fully Bayesian inference framework. It should be noted that the weighted partial covariance vector, which can be regarded as the output of the PVA, is expected to have a higher SNR than the array outputs and even the whole covariance vector when moderately many snapshots are collected. The enhanced SNR is expected to favor the learning process and improve the estimation precision of the proposed method when the input SNR is relatively low. Moreover, the proposed method can work without the knowledge of the number of sources.

The rest of paper is organized as follows. Section 2 presents the signal model for nonuniform noise and its relations. Next, array SNR of the partial covariance vector is studied in Section 3. The off-grid DOA estimation model and the proposed algorithm are illustrated in Section 4. The numerical experiments are presented in Section 5. Section 6 concludes this paper.

Notation: Bold-face letters are reserved for vectors and matrices. (·)*, (·)^T, (·)^H, vec( · ), Tr(·), ⊗ and ⊙ denote conjugate, transpose, conjugate transpose, vectorization, trace, Kronecker product and Khatri-Rao product operations, respectively. I_N represents a N × N identity matrix. ‖·‖₂ denotes the ℓ₂ norm. $\mathit{A}\in {\mathbb{R}}^{M\times N}$ and $\mathit{A}\in {\mathbb{C}}^{M\times N}$ denote a real and complex valued M × N matrix A, respectively. diag(A) denotes a column vector composed of the diagonal elements of a matrix A, and diag(x) is a diagonal matrix with x being its diagonal elements. x_i denotes the ith entry of x. A_i and A_{i, j} are the ith column and (i, j)th entry of a matrix A, respectively. ℜ(·) takes the real part of a complex variable.