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Synthesis of optimal digital fractional operator devices

Keywords:

M.R. Vyaselev – Dr. Sc. (Eng.), Professor, Radio Electronics and Information-Measuring Engineering Department, Kazan National Research Technical University named after A.N. Tupolev. E-mail: muruvya@riit.kstu-kai.ru
E.F. Bazlov – Ph.D. (Eng.), Associate Professor, Radio Electronics and Information-Measuring Engineering Department, Kazan National Research Technical University named after A.N. Tupolev. E-mail: evgeniy.bazlov@outlook.com
Y.K. Evdokimov – Dr. Sc. (Eng.), Professor, Head of Radio Electronics and Information-Measuring Engineering Department, Kazan National Research Technical University named after A.N. Tupolev. E-mail: evdokimov@tre.kstu-kai.ru
M.A. Titov – Post-graduate Student, Radio Electronics and Information-Measuring Engineering Department, Kazan National Research Technical University named after A.N. Tupole. E-mail:kaiman89@yandex.ru


The relevance of synthesis of the fractional operator devices (FOD), electrically realizing operations of fractional differentiation and integration has increased significantly, because increasing application of fractal methods and fractional operators in modeling and signal processing in electronics and other fields recently. In particular, a number of successful works on the development of analog FOD (based on lumped and distributed structures) having, in principle, a high rate of speed, small size, low cost. However, thanks to the rapid development of computer aids technology and its elemental base, digital FOD become quite competitive and the proper design can be comparable in performance with analog FOD. Therefore improvement and optimization of methods ofdigital FOD synthesis is represented quite actual. In this workthe method of synthesis of the optimal structure and parameters of functional circuit fractional operator devices was developed, which provides them with minimum complexity and maximum performance in the implementation of the fractional operator with the specified fractional exponent in the specified frequency range with a given accuracy, i.e., with a given permissible standard deviation (RMS) amplitude-and phase-frequency characteristics of their ideal values. It is known, that the functional properties of such digital devices is uniquely determined by the parameters of z-transformed impulse response. Therefore, the above synthesis was carried by developing a procedure for determining the parameters of the characteristics, which have minimum order. It’s not difficult to show that this FOD scheme will have the above optimal properties. Synthesis of fractional integrating and fractional differentiating digital FOD was carried out at the different specified initial data – at various fractional exponents and at different values of a frequency interval. Analysis of the frequency and phase response of the synthesized FOD showed correctness of synthesis in the implementation of all the initial requirements. It is also shown that in this case the upper bound of the specified frequency interval (frequency overlap) must not be less than 30 times the sampling frequency (at rated RMS magnitude and phase response of 1 %). It is important to note that the developed synthesis is conducted in the dimensionless (generalized) values of used sizes that makes the results more general, in particular – invariant in relation to the concrete working range of frequencies. For the specific conditions dimension parameters of the FOD characteristics can be easily determined by their generic values.
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