Editing Linear time-invariant system (section)

=== Important system properties ===
Some of the most important properties of a system are causality and stability.  Causality is a necessity for a physical system whose independent variable is time, however this restriction is not present in other cases such as image processing.

==== Causality ====
{{Main|Causal system}}
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A system is causal if the output depends only on present and past, but not future inputs.  A necessary and sufficient condition for causality is
<math display="block">h(t) = 0 \quad \forall t < 0,</math>

where <math>h(t)</math> is the impulse response.  It is not possible in general to determine causality from the [[two-sided Laplace transform]]. However, when working in the time domain, one normally uses the [[Laplace transform|one-sided Laplace transform]] which requires causality.

==== 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 every input satisfying
<math display="block">\ \|x(t)\|_{\infty} < \infty</math>

leads to an output satisfying
<math display="block">\ \|y(t)\|_{\infty} < \infty</math>

(that is, a finite [[Infinity norm|maximum absolute value]] of <math>x(t)</math> implies a finite maximum absolute value of <math>y(t)</math>), then the system is stable.  A necessary and sufficient condition is that <math>h(t)</math>, the impulse response, is in [[Lp space|L<sup>1</sup>]] (has a finite L<sup>1</sup> norm):
<math display="block">\|h(t)\|_1 = \int_{-\infty}^\infty |h(t)| \, \mathrm{d}t < \infty.</math>

In the frequency domain, the [[region of convergence]] must contain the imaginary axis <math>s = j\omega</math>.

As an example, the ideal [[low-pass filter]] with impulse response equal to a [[sinc function]] is not BIBO stable, because the sinc function does not have a finite L<sup>1</sup> norm.  Thus, for some bounded input, the output of the ideal low-pass filter is unbounded.  In particular, if the input is zero for <math>t < 0</math> and equal to a sinusoid at the [[cut-off frequency]] for <math>t > 0</math>, then the output will be unbounded for all times other than the zero crossings.{{dubious|date=September 2020}}