Mother wavelet functions generalized through q-exponentials

TL;DR

This paper introduces q-exponential based generalizations of mother wavelets, including the mexican hat and Morlet, aiming to improve signal analysis in systems with long-range interactions and fractal structures.

Contribution

It develops a new class of wavelets using q-exponentials from nonextensive statistical mechanics, extending traditional wavelets and enabling better analysis of complex signals.

Findings

Generalized wavelets reduce to classical ones as q approaches 1

Enhanced analysis of signals with long-range correlations

Application to mono- and multi-fractal signals using WTMM

Abstract

We generalize some widely used mother wavelets by means of the q-exponential function $e_q^x \equiv [1+(1-q)x]^{1/(1-q)}$ ($q \in {\mathbb R}$, $e_1^x=e^x$) that emerges from nonextensive statistical mechanics. Particularly, we define extended versions of the mexican hat and the Morlet wavelets. We also introduce new wavelets that are $q$-generalizations of the trigonometric functions. All cases reduce to the usual ones as $q \to 1$. Within nonextensive statistical mechanics, departures from unity of the entropic index q are expected in the presence of long-range interactions, long-term memory, multi-fractal structures, among others. Consistently the analysis of signals associated with such features is hopefully improved by proper tuning of the value of q. We exemplify with the WTMM Method for mono- and multi-fractal self-affine signals.