International Journal of Antennas and Propagation

Volume 2019, Article ID 7657898, 13 pages

https://doi.org/10.1155/2019/7657898

## Covariance Matrix Reconstruction for Direction Finding with Nested Arrays Using Iterative Reweighted Nuclear Norm Minimization

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Weijie Tan; moc.liamtoh@eijiewnat

Received 30 July 2018; Accepted 10 December 2018; Published 18 March 2019

Guest Editor: Raed A. Abd-Alhameed

Copyright © 2019 Weijie Tan and Xi’an Feng. 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

In this paper, we address the direction finding problem in the background of unknown nonuniform noise with nested array. A novel gridless direction finding method is proposed via the low-rank covariance matrix approximation, which is based on a reweighted nuclear norm optimization. In the proposed method, we first eliminate the noise variance variable by linear transform and utilize the covariance fitting criteria to determine the regularization parameter for insuring robustness. And then we reconstruct the low-rank covariance matrix by iteratively reweighted nuclear norm optimization that imposes the nonconvex penalty. Finally, we exploit the search-free DoA estimation method to perform the parameter estimation. Numerical simulations are carried out to verify the effectiveness of the proposed method. Moreover, results indicate that the proposed method has more accurate DoA estimation in the nonuniform noise and off-grid cases compared with the state-of-the-art DoA estimation algorithm.

#### 1. Introduction

Source localization is always a significant research direction in the past decades and today which is widely used in various fields including radar, sonar, wireless communication, and acoustics [1–4]. However, achieving superresolution is a great challenge for more sources with the least elements. Compared with the uniform linear arrays (ULAs), the sparse linear arrays (SLAs) can provide much more degree of freedoms (DOFs) and resolve more sources with the same number of physical elements. Meanwhile, the SLAs can efficiently reduce the cost and power consumption. Therefore, the study of SLAs has attracted more and more researchers’ great attention [5, 6].

So far, there are some works about SLA configurations. The minimum redundancy arrays (MRAs) are class optimum lag SLAs [5], which provide a complete set of spatial lags between pairs of elements with minimum redundancies. However, it is not easy to attain the optimum design of MRAs, especially for a large array; it is time-consuming to search the optimum structure. In the past decades, two potential array configurations, i.e., coprime arrays (CPA) and nested arrays (NA), have drawn the researchers’ attentions since they have exactly closed-form expressions for sensor locations and it is easy to predict the attainable DOFs [7, 8]. Coprime arrays can resolve up to sources with only elements; however, the CPAs do not generate a filled coarray since CPA has holes in the difference coarray [7]. So it cannot directly apply the augmentation techniques. By contrast, nested arrays can generate a filled difference coarray except the mentioned advantages [8]. Although the NAs and CPAs are not optimum lag arrays like MRAs, they are the most attractive array configuration because they are easy to build in the last decade.

The processing of SLA mainly has array interpolation (AI) [9–11] and the Toeplitz completion method [12–14]. The classical AI technique [10] can effectively obtain the element data of virtual ULAs, which imposes a linear interpolation process on the element data of a real SLA and selects interpolator coefficients by minimizing the interpolation error for a source coming from a certain angular sector. The drawback is that this technique needs to know the angular sector. In AI techniques, Wiener array interpolation (WAI) [15] is a practically attractive method, which exploits a maximum likelihood method to estimate the power of signal and noise and use the calibration angles to recover the array steering matrix; hence, it can approximate the MSE optimum solution. However, WAI requires the initial DoA estimation. In the classical Toeplitz completion method, the direct augmentation approach (DAA) is a widely used method for improving the covariance matrix of the SLA [12]. DAA constructs a Toeplitz matrix using the sample covariance matrix; since there is one-to-one correspondence between covariance lags and spatial lags, the diagonal elements could be obtained by redundant averaging. Unfortunately, Toeplitz completion method does not guarantee positive semidefinite augmented covariance matrix. In order to construct a positive definite augmented matrix, the authors in [13, 14] proposed an iterative DAA algorithm; however, the complicated iteration procedure cannot guarantee the global convergence. Besides, the coarray MUSIC algorithm is used to estimated angle parameters [16] which utilizes the Toeplitz properties of matrices.

Sparse signal representation (SSR) framework has attracted a great interest in direction finding [17–23], which exploits the spatial sparsity of the signal arriving angles. Despite these aforementioned SSR-based methods have some attractive features, there are still two problems need to be considered. The first problem is nonuniform noise [23, 24]; many of SSR-based methods assume that the element noises are spatially uniform, and the reason for this is the uniform noise assumption benefits to choose an appropriate regularization parameter in a sparse representation model. However, in some practical applications, due to the nonidealistic of the practical arrays, such as the nonideality of the receiving channel, the nonuniformity of the element response, and the mutual coupling between elements, the uniform white noise assumption among all elements may not be satisfied. Therefore, the diagonal elements in noise covariance matrix should be considered as arbitrary values which represent the noise levels. Once the assumption misfits the true noise levels, the performance of conventional SSR-based DoA estimation approaches may thereby degrade severely. The second problem is the off-grid problem (also called basis mismatch) [17, 25]; in fact, a great deal of SSR-based DoA estimation methods assume that the true target directions exactly lie on the prespecified angular grid. In practical application, the angle parameters are continuous variables, the on-grid assumption rarely holds, and the basis mismatch always exists, which leads to the accuracy degradation of DoA estimation.

In this paper, we propose a gridless direction finding method under the unknown nonuniform noise over with nested array, which is based on low-rank covariance matrix approximation by an iteratively reweighted nuclear norm minimization. The advantages of the proposed method are in three aspects: Firstly, we mitigate the noise variance variable by linear transform to reduce the effect of unknown nonuniform noise. Secondly, we utilize the covariance fitting criteria to determine the regularization parameter for the accuracy error fitting. Last but not least, we develop a novel reweighted nonconvex penalty objective function for exactly approximating the low-rank covariance matrix. After attaining the covariance matrix, we exploit the search-free DoA method to estimate parameters. Numerical simulation is carried out to verify the effectiveness of the proposed method.

The rest of this paper is organized as follows: In Section 2, we present the signal model with nested array. In Section 3, the DoA estimation with iteratively covariance matrix reconstruction (ICMR) is introduced to the nested array, which is based on an iteratively reweighted nuclear norm minimization algorithm. Section 4 validates the proposed method by numerical simulations and comparing the algorithm with state-of-the-art DoA estimation methods based on nested array. Section 5 concludes this paper.

*Notation. *We define a vector with a boldface lowercase and a matrix with a boldface uppercase. The symbol denotes the complex conjugate of vector or matrix , and the transpose and the Hermitian transpose of vector or matirx are expressed by and , respectively. denote the statistical expectation, denotes stacking element-wise in a column, and denote the rank and nuclear norm of the matrix, respectively, denotes the Kronecker product, denotes the element-wise multiplication between two matrices with the same dimensional, denotes the diagonal operator that takes a vector to a matrix with vector on the diagonal, is signum function, denote the eigenvalue of matrix , denote the -th largest eigenvalue value, denotes a dimensional identity matrix, denotes the norm, and denotes probability density function.

#### 2. Array Model for Nested Array

We consider a two-level nested array consisting of two ULA subarrays, where the first ULA subarray has elements with the interelement spacing , while the second ULA subarray has elements with the interelement spacing ; the total number of element is , as shown in Figure 1.