Design of Low-Rank differential beamformers with constrained directivity or robustness

Differential microphone arrays (DMAs) are recognized for their highly directive broadband beampatterns and have attracted significant interest in the design of compact microphone arrays. It has been shown that increasing the number of microphones in a DMA can improve array performance. However, when applying DMAs to embedded systems, this creates challenges due to the increased number of parameters, higher computational complexity, and the need to maintain the array's robustness. To address these challenges, this paper presents a method for designing robust low-rank (LR) differential beamformers. Initially, we extend traditional differential beamforming by introducing an LR differential beamforming framework, which represents a long filter as the Kronecker product of two sets of shorter filters, significantly reducing both the number of parameters and computational complexity. Next, we derive robust designs for the two sets of shorter filters by maximizing the directivity factor (DF) subject to a white noise gain (WNG) constraint, or by maximizing the WNG subject to a DF constraint. This results in two types of LR differential beamformers that achieve the desired DF or WNG levels. The optimization problems are formulated and transformed into quadratic eigenvalue problems (QEPs), leading to closed-form solutions for both the WNG-constrained and DF-constrained LR differential beamformers. Simulation results demonstrate the effectiveness of the proposed method, confirming its robustness and enhanced computational efficiency.