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Studies on fractional order differentiators and integrators: a survey. (English) Zbl 1203.94035
Summary: Studies on analysis, design and applications of analog and digital differentiators and integrators of fractional order is the main objective of this paper. Time and frequency domain analysis, different ways of realization of fractance device is presented. Active and passive realization of fractance device of order $$\frac 12$$ using continued fraction expansion is carried out. Later, time and frequency domain analysis of fractance based circuits is considered. The variations of rise time, peak time, settling time, time constant, percent overshoot with respect to fractional order $$\alpha$$ is presented.
Digital differentiators and integrators of fractional order can be obtained by using direct and indirect discretization techniques. The $$s$$ to $$z$$ transforms used for this purpose are revisited. In this paper by using indirect discretization technique fractional order differentiators and integrators of order $$\frac 12$$ and $$\frac 14$$ are designed. These digital differentiators and integrators are implemented in real time using TMS320C6713 DSP processor and tested using National instruments education laboratory virtual instrumentation system (NIELVIS). The designed fractional order differentiators have been used for the detection of QRS sequences as well as the occurrence of Sino Atrial Rhythms in an ECG signal and also for the detection of edges in an image. The obtained results are in comparison with the conventional techniques.

##### MSC:
 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 34A08 Fractional ordinary differential equations 26A33 Fractional derivatives and integrals
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