Optimal design of symmetric and asymmetric beampatterns with circular microphone arrays

This paper is devoted to the study of the beamforming problem with circular microphone arrays (CMAs). It presents an approach to the design of beamformers with asymmetric and symmetric frequency-invariant beampatterns. We first discuss how to express a desired target directivity pattern, either symmetric or asymmetric, into a linear weighted combination of sine and cosine functions as well as circular harmonics of different orders. Using the hypercardioid pattern as an example, we show how to determine the weighting coefficients by maximizing the directivity factor (DF) with different constraints. Then, by using the Jacobi-Anger expansion, an approximation of the beamformer's beampattern is presented. A linear system is subsequently formed by forcing the approximated beampattern to be equal to the target asymmetric or symmetric directivity pattern. The optimal beamforming filter is finally determined by identifying the linear system. In comparison with symmetric beampatterns, the use of asymmetric beampatterns can offer more flexibility for practical application, such as a high DF or better null positions for interference rejection, which is demonstrated by simulations.