A Procedure for Efficient Generation of 1/fβ Noise Sequences

  • Youcef Ferdi
  • Abdelmalik Taleb-Ahmed
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5099)

Abstract

This paper presents a simple, efficient and fast procedure for generation of 1/f β noise sequences. The proposed procedure is based on the impulse invariance method applied to the impulse response of the ideal fractional order integrator whose order α = β/2is between 0 and 1. First, an optimal value for the initial value of the impulse response is obtained by minimizing a least squares error criterion and then any of the well-established signal modeling techniques can be employed for the parameterization of the discrete impulse response by pole-zero models. For a given model order, the approximation accuracy depends on the signal modeling technique used. An illustrative example is presented to demonstrate the effectiveness of the method.

Keywords

1/f-noise fractional order integration impulse invariance method pole-zero model signal modeling 

References

  1. 1.
    Keshner, M.S.: 1/f Noise. Proceedings of the IEEE 70(3), 212–218 (1982)CrossRefGoogle Scholar
  2. 2.
    Wornell, G.W.: Signal processing with fractals: A wavelet-based approach. Prentice-Hall, Englewood Cliffs (1995)Google Scholar
  3. 3.
    Flandrin, P.: Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inform. Theory 38(2), 910–917 (1992)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Perrin, E., Harba, R., Jennane, R.: Fast and exact synthesis of 1D fractional Brownian motion. IEEE Signal Proc. Letters 9(11), 382–384 (2002)CrossRefGoogle Scholar
  5. 5.
    Guglielmi, M.: 1/F α signal synthesis with precision control. Signal Processing 86, 2548–2553 (2006)MATHCrossRefGoogle Scholar
  6. 6.
    Rodriguez, E., Echeverria, J.C., Ramirez, J.A.: 1/F α fractal signal generation from Grünwald-Letnikov formula. Chaos, Solitons and Fractals (in press, 2008)Google Scholar
  7. 7.
    Proakis, J.G., Manolakis, D.G.: Digital Signal Processing. Principles, Algorithms, and Applications. Prentice-Hall international, Englewood Cliffs (1996)Google Scholar
  8. 8.
    Steiglitz, K., McBride, L.E.: A Technique for the Identification of Linear Systems. IEEE Transactions on Automatic Control AC-10, 461–464 (1965)CrossRefGoogle Scholar
  9. 9.
    Ferdi, Y.: Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modeling. Nonlinear Dynamics 46(1-2), 1–15 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review Nonlinear Dynamics 38(1-2), 155-170 (2004)Google Scholar
  11. 11.
    Lee, S., Zhao, W., Narasimha, R., Rao, R.M.: Discrete-time models for statistically self-similar signals. IEEE Transactions on Signal Processing 51(5), 1221–1230 (2003)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)MATHGoogle Scholar
  13. 13.
    Oppenheim, A.V., Schafer, R.W.: Digital Signal Processing, pp. 198–203. Prentice-Hall, Englewood Cliffs (1975)MATHGoogle Scholar
  14. 14.
    Al-Alaoui, M.A.: Novel Digital Integrator and Differentiator. Electronics letters 29(4), 376–378 (1993)CrossRefGoogle Scholar
  15. 15.
    Tseng, C.-C.: Digital Integrator Design Using Simpson Rule and Fractional Delay Filter. IEE Proc.-VIS, Image Signal Process 153(1), 79–85 (2006)CrossRefGoogle Scholar
  16. 16.
    Steiglitz, K.: Computer-aided design, of recursive digital filters. IEEE Transactions on Audio and Electroacoustics AU-18, 123–129 (1970)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Youcef Ferdi
    • 1
  • Abdelmalik Taleb-Ahmed
    • 2
  1. 1.LRES Laboratory, Department of ElectrotechnicsUniversity of SkikdaSkikdaAlgeria
  2. 2.LAMIH UMR CNRS 8530 LaboratoryUniversity of Valenciennes and Hainaut CambresisValenciennesFrance

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