Fractional 90° phase-shift filtering based on the double-sided Grünwald–Letnikov differintegrator

A fractional 90° phase-shift filtering technique is investigated based on the double-sided Grünwald-Letnikov differintegrator. Thanks to the Grünwald-Letnikov definition of fractional calculus, first the left and the right-sided Grünwald-Letnikov differintegrators are presented, which are generalised magnitude-and-phase modulations. Then, parallel a left-sided Grünwald-Letnikov differintegrator with a right-sided Grünwald-Letnikov differintegrator to obtain a double-sided Grünwald-Letnikov differintegrator, which is essentially a fractional 90° phase-shift filter with the capability of noise immunity. A double-sided symmetrical convolution mask is constructed to implement the proposed fractional 90° phase shifter. Finally, a singularity detection example is illustrated to demonstrate that the proposed double-sided Grünwald-Letnikov differintegrator is forward 90°, which is the same as the conventional first-order differential, and possesses good noise immunity that can be tuned by adjusting the differintegral order.