Abstract
The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Padé, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Padé method which is equivalent to continued fraction expansion technique.
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Ferdi, Y. Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling. Nonlinear Dyn 46, 1–15 (2006). https://doi.org/10.1007/s11071-005-9000-1
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DOI: https://doi.org/10.1007/s11071-005-9000-1