A wavelet-based measurement of signal fractal dimensions
Armein Z. R. Langi, H.B. Nugraha · 2003
This paper describes a study of measuring signal fractal dimensions (especially in a form of Lipschitz exponents /spl gamma/) 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/sub f/(a, b), we find wavelet maxima lines l(a, b) and their corresponding wavelet maxima |W/sub f/(a, b)|. Suppose the signal f(t) has a Lipschitz exponent /spl gamma/ at t=b/sub 0/, and there is a maxima line l(a,b) reaching b/sub 0/ as a/spl rarr/0. The corresponding wavelet maxima in the line satisfy an inequality |W/sub f/(a, b)|/spl les/Ca/sup /spl gamma/+0.5/ for some constant C and a/spl rarr/0. A log-log plot on the inequality estimates the Lipschitz exponent /spl gamma/. We have performed an experiment of the procedure for f(t)=1-|0.5-t|/sup /spl gamma//, where the Lipschitz component /spl gamma/ varies from 0.1 to 0.9 at a 0.1 interval. The procedure provides relatively good estimates for 0.5/spl les//spl gamma//spl les/0.9, with relative errors less them 10%.