Randomized Continuous Frames in Time-Frequency Analysis

TL;DR

This paper extends the Monte Carlo approach to continuous frames in time-frequency analysis, demonstrating that the localizing time-frequency transform (LTFT) is computationally efficient and comparable to classical methods like STFT and CWT.

Contribution

It proves that LTFT is linear volume discretizable, enabling efficient Monte Carlo sampling similar to traditional time-frequency transforms.

Findings

LTFT is LVD, allowing efficient Monte Carlo processing.

LTFT has higher-dimensional coefficient space but same asymptotic complexity as STFT and CWT.

Monte Carlo method's complexity depends on the LVD property of the continuous frame.

Abstract

Recently, a Monte Carlo approach was proposed for processing highly redundant continuous frames. In this paper we present and analyze applications of this new theory. The computational complexity of the Monte Carlo method relies on the continuous frame being so called linear volume discretizable (LVD). The LVD property means that the number of samples in the coefficient space required by the Monte Carlo method is proportional to the resolution of the discrete signal. We show in this paper that the continuous wavelet transform (CWT) and the localizing time-frequency transform (LTFT) are LVD. The LTFT is a time-frequency representation based on a 3D time-frequency space with a richer class of time-frequency atoms than classical time-frequency transforms like the short time Fourier transform (STFT) and the CWT. Our analysis proves that performing signal processing with the LTFT has the same asymptotic complexity as signal processing with the STFT and CWT (based on FFT), even though the coefficient space of the LTFT is higher dimensional.