A class of non-linear adaptive time-frequency transform

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 controls are obtained for both transforms in \(\textit{L}^2({\mathbb {R}})\) ${\mathit{L}}^2(\mathbb{R})$ spaces, which extend the classical continuous frame norm control and guarantees well-definedness on \(\textit{L}^2\) ${\mathit{L}}^2$ . 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, together with corresponding focused transforms of real and synt-hetic signals. These can serve as starting point for concrete applications.