Calculus of the fractional order operators in a discrete time domain

Abstract

The article presents the elementary theory of differential and integral operators of the fractional order in a discrete time approach. A notion of a simple proper fraction operator has been introduced. It has been done for the time equivalent by applying the Taylor series. On this basis a new theory of a certain complexity operators has been formed which includes differential operators of the fractional order. Somewhat more general approach has been presented in the later part of the article by introducing a rational power of the convolution operator. Both approaches to the fractional operators are realized by non–recursive digital filters of infinite impulse responses. The stability of such filters is also being considered. The article also contains the application to the distributed parameters electrical circuits theorem.