Time-scale analysis of the self-similar properties of fractal signals

The self-similar properties of fractal signals are summarised in this paper when the fractal reveals respectively in probability measure, variance, time series, time-average autocorrelation, ensemble-average autocorrelation, time-average power spectrum, average power spectrum and distribution functions. Approaches to preserve the one-dimensional (1D)/two-dimensional (2D) self-similarities for fractal signals and fractional Brownian motions (fBm) by using the discrete wavelet transform (DWT) based on the perfect reconstruction-quadrature mirror filter structure are proposed. Furthermore, the CWT cases are summarised and put together with the results of DWT to point out the relationships of the self-similarities between the continuous wavelet transform and DWT. With the application of this work, an algorithm is derived to estimate the fractal dimensions of fractal signals. The three-section Cantor set, Sierpinski gasket, 1D and 2D fBm fields are provided to illustrate the results.