Lecture 21 - Notes

March 2, 2016

Z-Transform

Properties

The Z-transforms properties strictly apply to a certain region of convergence.

• Causal: $H(z)$ has a region of convergence representing the exterior of a circle including $z = \infty$.
• Anti-Causal: $H(z)$ has a region of convergence representing the interior of a circle including $z = 0$.
• Stable: The region of convergence of $H(z)$ includes the unit circle.

Multiple Regions of Convergence

Suppose,

$$\begin{aligned} \left|z_1\right| \le \left|z_2\right| \le \ldots \le \left|z_k\right| \end{aligned}$$

the in general the regions of convergence will be,

$$\begin{aligned} \text{ROC}_1 &\to z \lt \left|z_1\right| \\\\ \text{ROC}_2 &\to \left|z_1\right| \lt z \lt \left|z_2\right| \\\\ &~~\vdots \\\\ \text{ROC}_k &\to z \gt \left|z_k\right| \\\\ \end{aligned}$$