**JULIUS ORION SMITH III
and HARRISON FREEMAN SMITH**
*Center for Computer Research in Music
and Acoustics (CCRMA)*

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
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.

- Introduction
- Fractional Derivatives and Integrals
- Filter Interpretation
- Exponentially Distributed Real Pole-Zero Pairs
- Importance in Audio Signal Processing
- Summary of Results
- Outline of the Remainder

- Bode Plots
- Stick Diagram Design
- Approximating Arbitrary Slopes
- About this document ...

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University