Multistage approach for steerable differential beamforming with rectangular arrays

This paper presents a multistage rectangular approach for steerable differential beamforming. As a first step, we propose employing a two-dimensional (2-D) differentiation scheme that operates independently on the columns and rows of a uniform rectangular array (URA). This yields a differentials matrix controlled by two parameters, P_c and P_r, which indicate the number of differential stages for the URA columns and rows. Then, as a second step, we design a rectangular differential beamformer and apply it to the vector form of the differentials matrix. We show that the proposed differentiation scheme may significantly improve the directivity of the resulted beamformer at the expense of white noise amplification. This improvement is heavily tied to selecting the (P_c,P_r) configuration per the desired signal incident angle. Next, we propose four rectangular differential beamformers and analyze their performances in terms of the white noise gain (WNG) and directivity factor (DF) measures. In addition, we address reverberant scenarios with three distinct incident angles of the desired signal. We examine the performances of each beamformer in terms of four reduction factors calculated from the noisy and enhanced signals and investigate their quality and intelligibility. We demonstrate that the proposed rectangular differential beamformers outperform common linear and differential beamformers in these measures, mainly when the incident angle is far from the endfire direction. Finally, we compare the proposed approach to two existing rectangular differential beamforming approaches from the literature. We show the proposed method to be preferable, considering both the quality and intelligibility of the enhanced signals.