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[[File:Chebyshev mapped z plane.svg|right|thumb|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<sup>iωT</sup>'', 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'''<ref name=":3">{{cite book
| title = Signals and Systems with MATLAB
| author = Won Young Yang
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| page = 652
| url = https://books.google.com/books?id=VSlHxALK6OoC&pg=PA652
}}</ref> and abbreviated '''MPZ''' or '''MZT''',<ref name=":4">{{Cite journal|last=Al-Alaoui|first=M. A.|date=February 2007|title=Novel Approach to Analog-to-Digital Transforms|journal=IEEE Transactions on Circuits and Systems I: Regular Papers|volume=54|issue=2|pages=338–350|doi=10.1109/tcsi.2006.885982|s2cid=9049852|issn=1549-8328}}</ref> 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 [[Laplace transform|''s''-plane]] design to [[Z-transform|''z''-plane]] locations <math>z=e^{sT}</math>, for a sample interval <math>T=1 / f_\mathrm{s}</math>.<ref>
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:<math> H(z) = k_{\mathrm d} \frac{ \prod_{i=1}^M (1 - e^{\xi_iT}z^{-1})}{ \prod_{i=1}^N (1 - e^{p_iT}z^{-1})} </math>
The gain <math>k_{\mathrm d}</math> must be adjusted to normalize the desired gain, typically set to match the analog filter's gain at DC by [[Final value theorem|setting <math>s=0</math> and <math>z=1</math>]] and solving for <math>k_{\mathrm d}</math>.<ref name=":1" /><ref name=":2">{{Cite book
Since the mapping wraps the ''s''-plane's <math>j\omega</math> 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.<ref name=":0">{{Cite book|url=https://archive.org/details/theoryapplicatio00rabi/page/224|title=Theory and application of digital signal processing|
In the (common) case that the analog transfer function has more poles than zeros, the zeros at <math>s=\infty</math> may optionally be shifted down to the Nyquist frequency by putting them at <math>z=-1</math>, causing the transfer function to drop off as <math>z \rightarrow -1</math> in much the same manner as with the [[bilinear transform]] (BLT).<ref name=":3" /><ref name=":1" /><ref name=":2" /><ref name=":0" />
▲Since the mapping wraps the ''s''-plane's <math>j\omega</math> 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.<ref name=":0">{{Cite book|url=|title=Theory and application of digital signal processing|last=Rabiner|first=Lawrence R|last2=Gold|first2=Bernard|date=1975|publisher=Prentice-Hall|year=|isbn=0139141014|___location=Englewood Cliffs, New Jersey|pages=224-226|language=English|quote=In general, use of impulse invariant or bilinear transformation is to be preferred over the matched z transformation.}}</ref>
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 [[Controllability|controllable]] system; in general from an unstable (or nearby) ___location to a stable ___location.
▲This transform doesn't preserve time- or frequency-___domain properties, and so is rarely used.<ref name=":0" /> Alternative methods include the [[bilinear transform]] and [[impulse invariance]] methods. 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").<ref>{{Cite journal|last=Ojas|first=Chauhan|last2=David|first2=Gunness|date=2007-09-01|title=Optimizing the Magnitude Response of Matched Z-Transform Filters ("MZTi") for Loudspeaker Equalization|url=http://www.aes.org/e-lib/browse.cfm?elib=14198|journal=Audio Engineering Society|language=English|volume=|pages=|archive-url=http://www.khabdha.org/wp-content/uploads/2008/03/optimizing-the-magnitude-response-of-mzt-filters-mzti-2007.pdf|archive-date=2007|via=}}</ref>
[[File:Chebyshev responses.svg
==References==
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