Sine function and generalized digital fractional differentiators

The topics on fractional-order derivative are studied from the viewpoint of signal processing. Firstly, some basic concepts and fundamental properties of fractional derivative and its operator are discussed. Secondly, the ideal digital fractional differentiators are proposed in this paper, its basic conditions and integral formula of filtering coefficients are given. And next the concept of differentiator coefficient function which is used to generalize digital fractional differentiator is introduced. Finally, according to the differentiator coefficient function, the notions of the generalized digital fractional differentiator are introduced. In this paper we mainly investigates the relationships between the generalized digital differentiator coefficient functions and sine function, and the generalizing problems of digital differentiator. Some new concepts, such as the differentiator coefficient functions, odd differentiators, even differentiators, nonsymmetric differentiators, the generalized Hilbert transform, and so on are introduced firstly in this paper.