Matched Z-transform method

The s-plane poles and zeros of a 5th-order Chebyshev type II lowpass filter to be approximated as a discrete-time filter

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=e^iωT, 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=e^sT, 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

1. Won Young Yang (2009). Signals and Systems with MATLAB. Springer. p. 292. ISBN 978-3-540-92953-6.
2. Bong Wie (1998). Space vehicle dynamics and control. AIAA. p. 151. ISBN 978-1-56347-261-9.
3. Arthur G. O. Mutambara (1999). Design and analysis of control systems. CRC Press. p. 652. ISBN 978-0-8493-1898-6.
4. S. V. Narasimhan and S. Veena (2005). Signal processing: principles and implementation. Alpha Science Int'l Ltd. p. 260. ISBN 978-1-84265-199-5.