# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 1 Introduction  
Direction-of-arrival (DOA) estimation using sensor arrays plays an important role in various applications such as radar, sonar, and industrial Internet of Things (IIoT) [1, 2, 3, 4, 5]. Existing methods include subspace-based methods [6, 7], sparsity-based methods [8, 9, 10], beamforming-based methods [11, 12], statistical reconstruction-based methods [13, 14], and neural-network-based methods [15, 16]. Furthermore, advanced DOA estimation methods have mitigated the multi-path effects in practical environments, based on either iterative implementation approach [17, 18] or low-rank decomposition and sparse representation approach [19]. However, all the aforementioned methods typically work under the presumption that the deployed sensor array remains unimpaired. Nevertheless, in practical environments with complex electromagnetic propagation and harsh working conditions, the gain-phase deviation or position deviation of sensors commonly occurs [20, 21, 22]. Inmore severe cases, some sensors will be totally damaged and become nonfunctional, i.e., the sensor failure problem. The above-mentioned methods are not robust to the sensor failure scenario, and thus, the DOA estimation performance will dramatically drop. To achieve robust DOA estimation with sensor failure, matrix completion methods are developed to recover a complete signal matrix for the impaired array [23; 24; 25]. In particular, the principles of recursive least squares [23], nuclear-norm-based convex relaxation [26], and reweighted \(\ell_{2,1}\)-norm [24] can be incorporated with a low-rank regularization for signal matrix completion [27], leading to an effective DOA estimation. Meanwhile, co-array processing is adopted to help reconstruct an unimpaired array with improved performance [28]. To demonstrate the lower bound on the estimation accuracy under sensor failure, the Cramer-Rao bound (CRB) in various sensor failure scenarios is analysed for regular tetrahedral arrays [29]. Furthermore, the weighted sum of CRB for independent sensor failure event has been proposed for scenario of randomly impaired sensors [30]. Nevertheless, as the dimensions of sensor arrays expand, the matrix-based signal modelling and processing reflect at most two-dimensional (2-D) information, while complicated multi-way signal characteristics are ignored when it comes to multi-dimensional array processing. This phenomenon results in performance deterioration especially under sensor failure. Different from matrix, tensor is utilized to preserve the original high-level information of data with three-dimensional (3D) structure and beyond 3D. Meanwhile, numerous tensor decomposition techniques such as the canonical polyadic decomposition (CPD) [31] and Tucker decomposition [32] have been proposed to retrieve parameters from the tensor data [33; 34]. Based on tensor models and tensor decompositions, many tensor-based DOA estimation methods are proposed to enhance the estimation performance for multi-dimensional arrays [35; 36]. Specifically, the CPD is applied to multiple-invariance array signals for effective DOA estimation [37]. The tensor subspace method is developed by formulating high-order singular value decomposition (HOSVD)-based spectral function [38]. The coupled CPD is proposed to fully exploit the spatial relevance of segmented signals to enhance the DOA estimation accuracy [39]. Moreover, to improve computational efficiency while maintaining estimation performance, the tensor train decomposition is developed to replace the CPD [40]. To increase the degrees-of-freedom for DOA estimation, the strategies of cross-correlation tensor processing [41], co-array tensor processing [42], and co-array tensor completion [43] have been investigated. Targeting the problem of coherent source DOA estimation in a multi-path environment, the tensor spatial smoothing [44] and tensor reconstruction [45] approaches are further designed to obtain a high robustness to the coherency of signal statistics.