# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 1 Introduction
Unfortunately, none of these methods have considered the problem of sensor failure for multi-dimensional arrays. Therefore, it is of great significance to develop an effective tensor processing strategy tailored for robust DOA estimation with sensor failure. From the perspective of recovering damaged signals as the matrix completion methods do, a promising solution is to complete tensor signals corresponding to the impaired multi-dimensional array. With regard to the tensor signal model of an impaired array, the missing elements will concentrate on entire fibers or even slices. This structural-missing pattern (i.e., general cases of signal-missing pattern including the fiber-missing and slice-missing pattern [46]) prevents the conventional low-rank tensor completion to be directly implemented, because the effective low-rank regularization requires a randomly missing pattern to guarantee sufficient spatial relevance [47]. With respect to the need for completing structurally missing elements, the slice completion methods are proposed exploiting multi-way delay-embedding transformation and Tucker rank increment [48; 49]. Nevertheless, the complexity of rank-increment is high, and the Tucker model is unsuitable for the array tensor signal, which is normally modelled in a canonical polyadic (CP) format due to the deterministic source number. Furthermore, a CP approximation method has been proposed to reconstruct functional magnetic resonance imaging (fMRI) data with regular under-sampling patterns including both fiber-missing and slice-missing [50; 51]. However, such reconstruction changes the undamaged pixels in the fMRI tensor, whereas the observed elements in the tensor signal are expected to remain unchanged to avoid extra model deviations including unwanted noise and phase error. As such, all these tensor completion methods cannot be applied to the problem of robust DOA estimation with sensor failure. Finding a successful approach to completing the structural-missing tensor signal remains a challenging task. In this paper, we propose a structural-missing tensor completion algorithm for robust DOA estimation with sensor failure. In particular, the signal received at a uniform rectangular array (URA) is represented as a tensor, which contains whole fibers or slices of missing elements when several sensors are impaired. To enforce effective low-rank regularization on the structural-missing tensor signal, a spatio-temporal dimension augmentation method is developed to transform the original tensor into a dimensional-augmented Hankel tensor, whose missing elements are sufficiently dispersed and dimensional information is augmented. On this basis, the dimensional-augmented Hankel tensor can be optimized through low-rank completion, which is characterized by a convex tensor nuclear norm minimization problem. After solving this dimensional-augmented Hankel tensor completion problem, a completed tensor signal corresponding to an unimpaired array is obtained from an inverse Hankelization operation. As such, the completed covariance tensor can be derived and decomposed via CPD to achieve closed-form DOA estimation. According to our simulation results, the proposed algorithm shows high robustness to different simulated sensor failure scenarios compared to conventional methods. Although the proposed algorithm is designed for the impaired URA with a uniform inter-sensor spacing, it can be extended to deployments of sparse arrays such as coprime array [52]. On one hand, the physical sparse array can be considered as a special case of partially damaged URA. Hence, the proposed algorithm can be applied to the incomplete sparse array signals for a direct completion. Based on this, the subsequent signal statistics can be derived and used for DOA estimation. On the other hand, for the augmented discontinuous co-array of the coprime array [53; 54], the proposed algorithm can also be extended to complete the co-array signal.