Two-Stage Beamforming With Arbitrary Planar Arrays of Differential Microphone Array Units

Differential Microphone Arrays (DMAs) are of great interest in the literature on small-sized microphone arrays, due to their good directivity properties and nearly frequency-invariant spatial responses. Recently developed beamforming techniques combine multiple DMA units to form flexible two-stage spatial filtering systems, where the output of each DMA is fed into a higher-level filter, called virtual filter, for further processing. In this manuscript, we analyze and discuss some properties of a broad class of two-stage beamformers with arbitrary planar geometry. In this context, the DMA units are all assumed to have the same directivity pattern of arbitrary order and can be characterized by a variable number of omnidirectional sensors organized in an arbitrary geometry. For any given choice of the virtual array filter, we introduce a closed-form optimization procedure to design DMA filters that maximize the White Noise Gain (WNG) or the Directivity Factor (DF) of the resulting two-stage beamformer at any frequency. Based on this frequency-dependent design, we propose a frequency-invariant design of the two-stage beamformer and we compare the performance of the two approaches. Finally, we propose two possible computational schemes for the proposed generic two-stage spatial filtering system and discuss their efficiency in performing filtering, steering, and changing beampattern.