Computation of fractional derivatives using Fourier transform and digital FIR differentiator

In this paper, the computation of a fractional derivative using the Fourier transform and a digital FIR di!erentiator is investigated. First, the Cauchy integral formula is generalized to de"ne the fractional derivative of functions. Then the fractional di!erentiation property of the Fourier transform of functions is presented. Using this property, the fractional derivative of a function can be computed in the frequency domain. Next, we develop a least-squares method to design the fractional order digital di!erentiator. When a signal passes through the designed di!erentiator, the output will be its fractional derivative. One design example is included to illustrate the e!ectiveness of this approach. Finally, the designed fractional order di!erentiator is used to generate a random fractal process which is better than the process obtained by the conventional method. ( 2000 Elsevier Science B.V. All rights reserved.