The z-transform is the Discrete-Time counterpart of the Laplace Transform.
- Used in Digital Signal Processing
- Used to Define Frequency Response of Discrete-Time System.
- Used to Solve Difference Equations - use algebraic methods as we did for
differential equations with Laplace Transforms; it is easier
to solve the transformed equations since they are algebraic.
We will see that
- Lines on the s-plane map to circles on the z-plane.
- Role of -axis is replaced by unit circle, so
- The DT Fourier Transform exists for a signal if the ROC
includes the unit circle.
- A stable system must have an ROC that contains the unit circle.
- A causal and stable system must have poles inside the unit circle.
Aside: You can relate the z- transform and Laplace
transform directly when you are dealing with sampled signals:
Take a CT signal and sample it:
The Laplace transform of the sampled signal is
by the sifting property.
Let and , then
Thus, the z- transform with is
the same as the Laplace transform of a sampled
signal! Of course, if the signal is already discrete, the notion of
sampling is unnecessary for understanding and using the z- transform.