Design of 2D and 3D Differential Microphone Arrays With a Multistage Framework

Differential microphone arrays (DMAs) have demonstrated a great potential for high-fidelity acoustic and speech signal acquisition in a wide range of applications since such arrays are able to achieve frequency-invariant beampatterns with high directivity. Consequently, a great number of efforts have been devoted to the design of DMAs and the associated beamformers in the literature. However, most of the methods only work for arrays with particular topologies, e.g., linear, circular, concentric circular, and spherical ones. How to design general two-dimensional (2D) and three-dimensional (3D) DMAs that can measure the desired differential sound field and form the desired spatial response in the 3D space remains an unsolved problem. This article investigates this problem and presents a multistage design approach. The major contributions of this work are as follows. First, we reexamine the differentials of the acoustic pressure field in the 3D space and derive the general expression of the directivity patterns resulting from the spatial differential operation, which serves as the foundation for differential beamforming with 2D or 3D microphone arrays. Second, we present a multistage approach to the design of 2D and 3D DMAs, and deduce the relationship between the global beamformer and the beamformers at different stages as well as the relationship between their beampatterns. Third, several algorithms are presented for the design of differential as well as robust differential beamformers in this multistage framework. Simulation results validate the proposed approach and justifies its properties.