Analysis and design of non recursive digital differentiators in fractional domain for signal processing applications

Abstract

There is a universe of mathematics lying in between the complete differentiations and integrations. O. Heaviside. THE PURPOSE OF THE REPORTED WORK is to provide a comprehensive and unified introduction to the principles and applications of the differentiation property in the fractional Fourier transform domain. The fractional Fourier transform (FrFT) is a generalized definition of the classical Fourier transform (FT). The FrFT has proved to be a powerful tool for the analysis of time varying signals by representing rotation of a signal in the time frequency plane. In the areas where FT and the frequency domain concepts are used, the performance can be enhanced through the use of FrFT. In addition, it has a close relationship with other signal transforms like wavelet, linear canonical transform. The well known properties of the FrFT have been established in many of the rich literatures, which have many applications in the signal and image processing fields. In the reported work, the concept of fractional order differentiation property in the FrFT domain is established. Keeping an eye on the aspects of the fractional order calculus, the proposed work is carried out. The fractional order calculus (FOC) is a branch of mathematics is as old as the classical Newtonian calculus, and deals with the theory of differential and integral operators of non integer order and to the differential equations containing such operators. The origin of the FOC theory can be traced back to the end of the seventeenth century, the time when Newton and Leibniz developed the foundation of the differential and integral calculus. Even though the concept of the FOC theory is established long back, yet the subject came practically over the last few decades. The new models of the systems have been developed by engineers and scientists based on the fractional differential equations approach.