Matched Z-transform method: Difference between revisions

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[[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^iωT'', 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

{{cite book

| title = Signals and Systems with MATLAB

| author = Won Young Yang

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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=224-226|language=English|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.}}</ref>

In the (common) case that the analog transfer function has more poles than zeros, extrathe zeros canat <math>s=\infty</math> may optionally be addedshifted 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 properties, and so is rarely used.<ref name=":0" /> Alternative methods include the [[bilinear transform]] and [[impulse invariance]] methods. 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|left|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.]]

==References==