Variable-, fractional-order digital filter theoretical background

Abstract

Digital filtering is a subject of scientific investigations [1-3], [7,9], [11-13] from over than 50 years. Many proposed algorithms for digital filtering has already been recognized as a classic solutions. Here one can mention IIR and FIR filter types [l]. All filters algorithms base on the backward difference notion [5] used to its description. Since the early seventies of the last century a growing focus on the generalisation of the backward difference notion can be seen. The fractional-order backward difference [4] (FOBD) is a discrete counterpart of the fractional-order derivative [6,7]. In this work, a further generalization of the FOBD named variable-, fractional-order backward difference [10] (VFOBD) is applied to the digital filtering performed by the variable-, fractional-order digital filter (VFO-DF). A paper is organised as follows. First, in Section 2 fundamental definitions of the VFOBD is given. Next two equivalent descriptions of the VFODF are introduced. Finally some remarks concerning the order functions are given.[...]