Linear Lhase Low-Pass IIR Fractional Order Digital Differentiator

Abstract

The design of digital filters at low frequency range has become increasingly important as it can be used to design all types of filters. It is found in many applications from low frequency biomedical equipment to high frequency radars. 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. 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, low-pass IIR digital differentiators is developed by inverting the transfer functions of a class of numerical integrators, stabilizing the resulting transfer functions and compensating their magnitudes. The class of digital integrators is first obtained by interpolating the various numerical integrators. The designed digital differentiator is modelled to find the correct response by passing some test signal. The low order and high accuracy of the filters make them attractive for real time applications. A method for optimizing low-pass infinite impulse response (IIR) digital differentiators is presented in this dissertation. The wide band differentiator is cascaded with low-pass IIR filter resulting in linear phase low-pass IIR digital differentiator. Further the frequency response of IIR differentiator is improved by altering the gain and denominator coefficients of that differentiator. The genetic algorithm approach is used for optimizing the least square error that is defined by the fitness function. The low order optimized IIR differentiator in this paper is almost completely approximate the higher order finite impulse response (FIR) differentiator. A novel approach for designing fractional order low-pass infinite impulse response (IIR) digital differentiators is also introduced. First, the numerical differentiators are obtained by inverting the weighted transfer function results from interpolation of various numerical integration rules. The half-order numerical differentiator is expressed in terms of higher order by using continuous fraction expansion. This discretized half order differentiator is cascaded with appropriate low-pass IIR filter resulting in linear phase low-pass IIR fractional order digital differentiator. The simulation studies have shown that, the fractional order IIR differentiator gives better results as compared integer order IIR differentiators. Many methods have been developed to design all types of differentiators but there is still scope of improvement in terms of parameters optimization. The design problem of differentiators is a challenging task. Therefore, there is strong motivation to make design process easy and efficient.