Advertisement

Telecommunication Systems

, Volume 43, Issue 3–4, pp 207–217 | Cite as

Harmonic wavelet approximation of random, fractal and high frequency signals

  • Carlo Cattani
Article

Abstract

The analysis of a periodic signal with localized random (or high frequency) noise is given by using harmonic wavelets. Since they are orthogonal to the Fourier basis, by defining a projection wavelet operator the signal is automatically decomposed into the localized pulse and the periodic function. An application to the analysis of a self-similar non-stationary noise is also given.

Keywords

Harmonic wavelets Signal analysis Denoising Random Scale Self-similar Discrete Fourier series 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abry, P., Goncalves, P., & Lévy-Véhel, J. (2002). Lois d’éschelle, Fractales et ondelettes. Hermes. Google Scholar
  2. 2.
    Aubry, J.-M., & Jaffard, S. (2002). Random wavelet series. Communications Mathematical Physics, 227, 483–514. CrossRefGoogle Scholar
  3. 3.
    Borgnat, P., & Flandrin, P. (2003). On the chirp decomposition of Weierstrass-Mandelbrot functions, and their time-frequency interpretation. Applied and Computational Harmonic Analysis, 15, 134–146. CrossRefGoogle Scholar
  4. 4.
    Cattani, C. (2005). Harmonic wavelets towards solution of nonlinear PDE. Computers and Mathematics with Applications, 50(8–9), 1191–1210. CrossRefGoogle Scholar
  5. 5.
    Cattani, C. (2008). Wavelet extraction of a pulse from a periodic signal. In M. Gavrilova et al. (Eds.), LNCS. Proceedings of the international conference on computational science and its applications (ICCSA 2008) (pp. 1202–1211). Perugia (It), June 30–July 3, 2008. Berlin: Springer. CrossRefGoogle Scholar
  6. 6.
    Cattani, C. (2008). Shannon wavelets theory. Mathematical Problems in Engineering, 2008, 24. doi: 10.1155/2008/164808. Article ID 164808. Google Scholar
  7. 7.
    Cattani, C. (2009). Fractals based on harmonic wavelets. In O. Gervasi et al. (Eds.), LNCS : Vol. 5592. Proceedings of the international conference on computational science and its applications (ICCSA 2009) (pp. 729–744). Seoul (Kr), June 29–July 2, 2009. Berlin: Springer. CrossRefGoogle Scholar
  8. 8.
    Cattani, C., & Rushchitsky, J. J. (2007). Wavelet and wave analysis as applied to materials with micro or nanostructure. Series on advances in mathematics for applied sciences. Singapore: World Scientific. Google Scholar
  9. 9.
    Chui, C. K. (1992). An introduction to wavelets. New York: Academic Press. Google Scholar
  10. 10.
    Coifman, R., & Meyer, Y. (1991). Remarques sur l’analyse de Fourier á fenetre. Comptes Rendus de l’Académie des Sciences de Paris, 312, 259–261. Google Scholar
  11. 11.
    Daubechies, I. (1992). Ten lectures on wavelets. Philadelphia: SIAM. Google Scholar
  12. 12.
    Härdle, W., Kerkyacharian, G., Picard, D., & Tsybakov, A. (1998). Lecture notes in statistics : Vol. 129. Wavelets, approximation, and statistical applications. Berlin: Springer. Google Scholar
  13. 13.
    Lepik, Ü. (2003). Exploring irregular vibrations and chaos by the wavelet method. Proceedings of the Estonian Academy on Sciences Engineering, 9, 109–135. Google Scholar
  14. 14.
    Li, M., Zhao, W., Long, D., & Chi, Ch. (2003). Modeling autocorrelation functions of self-similar teletraffic in communication networks based on optimal approximation in Hilbert space. Applied Mathematical Modelling, 27(3), 155–168. CrossRefGoogle Scholar
  15. 15.
    Mallat, S. (1998). A wavelet tour of signal processing. San Diego: Academic Press. Google Scholar
  16. 16.
    Meyer, Y. (1990). Ondelettes et opérateurs. Paris: Hermann. Google Scholar
  17. 17.
    Mouri, H., & Kubotani, H. (1995). Real-valued harmonic wavelets. Physics Letters A, 201, 53–60. CrossRefGoogle Scholar
  18. 18.
    Muniandy, S. V., & Moroz, I. M. (1997). Galerkin modelling of the Burgers equation using harmonic wavelets. Physics Letters A, 235, 352–356. CrossRefGoogle Scholar
  19. 19.
    Newland, D. E. (1993). Harmonic wavelet analysis. Proceedings of the Royal Society of London, Series A, 443, 203–222. CrossRefGoogle Scholar
  20. 20.
    Percival, D. B., & Walden, A. T. (2000). Wavelet methods for time series analysis. Cambridge: Cambridge University Press. Google Scholar
  21. 21.
    Weierstrass, K. (1967) Über continuirliche Functionen eines reelles Arguments, die für keinen Werth des letzteren einen Bestimmten Differentialquotienten besitzen. In König. Akad. der Wissenschaften, Berlin, July 18, 1872; Reprinted in: K. Weierstrass, Mathematische Werke II (pp. 71–74). New York: Johnson. Google Scholar
  22. 22.
    Wojtaszczyk, P. A. (1997). London mathematical society student texts : Vol. 37. A mathematical introduction to wavelets. Cambridge: Cambridge University Press. 2nd ed. 2003. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.DiFarmaUniversity of SalernoFiscianoItaly

Personalised recommendations