Content deleted Content added
May as well express it with the math symbols. |
Citation bot (talk | contribs) Alter: url. URLs might have been anonymized. Add: s2cid, authors 1-1. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by AManWithNoPlan | #UCB_webform 857/2200 |
||
| (29 intermediate revisions by 12 users not shown) | |||
Line 3:
[[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
Line 11 ⟶ 10:
| isbn = 978-3-540-92953-6
| page = 292
| url =
}}</ref><ref>
{{cite book
Line 20 ⟶ 19:
| isbn = 978-1-56347-261-9
| page = 151
| url =
}}</ref> or '''pole–zero matching method''',<ref name=":1">{{cite book
| title = Design and analysis of control systems
| author = Arthur G. O. Mutambara
Line 29 ⟶ 27:
| isbn = 978-0-8493-1898-6
| page = 652
| url =
}}</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
{{cite book
| title = Signal processing: principles and implementation
Line 40 ⟶ 38:
| isbn = 978-1-84265-199-5
| page = 260
| url =
}}</ref> So an analog filter with transfer function:
:<math>H(s) = k_{\mathrm a} \frac{\prod_{i=1}^M (s-\xi_i) }{\prod_{i=1}^N (s-p_i) }</math>
is transformed into the digital transfer function
:<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|title=Feedback control of dynamic systems|last=Franklin|first=Gene F.|date=2015|publisher=Pearson|others=Powell, J. David, Emami-Naeini, Abbas|isbn=978-0133496598|edition=Seventh|___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=https://archive.org/details/theoryapplicatio00rabi/page/224|title=Theory and application of digital signal processing|last1=Rabiner|first1=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=en|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>, 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" />
While this transform preserves [[BIBO stability|stability]] and [[minimum phase]], it preserves neither time- nor frequency-___domain response and so is not widely used.<ref>{{Cite book|url=https://books.google.com/books?id=VZ8uabI1pNMC&pg=PA262|title=Digital Filters and Signal Processing|last=Jackson|first=Leland B.|date=1996|publisher=Springer Science & Business Media|isbn=9780792395591|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 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|last1=Ojas|first1=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=en|archive-url=https://web.archive.org/web/20190727193622/http://www.aes.org/e-lib/browse.cfm?elib=14198|archive-date=July 27, 2019}} [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.
[[File:Chebyshev responses.svg
==References==
| |||