Lesson 12

The *z*-Transform

The* z*-transform is the Discrete-Time counterpart of the Laplace Transform.

It is

- 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.

- The DT Fourier Transform exists for a signal if the ROC
includes 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.