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 differentiator is investigated. First, the Cauchy integral formula is generalized to define the fractional derivative of functions. Then the fractional differentiation 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 differentiator. When a signal passes through the designed differentiator, the output will be its fractional derivative. One design example is included to illustrate the effectiveness of this approach. Finally, the designed fractional order differentiator is used to generate a random fractal process which is better than the process obtained by the conventional method.