Optimal Real-Weighted Beamforming With Application to Linear and Spherical Arrays

TL;DR

This work develops real-valued beamforming techniques that maximize directivity for both linear and spherical (phase-mode) microphone arrays, yielding closed-form solutions and robust variants. By replacing complex weights with real ones, the authors achieve reduced computational complexity while carefully analyzing the consequences of symmetry and the resulting parasitic lobes in open-array geometries. They derive exact real-valued solutions, formulate bounded-sensitivity versions, and establish lower bounds on sensitivity, with clear parallels between linear and spherical cases. Simulation and experimental results show that real-valued beamformers approach the performance of complex-valued counterparts, especially near spatial Nyquist, while enabling hardware-friendly implementations and flexible steering via phase-mode processing.

Abstract

One of the uses of sensor arrays is for spatial filtering or beamforming. Current digital signal processing methods facilitate complex-weighted beamforming, providing flexibility in array design. Previous studies proposed the use of real-valued beamforming weights, which although reduce flexibility in design, may provide a range of benefits, e.g., simplified beamformer implementation or efficient beamforming algorithms. This paper presents a new method for the design of arrays with real-valued weights, that achieve maximum directivity, providing closed-form solution to array weights. The method is studied for linear and spherical arrays, where it is shown that rigid spherical arrays are particularly suitable for real-weight designs as they do not suffer from grating lobes, a dominant feature in linear arrays with real weights. A simulation study is presented for linear and spherical arrays, along with an experimental investigation, validating the theoretical developments.

Figures (5)

• Figure 1: Comparison between two maximum directivity beampatterns - the real-weighted and the general complex-weighted, for a linear array with parameters: $d=10$ cm, $M=25$, at $f=1715$ Hz.
• Figure 2: Comparison of the performance measures of complex-valued and real-valued optimal beamformers: (a) - directivity index, (b) - sensitivity.
• Figure 3: Maximum directivity beampatterns with real-valued weights for a spherical array of order $N=10$, at $kr=10$.
• Figure 4: Spherical array beam patterns using various cost functions: (a)-linear, (b)-uniform, (c)-step function.
• Figure 5: Spatial sound-field analysis at $2400$ Hz ($N=4$, $r=9$ cm) using various beamformers: (a) complex-valued max. directivity (b) real-valued max. directivity (c) real-valued with linear cost function.