Time-Frequency Fading (TFF)

Abstract

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. 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, ie 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 specific region 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.

Link to the papers and code (Github)

1. Time-Frequency Fading Algorithms Based on Gabor Multipliers A. Marina Kreme, Valentin Emiya, Caroline Chaux. IEEE Journal of Selected Topics in Signal Processing, 2021
2. Filtering out time-frequency areas using Gabor multipliers A. Marina Kreme, Valentin Emiya, Caroline Chaux. ICASSP: 45th International Conference on Acoustics, Speech, and Signal Processing, May 2020.
3. tff2020 (Github)

Data Presentation

All the audio signals considered here have been sampled at 8 kHz and their length is L = 16384 samples (about 2 seconds) The observed signal Mix = Target + Perturbation is constructed as the sum of an interest signal and a perturbation signal. The interest signals have a wide band spectrogram while the pertubation signals are well localized in time-frequency. We present listening tests of the methods presented in reference 1 :

• Zero-fill : a reference method consisting of reconstructing the signal after filling in the regions masked by zeros
• Interp : reference method consisting in performing a linear interpolation along the frequency axis of the amplitude of the time-frequency matrix of the observations, and in generating the associated phases in a uniform and random way
• TFF-1 : time-frequency fading method considering the mask as a single region (Algorithm 6 of the reference 1).