Closed Form Fractional Integration and Differentiation via Real   Exponentially Spaced Pole-Zero Pairs
  We derive closed-form expressions for the poles and zeros of approximate fractional integrator/differentiator filters, which correspond to spectral roll-off filters having any desired log-log slope to a controllable degree of accuracy over any bandwidth. The filters can be described as a uniform exponential distribution of poles along the negative-real axis of the s plane, with zeros interleaving them. Arbitrary spectral slopes are obtained by sliding the array of zeros relative to the array of poles, where each array maintains periodic spacing on a log scale. The nature of the slope approximation is close to Chebyshev optimal in the interior of the pole-zero array, approaching conjectured Chebyshev optimality over all frequencies in the limit as the order approaches infinity. Practical designs can arbitrarily approach the equal-ripple approximation by enlarging the pole-zero array band beyond the desired frequency band. The spectral roll-off slope can be robustly modulated in real time by varying only the zeros controlled by one slope parameter. Software implementations are provided in matlab and Faust. 
Julius Orion Smith and Harrison Freeman Smith
cs.CE cs.SD cs.SY