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

1. Lines on the s-plane map to circles on the z-plane.

2. Role of -axis is replaced by unit circle, so
1. The DT Fourier Transform exists for a signal if the ROC includes the unit circle.

2. A stable system must have an ROC that contains the unit circle.
3. 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.