BeginnerTopic 30 of 101Signals, Control & Comms
Z-transform
Discrete counterpart of Laplace; poles inside the unit circle mean stability.
In depth
X(z)=Σx[n]z^{-n}: the discrete analogue of Laplace. Poles inside the unit circle mean stability; H(z)=Y/X gives difference-equation structure; the frequency response is H(e^{jω}). Region of convergence defines causality. Inverse via partial fractions or long division.
Key points
- z⁻¹ is one-sample delay
- Unit circle = jω axis
- Bilinear transform links s and z
ExampleH(z)=1/(1-0.5z⁻¹) is stable with a pole at z=0.5.
What is Z-transform?
Discrete counterpart of Laplace; poles inside the unit circle mean stability.
Why is it important to understand Z-transform?
The z-transform describes sampled systems, so it is used to design and check digital filters and controllers.
When is a discrete system stable?
When all poles lie inside the unit circle in the z-plane, so the response decays rather than grows.
How does z relate to s?
z equals e to the power of s times the sampling period, which links continuous and discrete designs.
How is Z-transform used in practice?
Typical applications include analysing the stability of a digital control loop from its poles, designing a difference equation for an IIR filter and converting a continuous controller into a discrete one for a microcontroller.
What should I learn before Z-transform?
Study "Windowing" first (topic 29 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Z-transform?
z⁻¹ is one-sample delay. Unit circle = jω axis. Bilinear transform links s and z.
Can you give an example of Z-transform?
H(z)=1/(1-0.5z⁻¹) is stable with a pole at z=0.5.
What comes after Z-transform?
The next topic is "FIR filters", which builds directly on this one.
How difficult is Z-transform?
It is classed as Beginner in Signals, Control & Comms (topic 30 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.