Phase delay

Whenever we put a sinusoid into a filter, a sinusoid must come out. The only things that can change between input and output are the amplitude and the phase. Comparing a zero crossing of the input to a zero crossing of the output measures the so-called phase delay. To quantify this, define an input, $\sin \omega t$,and an output, $\sin (\omega t - \phi )$.Then the phase delay t_p is found by solving  

$$\begin{array} {rcl} \sin( \omega t - \phi ) &=& \sin \omega ... ... \omega t_p \nonumber \ t_p &=& \frac{\phi}{\omega}\end{array}$$

A problem with phase delay is that the phase can be ambiguous within an additive constant of $2\pi N$, where N is any integer. In wave-propagation theory, ``phase velocity" is defined by the distance divided by the phase delay. There it is hoped that the $2\pi N$ ambiguity can be resolved by observations tending to zero frequency or physical separation.