Signal-Dependent Penalty Functions for Robust Acoustic Multi-Channel Equalization

Abstract

Acoustic multi-channel equalization techniques, which aim to achieve dereverberation by reshaping the room impulse responses (RIRs) between the source and the microphone array, are known to be highly sensitive to RIR perturbations. In order to increase the robustness against RIR perturbations, several signal-independent methods have been proposed, which only rely on the available perturbed RIRs and do not incorporate any knowledge about the output signal. This paper presents a novel signal-dependent method to increase the robustness of equalization techniques by enforcing the output signal to exhibit spectro-temporal characteristics of a clean speech signal. Motivated by the sparse nature of clean speech, we propose to extend the cost function of state-of-the-art least squares equalization techniques, i.e., the multiple-input/output inverse theorem (MINT), relaxed multi-channel least squares (RMCLS), and partial multi-channel equalization based on MINT (PMINT), with a signal-dependent penalty function promoting sparsity of the output signal in the short-time Fourier transform domain. Three conventionally used sparsity-promoting penalty functions are investigated, i.e., the $l_0$ -norm, the $l_1$ -norm, and the weighted $l_1$ -norm, and the sparsity-promoting reshaping filters are iteratively computed using the alternating direction method of multipliers. Simulation results for several acoustic systems and RIR perturbations demonstrate that incorporating sparsity-promoting penalty functions significantly increases the robustness of MINT, RMCLS, and PMINT, with the weighted $l_1$ -norm typically outperforming the $l_0$ -norm and the $l_1$ -norm. Furthermore, it is shown that the weighted $l_1$ -norm sparsity-promoting PMINT technique outperforms the other sparsity-promoting techniques in terms of perceptual speech quality. Finally, it is shown that the signal-dependent weighted $l_1$ -norm sparsity-promoting PMINT technique yields a similar or better dereverberation performance than the signal-independent regularized PMINT technique, confirming the advantage of using signal-dependent penalty functions for robust dereverberation filter design.