  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.
