Transfer Functions of Fractional-Order Band-Pass Filter with Arbitrary Magnitude Slope in Stopband.

Four band-pass (BP) filter transfer functions of the fractional order 1<(1+$\alpha$)<2 are presented. Their coefficients are found using numerical least squares optimization that aims at minimizing the magnitude errors between these transfer functions and the specimen functions that correspond to $2^{nd_{-}}$ order Butterworth filter responses. The fractional-order (FO) BP transfer functions with the proposed coefficients and coefficients previously presented in the literature for Butterworth-like FO low-pass (LP) transfer functions are compared. This comparison highlights that using the coefficients determined for FO LP transfer functions in FO BP functions causes distorted characteristics in some cases. The FO BP functions with the proposed coefficients match the specimen functions very well reducing this distortion.