On the relationship between the self-similarities of fractal signals and wavelet transforms

Since many natural phenomena are occasionally defined as stochastic processes and the corresponding fractal characteristics are hidden from their correlation functions or power spectra, the topic would become very interest in signal processing. In this paper, we summarize the fractal dimensions and the relationship of the fractal in probability measure, variance, time series, time-averaging autocorrelation, ensemble-averaging autocorrelation, time-averaging power spectrum, average power spectrum and distribution functions for stationary and nonstationary processes. We also propose that the preservation of the one-dimensional self-similarity of a fractal signal is obtained by using the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT) with the perfect reconstruction - quadrature mirror filter structure. Moreover, we extend the results to the two-dimensional case and point out the relationship of the self-similarities between the CWT and DWT of the fractal signals. A fractional Brownian motion process is provided as an example to show the results of this paper.