Nonlinear Kronecker product filtering for multichannel noise reduction

Multichannel noise reduction in the frequency domain is a fundamental problem in the areas of speech processing and speech recognition. In this paper, we address this problem and propose an alternative approach to retrieve a speech signal out of microphone array noisy observations. We focus on the spectral amplitude of the speech signal and assume that the spectral phase is less significant. The estimate of the spectral amplitude squared, that is the spectral power, is obtained by applying a complex linear filter to a modified version of the observations vector. This modified version is obtained as a Kronecker product of the complex conjugate of the observations vector and the original observations vector. The complex speech signal estimate is obtained by multiplying the spectral amplitude estimate with a complex exponential whose phase may be extracted from the minimum variance distortionless response beamformer. We present a modified optimization criterion according to which the proposed filters may be derived, and compare their performances to conventional multichannel noise reduction filters. We show that the new approach is preferable, in particular when the input signal-to-noise ratio (SNR) is low or the number of sensors is small.