A General Framework for Trigonometric Beamformers: Theoretical Analysis, Efficient Design and Implementation

Polynomial beamformers for microphone arrays have gained interest thanks to their efficient and flexible beam-steering capability. Recent findings indicate that polynomial beamformers for circular arrays can experience significant performance degradation due to the Runge's phenomenon. To address this issue, trigonometric beamformers that use trigonometric interpolation instead of polynomial interpolation have been proposed. However, the trigonometric beamformers require a large beamforming filter length to ensure their performance, which leads to high design and implementation complexity. In this paper, we propose a general framework for the design and implementation of trigonometric beamformers using the Laguerre filters, which include the existing trigonometric beamformers as a special case. Under the general framework, the performance of the trigonometric beamformers can be greatly improved for short beamforming filter length by exploiting the degrees of freedom of the Laguerre filter poles (LFPs). We theoretically derive the properties of the beamformer coefficients and LFPs of the generalized trigonometric beamformers. Based on the derived properties, we then develop efficient design approach and implementation structure for the generalized trigonometric beamformers, which significantly reduces the design and implementation complexity. Simulation results have demonstrated the superior performance of the generalized trigonometric beamformers and the high efficiency of the proposed design and implementation schemes.