Fractional Euler analog-to-digital transform

In this paper, the design of fractional Euler transform which is the regular Euler transform where the digital delay operator z−1 is replaced by the ideal digital fractional delay operator z−α is presented. This introduced transform will be used in analog-to-digital transform to improve the accuracy of a digital filter equivalent to a given analog filter. However, because of the fractional delay operator z−α the proposed fractional Euler transform cannot be exactly implemented; only a limited implementation by means of infinite impulse response (IIR) digital filter can be achieved using approximation techniques. First, the fractional order α of the digital fractional delay operator z−α, for 0 < α < 0.5, is selected such that the relative error between the analog differentiator s and its equivalent digital one obtained by the fractional Euler transform is minimum. Then, the ideal digital fractional delay operator z−α is approximated by a digital IIR filter based on approximation of analog fractional order system leading to an IIR digital filter implementation of the fractional Euler transform. Illustrative examples are given to show the effectiveness of the fractional Euler analog-to-digital transform design. Finally, the design of low order digital differentiator using the proposed fractional Euler transform has been compared to the most recent designed analog-to-digital transforms.