Distributed-Order Fractional Signal Processing

Part of the Signals and Communication Technology book series (SCT)

Abstract

Chapter 7 studies the impulse response functions of the distributed-order integrator/differentiator, fractional-order distributed low-pass filter, and the fractional-order distributed parameter low-pass filter from the complex path integral expressed in the definite integral form. Based on these results, we obtained some asymptotic properties, and we can accurately compute the integrals on the whole time domain. Moreover, for practical applications, we presented a technique known as “impulse-response-invariant discretization” to perform the discretization of the above three distributed-order filters. Lastly, it was shown that the distributed-order fractional filters had some unique features compared with the classical integer-order or constant-order fractional filters.

References

  1. 2.
    Adams, J.L., Hartley, T.T., Lorenzo, C.F.: Identification of complex order-distributions. J. Vib. Control 14(9–10), 1375–1388 (2008) MathSciNetCrossRefGoogle Scholar
  2. 9.
    Atanackovic, T.M., Budincevic, M., Pilipovic, S.: On a fractional distributed-order oscillator. J. Phys. A, Math. Gen. 38(30), 6703–6713 (2005) MathSciNetMATHCrossRefGoogle Scholar
  3. 10.
    Atanackovic, T.M., Pilipovic, S., Zorica, D.: Existence and calculation of the solution to the time distributed order diffusion equation. Phys. Scr. T136, 014012 (2009) (6 pp.) CrossRefGoogle Scholar
  4. 11.
    Atanackovic, T.M., Oparnica, L., Pilipovic, S.: On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328(1), 590–608 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 12.
    Atanackovic, T.M., Pilipovic, S., Zorica, D.: Time distributed-order diffusion-wave equation. I. Volterra-type equation. Proc. R. Soc. Lond. Ser. A 465, 1869–1891 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 13.
    Atanackovic, T.M., Pilipovic, S., Zorica, D.: Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations. Proc. R. Soc. Lond. Ser. A 465, 1893–1917 (2009) MathSciNetMATHCrossRefGoogle Scholar
  7. 15.
    Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations (Parts I, II). Int. J. Appl. Math. 2, 865–882, 965–987 (2000) Google Scholar
  8. 17.
    Barbosa, R.S., Machado, J.A.T.: Implementation of discrete-time fractional-order controllers based on LS approximations. Acta Polytech. Hung. 3(4), 5–22 (2006) Google Scholar
  9. 26.
    Bohannan, G.: Application of fractional calculus to polarization dynamics in solid dielectric materials. Ph.D. Dissertation, Montana State University (November 2000) Google Scholar
  10. 45.
    Caputo, M.: Elasticità e Dissipazione. Zanichelli, Bologna (1969) Google Scholar
  11. 46.
    Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara 41(1), 73–84 (1995) MathSciNetMATHGoogle Scholar
  12. 47.
    Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4(4), 421–442 (2001) MathSciNetMATHGoogle Scholar
  13. 48.
    Carlson, G., Halijak, C.: Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process. IEEE Trans. Circuit Theory 11(2), 210–213 (1964) Google Scholar
  14. 53.
    Chen, W., Sun, H., Zhang, X., Korosak, D.: Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59(5), 1754–1758 (2009) MathSciNetCrossRefGoogle Scholar
  15. 54.
    Chen, Y.Q.: Low-pass IIR digital differentiator design. http://www.mathworks.com/matlabcentral/fileexchange/3517 (2003)
  16. 55.
    Chen, Y.Q.: A new IIR-type digital fractional order differentiator. http://www.mathworks.com/matlabcentral/fileexchange/3518 (2003)
  17. 57.
    Chen, Y.Q.: Impulse response invariant discretization of fractional order integrators or differentiators. http://www.mathworks.com/matlabcentral/fileexchange/21342 (2008)
  18. 58.
    Chen, Y.Q.: Impulse response invariant discretization of fractional order low-pass filters. http://www.mathworks.com/matlabcentral/fileexchange/21365 (2008)
  19. 59.
    Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(3), 363–367 (2002) MathSciNetCrossRefGoogle Scholar
  20. 62.
    Chen, Y.Q., Vinagre, B.M.: A new IIR-type digital fractional order differentiator. Signal Process. 83(11), 2359–2365 (2003) MATHCrossRefGoogle Scholar
  21. 63.
    Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dyn. 38(16), 155–170 (2004) MathSciNetMATHCrossRefGoogle Scholar
  22. 69.
    Connolly, J.A.: The numerical solution of fractional and distributed order differential equations. Thesis, University of Liverpool (December 2004) Google Scholar
  23. 81.
    Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225(1), 96–104 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 95.
    Ferdi, Y.: Impulse invariance-based method for the computation of fractional integral of order 0<α<1. Comput. Electr. Eng. 35(5), 722–729 (2009) MATHCrossRefGoogle Scholar
  25. 111.
    Hartley, T.T., Lorenzo, C.F.: Fractional-order system identification based on continuous order-distributions. Signal Process. 83(11), 2287–2300 (2003) MATHCrossRefGoogle Scholar
