Editing BIBO stability (section)

== Frequency-domain condition for linear time-invariant systems==
=== Continuous-time signals ===

For a [[rational function|rational]] and [[continuous function|continuous-time system]], the condition for stability is that the [[region of convergence]] (ROC) of the [[Laplace transform]] includes the [[complex plane|imaginary axis]]. When the system is [[Causal system|causal]], the ROC is the [[open region]] to the right of a vertical line whose [[abscissa]] is the [[real part]] of the "largest pole", or the [[pole (complex analysis)|pole]] that has the greatest real part of any pole in the system. The real part of the largest pole defining the ROC is called the [[abscissa of convergence]]. Therefore, all poles of the system must be in the strict left half of the [[s-plane]] for BIBO stability.

This stability condition can be derived from the above time-domain condition as follows:

:<math>
\begin{align}
\int_{-\infty}^\infty \left|h(t)\right| \, dt
& = \int_{-\infty}^\infty \left|h(t)\right| \left| e^{-j \omega t }\right| \, dt \\
& = \int_{-\infty}^\infty \left|h(t) (1 \cdot e)^{-j \omega t} \right| \, dt \\
& = \int_{-\infty}^\infty \left|h(t) (e^{\sigma + j \omega})^{- t} \right| \, dt \\
& = \int_{-\infty}^\infty \left|h(t) e^{-s t} \right| \, dt
\end{align}
</math>

where <math>s = \sigma + j \omega</math> and <math>\operatorname{Re}(s) = \sigma = 0.</math>

The [[region of convergence]] must therefore include the [[complex plane|imaginary axis]].

=== 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]].