Performance Evaluation of Fractional Order Digital Filter for Signal and Image Processing Applications

Abstract

Fractional-order calculus (FOC), which is often contemplated as a branch of pure mathematics, has encapsulated the attention of eminent researchers from several backgrounds due to its profound applications in various fields of science and engineering. The mathematical tool of FOC, which is, in fact, the generalization of integer-order calculus, is concerned with differentiation and integration of non-integer orders. FOC concept emerged from the generalization of integer-order calculus in two most prominent ways: One based upon the direct generalization of the limit of difference quotients defining integer-order derivatives and another one based upon the inversion of the generalization of n-fold integer-order successive integration rule (a.k.a. Cauchy s formula for repeated integration). These generalizations provided two different ways to define fractional-order derivatives and hence, referred to as Gr¨unwald-Letnikov (GL) and Riemann-Liouville (RL) fractional-order derivatives, respectively.From signal and image processing viewpoint, the definitions of fractional-order derivatives can be viewed in terms of convolution, where the filter or kernel function is of power-law type. A natural query that arises is: Why the kernel is of power-law type? This is because of the association of power-law in Cauchy s integral formula. Fractional-order operators exhibiting power-law kernel have found tremendous applications in natural and man-made phenomena. However, it has been pointed out recently that there exists an artificial singularity at the initial point of the power-law kernel, due to which many real-life phenomena cannot be implemented exactly. Therefore, several mathematicians made an attempt to replace the power-law kernel with other appropriate kernels (such as, exponential kernel, Mittag-Leffler kernel) and hence, led to the development of various fractional-order operators based upon different kernel functions.On the other hand, a significant attention has been paid by the research community on the study