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$](img12.png)
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 F
AUST.