Stability estimates for adaptive focused time-frequency transforms

TL;DR

This paper establishes stability results for adaptive, focus-based time-frequency transforms, particularly the STFT, demonstrating how the focus function influences the transform's stability and continuity.

Contribution

It provides new stability estimates for adaptive time-frequency transforms and analyzes focus functions based on regularized Rényi entropies.

Findings

Stability results with respect to the focus function

Continuity of the transform under focus function variations

Analysis of regularized Rényi entropy-based focus functions

Abstract

This contribution is a follow-up of a recent paper by the authors on adaptive, non-linear time-frequency transforms, focusing on the STFT based transforms. The adaptivity is provided by a focus function, that depends on the analyzed function or signal, and that adapts dynamically the time-frequency resolution of the analysis. Sticking to the continuous case setting, this work provides new stability results for the transform (stability with respect with the focus function). It also investigates in some details focus functions based upon regularized R{\'e}nyi entropies and show corresponding continuity results.