The
z-plane poles and zeros of the discrete-time Chebyshev filter, as mapped into the z-plane using the matched Z-transform method with
T = 1/10 second. The labeled frequency points and band-edge dotted lines have also been mapped through the function
z=eiωT, to show how frequencies along the
iω axis in the
s-plane map onto the unit circle in the
z-plane.
The matched Z-transform method, also called the pole–zero mapping[1][2] or pole–zero matching method,[3] is a technique for converting a continuous-time filter design to a discrete-time filter (digital filter) design.
The method works by mapping all poles and zeros of the s-plane design to z-plane locations z=esT, for a sample interval T.[4]
Alternative methods include the bilinear transform and impulse invariance methods.
Responses of the filter (dashed), and its discrete-time approximation (solid), for nominal cutoff frequency of 1 Hz, sample rate 1/T = 10 Hz. The discrete-time filter does not reproduce the Chebyshev equiripple property in the stopband due to the interference from cyclic copies of the response.
References
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