A class of non linear adaptive time-frequency transforms
Abstract
This paper introduces a couple of new time-frequency transforms, designed to adapt their scale to specific features of the analyzed function. Such an adaptation is implemented via so-called focus functions, which control the window scale as a function of the time variable, or the frequency variable. In this respect, these transforms are non-linear, which makes the analysis more complex than usual.Under appropriate assumptions, some norm control can be obtained for both transforms in L^2(R) spaces, which extend the classical continuous frame norm control and guarantees well-definedness on L^2. Given the non-linearity of the transforms, the existence of inverse transforms is not guaranteed anymore, and is an open question. However, the results of this paper represent a first step towards a more general theory.Besides mathematical results, some elementary examples of time and frequency focus functions are provided, which can serve as staring point for concrete applications.
Cite
@article{arxiv.2306.14550,
title = {A class of non linear adaptive time-frequency transforms},
author = {Pierre Warion and Bruno Torrésani},
journal= {arXiv preprint arXiv:2306.14550},
year = {2024}
}