Singular block Toeplitz matrix approximation and application to multi-microphone speech dereverberation

We consider the blind multichannel dereverberation problem for a single source. We have shown before [5] that the single-input multi-output (SIMO) reverberation filter can be equalized blindly by applying MIMO Linear Prediction (LP) to its output (after SISO input pre-whitening). In this paper, we investigate the LP-based dereverberation in a noisy environment, and/or under acoustic channel length underestimation. Considering ambient noise and late reverberation as additive noises, we propose to introduce a postfilter that transforms the MIMO prediction filter into a somewhat longer equalizer. The postfilter allows to equalize to non-zero delay. Both MMSE-ZF and MMSE design criteria are considered here for the postfilter.We also focus here on computationally efficient (FFT based) block Toeplitz covariance matrix enhancement that enforces the SIMO filtered source plus white noise structure before applying MIMO LP. A second suggested refinement is an iterative refinement between SISO and MIMO LP. Simulations show that the proposed scheme is robust in noisy environments, and performs better compared to the classic Delay-&-Predict equalizer and the Delay-&-Sum beamformer.