  26. 141.
    Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340(1), 252–281 (2008) MathSciNetMATHCrossRefGoogle Scholar
  27. 167.
    Li, Y., Sheng, H., Chen, Y.Q.: Impulse response invariant discretization of distributed order low-pass filter. http://www.mathworks.com/matlabcentral/fileexchange/authors/82211 (2010)
  28. 168.
    Li, Y., Sheng, H., Chen, Y.Q.: On distributed order integrator/differentiator. Signal Process. 91(5), 1079–1084 (2010) CrossRefGoogle Scholar
  29. 171.
    Li, Y., Sheng, H., Chen, Y.Q.: Analytical impulse response of a fractional second order filter and its impulse response invariant discretization. Signal Process. 91(3), 498–507 (2011) MATHCrossRefGoogle Scholar
  30. 179.
    Lorenzo, C.F., Hartley, T.T.: Initialization, conceptualization, and application in the generalized fractional calculus. NASA technical paper, NASA/TP 1998-208415 (1998) Google Scholar
  31. 180.
    Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002) MathSciNetMATHCrossRefGoogle Scholar
  32. 182.
    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986) MathSciNetMATHCrossRefGoogle Scholar
  33. 185.
    Machado, J.A.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27(2–3), 107–122 (1997) MATHGoogle Scholar
  34. 189.
    Mainardi, F., Mura, A., Gorenflo, R., Stojanovic, M.: The two forms of fractional relaxation of distributed order. J. Vib. Control 9, 1249–1268 (2007) MathSciNetCrossRefGoogle Scholar
  35. 190.
    Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Time-fractional diffusion of distributed order. http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0701132 (2007)
  36. 191.
    Mainardi, F., Pagnini, G.: The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. J. Comput. Appl. Math. 207(2), 245–257 (2007) MathSciNetMATHCrossRefGoogle Scholar
  37. 225.
    Oustaloup, A.: Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulation. IEEE Trans. Circuits Syst. 28(10), 1007–1009 (1981) CrossRefGoogle Scholar
  38. 227.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47(1), 25–39 (2000) CrossRefGoogle Scholar
  39. 241.
    Radwan, A.G., Elwakil, A.S., Soliman, A.M.: Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits Syst. I, Regul. Pap. 55(7), 2051–2063 (2008) MathSciNetCrossRefGoogle Scholar
  40. 242.
    Radwan, A.G., Soliman, A.M., Elwakil, A.S.: Design equations for fractional-order sinusoidal oscillators: practical circuit examples. In: Proceedings of the International Conference on Microelectronics, pp. 89–92, December 2007 Google Scholar
  41. 265.
    Sheng, H.: Impulse Response Invariant Discretization of Distributed Order Integrator. http://www.mathworks.com/matlabcentral/fileexchange/26380 (2010) Google Scholar
  42. 273.
    Signal Processing Toolbox 6.12: http://www.mathworks.com/products/signal
  43. 275.
    Smith, J.O.: Physical Audio Signal Processing (2008). http://ccrma.stanford.edu/jos/pasp/ Online book Google Scholar
  44. 276.
    Sokolov, I.M., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0401146 (2004)
  45. 280.
    Srokowski, T.: Lévy flights in nonhomogeneous media: distributed-order fractional equation approach. Phys. Rev. E 78(3), 031135 (2008) CrossRefGoogle Scholar
  46. 283.
    Steiglitz, K., McBride, L.: A technique for the identification of linear systems. IEEE Trans. Autom. Control 10(4), 461–464 (1965) CrossRefGoogle Scholar
  47. 290.
    Sun, H., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A, Stat. Mech. Appl. 388(21), 4586–4592 (2009) CrossRefGoogle Scholar
  48. 291.
    Sun, H., Chen, W., Sheng, H., Chen, Y.Q.: On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys. Lett. A 374(7), 906–910 (2010) CrossRefGoogle Scholar
  49. 303.
    Tsao, Y.Y.: Fractal concepts in the analysis of dispersion or relaxation processes. Ph.D. Dissertation, Drexel University (June 1987) Google Scholar
  50. 307.
    Umarov, S., Steinberg, S.: Random walk models associated with distributed fractional order differential equations. In: IMS Lecture Notes Monogr. Ser., vol. 51, pp. 117–127 (2006) Google Scholar
  51. 311.
    Vinagre, B.M., Chen, Y.Q., Petras, I.: Two direct Tustin discretization methods for fractional-order differentiator/integrator. J. Franklin Inst. 340(5), 349–362 (2003) MathSciNetMATHCrossRefGoogle Scholar
  52. 325.
    Xu, M., Tan, W.: Intermediate processes and critical phenomena: theory, method and progress of fractional operators and their applications to modern mechanics. Sci. China Ser. G, Phys. Astron. 49(3), 257–272 (2006) MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.School of Electronic and Information EngineeringDalian Jiaotong UniversityDalianPeople’s Republic of China
  2. 2.Department of Electrical and Computer Engineering, CSOISUtah State UniversityLoganUSA
  3. 3.School of Electronic and Information EngineeringDalian University of TechnologyDalianPeople’s Republic of China

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