Matched Z-transform method

The s-plane poles and zeros of a 5th-order Chebyshev type II lowpass filter to be approximated as a discrete-time filter
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=eiωT, 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[1][2] or pole–zero matching method,[3] and abbreviated MPZ or MZT,[4] 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 s T {\displaystyle z=e^{sT}} , for a sample interval T = 1 / f s {\displaystyle T=1/f_{\mathrm {s} }} .[5] So an analog filter with transfer function:

H ( s ) = k a ∏ i = 1 M ( s − ξ i ) ∏ i = 1 N ( s − p i ) {\displaystyle H(s)=k_{\mathrm {a} }{\frac {\prod _{i=1}^{M}(s-\xi _{i})}{\prod _{i=1}^{N}(s-p_{i})}}}

is transformed into the digital transfer function

H ( z ) = k d ∏ i = 1 M ( 1 − e ξ i T z − 1 ) ∏ i = 1 N ( 1 − e p i T z − 1 ) {\displaystyle H(z)=k_{\mathrm {d} }{\frac {\prod _{i=1}^{M}(1-e^{\xi _{i}T}z^{-1})}{\prod _{i=1}^{N}(1-e^{p_{i}T}z^{-1})}}}

The gain k d {\displaystyle k_{\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 {\displaystyle s=0} and z = 1 {\displaystyle z=1} and solving for k d {\displaystyle k_{\mathrm {d} }} .[3][6]

Since the mapping wraps the s-plane's j ω {\displaystyle j\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.[7]

In the (common) case that the analog transfer function has more poles than zeros, the zeros at s = ∞ {\displaystyle s=\infty } may optionally be shifted down to the Nyquist frequency by putting them at z = − 1 {\displaystyle z=-1} , causing the transfer function to drop off as z → − 1 {\displaystyle z\rightarrow -1} in much the same manner as with the bilinear transform (BLT).[1][3][6][7]

While this transform preserves stability and minimum phase, it preserves neither time- nor frequency-domain response and so is not widely used.[8][7] More common methods include the BLT and impulse invariance methods.[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").[9]

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.

Responses of the filter (dashed), and its discrete-time approximation (solid), 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

  1. ^ a b Won Young Yang (2009). Signals and Systems with MATLAB. Springer. p. 292. ISBN 978-3-540-92953-6.
  2. ^ Bong Wie (1998). Space vehicle dynamics and control. AIAA. p. 151. ISBN 978-1-56347-261-9.
  3. ^ a b c Arthur G. O. Mutambara (1999). Design and analysis of control systems. CRC Press. p. 652. ISBN 978-0-8493-1898-6.
  4. ^ a b Al-Alaoui, M. A. (February 2007). "Novel Approach to Analog-to-Digital Transforms". IEEE Transactions on Circuits and Systems I: Regular Papers. 54 (2): 338–350. doi:10.1109/tcsi.2006.885982. ISSN 1549-8328. S2CID 9049852.
  5. ^ S. V. Narasimhan and S. Veena (2005). Signal processing: principles and implementation. Alpha Science Int'l Ltd. p. 260. ISBN 978-1-84265-199-5.
  6. ^ a b Franklin, Gene F. (2015). Feedback control of dynamic systems. Powell, J. David, Emami-Naeini, Abbas (Seventh ed.). Boston: Pearson. pp. 607–611. ISBN 978-0133496598. OCLC 869825370. Because physical systems often have more poles than zeros, it is useful to arbitrarily add zeros at z = -1.
  7. ^ a b c Rabiner, Lawrence R; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, New Jersey: Prentice-Hall. pp. 224–226. ISBN 0139141014. 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.
  8. ^ Jackson, Leland B. (1996). Digital Filters and Signal Processing. Springer Science & Business Media. p. 262. ISBN 9780792395591. 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.
  9. ^ Ojas, Chauhan; David, Gunness (2007-09-01). "Optimizing the Magnitude Response of Matched Z-Transform Filters ("MZTi") for Loudspeaker Equalization". Audio Engineering Society. Archived from the original on July 27, 2019. Alt URL
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