A class of non linear adaptive time-frequency transforms

TL;DR

This paper introduces new non-linear, adaptive time-frequency transforms that adjust their scale based on features of the analyzed function, extending classical analysis and opening new research directions.

Contribution

It proposes a novel class of non-linear adaptive transforms with focus functions controlling scale, providing initial mathematical analysis and examples for potential applications.

Findings

Norm control in L^2 spaces established

Transforms are well-defined under certain conditions

Inverse transform existence remains an open question

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.