Analog filters based on the Mittag-Leffler functions

Abstract

We propose and study a new class of filters (named hereinafter the Mittag-Leffler filters) based on the Mittag-Leffler function $E_{\alpha ,\beta }\left(z\right)$ in its single-parameter or double-parameter forms by transposing its argument to the frequency-domain; i.e. $z=-s=-j\omega$. A unique feature of these filters is that their impulse response is a Gaussian-like (delta-like) deformed and delayed impulse function for which we derive exact expressions using the $H-$Fox function. We also study the frequency response of this class of filters and obtain lower-order, realizable integer-order approximations of its transfer functions. A second-order curve-fitting approximation is then used to perform experimental results using a Field Programmable Analog Array platform to verify the theory.