On Limitations and Improvement of Differential Beam Forming Via Quadratic Eigenvalue Optimization

Differential microphone arrays (DMAs) have attracted considerable attention for their high spatial gains and frequency-invariant spatial responses. However, they often face significant white noise amplification at low frequencies. One approach to mitigating this challenge is by increasing the number of microphones while fixing the order of the DMA, leveraging additional degrees of freedom to optimize the white noise gain (WNG). But this compensation of WNG can lead to beampattern distortion at mid and high frequencies. To address this issue, we recently explored an approach to designing differential beamformers with predefined WNG levels. This involves formulating beamforming as a quadratic eigenvalue problem (QEP) to efficiently derive optimal solutions without iterative processes, leading to the QEP-based differential beamformers. While it is successful in controlling WNG, this method is found to exhibit great and atypical performance degradation at lower frequencies in certain scenarios. In this paper, we illustrate this phenomenon using a two-stage structured beamformer as a case study and offer insights into why this occurs, along with proposing a solution.