filters - What is the difference between phase delay and group delay?

I'm studying some DSP and I'm having trouble understanding the difference between phase delay and group delay.

It seems to me that they both measure the delay time of sinusoids passed through a filter.

• Am I correct in thinking this?


• If so, how do the two measurements differ?


• Could someone give an example of a situation in which one measurement would be more useful than the other?

UPDATE

Reading ahead in Julius Smith's Introduction to Digital Filters, I've found a situation where the two measurements at least give different results: affine-phase filters. That's a partial answer to my question, I guess.

Answer

First of all the definitions are different:

• Phase delay: (the negative of) Phase divided by frequency


• Group delay: (the negative of) First derivative of phase vs frequency

In words that means:

• Phase delay: Phase angle at this point in frequency


• Group delay: Rate of change of the phase around this point in frequency.

When to use one or the other really depends on your application. The classical application for group delay is modulated sine waves, for example AM radio. The time that it takes for the modulation signal to get through the system is given by the group delay not by the phase delay. Another audio example could be a kick drum: This is mostly a modulated sine wave so if you want to determine how much the kick drum will be delayed (and potentially smeared out in time) the group delay is the way to look at it.