On Differential Beamforming With Nonuniform Linear Microphone Arrays

While differential beamforming with uniform linear arrays (ULAs) has been widely studied, there is little work so far regarding the design of differential beamformers with nonuniform linear arrays (NULAs). This article attempts to shed some light on the principles of differential beamforming with NULAs. We define spatial difference operators with NULAs, where any order of the spatial difference of the observation signals can be represented as the product of a nonuniform spatial difference operator matrix and the observation vector. Consequently, the design of differential beamformers is performed in two stages. In the first one, a nonuniform spatial difference operator matrix is applied to the array observations, thereby yielding differential signals. In the second stage, beamformers are designed and applied to the obtained differential signals to optimize the array performance. Based on the defined spatial difference operators, we derive from some performance metrics a family of differential beamformers with NULAs, which include the maximum directivity factor (DF), the maximum white noise gain (WNG), and the maximum front-to-back ratio (FBR) differential beamformers. To compromise between the DF and array robustness, we also derive the parameterized maximum DF and parameterized maximum FBR differential beamformers. The null-constraint maximum DF and WNG differential beamformers are also developed so that some nulls can be placed in specified directions for interference suppression. Simulation results validate the theoretical analysis and justify the properties of the proposed methods.