In this letter, we propose a coprime array interpolation approach to provide an off-grid direction-of-arrival (DOA) estimation. Through array interpolation, a uniform linear array (ULA) with the same aperture is generated from the deterministic non-uniform coprime array. Taking the observed correlations calculated from the signals received at the coprime array, a gridless convex optimization problem is formulated to recover all the rows and columns of the unknown correlation matrix entries corresponding to the interpolated sensors. The optimized Hermitian positive semidefinite Toeplitz matrix functions as the covariance matrix of the interpolated ULA, which enables to resolve off-grid sources. Simulation results demonstrate that the proposed array interpolation-based DOA estimation algorithm achieves improved performance as compared to existing coarray-based DOA estimation algorithms in terms of the number of achievable degrees-of-freedom and estimation accuracy.

Sparse arrays are known to achieve an increased number of degrees-of-freedom (DOFs) for direction-of-arrival (DOA) estimation, where an augmented virtual uniform array calculated from the correlations of sub-Nyquist spatial samples is processed to retrieve the angles unambiguously. Nevertheless, the geometry of the derived virtual array is dominated by the specific physical array configurations, as well as the deviation caused by the practical unforeseen circumstances such as detection malfunction and missing data, resulting in a quite sensitive model for virtual array signal processing. In this paper, we propose a novel sparse array DOA estimation algorithm via structured correlation reconstruction, where the Nyquist spatial filling is implemented on the physical array with a compressed transformation related to its equivalent filled array to guarantee the general applicability. While the unknown correlations located in the whole rows and columns of the augmented covariance matrix lead to the fact that strong incoherence property is no longer satisfied for matrix completion, the structural information is introduced as a prior to formulate the structured correlation reconstruction problem for matrix reconstruction. As such, the reconstructed covariance matrix can be effectively processed with full utilization of the achievable DOFs from the virtual array, but with a more flexible constraint on the array configuration. The described estimation problem is theoretically analyzed by deriving the corresponding Cram´er-Rao bound (CRB). Moreover, we compare the derived CRB with the performance of the virtual array interpolation-based algorithm. Extensive simulations are conducted to demonstrate the effectiveness of the proposed DOA estimation algorithm in terms of achievable DOFs, resolution, and estimation accuracy.

Augmented coarrays can be derived from spatially undersampled signals of sparse arrays for underdetermined direction-of-arrival (DOA) estimation. With the extended dimension of sparse arrays, the sampled signals can be modeled as sub-Nyquist tensors, thereby enabling coarray tensor processing to enhance the estimation performance. The existing methods, however, are not applicable to generalized multi-dimensional sparse arrays, such as sparse planar array and sparse cubic array, and have not fully exploited the achievable source identifiability. In this paper, we propose a coarray tensor DOA estimation algorithm for multi-dimensional structured sparse arrays and investigate an optimal coarray tensor structure for source identifiability enhancement. Specifically, the cross-correlation tensor between sub-Nyquist tensor signals is calculated to derive a coarray tensor. Based on the uniqueness condition for coarray tensor decomposition, the achievable source identifiability is analysed. Furthermore, to enhance the source identifiability, a dimension increment approach is proposed to embed shifting information in the coarray tensor. The shifting-embedded coarray tensor is subsequently reshaped to optimize the source identifiability. The resulting maximum number of degrees-of-freedom is theoretically proved to exceed the number of physical sensors. Hence, the optimally reshaped coarray tensor can be decomposed for underdetermined DOA estimation with closed-form solutions. Simulation results demonstrate the effectiveness of the proposed algorithm in both underdetermined and overdetermined cases.

Conventional canonical polyadic decomposition (CPD) approach for tensor-based sparse array direction-of-arrival (DOA) estimation typically partitions the coarray statistics to generate a full-rank coarray tensor for decomposition. However, such an operation ignores the spatial relevance among the partitioned coarray statistics. In this letter, we propose a coupled coarray tensor CPD-based two-dimensional DOA estimation method for a specially designed coprime L-shaped array. In particular, a shifting coarray concatenation approach is developed to factorize the partitioned fourth-order coarray statistics into multiple coupled coarray tensors. To make full use of the inherent spatial relevance among these coarray tensors, a coupled coarray tensor CPD approach is proposed to jointly decompose them for high-accuracy DOA estimation in a closed-form manner. According to the uniqueness condition analysis on the coupled coarray tensor CPD, an increased number of degrees-of-freedom for the proposed method is guaranteed.

A coprime array is capable of achieving more degrees-of-freedom for direction-of-arrival (DOA) estimation than a uniform linear array when utilizing the same number of sensors. However, existing algorithms exploiting coprime array usually adopt predefined spatial sampling grids for optimization problem design or include spectrum peak search process for DOA estimation, resulting in the contradiction between estimation performance and computational complexity. To address this problem, we introduce the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) to the coprime coarray domain, and propose a novel coarray ESPRIT-based DOA estimation algorithm to efficiently retrieve the off-grid DOAs. Specifically, the coprime coarray statistics are derived according to the received signals from a coprime array to ensure the degrees-of-freedom (DOF) superiority, where a pair of shift invariant uniform linear subarrays is extracted. The rotational invariance of the signal subspaces corresponding to the underlying subarrays is then investigated based on the coprime coarray covariance matrix, and the incorporation of ESPRIT in the coarray domain makes it feasible to formulate the closed-form solution for DOA estimation. Theoretical analyses and simulation results verify the efficiency and the effectiveness of the proposed DOA estimation algorithm.