Differential beamformers derived from approximate performance measures

Differential microphone arrays (DMAs) have attracted great interest over the past two decades, since this type of arrays can form frequency-invariant beampatterns and achieve maximum directional gains with a given number of sensors. Generally, the design of DMA beamformers involves optimization of some performance measures such as the directivity factor (DF), front-to-back ratio (FBR), white noise gain (WNG), etc. In this paper, we develop approximate performance measures, which basically approximate the integral part in the exact performance measures with a weighted sum. This approximation gets finer and finer as more points are used. When applied to the design of DMAs, the major advantages of using these approximate measures is that the design problem is simplified and many differential beamformers, including commonly-used standard ones, can be easily derived.