Design of digital differentiators

A digital differentiator simply involves the derivation of an input signal. This work includes the presentation of first-degree and second-degree differentiators, which are designed as both infinite-impulse-response (IIR) filters and finite-impulse-response (FIR) filters. The proposed differentiators have low-pass magnitude response characteristics, thereby rejecting noise frequencies higher than the cut-off frequency. Both steady-state frequency-domain characteristics and Time-domain analyses are given for the proposed differentiators. It is shown that the proposed differentiators perform well when compared to previously proposed filters. When considering the time-domain characteristics of the differentiators, the processing of quantized signals proved especially enlightening, in terms of the filtering effects of the proposed differentiators. The coefficients of the proposed differentiators are obtained using an optimization algorithm, while the optimization objectives include magnitude and phase response. The low-pass characteristic of the proposed differentiators is achieved by minimizing the filter variance, which provides a measure of the ability of the filter to attenuate noise corruption on an input signal. The low-pass differentiators designed show the steepest roll-off when compared with a number of other previously proposed differentiators of the same order, as well as having highly accurate magnitude response in the pass-band. The proposed differentiators can be designed for a particular group delay or roll-off characteristics, by choosing an application-dependent weight vector when defining the goals. While having a history of over three hundred years, the design of fractional differentiator has become a ‘hot topic’ in recent decades, because many authors have pointed out that these mathematical phenomena allow for the description of a real object more accurately than the classical integer-degree model. One challenging problem in this area is that there are many different definitions to describe the fractional model, such as the Riemann-Liouville and Caputo definitions. Some new fractional degree differentiators are designed using both of the above definitions. Through use of a feedback structure, based on the Riemann-Liouville definition. It is shown that the performance of the fractional differentiator can be improved in both the frequency-domain and time-domain. Further time-domain analyses show that the Caputo differentiators have better performance than other Fractional degree (FD differentiators when the initial value of the input signal is not zero. Two applications based on the proposed differentiators are described in the thesis. Specifically, the first of these involves the application of second degree differentiators in the estimation of the frequency components of a power system. A linear phase FIR second degree differentiator replaces a previously proposed IIR differentiator in a frequency-estimation algorithm, to avoid the error caused by the variable group delay. The second example concerns for an image processing, edge detection application. In this application, it is shown that IIR FD differentiators can be used to advantage when applied to the edge detection application, with the Caputo-based FD differentiators showing particular advantages.