Robust and steerable kronecker product differential beamforming with rectangular microphone arrays

Differential microphone arrays (DMAs), a class of welldesigned small-size arrays combined with differential beamforming, are very useful for processing broadband acoustic, audio, and speech signals in a wide range of applications. However, most efforts in the literature so far have been devoted to linear, circular, and spherical arrays. In this paper, we consider rectangular shapes of planar microphone arrays. Instead of adopting the traditional differential beamforming methods developed in the literature, we present a differential beamforming method based on the so-called Kronecker product. We first decompose the entire rectangular array into two virtual rectangular sub-arrays so that the steering vector of the entire array is the Kronecker product of the steering vectors of the two smaller virtual rectangular sub-arrays. We use the first virtual rectangular array, which is much smaller in size than the entire array but well satisfies the basic requirements for differential beamforming, to design a steerable differential beamformer. For the second virtual rectangular array, we can design either the delay-and-sum (DS) beamformer, which helps to improve the robustness of the global differential beamformer, or an adaptive beamformer, which makes the global differential beamformer adaptive. This method has many interesting properties, particularly the designed beamformer is fully steerable, and its robustness and the array gain can be easily controlled.