Matched Z-transform method: Difference between revisions

Revision as of 20:02, 19 July 2020

Revision as of 08:14, 25 August 2020

Line 51:

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|last=Rabiner|first=Lawrence R|last2=Gold|first2=Bernard|date=1975|publisher=Prentice-Hall|isbn=0139141014|___location=Englewood Cliffs, New Jersey|pages=[https://archive.org/details/theoryapplicatio00rabi/page/224 224–226]|language=English|quote=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>, dropping off like the [[bilinear transform]] (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/?id=VZ8uabI1pNMC&lpg=PA262&pg=PA262|title=Digital Filters and Signal Processing|last=Jackson|first=Leland B.|date=1996|publisher=Springer Science & Business Media|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" /> More 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|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=https://web.archive.org/web/20190727193622/http://www.aes.org/e-lib/browse.cfm?elib=14198|archive-date=July 27, 2019|via=}} [http://www.khabdha.org/wp-content/uploads/2008/03/optimizing-the-magnitude-response-of-mzt-filters-mzti-2007.pdf 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.