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 = 0.1 second. The labeled frequency points and band-edge dotted lines have also been mapped through the function exp(sT), 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 = exp(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 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 9783540929536. http://books.google.com/books?id=GnfpELDfzmEC&pg=PA292. 
2. ^ Bong Wie (1998). Space vehicle dynamics and control. AIAA. p. 151. ISBN 9781563472619. http://books.google.com/books?id=n97tEQvNyVgC&pg=PA151. 
3. ^ Arthur G. O. Mutambara (1999). Design and analysis of control systems. CRC Press. p. 652. ISBN 9780849318986. http://books.google.com/books?id=VSlHxALK6OoC&pg=PA652. 
4. ^ S. V. Narasimhan and S. Veena (2005). Signal processing: principles and implementation. Alpha Science Int'l Ltd.. p. 260. ISBN 9781842651995. http://books.google.com/books?id=8UbV8vq8uV0C&pg=PA260.