Virtual Augmentation of the Beamforming Array Based on a Sub-cross-spectral Matrix Computation for Localizing Stationary Signal Noise Sources

Abstract

This paper presents a generalized algorithm called the sub-cross-spectral matrix (SCSM) beamforming technique for the virtual augmentation of an N-channel beamforming array based on sequential computation of the cross-spectral matrix (CSM) terms for localizing stationary signal sources. To this end, first, the diagonal sub-cross-spectral matrices (SCSMs) of the N-channel array pertaining to M different spatial locations were obtained. Next, the off-diagonal SCSMs were systematically computed by directly evaluating the cross-spectral terms between some microphones placed in the array at \(i{\text{th}}\) location \((1 \le i \le M)\) and the remaining microphones placed in the array at \(j{\text{th}}\) location \((j \ne i, \, 1 \le j \le M)\). As a proof of concept, the SCSM beamforming was used to virtually construct a 32-channel planar Underbrink spiral array by sequentially measuring data using \(\left( {\begin{array}{*{20}c} {32} \\ 2 \\ \end{array} } \right)\) microphone pairs. The resultant 2-D beamforming map of a loudspeaker source was found to be nearly identical to the counterpart result produced when data from 32-channel simultaneous measurements were used. The SCSM technique was then extended to increase the density and aperture of a planar array by constructing a virtual 64-channel planar array from 32-channel simultaneous measurements. For the former case, the source maps were found to be identical to the counterpart results obtained from the existing geometric mean and combined CSM algorithms. However, for the latter case, the SCSM beamforming delivered a noticeably improved focal-resolution along the direction in which there was a virtual increase in aperture. For localizing loudspeaker source(s) in a 3-D domain, the SCSM beamforming implemented using two orthogonal Underbrink arrays was shown to deliver a significantly improved resolution (focal lobe) and unambiguous localization because it considers the complete CSM unlike the multiplicative beamforming and combined CSM algorithms which do not account for the phase-information between the two orthogonal arrays.