A Finite Difference Framework for Differential Beamforming With Planar Microphone Arrays

Differential microphone arrays (DMAs) have demonstrated great advantages in high-fidelity acoustic signal acquisition and enhancement due to their capability to capture both the acoustic pressure and differential fields. While considerable progress has been made in differential beamforming with planar DMAs, most existing methods are rooted in beampattern approximation. In such approaches, the connection between differential beamformers and physical differential operations is usually implicit or confined to specific array topologies. To address this, this paper presents a finite difference framework for the analysis and design of planar DMAs. The main contributions of this work are threefold. First, we revisit the fundamentals of DMAs from an acoustic physics perspective, demonstrating how finite differences can be applied to planar arrays to approximate the directional derivatives of the acoustic pressure field. Second, based on this finite difference approximation, we extend the spatial difference operator (SDO) for linear DMAs to the two-dimensional (2D) case. The resulting 2D SDO is applied to the observation signal vector to extract differential signals, which are then filtered to optimize various performance metrics, enabling the design of different differential beamformers. Finally, we show that the 2D SDO can be obtained by solving a linear system of equations, offering a clear and systematic design procedure.