Approximations of higher-order fractional differentiators and integrators using indirect discretization

This paper describes new approximations of fractional order integrators (FOIs) and fractional order differentiators (FODs) by using a continued fraction expansion-based indirect discretization scheme. Different tenth-order fractional blocks have been derived by applying three different s-to-z transforms described earlier by Al-Alaoui, namely new two-segment, four-segment, and new optimized four-segment operators. A new addition has been done in the new optimized four-segment operator by modifying it by the zero reflection method. All proposed half (s(+/- 1/2)) and one-fourth (s(+/- 1/4)) differentiator and integrator models fulfill the stability criterion. The tenth-order fractional differ-integrators (s(+/-alpha)) based on the modified new optimized four-segment rule show tremendously improved results with relative magnitude errors (dB) of <= -15 dB for alpha = 1/2 and <= -20 dB for alpha = 1/4 in the full range of Nyquist frequency so these have been further analyzed. The main contribution of this paper lies in the reduction of these tenth-order blocks into four new fifth-order blocks of half and one-fourth order models of FODs and FOIs. The analyses of magnitude and phase responses show that the proposed new fifth-order half and one-fourth differ-integrators closely approximate their ideal counterparts and outperform the existing ones.