Editing BIBO stability (section)

== Time-domain condition for linear time-invariant systems==
===Continuous-time necessary and sufficient condition===
For a [[continuous function|continuous time]] [[LTI system theory|linear time-invariant (LTI)]] system, the condition for BIBO stability is that the [[impulse response]], <math> h(t)</math> , be [[P-integrable function|absolutely integrable]], i.e., its [[Lp space|L<sup>1</sup> norm]] exists.

: <math> \int_{-\infty}^\infty \left|h(t)\right|\,\mathord{\operatorname{d}}t = \| h \|_1 \in \mathbb{R}</math>

===Discrete-time sufficient condition===
For a [[discrete time]] LTI system, the condition for BIBO stability is that the [[impulse response]] be [[P-integrable function|absolutely summable]], i.e., its <math>\ell^1</math> [[Lp space|norm]] exists.

:<math>\ \sum_{n=-\infty}^\infty |h[n]| = \| h \|_1 \in \mathbb{R}</math>

====Proof of sufficiency====
Given a [[discrete mathematics|discrete]] time LTI system with [[impulse response]] <math>\ h[n]</math> the relationship between the input <math>\ x[n]</math> and the output <math>\ y[n]</math> is

:<math>\ y[n] = h[n] * x[n]</math>

where <math>*</math> denotes [[convolution]]. Then it follows by the definition of convolution

:<math>\ y[n] = \sum_{k=-\infty}^\infty h[k] x[n-k]</math>

Let <math>\| x \|_{\infty}</math> be the maximum value of <math>\ |x[n]|</math>, i.e., the [[Supremum norm|<math>L_{\infty}</math>-norm]].

:<math>\left|y[n]\right| = \left|\sum_{k=-\infty}^\infty h[n-k] x[k]\right|</math>

::<math>\le \sum_{k=-\infty}^\infty \left|h[n-k]\right| \left|x[k]\right|</math> (by the [[triangle inequality]])

: <math>
\begin{align}
& \le \sum_{k=-\infty}^\infty \left|h[n-k]\right| \| x \|_\infty \\
& = \| x \|_{\infty} \sum_{k=-\infty}^\infty \left|h[n-k]\right| \\
& = \| x \|_{\infty} \sum_{k=-\infty}^\infty \left|h[k]\right|
\end{align}
</math>

If <math>h[n]</math> is absolutely summable, then <math>\sum_{k=-\infty}^{\infty}{\left|h[k]\right|} = \| h \|_1  \in \mathbb{R}</math> and

:<math>\| x \|_\infty \sum_{k=-\infty}^\infty \left|h[k]\right| = \| x \|_\infty \| h \|_1</math>

So if <math>h[n]</math> is absolutely summable and <math>\left|x[n]\right|</math> is bounded, then <math>\left|y[n]\right|</math> is bounded as well because <math>\| x \|_{\infty} \| h \|_1 \in \mathbb{R}</math>.

The proof for continuous-time follows the same arguments.