Second order approximation of the fractional laplacian operator for equal-ripple response

In this paper we propose a modification to a second order approximation of the fractional-order Laplacian operator, sα, where 0 < α < 1. We show how this proposed modification can be used to change the ripple error of both the magnitude and phase responses of the approximation when compared to the ideal case. Equal-ripple magnitude and phase responses that have both less cumulative error and less maximum ripple deviation are presented using this modification. A 1st order lowpass filter with fractional step of 0.8, that is of order 1.8, is implemented using the proposed approximation. Experimental results verify the operation of this approximation in the realization of the fractional step filter.