matched Z-transform method

File:Chebyshev s plane.svg to be approximated as a discrete-time filter]]

File:Chebyshev mapped z plane.svg

The matched Z-transform method, also called the pole–zero mapping{{cite book

| title = Signals and Systems with MATLAB

| author = Won Young Yang

| publisher = Springer

| year = 2009

| isbn = 978-3-540-92953-6

| page = 292

| url = https://books.google.com/books?id=GnfpELDfzmEC&pg=PA292

}}

{{cite book

| title = Space vehicle dynamics and control

| author = Bong Wie

| publisher = AIAA

| year = 1998

| isbn = 978-1-56347-261-9

| page = 151

| url = https://books.google.com/books?id=n97tEQvNyVgC&pg=PA151

}} or pole–zero matching method,{{cite book

| title = Design and analysis of control systems

| author = Arthur G. O. Mutambara

| publisher = CRC Press

| year = 1999

| isbn = 978-0-8493-1898-6

| page = 652

| url = https://books.google.com/books?id=VSlHxALK6OoC&pg=PA652

}} and abbreviated MPZ or MZT,{{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}} 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 s-plane design to z-plane locations $z=e^\{sT\}$, for a sample interval $T=1\; /\; f\_\backslash mathrm\{s\}$.

{{cite book

| title = Signal processing: principles and implementation

| author = S. V. Narasimhan and S. Veena

| publisher = Alpha Science Int'l Ltd.

| year = 2005

| isbn = 978-1-84265-199-5

| page = 260

| url = https://books.google.com/books?id=8UbV8vq8uV0C&pg=PA260

}} So an analog filter with transfer function:

:$H(s)\; =\; k\_\{\backslash mathrm\; a\}\; \backslash frac\{\backslash prod\_\{i=1\}^M\; (s-\backslash xi\_i)\; \}\{\backslash prod\_\{i=1\}^N\; (s-p\_i)\; \}$

is transformed into the digital transfer function

:$H(z)\; =\; k\_\{\backslash mathrm\; d\}\; \backslash frac\{\; \backslash prod\_\{i=1\}^M\; (1\; -\; e^\{\backslash xi\_iT\}z^\{-1\})\}\{\; \backslash prod\_\{i=1\}^N\; (1\; -\; e^\{p\_iT\}z^\{-1\})\}$

The gain $k\_\{\backslash mathrm\; d\}$ must be adjusted to normalize the desired gain, typically set to match the analog filter's gain at DC by setting $s=0$ and $z=1$ and solving for $k\_\{\backslash mathrm\; d\}$.{{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.}}

Since the mapping wraps the s-plane's $j\backslash omega$ 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.{{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}}

In the (common) case that the analog transfer function has more poles than zeros, the zeros at $s=\backslash infty$ may optionally be shifted down to the Nyquist frequency by putting them at $z=-1$, causing the transfer function to drop off as $z\; \backslash rightarrow\; -1$ in much the same manner as with the bilinear transform (BLT).

While this transform preserves stability and minimum phase, it preserves neither time- nor frequency-domain response and so is not widely used.{{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.}} More common methods include the BLT 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").{{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]

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 controllable system; in general from an unstable (or nearby) location to a stable location.

File:Chebyshev responses.svg