AdvancedTopic 101 of 104Engineering Mathematics
Z-transform and difference equations
The discrete-time counterpart of the Laplace transform for sampled systems.
In depth
The Z-transform X(z)=Σx[n]z^{-n} handles sampled signals and linear difference equations such as y[n]=a y[n-1]+x[n]. Stable systems have poles inside the unit circle. It is the discrete counterpart of the Laplace transform.
Key points
- Delay by one sample = z^{-1}
- Transfer function H(z)
- Frequency response on the unit circle
ExampleA first-order digital low-pass filter is y[n]=0.9y[n-1]+0.1x[n].
What is Z-transform and difference equations?
The discrete-time counterpart of the Laplace transform for sampled systems.
Why is it important to understand Z-transform and difference equations?
The z-transform is the discrete-time counterpart of the Laplace transform, so it is needed for digital filters and controllers.
What is the z-transform used for?
It describes discrete systems and difference equations, and stability is determined by whether poles lie inside the unit circle.
How does z relate to s?
z equals e to the power of s times the sampling period. This mapping is the basis of many digital design methods.
How is Z-transform and difference equations used in practice?
Typical applications include designing a digital low-pass filter from an analogue prototype, analysing the stability of a sampled-data control loop and solving a difference equation for a discrete system.
What should I learn before Z-transform and difference equations?
Study "Graph theory" first (topic 100 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Z-transform and difference equations?
Delay by one sample = z^{-1}. Transfer function H(z). Frequency response on the unit circle.
Can you give an example of Z-transform and difference equations?
A first-order digital low-pass filter is y[n]=0.9y[n-1]+0.1x[n].
What comes after Z-transform and difference equations?
The next topic is "Tensors", which builds directly on this one.
How difficult is Z-transform and difference equations?
It is classed as Advanced in Engineering Mathematics (topic 101 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.