TY - JOUR
T1 - Time-scale analysis of the self-similar properties of fractal signals
AU - Wu, Bing-Fei
AU - Su, Yu Lin
PY - 1997/12/1
Y1 - 1997/12/1
N2 - 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.
AB - 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.
KW - Fractals
KW - Fractional Brownian motion
KW - Self-similarity
KW - Wavelet transforms
UR - http://www.scopus.com/inward/record.url?scp=8644224195&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:8644224195
VL - 4
SP - 190
EP - 205
JO - Eurasip Journal on Advances in Signal Processing
JF - Eurasip Journal on Advances in Signal Processing
SN - 1687-6172
IS - 4
ER -