Ellipse in S plane to Z plane.

Cuthbert Nyack
Butterworth filters have poles on a unit semicircle in the left half of the s plane while Chebyshev filters have poles on a semiellipse.
The applet below shows how a semicircle or semiellipse in the -ve s plane is mapped to the unit circle in the Z plane.

The unit semicircle in the s plane is shown in magenta and its map in the Z plane is shown in orange. The semiellipse is shown in cyan and its Z plane map in red. The vertical "radius" is a and the horizontal "radius" is b. ws is the sampling frequency.
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© 1999 Cuthbert A. Nyack.