Sub-Nyquist Tensor Beamformer: A Coprimality Constrained Design

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 article, 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 sub-beamformer 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.