Comparison of Different Methods for Digital Fractional-Order Differentiator and Integrator Design

Abstract

The main goal of this study is comparative analysis of different methods used in design of digital fractional-order differentiator and integrator. The fractional-order digital differentiator or integrator can be described (in continuous time domain) with a transfer function H(s)=s^ɑ, where ɑ is a real number. To implement digital differentiators and integrators of arbitrary order the main step is the discretization. There are two common approaches of discretization. In this paper the direct and indirect discretization are presented but the emphasis will be on the indirect method, where the generating functions can be obtained through bilinear transformation, Al-Alaoui operator, Euler's backward operator and stable Simspon operator. The main differences between alternatives will be provided through analysis and comparison of their frequency responses - magnitude-frequency response, phase-frequency response and Nyquist diagrams.