Matched Z-transform method: Difference between revisions

Revision as of 04:13, 13 November 2017

Revision as of 05:22, 14 November 2017

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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=607-611607–611|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=|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=224-226224–226|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.}}</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 BLT.<ref name=":3" /><ref name=":1" /><ref name=":2" /><ref name=":0" />