Editing Group delay and phase delay (section)

=== LTI system response to complex sinusoid ===
More generally, it can be shown that for an LTI system with transfer function <math>\displaystyle H(s)</math> driven by a [[phasor|complex sinusoid]] of unit amplitude,

: <math> x(t) = e^{i \omega t} \ </math>

the output is

: <math> \begin{align}
 y(t) & = H(i \omega) \ e^{i \omega t} \ \\
      & = \left( \big| H(i \omega) \big| e^{i \phi(\omega)} \right) \ e^{i \omega t} \ \\
      & = \big| H(i \omega) \big| \ e^{i \left(\omega t + \phi(\omega) \right)} \ \\
\end{align} \ </math>

where the phase shift <math>\displaystyle \phi</math> is

: <math> \phi(\omega) \ \stackrel{\mathrm{def}}{=}\ \arg \left\{ H(i \omega) \right\}  \;. </math>