Design of discrete Fractional Hilbert transformer in time domain

We propose a design of discrete Fractional Hilbert transformer. Through the time domain analysis of the ideal input and output signal of a LTI system, we derived equations which can be solved numerically for the filter´s impulse response. The conventional Hilbert transformer is derived as a special case of the fractional Hilbert transformer. We also show that the difference between the designed system and the ideal system can be represented by the Peano kernel of the system. Some designing examples of the proposed filter are also illustrated.