Design of Non-integer Order Digital Differentiator

Abstract

The design of digital filters at low frequencies range has become increasingly important as it can be used to design all types of filters. Use of Integer order calculus for the purpose of design often results in narrow bandwidth for the low pass. With the development of fractional order calculus in recent years the response becomes more ideal. But the drawback of design is that due to multiple variables, the response is bit slow. The major objective of this work is to understand the different design strategy for digital differentiators and compare their response for various orders and to explore new design techniques for designing fractional order IIR differentiator. A stable minimum phase, second-order, low-pass IIR digital differentiators is developed by inverting the transfer functions of a class of second-order integrators, stabilizing the resulting transfer functions, and compensating their magnitudes. The class of second-order integrators are obtained by interpolating the traditional Simpson and trapezoidal integrators. The designed digital differentiator is modelled to find the correct response by passing some test signal. The designed differentiators extend the frequency range of operation beyond that possible by using either of the two traditional integrators. The low order and high accuracy of the filters make them attractive for real time applications. Al-Alaoui operator and new mapping function (NMF)) models are used for the design of fractional order digital differentiator. The proposed low-pass differentiators are shown to have shorter transition regions, and thus better ability to suppress high frequency noise, for much lower order filters.