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 as the pole-zero density and span along the negative real axis increase.
Any needed integer part for can be trivially provided using zeros or poles at/near the origin of the complex plane.
The proposed filter structure is furthermore robust for time-varying , because the poles are fixed, and only the zeros need to slide left or right along the negative real axis of the plane as is changed. On a log scale, the spacing of the zeros does not change as they are slid left and right.
The fractional order sets the spacing of the zeros array relative to the poles array along the negative-real axis. For , the zeros lie on the midpoints between the poles. For , the array of zeros slides to right so as to cancel all of the poles, leaving the trivial filter , as desired. At , all poles are canceled except the first to the left along the negative real axis, leaving a normal integrator , as desired. At , all poles are canceled except the last, and one zero is exposed near , yielding a normal ``leaky differentiator with high-frequency leveling'' for frequencies interior to the interval . In the limit as the number of poles goes to infinity, and as , we obtain the ideal differentiator , as desired.