Design of frequency-invariant uniform concentric circular arrays with first-order directional microphones

Most of the existing frequency-invariant beamformers (FIBs) with circular arrays are implemented with omnidirectional microphones, where uniform circular arrays (UCAs) and uniform concentric circular arrays (UCCAs) have been widely utilized. However, the resulting FIB usually suffers from the deep nulls problem in the white noise gain (WNG) and the white noise amplification problem at low frequencies even if the order of the target beampattern is one. Due to its simple implementation, first-order directional microphones are adopted to improve the robustness of the FIB with UCCAs (FIB-UCCAs) in this paper. We exploit the circular harmonics decomposition rather than the Jacobi-Anger expansion to approximate the beampattern of the FIB-UCCAs with first-order directional microphones, because the directivity pattern of directional microphones is related to the direction of the signal. We show that the FIB-UCCAs can completely avoid the deep nulls problem in the WNG as long as the ring with the largest radius utilizes first-order directional microphones. In addition, the white noise amplification problem at low frequencies can be greatly alleviated if first-order directional microphones are utilized. Specifically, we show that the inherent white noise amplification problem at low frequencies can be completely circumvented when the order of the desired beampattern is one.