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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|url=https://www.worldcat.org/oclc/869825370|title=Feedback control of dynamic systems|last=Franklin|first=Gene F.|date=2015|publisher=Pearson|others=Powell, J. David, Emami-Naeini, Abbas|year=|isbn=0133496597|edition=Seventh edition|___location=Boston|pages=607-611|oclc=869825370|quote=Because physical systems often have more poles than zeros, it is useful to arbitrarily add zeros at z = -1.}}</ref>
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. ... 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.}}</ref>
In the (common) case that the analog transfer function has more poles than zeros, extra zeros can be added at <math>z=-1</math>.<ref name=":2" /><ref name=":0" />
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>
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