Butterworth and Chebyshev Pole Locations

Cuthbert Nyack

The Applets below shows the relative pole location for Butterworth and Chebyshev low pass filters in the complex s plane.

For the Butterworth approximation, the green and red semicircles together show the unit circle on which the Butterworth poles lie. The red poles on the unit circle produce unstable responses and are not used to implement the filter. The filter transfer function is derived from the green poles on the left of the imaginary axis in the s plane.

For the Chebyshev approximation, the poles lie on the (cyan + orange) ellipse and the filter transfer function is calculated from the cyan poles on the left of the imaginary axis.

In the case where the Chebyshev poles are calculated to produce an attenuation equal to the ripple at the cutoff frequency, the pole locations are shown below. In this case the major radius of the ellipse can be larger than unity.

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