Lecture 18 - Notes

February 24, 2016

Region of Convergence

defintion: The region of convergence, $\mathcal S \subset \mathbb C$ of the Z-transform of $x[n]$ is,

$$\begin{aligned} \mathcal S = \left\{ z \in \mathbb C: \sum_{n = -\infty}^{\infty} \left| x[n] z^{-n}\right| \lt \infty \right\} \end{aligned}$$

Example

Compute the Z-Transform for $x[n] = a^n u[n]$

Consider,

$$\begin{aligned} x[n] &= a^n u[n] \\\\ X(z) &= \sum_{n = -\infty}^{\infty} a^n u[n] z^{-n} \\\\ &= \sum_{n = 0}^{\infty} a^n z^{-n} \\\\ &= \sum_{n = 0}^{\infty} \left( a z^{-1} \right)^n \\\\ &= \frac{1}{1 - a z^{-1}} \\\\ &= \frac{z}{z - a} \end{aligned}$$

if $\left| az^{-1} \right| \lt 1$.

Causality

$x[n]$ is causal if and only if the Region of Convergence of $X(z)$ goes out to infinity. If the ROC goes in to zero then then $x[n]$ is anti-causal.

Absolute Summability

$x[n]$ is absolutely summable (i.e. stable) if and only if the Region of Convergence of $X(z)$ includes the unit circle ($\left|z\right| = 1$).