Designs of Discrete-Time Generalized Fractional Order Differentiator, Integrator and Hilbert Transformer

In this paper, the design of a generalized fractional order differentiator (FOD) whose magnitude and phase responses can be controlled independently is investigated. First, a relation between conventional FOD and generalized FOD is studied such that the design tools of conventional FOD in the literature can be used to design variable generalized FOD directly. Then, the similar method is applied to design generalized fractional order integrator (FOI). Next, the proposed generalized FOD and FOI are used to generate a secure single side band (SSB) signal for saving the transmission bandwidth. The parameters of variable generalized FOD and FOI can be used as the secure keys for construction and reconstruction. Finally, the relation between fractional Hilbert transformer and generalized FOD is studied and the edge detection application is demonstrated to show the flexibility and effectiveness of the proposed generalized FOD.