Editing BIBO stability (section)

=== Discrete-time signals ===

For a [[rational function|rational]] and [[discrete signal|discrete time system]], the condition for stability is that the [[Laplace transform#Region of convergence|region of convergence]] (ROC) of the [[z-transform]] includes the [[unit circle]]. When the system is [[Causal system|causal]], the ROC is the [[open region]] outside a circle whose radius is the magnitude of the [[pole (complex analysis)|pole]] with largest magnitude. Therefore, all poles of the system must be inside the [[unit circle]] in the [[Z-transform|z-plane]] for BIBO stability.

This stability condition can be derived in a similar fashion to the continuous-time derivation:

:<math>
\begin{align}
\sum_{n = -\infty}^\infty \left|h[n]\right|
& = \sum_{n = -\infty}^\infty \left|h[n]\right| \left| e^{-j \omega n} \right| \\
& = \sum_{n = -\infty}^\infty \left|h[n] (1 \cdot e)^{-j \omega n} \right| \\
& =\sum_{n = -\infty}^\infty \left|h[n] (r e^{j \omega})^{-n} \right| \\
& = \sum_{n = -\infty}^\infty \left|h[n] z^{- n} \right|
\end{align}
</math>

where <math>z = r e^{j \omega}</math> and <math>r = |z| = 1</math>.

The [[Laplace transform#Region of convergence|region of convergence]] must therefore include the [[unit circle]].