Compressive beamforming and directional sound reconstruction using the Kronecker array transform

The Kronecker array transform (KAT) was introduced to reduce the computational burden in acoustic image estimation and other 2-dimensional array processing applications. The KAT can be applied only when the planar microphone array is separable. In this paper we study the performance of separable arrays using compressive beamforming algorithms to estimate the direction of arrival (DOA) of far-field sources and to recover the signal arriving from a particular direction. Our analysis extends previous studies based on mutual coherence to separable arrays, and shows that non-redundant separable arrays have better mutual coherence than can be obtained using 2D random arrays. We also derive the space discretization that yields the minimum coherence, and study the influence of frequency and the spatial resolution on the coherence. In order to verify the results, we present the performance of a separable array in two problems: DOA estimation and signal recovery using sparse reconstruction, and compare its performance with classical beamforming techniques. The sparsity-based approach shows better performance in DOA estimation and great improvement in time-domain signal recovery.