A Perspective on Fully Steerable Differential Beamformers for Circular Arrays

Both the null-constrained method and the Jacobi-Anger expansion method are commonly adopted to design the fully steerable circular differential microphone arrays (CDMAs). So far, these two methods have been independently studied, and it is unclear why they exhibit pronounced performance difference and whether they have a potential connection. In this letter, we develop a unified framework for the design of the fully steerable CDMA, which jointly utilizes the null constraints and the Jacobi-Anger expansion. We show that the existing two methods can be treated as special cases of the proposed approach and they can indeed be connected. We then reveal that the null-constrained CDMA can be viewed as the regularized Jacobi-Anger expansion-based CDMA, which theoretically explains why the former can alleviate the deep nulls problem in white noise gain in certain cases and why it still suffers from the severe deep nulls in other cases. We further explain why the magnitude of the synthesized beampattern of the Jacobi-Anger expansion-based method in nulls' direction can be much larger than that of the null-constrained method. Simulations support the theoretical results well.