A wavelet-based measurement of signal fractal dimensions

This paper describes a study of measuring signal fractal dimensions (especially in a form of Lipschitz exponents γ) using wavelets. The procedure is as follows. Given a one-dimensional signal f(t) and its corresponding wavelet transform (at a scale a and a position b) as W_f(a, b), we find wavelet maxima lines l(a, b) and their corresponding wavelet maxima |W_f(a, b)|. Suppose the signal f(t) has a Lipschitz exponent γ at t=b₀, and there is a maxima line l(a,b) reaching b₀ as a→0. The corresponding wavelet maxima in the line satisfy an inequality |W_f(a, b)|≤Ca^γ+0.5 for some constant C and a→0. A log-log plot on the inequality estimates the Lipschitz exponent γ. We have performed an experiment of the procedure for f(t)=1-|0.5-t|^γ, where the Lipschitz component γ varies from 0.1 to 0.9 at a 0.1 interval. The procedure provides relatively good estimates for 0.5≤γ≤0.9, with relative errors less them 10%.