Matched Z-transform method: Difference between revisions

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}}</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|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=0133496597978-0133496598|edition=Seventh edition|___location=Boston|pages=607-611607–611|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=https://archive.org/details/theoryapplicatio00rabi/page/224|title=Theory and application of digital signal processing|lastlast1=Rabiner|firstfirst1=Lawrence R|last2=Gold|first2=Bernard|date=1975|publisher=Prentice-Hall|year=|isbn=0139141014|___location=Englewood Cliffs, New Jersey|pages=[https://archive.org/details/theoryapplicatio00rabi/page/224-226 224–226]|language=Englishen|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. ... In general, use of impulse invariant or bilinear transformation is to be preferred over the matched z transformation.|url-access=registration}}</ref>

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>, droppingcausing the transfer function to drop off likeas <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" />

ThisWhile this transform doesn'tpreserves [[BIBO stability|stability]] and [[minimum phase]], it preserves preserveneither time- ornor frequency-___domain properties,response and so is rarelynot widely used.<ref>{{Cite book|url=https://books.google.com/books?id=VZ8uabI1pNMC&lpg=PA262&ots=gSD3om4Hy4&pg=PA262|title=Digital Filters and Signal Processing|last=Jackson|first=Leland B.|date=1996|publisher=Springer Science & Business Media|year=|isbn=9780792395591|___location=|pages=262|language=en|quote=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.}}</ref><ref name=":0" /> AlternativeMore common methods include the [[bilinear transform]]BLT and [[impulse invariance]] methods.<ref name=":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").<ref>{{Cite journal|lastlast1=Ojas|firstfirst1=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=Englishen|volumearchive-url=|pageshttps://web.archive.org/web/20190727193622/http://www.aes.org/e-lib/browse.cfm?elib=14198|archive-urldate=July 27, 2019}} [http://www.khabdha.org/wp-content/uploads/2008/03/optimizing-the-magnitude-response-of-mzt-filters-mzti-2007.pdf|archive-date=2007|via=}} Alt URL]</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.

[[File:Chebyshev responses.svg|thumb|350px|Responses of the filter (solid), and its discrete-time approximation (dashed), 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.]]

[[File:Chebyshev responses.svg|thumb|350px|Responses of the filter (soliddashed), and its discrete-time approximation (dashedsolid), 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==