Matched Z-transform method

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 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=e^iωT, to show how frequencies along the iω axis in the s-plane map onto the unit circle in the z-plane.

The method works by mapping all poles and zeros of the s-plane design to z-plane locations ${\displaystyle z=e^{sT}}$, for a sample interval ${\displaystyle T=1/f_{\mathrm {s} }}$.^[5] So an analog filter with transfer function:

${\displaystyle H(s)=k_{\mathrm {a} }{\frac {\prod _{i=1}^{M}(s-\xi _{i})}{\prod _{i=1}^{N}(s-p_{i})}}}$

is transformed into the digital transfer function

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

The gain ${\displaystyle k_{\mathrm {d} }}$ must be adjusted to normalize the desired gain, typically set to match the analog filter's gain at DC by setting ${\displaystyle s=0}$ and ${\displaystyle z=1}$ and solving for ${\displaystyle k_{\mathrm {d} }}$.^[3][6]

Since the mapping wraps the s-plane's ${\displaystyle j\omega }$ 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 ${\displaystyle s=\infty }$ may optionally be shifted down to the Nyquist frequency by putting them at ${\displaystyle z=-1}$, causing the transfer function to drop off as ${\displaystyle z\rightarrow -1}$ 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.