Time-Frequency Fading Algorithms Based on Gabor Multipliers

In this paper, we address a particular instance of time-frequency filter design, which we call Time-Frequency Fading (TFF). In TFF the only available information concerns the time-frequency localization of the component to be filtered out or attenuated: the signal of interest is supposed to be spread out in the time-frequency plane, whereas the perturbation signal is concentrated within a specified time-frequency region Omega. The problem is formulated as an optimization problem designed to fade out the perturbation with accurate control on the fading level. The corresponding objective function involves a data fidelity term that aims to match the TF coefficients of the estimated signal to those of the observed signal outside the perturbation support. It also involves a penalty term that controls the energy of the reconstructed signal, within that region. We obtain the closed-form solution of the problem which involves Gabor multipliers, i.e. linear operators of the pointwise product by a time-frequency transfer function called a mask. We study the TF localization properties of dominant eigenvectors of these Gabor multipliers, with particular attention to the case where the region Omega is a disjoint union of several sub-regions. The decay properties of eigenvalues naturally lead to reduced-rank approximations, and further approximations are obtained in the multiply connected region case. Also, we exploit random projection methods to speed up eigenvalue decompositions and rank reduction. This is implemented in two TFF algorithms, that cover the cases of single or multiple regions. The efficiency of the proposed approach is demonstrated on several audio signals where the perturbations are filtered while leading to a good quality of signal reconstruction.