Editing Group delay and phase delay (section)

=== LTI system response to wave packet ===
Suppose that such a system is driven by a wave packet formed by a [[Sine wave|sinusoid]] multiplied by an amplitude envelope <math>\displaystyle A_\text{env}(t)>0</math>, so the input <math>\displaystyle x(t)</math> can be expressed in the following form:

: <math> x(t) = A_\text{env}(t) \cos(\omega t + \theta) \, . </math>

Also suppose that the envelope <math>\displaystyle A_\text{env}(t)</math> is slowly changing relative to the sinusoid's frequency <math>\displaystyle \omega</math>. This condition can be expressed mathematically as:

: <math> \left| \frac{d}{dt} \log \big( A_\text{env}(t) \big) \right| \ll \omega \ .</math>

Applying the earlier convolution equation would reveal that the output of such an LTI system is very well approximated{{Clarification needed|reason=Some of the missing math steps (maybe as a footnote) would be nice here to show how the earlier condition allows for this to be "very well approximated".|date=June 2023}} as:

: <math> y(t) = \big| H(i \omega) \big| \ A_\text{env}(t - \tau_g) \cos \big( \omega (t - \tau_\phi) + \theta \big) \; .</math>

Here <math>\displaystyle \tau_g</math> is the group delay and <math>\displaystyle \tau_\phi</math> is the phase delay, and they are given by the expressions below (and potentially are functions of the [[angular frequency]] <math>\displaystyle \omega</math>). The phase of the sinusoid, as indicated by the positions of the zero crossings, is delayed in time by an amount equal to the phase delay, <math>\displaystyle \tau_\phi</math>. The envelope of the sinusoid is delayed in time by the group delay, <math>\displaystyle \tau_g</math>.