Principles of fractional signal processing

A general framework for fractional signal processing is described and used to derive several interesting formulations. This scheme is based on the Liouville approach that gave rise to the classic Riemann-Liouville and Liouville-Caputo derivatives, here dismissed. Liouville's idea consisted of fractionalizing the transfer function of the basic definition of derivative. Various coherent formulations are introduced from suitable derivative definitions and the corresponding ARMA-type linear systems are obtained. In particular, the Euler discrete-time, classic continuous-time, bilinear (Tustin), and scale-invariant systems are introduced and studied as applications of the proposed scheme. The two-sided derivatives and systems are also considered.