Editing Group delay and phase delay (section)

=== Mathematical definition of group delay and phase delay ===
The '''group delay''', <math>\displaystyle \tau_g</math>, and '''phase delay''', <math>\displaystyle \tau_\phi</math>, are (potentially) frequency-dependent<ref name="Ambardar1999" /> and can be computed from the [[phase unwrapping|unwrapped]] phase shift <math>\displaystyle \phi( \omega )</math>. The '''phase delay''' at each frequency equals the negative of the phase shift at that frequency divided by the value of that frequency:

: <math> \tau_\phi(\omega) = - \frac{\phi(\omega)}{\omega} \, . </math>
The '''group delay''' at each frequency equals the negative of the ''slope'' (i.e. the [[derivative]] with respect to frequency) of the phase at that frequency:<ref name="OppenheimWillskyNawab1997" />
: <math> \tau_g(\omega) = - \frac{d \phi(\omega)}{d \omega} \, . </math>

In a [[linear phase]] system (with non-inverting gain), both <math>\displaystyle \tau_g</math> and <math>\displaystyle \tau_\phi</math> are constant (i.e., independent of <math>\displaystyle \omega</math>) and equal, and their common value equals the overall delay of the system; and the unwrapped [[Phase (waves)|phase shift]] of the system (namely <math>\displaystyle -\omega \tau_\phi</math>) is negative, with magnitude increasing linearly with frequency <math>\displaystyle \omega</math>.