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Exponentially Distributed Real Pole-Zero Pairs

In contrast to exact integral forms or general-purpose filter designs for fractional differintegrals, we will develop them in closed form as exponentially distributed pole-zero pairs. (On a log scale, the poles and zeros are uniformly spaced.) It appears that such filters approach the Chebyshev optimal approximation (in terms of log-log slope error) for any $ \alpha$ as the pole-zero density and span along the negative real axis increase.

Any needed integer part $ \pm M$ for $ \alpha$ can be trivially provided using $ M$ zeros or poles at/near the origin $ s=0$ of the complex plane.

The proposed filter structure is furthermore robust for time-varying $ \alpha$ , because the poles are fixed, and only the zeros need to slide left or right along the negative real axis of the $ s$ plane as $ \alpha$ is changed. On a log scale, the spacing of the zeros does not change as they are slid left and right.

The fractional order $ \alpha$ sets the spacing of the zeros array relative to the poles array along the negative-real axis. For $ \alpha=1/2$ , the zeros lie on the midpoints between the poles. For $ \alpha=0$ , the array of zeros slides to right so as to cancel all of the poles, leaving the trivial filter $ H_0(s)=1$ , as desired. At $ \alpha=-1$ , all poles are canceled except the first to the left along the negative real axis, leaving a normal integrator $ H_{-1}(s)=1/s$ , as desired. At $ \alpha=+1$ , all poles are canceled except the last, and one zero is exposed near $ s=0$ , yielding a normal ``leaky differentiator with high-frequency leveling'' $ H_{1}(s)=p_N(s+\epsilon)/(s+p_N)\approx j\omega$ for frequencies interior to the interval $ \omega\in[\epsilon,\vert p_N\vert]$ . In the limit as the number of poles $ N$ goes to infinity, and as $ \epsilon\to 0$ , we obtain the ideal differentiator $ H_1(s)=s$ , as desired.


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``Spectral Tilt Filters'', by Julius O. Smith IIIand Harrison F. Smith, adapted from arXiv.org publication arXiv:1606.06154 [cs.CE].
Copyright © 2021-06-18 by Julius O. Smith IIIand Harrison F. Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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