![Chebyshev s plane](https://upload.wikimedia.org/wikipedia/commons/6/68/Chebyshev_s_plane.svg)
The matched Z-transform method, also called the pole–zero mapping
or pole–zero matching method,
and abbreviated MPZ or MZT,
is a technique for converting a
continuous-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
filter design to a
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
filter (
digital filter
In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...
) design.
The method works by mapping all poles and zeros of the
''s''-plane design to
''z''-plane locations
, for a sample interval
. So an analog filter with transfer function:
:
is transformed into the digital transfer function
:
The gain
must be adjusted to normalize the desired gain, typically set to match the analog filter's gain at DC by
setting and and solving for
.
Since the mapping wraps the ''s''-plane's
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.
In the (common) case that the analog transfer function has more poles than zeros, the zeros at
may optionally be shifted down to the Nyquist frequency by putting them at
, causing the transfer function to drop off as
in much the same manner as with the
bilinear transform
The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear t ...
(BLT).
While this transform preserves
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
**Asymptotic stability
**Linear stability
**Lyapunov stability
**Orbital stability
**Structural stabilit ...
and
minimum phase
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The most general causal LTI transfer function can be uniquely factored into a series of a ...
, it preserves neither time- nor frequency-domain response and so is not widely used.
More common methods include the BLT and
impulse invariance
Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discre ...
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").
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A specific application of the ''matched Z-transform method'' in the digital control field is with the Ackermann's formula
In control theory, Ackermann's formula is a control system design method for solving the pole allocation problem for invariant-time systems by Jürgen Ackermann. One of the primary problems in control system design is the creation of controller ...
, which changes the poles of the controllable Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observabi ...
system; in general from an unstable (or nearby) location to a stable location.
References
{{DSP
Control theory
Digital signal processing
Filter theory