Editing Linear system (section)

==The convolution integral==

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:
<math display="block"> y(t) = \int_{-\infty}^{t}  h(t,t') x(t')dt' = \int_{-\infty}^{\infty}  h(t,t') x(t') dt' </math>

If the properties of the system do not depend on the time at which it is operated then it is said to be '''time-invariant''' and {{mvar|h}} is a function only of the time difference {{math|1=''τ'' = ''t'' − ''t' ''}} which is zero for {{math|''τ'' < 0}} (namely {{math|''t'' < ''t' ''}}). By redefinition of {{mvar|h}} it is then possible to write the input-output relation equivalently in any of the ways,
<math display="block"> y(t) = \int_{-\infty}^{t}  h(t-t') x(t') dt' = \int_{-\infty}^{\infty}  h(t-t') x(t') dt' = \int_{-\infty}^{\infty}  h(\tau) x(t-\tau) d \tau  = \int_{0}^{\infty}  h(\tau) x(t-\tau) d \tau </math>

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called  the ''transfer function'' which is:
<math display="block">H(s) =\int_0^\infty  h(t) e^{-st}\, dt.</math>

In applications this is usually a rational algebraic function of {{mvar|s}}. Because {{math|''h''(''t'')}} is zero for negative {{mvar|t}}, the integral may  equally be written over the doubly infinite range and putting {{math|1=''s'' = ''iω''}} follows the formula for the ''frequency response function'':
<math display="block"> H(i\omega) = \int_{-\infty}^{\infty}  h(t) e^{-i\omega t} dt </math>