Discrete wavelet analysis of multifractal measures and multi-affine signals

Abstract

We show that the discrete wavelet transform can be used to analyze multifractal properties of singular signals. In particular, we show that the discrete wavelet transform implemented with the pyramid algorithm by Mallat provides a very fast and efficient tool for determining simultaneously and with sufficient accuracy the singularity spectrum of the multifractal measures and the multi-affine function of self-affine multifractals. To illustrate our algorithm, we discuss applications of our wavelet analysis with a family of Daubechies wavelets to the fractal measures from the two-scale Canter set and from the critical golden mean orbit of the circle map and to the multi-affine signals from a simple deterministic model.