Estimation of multiple sound sources by inverting one eigenpair of cross-spectral matrix with dimensionality reduction

This paper addresses the problem of identifying multiple sound sources from pressure signals measured by a microphone array. The proposed method uses only one eigenvalue and the associated eigenvector of the cross-spectral matrix (CSM) to match their theoretical model to estimate the locations and powers of multiple sound sources, which ends up with searching the zeros of only $S+1$ equations ($S$ is the number of sources). This is possible because (i) we find an alternative $S$-by-$S$ matrix of the $M$-by-$M$ CSM ($M$ is the number of microphones and $M\gg S$) in the sense of having identical eigenvalues and equivalent eigenvectors; (ii) we prove that each eigenvalue and eigenvector pair of the CSM or its dimension-reduced alternative uniquely decides all the sound source parameters. As a result, the proposed method can accurately estimate all the parameters of multiple sound sources with super-resolution and it is easy to solve without any optimization problems, which are demonstrated via both numerical and experimental data.