# Matched Z-transform method

The matched Z-transform method, also called the pole–zero mapping[1][2] or pole–zero matching method,[3] and abbreviated MPZ or MZT,[4] 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 ${\displaystyle z=e^{sT}}$, for a sample interval ${\displaystyle T=1/f_{\mathrm {s} }}$.[5] So an analog filter with transfer function:

${\displaystyle H(s)=k_{\mathrm {a} }{\frac {\prod _{i=1}^{M}(s-\xi _{i})}{\prod _{i=1}^{N}(s-p_{i})}}}$

is transformed into the digital transfer function

${\displaystyle H(z)=k_{\mathrm {d} }{\frac {\prod _{i=1}^{M}(1-e^{\xi _{i}T}z^{-1})}{\prod _{i=1}^{N}(1-e^{p_{i}T}z^{-1})}}}$

The gain ${\displaystyle k_{\mathrm {d} }}$ must be adjusted to normalize the desired gain, typically set to match the analog filter's gain at DC by setting ${\displaystyle s=0}$ and ${\displaystyle z=1}$ and solving for ${\displaystyle k_{\mathrm {d} }}$.[3][6]

Since the mapping wraps the s-plane's ${\displaystyle j\omega }$ axis around the z-plane's unit circle repeatedly, any zeros (or poles) greater than the Nyquist frequency will be mapped to an aliased location.[7]

In the (common) case that the analog transfer function has more poles than zeros, the zeros at ${\displaystyle s=\infty }$ may optionally be shifted down to the Nyquist frequency by putting them at ${\displaystyle z=-1}$, dropping off like the BLT.[1][3][6][7]

This transform doesn't preserve time- or frequency-domain response (though it does preserve stability and minimum phase), and so is not widely used.[8][7] More common methods include the bilinear transform and impulse invariance methods.[4] MZT does provide less high frequency response error than the BLT, however, making it easier to correct by adding additional zeros, which is called the MZTi (for "improved").[9]

A specific application of the matched Z-transform method in the digital control field, is with the Ackermann's formula, which changes the poles of the controllable system; in general from an unstable (or nearby) location to a stable location.

## 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. Al-Alaoui, M. A. (February 2007). "Novel Approach to Analog-to-Digital Transforms". IEEE Transactions on Circuits and Systems I: Regular Papers. 54 (2): 338–350. doi:10.1109/tcsi.2006.885982. ISSN 1549-8328.
5. 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.
6. Franklin, Gene F. (2015). Feedback control of dynamic systems. Powell, J. David, Emami-Naeini, Abbas (Seventh ed.). Boston: Pearson. pp. 607–611. ISBN 978-0133496598. OCLC 869825370. Because physical systems often have more poles than zeros, it is useful to arbitrarily add zeros at z = -1.
7. Rabiner, Lawrence R; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, New Jersey: Prentice-Hall. pp. 224–226. ISBN 0139141014. The expediency of artificially adding zeros at z = —1 to the digital system has been suggested ... but this ad hoc technique is at best only a stopgap measure. ... In general, use of impulse invariant or bilinear transformation is to be preferred over the matched z transformation.
8. Jackson, Leland B. (1996). Digital Filters and Signal Processing. Springer Science & Business Media. p. 262. ISBN 9780792395591. although perfectly usable filters can be designed in this way, no special time- or frequency-domain properties are preserved by this transformation, and it is not widely used.
9. Ojas, Chauhan; David, Gunness (2007-09-01). "Optimizing the Magnitude Response of Matched Z-Transform Filters ("MZTi") for Loudspeaker Equalization" (PDF). Audio Engineering Society. Archived from the original on 2007.