Differential Beamforming on Graphs

In this article, we study differential beamforming from a graph perspective. The microphone array used for differential beamforming is viewed as a graph, where its sensors correspond to the nodes, the number of microphones corresponds to the order of the graph, and linear spatial difference equations among microphones are related to graph edges. Specifically, for the first-order differential beamforming with an array of M microphones, each pair of adjacent microphones are directly connected, resulting in M-1 spatial difference equations. On a graph, each of these equations corresponds to a 2-clique. For the second-order differential beamforming, each three adjacent microphones are directly connected, resulting in M-2 second-order spatial difference equations, and each of these equations corresponds to a 3-clique. In an analogous manner, the differential microphone array for any order of differential beamforming can be viewed as a graph. From this perspective, we then derive a class of differential beamformers, including the maximum white noise gain beamformer, the maximum directivity factor one, and optimal compromising beamformers. Simulations are presented to demonstrate the performance of the derived differential beamformers.