Fractals and the analysis of waveforms

Abstract

Waveforms are planar curves—ordered collections of ( x, y) point pairs—where the x values increase monotonically. One technique for numerically classifying waveforms assesses their fractal dimensionality, D. For waveforms: D = log(n) ( log(n) + log( d L )) , with n = number of steps in the waveform (one less than the number of ( x, y) point pairs), d = planar extent (diameter) of the waveform, and L = total length of the waveform. Under this formulation, fractal dimensions range from D = 1.0, for strainght lines through approximately D = 1.15 for random-walk waveforms, to D approaching 1.5 for the most convoluted waveforms. The fractal characterization may be especially useful for analyzing and comparing complex waveforms such as electroencephalograms (EEGs).