Continuous warped time-frequency representations - Coorbit spaces and discretization

TL;DR

This paper introduces a flexible family of continuous time-frequency transforms based on nonlinear warping functions, enabling the creation of generalized coorbit spaces and atomic decompositions, unifying classical and novel representations.

Contribution

It develops a new framework for warped time-frequency representations, extending coorbit space theory and providing conditions for discretization and frame properties.

Findings

The transforms are continuous and invertible.

Subsampled systems can form atomic decompositions and Banach frames.

Classical transforms like STFT and wavelets are special cases.

Abstract

We present a novel family of continuous, linear time-frequency transforms adaptable to a multitude of (nonlinear) frequency scales. Similar to classical time-frequency or time-scale representations, the representation coefficients are obtained as inner products with the elements of a continuously indexed family of time-frequency atoms. These atoms are obtained from a single prototype function, by means of modulation, translation and warping. By warping we refer to the process of nonlinear evaluation according to a bijective, increasing function, the warping function. Besides showing that the resulting integral transforms fulfill certain basic, but essential properties, such as continuity and invertibility, we will show that a large subclass of warping functions gives rise to families of generalized coorbit spaces, i.e. Banach spaces of functions whose representations possess a certain localization. Furthermore, we obtain sufficient conditions for subsampled warped time-frequency systems to form atomic decompositions and Banach frames. To this end, we extend results previously presented by Fornasier and Rauhut to a larger class of function systems via a simple, but crucial modification. The proposed method allows for great flexibility, but by choosing particular warping functions $\Phi$ we also recover classical time-frequency representations, e.g. $\Phi(t) = ct$ provides the short-time Fourier transform and $\Phi(t) = log_a(t)$ provides wavelet transforms. This is illustrated by a number of examples provided in the manuscript.