Directivity pattern or beampattern is an important performance measure in all fixed beamformers. Given a microphone array, how to design the beamforming filter so that the resulting directivity pattern is close to the desired one is a critical issue. In this paper, we study the design of such patterns for endfire uniform linear microphone arrays. By considering the frequency-independent Chebyshev pattern as the desired one, we derive an optimal beamforming filter based on the minimization of the mean-squared error (MSE) under the distortionless constraint. It is shown that the proposed beamformer design can generate beampatterns that are very close to the desired ones and, the larger is the number of microphones, the better is the designed beampattern.

Differential microphone arrays (DMAs), due to their small size and enhanced directivity, are quite promising in speech enhancement applications. However, it is well known that differential beamformers have the drawback of white noise amplification, which is a major issue in the processing of wideband signals such as speech. In this paper, we focus on the design of robust DMAs. Based on the Maclaurin’s series approximation and frequency-independent beampatterns, the robust first-, second-, and third-order DMAs are proposed by using more microphones than the order plus one, and the corresponding minimum-norm filters are derived. Compared to the traditional DMAs, the proposed designs are more robust with respect to white noise amplification while they are capable of achieving similar directional gains.