Two-dimensional grid-free compressive beamforming with spherical microphone arrays

Abstract

Compressive beamforming based on microphone array measurements and compressive sensing theory is an emerging and practical acoustic source identification technology. The grid-free compressive beamforming that treats the target source region as a continuum has attracted tremendous attention due to the advantage of circumventing the basis mismatch defect caused by the gridding. The grid-free methods have been developed for the linear and the planar microphone array measurements to identify sources in a local region. In contrast, for the spherical microphone array measurements suitable for 360° panoramic identification, the current research mainly focuses on grid-based methods. In this paper, we propose a two-dimensional grid-free compressive beamforming method with spherical microphone arrays. First of all, the relationship between the spherical harmonic function vector and the two-dimensional Fourier basis function vector is determined. Then, the mathematical model relating the measured sound pressures to the source distribution is built under a continuous setting. Next, a source sparsity constraint is imposed based on the total variation norm to solve the mathematical model, and its dual problem is deduced. Subsequently, the dual problem is solved by positive semidefinite programming. Finally, the DOAs of sources are estimated through polynomial rooting and utilizing the dual optimal variables, and the source strengths are quantified via least-square fitting. An experimental case is conducted to examine the practicability and superiority of the proposed method to the existing grid-based compressive beamforming method. Plenty of Monte Carlo simulations are carried out to disclose the influence of the source coherence, the angular distance between sources, the noise, the number of snapshots and the number of microphones. This study offers a new approach for the accurate and 360° panoramic identification of acoustic sources.