Consecutive differences as a method of signal fractal analysis

Abstract

We propose a new method for calculating fractal dimension (DF) of a signal y(t), based on coefficients m(y)((n)), mean absolute values of its nth order derivatives (consecutive finite differences for sampled signals). We found that logarithms of m(y)((n)), = 2, 3,..., n(max), exhibited linear dependence on n: log (m(y)((n))) = (slope)n + Y(int) with stable slopes and Y-intercepts proportional to signal DF values. Using a family of Weierstrass functions, we established a link between Y-intercepts and signal fractal dimension: DF = A(n(max))Y(int) + B(n(max)), and calculated parameters A(n(max)) and B(n(max)) for n(max) = 3,..., 7. Compared to Higuchi's algorithm, advantages of this method include greater speed and eliminating the need to choose value for k(max), since the smallest error was obtained with n(max) = 3.