ArticleslgStudy

science

Matched Z-transform method

Matched Z-transform method is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Matched Z-transform method rather than just read about it. In short: 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 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…

Matched Z-transform method — main illustration
Matched Z-transform method — illustration

Key takeaways

  • Matched Z-transform method belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Matched Z-transform method to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Matched Z-transform method from memory before moving on to harder problems.

Reference excerpt

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 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} }} . 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} }} . 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. 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). While this transform preserves stability and minimum phase, it preserves neither time- nor frequency-domain response and so 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"). 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.

References

Illustrations

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 s-plane poles and zeros of a 5th-order Chebyshev type II lowpass filter to be approximated as a discrete-time filter
Matched Z-transform method: 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 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.
Matched Z-transform method: 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.
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.

Worked examples

Example 1 — a first encounter with Matched Z-transform method

Start with the simplest possible case. Write down what Matched Z-transform method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Matched Z-transform method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Matched Z-transform method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Matched Z-transform method

In research
Matched Z-transform method appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Matched Z-transform method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Matched Z-transform method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Digital signal processing, Filter theory, so understanding it makes those chapters shorter.
In everyday life
Look for Matched Z-transform method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Matched Z-transform method in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Matched Z-transform method means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Matched Z-transform method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Matched Z-transform method in simple terms?

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 filter design to a discrete-time filter (digital filter) design. The method works by mapping all poles and zeros of the s-plane…

Why does Matched Z-transform method matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Matched Z-transform method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Matched Z-transform method.

Tags

  • Control theory
  • Digital signal processing
  • Filter theory

Keep exploring