The Short-Time Multifractal Formalism: Definition and Implement

  • Xiong Gang
  • Yang Xiaoniu
  • Zhao Huichang
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 15)


Although multifractal descirbles the singularity distribution of SE, there is no time information in the multifractal formalism, and the time-varying singularity distribution indicates the spatial dynamics character of system. Therefore, the definition and implement of the short-time multifractal formalism is proposed, which is the prelude of time time-singularity spectra distribution. In this paper, the singularity analysis of windowed signal was given, further the short-time hausdorff spectum was deduced. The Partition Function and Short-time Legendre Spectrum was fractal statistical distribution of SE. WTMM method is popular in implement of MFA, and in section IV,Short-time multifractal spectra based on WTMM is brough forward.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arneodo, A., Audit, B., Bacry, E., Manneville, S., Muzy, J.F., Roux, S.G.: Thermodynamics of Fractal Signals Based on Wavelet Analysis: Application to Fully Developed Turbulence Data and DNA Sequences. Physica A 254, 24–45 (1998)CrossRefGoogle Scholar
  2. 2.
    Arneodo, A., Bacry, E., Muzy, J.F.: The Thermodynamics of Fractals Revisited with Wavelets. Physica A 213(1-2), 232–275 (1995)CrossRefGoogle Scholar
  3. 3.
    Bacry, E.: Lastwave Pakage. Web Document, Febraury 28, 2005 (1997),
  4. 4.
    Donoho, D., Duncan, M.R., Huo, X.: WaveLab Documents, (Febraury 28, 2005) [Online] (1999),
  5. 5.
    Faghfouri, A., Kinsner, W.: 1D Mandelbrot Singularity Spectrum, Ver. 1.0, (Febraury 28, 2005) [Online] (2005),
  6. 6.
    Grassberger, P., Procaccia, I.: Dimensions and Entropies of Strange Aattractors from a Fluctuating Dynamics Approach. Physica D 13(1-2), 34–54 (1984)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hentschel, H., Procaccia, I.: The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors. Physica D 8D, 435–444 (1983)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Kinsner, W.: Fractal: Chaos Engineering Course Notes. Winnipeg, MB: Dept. Electrical & Computer Eng., University of Manitoba (2003)Google Scholar
  9. 9.
    Mallat, S.G., Hwang, W.L.: Singularity Detection and Processing with Wavelets. IEEE Trans. Infor. Theory 38, 617–643 (1992)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, Chestnut Hill (2001)Google Scholar
  11. 11.
    Mandelbrot, B.B., Fractals, Multifractals.: Noise, Turbulence and Galaxies. Springer, New York (1989)Google Scholar
  12. 12.
    Muzy, J.F., Bacry, E., Arneodo, A.: Wavelets and Multifractal Formalism for Singular Signals: Application to turbulence data. Phys. Rev. Lett. 67(25), 3515–3518 (1991)CrossRefGoogle Scholar
  13. 13.
    Muzy, J.F., Bacry, E., Arneodo, A.: Multifractal Formalism for Fractal Signals: The Structure Function Approach Versus the Wavelet-transform Modulus-maxima Method. Phys. Rev. E 47(2), 875–884 (1993)CrossRefGoogle Scholar
  14. 14.
    Muzy, J.F., Bacry, E., Arneodo, A.: The Multifractal Formalism Revisited with Wavelets. Int. Jrnl. Bif. Chaos 4(2), 245–302 (1994)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Oppenheim, A.V., Schafer, R.W., Buck, J.R.: Discrete-Time Signal Processing, 2nd edn. Prentice Hall, Englewood Cliffs (1999)Google Scholar
  16. 16.
    Proakis, J.G., Manolakis, D.G.: Digital Signal Processing: Principles, Algorithms and Applications, 2nd edn. Macmillan, New York (1996)Google Scholar
  17. 17.
    Van den Berg, J.: Wavelets in physics, 2nd edn. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Xiong Gang
    • 1
    • 2
  • Yang Xiaoniu
    • 1
  • Zhao Huichang
    • 2
  1. 1.NO.36 Research Institute of CETC, National Laboratory of Information Control Technology For Communication System, JiaxingZhe-JiangChina
  2. 2.Electronic engineering dept.NJUSTNanjingChina

Personalised recommendations