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.

Existing tensor-based coherent direction-of-arrival (DOA) estimation methods adopting spatial smoothing to decorrelate the coherent tensor statistics usually lead to a poor decorrelation performance. In this letter, we propose a structured tensor reconstruction method for two-dimensional coherent DOA estimation, which then avoids the inefficient spatial smoothing. In particular, after investigating the structural property of the four-dimensional incoherent covariance tensor, we propose a tensorial Hermitian Toeplitz mapping rule to reconstruct a structured covariance tensor from the rank-deficient coherent covariance tensor statistics. It is theoretically proved that, the reconstructed covariance tensor admits a decorrelated canonical polyadic model with a tensorial Hermitian Toeplitz structure, whose decomposition ensures a closed-form coherent DOA estimation. The effectiveness of the proposed method is verified by simulations.

Sparse array direction-of-arrival (DOA) estimation using tensor model has been developed to handle multi-dimensional sub-Nyquist sampled signals. Furthermore, augmented virtual arrays can be derived for Nyquist-matched coarray tensor processing. However, the partially augmentable sparse array corresponds to a discontinuous virtual array, whereas the existing methods can only utilize its continuous part. Conventional virtual linear array interpolation techniques complete coarray covariance matrices with dispersed missing elements, but fail to complete the coarray tensor with whole missing slices. In this paper, we propose a coarray tensor completion algorithm for two-dimensional DOA estimation, where the coarray tensor statistics can be entirely exploited. In particular, in order to impose an effective low-rank regularization on the slice-missing coarray tensor, we propose shift dimensional augmenting and coarray tensor reshaping approaches to reformulate a structured coarray tensor with sufficiently dispersed missing elements. Furthermore, the shape of the reformulated coarray tensor is optimized by maximizing the dispersion-to-percentage ratio of missing elements. As such, a coarray tensor nuclear norm minimization problem can be designed to optimize the completed coarray tensor corresponding to a filled virtual array, based on which the closed-form DOA estimation is achieved. Meanwhile, the global convergence of the coarray tensor completion is theoretically proved. Simulation results demonstrate the effectiveness of the proposed algorithm compared to other matrix-based and tensor-based methods.

Among various sensor array configurations, the L -shaped nested array offers improved performance for two-dimensional (2D) direction-of-arrival (DOA) estimation through co-array processing. However, conventional methods overlook the multi-dimensional signal structure and fail to eliminate the cross term generated from the correlated co-array signal and noise components. It leads to significant degradation in DOA estimation performance. To deal with this problem, an iterative 2D DOA estimation algorithm based on tensor modeling is proposed. It is capable to eliminate the cross term. Specifically, the co-array signals of virtual subarrays on orthogonal directions are derived and concatenated to construct a higher-order tensor, whose factor matrices have the Vandermonde structure and preserve the interconnected azimuth and elevation information. A computationally efficient tensor decomposition method is then developed to independently estimate the azimuth and elevation angles, which are effectively paired using the spatial cross-correlation matrix. Furthermore, after investigating the cross term effect, a two-step iterative algorithm is proposed to sequentially estimate and remove the cross term based on the initial estimates obtained from the high-order tensor decomposition. Consequently, the 2D DOA estimation with enhanced estimation accuracy, resolution and moderate computational complexity is achieved for the L -shaped nested array. Simulation results demonstrate the superiority of the proposed algorithm over competing methods.

Adaptive beamforming using sparse arrays can alleviate system burden with a sub-Nyquist sampling rate while achieving high resolution. To process multi-dimensional signals without losing structural information, tensor models can be incorporated in the beamformer design. Unfortunately, existing tensor beamformers are only suitable for uniform arrays and cannot handle ambiguous sidelobes caused by sparse sensor deployment. In this paper, we propose a sub-Nyquist tensor beamformer based on a coprimality constraint. Specifically, the signals received by the sparse subarrays of a coprime planar array are modeled as two sub-Nyquist tensors. To enhance the desired component of the sub-Nyquist tensor signals, we formulate a pair of tensor beamformer weights and investigate the principle of tensorial signal filtering. A coprimality-based combined distortionless response constraint is then imposed to jointly optimize the tensor beamformer weights, which eliminates spatial aliasing. Moreover, to solve the joint coprime tensor beamformer weights optimization problem with non-convex tensor-based objective and constraint, we decompose it into interconnected bidirectional subbeamformer optimization problems, which are further relaxed to ensure the convexity. The relaxed problems are solved by an alternating minimization approach with global convergence. Simulation results demonstrate the superiority of the proposed sub-Nyquist tensor beamformer over conventional beamformers in terms of mainlobe enhancement and sidelobe attenuation.