Editing Linear time-invariant system (section)

==== Stability ====
{{Main|BIBO stability}}
A system is '''bounded input, bounded output stable''' (BIBO stable) if, for every bounded input, the output is finite.  Mathematically, if
<math display="block">\|x[n]\|_{\infty} < \infty</math>

implies that
<math display="block">\|y[n]\|_{\infty} < \infty</math>

(that is, if bounded input implies bounded output, in the sense that the [[Infinity norm|maximum absolute values]] of <math>x[n]</math> and <math>y[n]</math> are finite), then the system is stable.  A necessary and sufficient condition is that <math>h[n]</math>, the impulse response, satisfies
<math display="block">\|h[n]\|_1 \mathrel{\stackrel{\text{def}}{=}} \sum_{n = -\infty}^\infty |h[n]| < \infty.</math>

In the frequency domain, the [[region of convergence]] must contain the [[unit circle]] (i.e., the [[locus (mathematics)|locus]] satisfying <math>|z| = 1</math> for complex ''z'').