Editing Linear time-invariant system (section)

=== Important system properties ===
The input-output characteristics of discrete-time LTI system are completely described  by its impulse response <math>h[n]</math>.
Two of the most important properties of a system are causality and stability. Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time. Unstable systems can also be analyzed and built, but are only useful as part of a larger system whose overall transfer function ''is'' stable.

==== Causality ====
{{Main|Causal system}}
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A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input.<ref>Phillips 2007, p. 508.</ref>   A necessary and sufficient condition for causality is
<math display="block">h[n] = 0 \ \forall n < 0,</math>
where <math>h[n]</math> is the impulse response.  It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique{{dubious|date=September 2020}}.  When a [[region of convergence]] is specified, then causality can be determined.

==== 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'').