Lesson 14

*z*-Transform Properties

The previous sections focused on the unilateral *z-* transform in
discussing properties, but as it turns out most of the properties
are similar for the two cases. Since they are easier and more
general for the bilateral case, we'll focus on those. We discuss
the most important ones here:

- Linearity:

where the new ROC .

- Time shift:

where the new ROC is the same as with the possible addition or deletion of the origin or infinity.

- Convolution:

where the new ROC .

Linearity and the time shift property will be useful for LCCDE systems, and the convolution property lets us avoid discrete-time convolutions. We'll use these properties a lot.

Convolution in Time

because we have a

Now, let and we get:

The new ROC will depend on both the poles in
and , giving since the ROC cannot include
poles. However, if one transform has a zero that cancels a pole of
the other then the ROC can be bigger, hence
.