Circuits, Systems, and Signal Processing

, Volume 32, Issue 4, pp 1875–1889 | Cite as

Closed-Form Analytical Expression of Fractional Order Differentiation in Fractional Fourier Transform Domain

  • Sanjay Kumar
  • Kulbir Singh
  • Rajiv Saxena


In this paper, a closed-form analytical expression for fractional order differentiation in the fractional Fourier transform (FrFT) domain is derived by utilizing the basic principles of fractional order calculus. The reported work is a generalization of the differentiation property to fractional (noninteger or real) orders in the FrFT domain. The proposed closed-form analytical expression is derived in terms of the well-known confluent hypergeometric function. An efficient computation method has also been derived for the proposed algorithm in the discrete-time domain, utilizing the principles of the discrete fractional Fourier transform algorithm. An application example of a low-pass finite impulse response (FIR) fractional order differentiator in the FrFT domain has also been investigated to show the practicality of the proposed method in signal processing applications.


Fractional order calculus Fractional order differentiation Fractional Fourier transform Grünwald–Letnikov fractional derivative Kummer confluent hypergeometric function Riemann–Liouville fractional derivative 



The authors express their sincere thanks to the Editor-in-Chief and to the learned reviewers for their valuable comments and suggestions in developing this article into its present form.


  1. 1.
    L.B. Almeida, The fractional Fourier transform and time–frequency representation. IEEE Trans. Signal Process. 42(11), 3084–3093 (1994) CrossRefGoogle Scholar
  2. 2.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55 (US Government Printing Office, Washington, DC, 1964) MATHGoogle Scholar
  3. 3.
    N.K. Bose, C.R. Rao, Digital differentiators, in Handbook of Statistics 10: Signal Processing and Its Applications (Elsevier, Amsterdam, 1993). Chapter 6 Google Scholar
  4. 4.
    S. Kumar, K. Singh, R. Saxena, Analysis of Dirichlet and generalized “Hamming” window functions in the fractional Fourier transform domains. Signal Process. 91(3), 600–606 (2011) MATHCrossRefGoogle Scholar
  5. 5.
    J.C. Lin, Edge detection for image processing using second directional derivative, in IEEE/IAS Conference on Industrial Automation and Control: Emerging Technologies (1995), pp. 669–672 Google Scholar
  6. 6.
    G. Maione, A. Digital, Noninteger order, differentiator using Laguerre orthogonal sequences. Int. J. Intell. Control Syst. 11(2), 77–81 (2006) Google Scholar
  7. 7.
    A.C. McBride, F.H. Kerr, On Namias’ fractional Fourier transforms. IMA J. Appl. Math. 39(2), 159–175 (1987) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    D. Middleton, An Introduction to Statistical Communication Theory (IEEE Press, Piscataway, 1996) MATHGoogle Scholar
  9. 9.
    V. Namias, The fractional order Fourier transform and its applications to quantum mechanics. J. Inst. Math. Appl. 25(3), 241–265 (1980) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974) MATHGoogle Scholar
  11. 11.
    A. Oustaloup, F. Levron, B. Mathieu, F.M. Nanot, Frequency–band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47(1), 25–39 (2000) CrossRefGoogle Scholar
  12. 12.
    S.C. Pei, J.J. Ding, Closed-form discrete fractional and affine Fourier transforms. IEEE Trans. Signal Process. 48(5), 1338–1353 (2000) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    M.V.N.V. Prasad, K.C. Ray, A.S. Dhar, FPGA implementation of discrete fractional Fourier transform, in International Conference on Signal Processing and Communications (SPCOM), 18–21 Jul. (2010), pp. 1–5 Google Scholar
  14. 14.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999) MATHGoogle Scholar
  15. 15.
    A.K. Singh, R. Saxena, On convolution and product theorems for FrFT. Wirel. Pers. Commun. 65(1), 189–201 (2012) MathSciNetCrossRefGoogle Scholar
  16. 16.
    S.N. Sharma, R. Saxena, S.C. Saxena, Tuning of FIR filter transition bandwidth using fractional Fourier transform. Signal Process. 87(12), 3147–3154 (2007) MATHCrossRefGoogle Scholar
  17. 17.
    M.I. Skolnik, Introduction to Radar Systems (McGraw–Hill, New York, 1980) Google Scholar
  18. 18.
    C.C. Tseng, S.C. Pei, S.C. Hsia, Computation of fractional derivatives using Fourier transform and digital FIR differentiator. Signal Process. 80(1), 151–159 (2000) MATHCrossRefGoogle Scholar
  19. 19.
    C.C. Tseng, Design of fractional order digital FIR differentiators. IEEE Signal Process. Lett. 8(3), 77–79 (2001) CrossRefGoogle Scholar
  20. 20.
    S. Usui, I. Amidror, Digital low-pass differentiation for biological signal processing. IEEE Trans. Biomed. Eng. 29(10), 686–693 (1982) CrossRefGoogle Scholar
  21. 21.
    A.I. Zayed, A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Electronics and Communication EngineeringThapar UniversityPatialaIndia
  2. 2.Department of Electronics and Communication EngineeringJaypee University of Engineering and TechnologyRaghogarh, GunaIndia

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