Advertisement

The Fractional Quantum Derivative and the Fractional Linear Scale Invariant Systems

  • Manuel Duarte OrtigueiraEmail author
Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 84)

Abstract

The normal way of introducing the notion of derivative is by means of the limit of an incremental ratio that can assume three forms, depending the used translations as we saw in Chaps. 1 and 4. On the other hand, in those derivatives the limit operation is done over a set of points uniformly spaced: a linear scale was used. Here we present an alternative derivative, that is valid only for t > 0 or t < 0 and uses an exponential scale

Keywords

Impulse Response Fractional Derivative Fractional Order System Integer Order Incremental Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Kac V, Cheung P (2002) Quantum calculus. Springer, New YorkCrossRefzbMATHGoogle Scholar
  2. 2.
    Ash JM, Catoiu S, Rios-Collantes-de-Terán R (2002) On the nth quantum derivative. J Lond Math Soc 2(66):114–130CrossRefGoogle Scholar
  3. 3.
    Bertrand J, Bertrand P, Ovarlez JP (2000) The mellin transform. In: Poularikas AD (ed) The Transforms and Applications Handbook, 2nd edn. CRC Press, Boca RatonGoogle Scholar
  4. 4.
    Poularikas AD (ed) (2000) The transforms and applications handbook. CRC Press, Boca RatonGoogle Scholar
  5. 5.
    Koornwinder TH (1999) Some simple applications and variants of the q-binomial formula. Informal note, Universiteit van AmsterdamGoogle Scholar
  6. 6.
    Al-Salam W (1966) Some fractional q-integrals and q-derivatives. Proc Edin Math Soc 15:135–140MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Henrici P (1991) Applied and computational complex analysis, vol 2. Wiley, New York, pp 389–391zbMATHGoogle Scholar
  8. 8.
    Braccini C, Gambardella G (1986) Form-invariant linear filtering: theory and applications. IEEE Trans Acoust Speech Signal Process ASSP-34(6):1612–1628CrossRefGoogle Scholar
  9. 9.
    Yazici B, Kashyap RL (1997) Affine stationary processes with applications to fractional Brownian motion. In: Proceedings of 1997 International Conference on Acoustics, Speech, and Signal Processing, vol 5. Munich, Germany, pp 3669–3672, April 1997Google Scholar
  10. 10.
    Yazici B, Kashyap RL (1997) A class of second-order stationary self-similar processes for 1/f phenomena. IEEE Trans Signal Process 45(2):396–410CrossRefGoogle Scholar
  11. 11.
    Abramowitz M, Stegun I (1972) Stirling Numbers of the First Kind. Sect. 24.1.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York, p 824Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Faculdade de Ciências/Tecnologia da UNLUNINOVA and DEECaparicaPortugal

Personalised recommendations