Fractional 90-degree phase-shift filtering based on Liouville-Weyl differintegral

In this brief, fractional 90-degree phase-shift filtering technique is investigated. First, thanks to the two commonly used Liouville and Weyl fractional integral definitions, we present Liouville and Weyl fractional differintegrators which are generalized magnitude-and-phase modulations. Then, paralleling a Liouville fractional differintegrator with a Weyl fractional differ-integrator to obtain a Liouville-Weyl fractional differintegrator, which is essentially a 90-degree phase shifter with noise immunity capability (low-pass). Finally, an inflexion point detection example is illustrated to demonstrate that the Liouville-Weyl fractional differintegrator is constantly forward 90-degree which is the same to the conventional first-order differential and bears good noise immunity which can be tuned by adjusting the differintegral order.