Advertisement

Journal of Theoretical Probability

, Volume 32, Issue 1, pp 202–215 | Cite as

Optimal Scale Invariant Wigner Spectrum Estimation of Gaussian Locally Self-Similar Processes Using Hermite Functions

  • Yasaman MalekiEmail author
Article
  • 27 Downloads

Abstract

This paper investigates the mean square error optimal estimation of scale invariant Wigner spectrum for the class of Gaussian locally self-similar processes, by the multitaper method. In this method, the spectrum is estimated as a weighted sum of scale invariant windowed spectrograms. Moreover, it is shown that the optimal multitapers are approximated by the quasi Lamperti transformation of Hermite functions, which is computationally more efficient. Finally, the performance and accuracy of the estimation is studied via simulation.

Keywords

Locally self-similar processes Scale invariant Wigner spectrum Multitaper method Hermite functions Time–frequency analysis 

Mathematics Subject Classification (2010)

60G18 60G99 

References

  1. 1.
    Martin, W.: Time-frequency analysis of random signals. In Proceedings of the ICASSP 82, vol. 3, pp. 1325-1328 (1982)Google Scholar
  2. 2.
    Bayram, M., Baraniuk, R.G.: Multiple window time-frequency analysis. In: IEEE International Symposium Time-Frequency and Time-Scale Analysis, Paris, France, pp. 173–176 (1996)Google Scholar
  3. 3.
    Sayeed, A.M., Jones, D.L.: Optimal kernels for nonstationary spectral estimation. IEEE Trans. Signal Process. 43, 478–491 (1995)CrossRefGoogle Scholar
  4. 4.
    Wahlberg, P., Hansson, M.: Kernels and multiple windows for estimation of the Wigner–Ville spectrum of gaussian locally stationary processes. IEEE Trans. Signal Process. 55(1), 73–84 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hansson-Sandsten, M.: Optimal multitaper wigner spectrum estimation of a class of locally stationary processes using Hermite functions. EURASIP (2011).  https://doi.org/10.1155/2011/980805 zbMATHGoogle Scholar
  6. 6.
    Flandrin, P., Borgnat, P., Amblard, P.O.: From stationarity to self-similarity and back: variations on the Lamperti transformation. Lecture Notes on Physics 621, pp. 88–117. Springer (2003)Google Scholar
  7. 7.
    Silverman, R.A.: Locally stationary random processes. IRE Trans. Inf. Theory 3(3), 182–187 (1957)CrossRefGoogle Scholar
  8. 8.
    Flandrin, P.: Scale-invariant wigner spectra and self-similarity. In: Torres, L., et al. (eds.) Signal Processing v: Theories and Applications, pp. 149–152. Elsevier, Amsterdam (1990)Google Scholar
  9. 9.
    Borgnat, P., Amblard, P.O., Flandrin, P.: Scale invariances and lamperti transformations for stochastic processes. J. Phys. A 38(10), 2081–2101 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Maleki, Y., Rezakhah, S.: The scale invariant wigner spectrum estimation of Gaussian locally self-similar processes. Commun. Stat. Theory Methods (2012).  https://doi.org/10.1080/03610926.2012.746987
  11. 11.
    Feichtinger, H.G., Zimmermann, G.: A Banach space of test functions for Gabor analysis. In: Feichtinger, H.G., Strohmer, T. (eds.) Gabor Analysis and Algorithms Theory and Applications, pp. 123–170. Birkhuser, Cambridge (1998)Google Scholar
  12. 12.
    Feichtinger, H.G.: On a new Segal algebra. Monatshefte fr Mathematik 92, 269–289 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wahlberg, P.: The random Wigner distribution of Gaussian stochastic processes with covariance in \(S_0({\cal{R}}^{2d})\). J. Funct. Spaces Appl. 3(2), 163–181 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wahlberg, P.: Regularization of kernels for estimation of the Wigner spectrum of Gaussian stochastic processes. Probab. Math. Stat. 30(2), 369–381 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wahlberg, P.: The Wigner distribution of Gaussian weakly harmonizable stochastic processes. In: Advances and Applications, Birkhuser, Proceedings of the International Conference on Pseudo-Differential Operators and Related Topics Series Operator Theory (2006)Google Scholar
  16. 16.
    Loeve, M.: Probability Theory, 3rd edn. D. Van Nostrand Co., London (1963)Google Scholar
  17. 17.
    Picinbono, B., Bondon, P.: Second-order statistics of complex signals. IEEE Trans. Signal Process. 45(2), 411–420 (1997)CrossRefGoogle Scholar
  18. 18.
    Schreier, P.J., Scharf, L.L.: Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters. IEEE Trans. Signal Process. 51(12), 3071–3079 (2003).  https://doi.org/10.1109/TSP.2003.818911 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zagier, D.: Personal Communication (2014)Google Scholar
  20. 20.
    Cohen, L.: Time-frequency distributions—a review. Proc. IEEE 77, 941–981 (1989)CrossRefGoogle Scholar
  21. 21.
    Thomson, D.J.: Spectrum estimation and harmonic analysis. Proc. IEEE 70(9), 1055–1096 (1982)CrossRefGoogle Scholar
  22. 22.
    Reed, B., Simon, M.: Methods of Modern Mathematical Physics I. Wiley, New York (1975)zbMATHGoogle Scholar
  23. 23.
    Modarresi, N., Rezakhah, S.: Spectral analysis of multi-dimensional self-similar Markov processes. J. Phys. A Math. Theor. 43(12), 125004 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Alzahra UniversityTehranIslamic Republic of Iran

Personalised recommendations