Properties and Limits of the Minimum-norm Differential Beamformers with Circular Microphone Arrays

Small aperture circular microphone arrays (CMAs) have been widely used in many applications such as teleconferencing, smart-speakers, and robotics. A critical component of such arrays is the differential beamformer, which can achieve relatively high spatial gains with the same beampatterns at most frequencies. Among different differential beamforming approaches that were developed in the literature, the minimum-norm one has attracted much interest as it can deal better with sensors' self noise, sensor mismatch, and beamformer's irregularity at some frequencies due to the zeros of the Bessel functions. In our previous study, we have investigated the performance of the minimum-norm differential beamformer with uniform CMAs (UCMAs) in the 2-dimensional (2D) space where the sound sources and the sensors are assumed to be in the same plane. But in practice, this assumption is generally not true. So, in this paper, we investigate the properties and limitations of the minimum-norm differential beamformer in the 3-dimensional (3D) space. Through theoretical study as well as simulations, we show that the minimum-norm differential beamformer is effective in dealing with the problem of white noise amplification and irregularity of the beampatterns and the directivity factor ({\text{DF}}) if the steering angles are within or near the sensor plane, but it becomes less and less effective as the beamformer is steered away from this plane.