Quasi Monte Carlo Time-Frequency Analysis

TL;DR

This paper introduces a quasi-Monte Carlo approach for efficient and even sampling of redundant higher-dimensional time-frequency representations in signal processing, improving accuracy and computational efficiency over traditional methods.

Contribution

The paper proposes a novel QMC method for sampling redundant time-frequency transforms, enabling efficient, low-discrepancy sampling with weak dependence on dimensionality.

Findings

QMC sampling requires log-linear samples relative to signal resolution.

QMC samples are evenly spread with low discrepancy.

Method improves time-frequency processing tasks like phase vocoder.

Abstract

We study signal processing tasks in which the signal is mapped via some generalized time-frequency transform to a higher dimensional time-frequency space, processed there, and synthesized to an output signal. We show how to approximate such methods using a quasi-Monte Carlo (QMC) approach. We consider cases where the time-frequency representation is redundant, having feature axes in addition to the time and frequency axes. The proposed QMC method allows sampling both efficiently and evenly such redundant time-frequency representations. Indeed, 1) the number of samples required for a certain accuracy is log-linear in the resolution of the signal space, and depends only weakly on the dimension of the redundant time-frequency space, and 2) the quasi-random samples have low discrepancy, so they are spread evenly in the redundant time-frequency space. One example of such redundant representation is the localizing time-frequency transform (LTFT), where the time-frequency plane is enhanced by a third axis. This higher dimensional time-frequency space improves the quality of some time-frequency signal processing tasks, like the phase vocoder (an audio signal processing effect). Since the computational complexity of the QMC is log-linear in the resolution of the signal space, this higher dimensional time-frequency space does not degrade the computation complexity of the proposed QMC method. The proposed QMC method is more efficient than standard Monte Carlo methods, since the deterministic QMC sample points are optimally spread in the time-frequency space, while random samples are not.