Low Complexity Modeling of Cross-Spectral Matrix and Its Application in the Non-Synchronous Measurements of Microphones Array

The resolution related with the image quality of acoustic imaging using a microphone array is limited by the size and density of the array. However, non-synchronous measurements can exceed the constraints defined by measurements with a single fixed array. The non-synchronous measurements can achieve a larger and denser aperture array by moving the single array. The key problem of archiving the non-synchronous measurements of a microphone array is the matrix completion of a block diagonal spectral matrix. A comprehensive investigation of the block diagonal spectral matrix completion is still missing in the microphone array signal processing literature since it is usually considered as a difficult problem if without the integrated physical information of the sound field. In this paper, the mathematical relation between the max-norm and the nuclear-norm is pointed out, and the max-norm minimization of the spectral matrix is discussed, which is an effective model with integrated spatial representation of the sound field for the spectral matrix completion. First, the max-norm is utilized for the low complexity modeling and the corresponding Proximal-Point method for Max-norm based Spectral matrix completion (PPMSMC) is proposed; Second, the rank, nuclear-norm and max-norm have been investigated as complexity measures in the context of spectral matrix completion, and their performances are compared. The proposed low-rank methods have constructed a concrete theorem foundation for the non-synchronous measurements of the microphone array.