Matched Z-transform method: Difference between revisions

Revision as of 05:22, 14 November 2017

Revision as of 01:07, 17 November 2017

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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|url=http://ieeexplore.ieee.org/document/4089107/|journal=IEEE Transactions on Circuits and Systems I: Regular Papers|volume=54|issue=2|pages=338–350|doi=10.1109/tcsi.2006.885982|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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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>, dropping off like the BLT.<ref name=":3" /><ref name=":1" /><ref name=":2" /><ref name=":0" />

This transform doesn't preserve time- or frequency-___domain response (though it does preserve [[BIBO stability|stability]] and [[minimum phase]]), and so is not 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]] 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|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|thumb|350px|Responses of the filter (dashed), and its discrete-time approximation (solid), 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.]]