Discretization of Fractional Order Differentiator over Paley-Wiener Space

The Paley-Wiener space consists of functions whose Fourier transform is compactly supported in the frequency domain. In the context of signal processing, such functions are also known as bandlimited signals, which represent a large class of signals in signal processing. An analog fractional order differentiator is representable by way of the Cauchy integral formula, special functions, as well as the Fourier/Laplace transformer, while a digital differentiator of fractional order can be obtained through direct or indirect discretization techniques. In this paper, we present the design of a finite impulse response (FIR) filter that discretizes the fractional order differentiator over functions in Paley-Wiener space. The proposed FIR model has some meritorious properties that are preferred in applications: the filter coefficients are independent of the signal samples; it is capable of interpolating or extrapolating at an arbitrary point in the sampling domain; it is adaptive to uniform or non-uniform sampling scenarios. We present explicit formulas on the matrices that lead to computing the filter coefficients. A closed form formula on the error of approximation is derived to demonstrate the accuracy of the proposed discretization model of fractional derivatives. Numerical results also show that the proposed method is more computationally efficient than the well-known methods such as the Grünwald-Letnikov approximation.