Hi Greg,
In the image below the red and pink curves show phase (degrees) , goup delay (mS) and group delay (periods) for a 16th order LR allpass.
The purple curve is the spectrum of a tone burst that we will be using for a test signal.
The blue curves in the group delay charts are actually phase delay for the 16th order LR allpass filter so you can compare it to the group delay (red curves) in the same charts.
Phase Delay=-Phase(periods)/Frequency(hz)
Note since phase delay is (-phase/frequency) and delay in periods is (delay*frequency) phase delay in periods is simply (-phase).
Looking at the charts you can see considerable divergence between group delay and phase delay.
I used a higher order allpass filter (16th) to make the difference very clear.
The importance of delay to an audio system is in the relative timing of one area of the spectrum vs. another.
Delay of the entire spectrum is meaningless.
It doesn't matter when you play your favorite Brittney Spears tunes if sound comes from the speakers immediatly upon pressing PLAY or 10 seconds later so long as the entire spectrum is syncronized.
Sounds from a single instrument can be very wideband and the envelope of it's entire spectrum must be syncronized.
You could also have multiple narrowband instruments playing notes in different octaves but in unison and their timing must remain intact.
To get things started lets look at our tone burst passing through our allpass filter (red curve).
It's spectrum is centered at the allpass filters Fc (632hz).
On the same chart we will look at the tone burst not passing through the allpass filter but instead delayed by the calculated delay (3.15mS) using the phase delay approach (blue curve).
The pink curve and the light blue curve are the envelopes of the tone bursts shown to make the timing and shape of the bursts more clear.
Well the good news is that the phase lines up perfectly between the tone burst from the AP filter and the PD (phase delay method) delayed tone burst.
The bad news is that tone burst passing through the AP filter did not arrive when predicted by the PD method and the whole idea is to be able to indicate the timing for each area within the spectrum.
Lets do that again but this time the tone burst that does not pass through the AP filter will be delayed by the time calculated using the GD (group delay) method.
The red curve is still the tone burst passing through the AP filter.
The green curve is the tone burst not passing through the AP filter but delayed by the time calculated using the GD method (4.5mS).
Again the light colored curves are the envelopes for clarity.
Good news this time is that the tone burst passing through the AP filter arrived when predicted by the GD method.
The bad news is that the phase between the tone burst passing through the AP filter and the GD delayed tone burst don't match up.
Fortunately phase is not relevant to our purpose as discussed in my last post where the operation of the ear is explained.
Your ears respond to the envelope of a given frequency band phase does not matter.
You can now make the obsevation that when passing through a filter with non-linear phase the phase of a signal can be converted by the filter to a value that does not directly correlate with the amount of time signals passing through the filter are delayed in every part of the spectrum.
Group delay gives an estimate of the time delay of the envelope of a narrowband, modulated signal.
Not exactly true although there are specific reasons why this is said.
First, group delay is not an estimate but actually an exact amount of time that signals are delayed for each area of the spectrum.
The reason you would say estimate is that if phase is not linear in the part of the spectrum you are considering then group delay will be changing with frequency.
If group delay is changing over the bandwidth of a particular signal then the parts of the signal will arrive at different times depending upon frequency and the signal will be altered.
You could pick a time in the middle of the range of times over the bandwidth of the signal and call it an estimate but really the signal has not been delayed one specific amount of time, but has in fact been delayed over range of times given exactly by the group delay at each frequency.
Again if group delay changes over the bandwidth of a signal that signal will be altered.
You can see in the chart above that compares the GD method delayed tone burst (green) to the tone burst passing through the AP filter (red) that the AP filter has slightly altered the shape of the tone burst.
The center of the red tone bursts envelope is delayed less than the center of the envelope of the green tone burst but the ends of the red tone bursts envelope actually have more delay than the ends of the envelope of the green tone burst.
The AP filtered signal has been altered although very slightly so it will not line up exactly with the delayed signal with any one amount of delay because they no longer are exactly the same.
Group delay, on the other hand, is a derived quantity that really only has application to the envelopes of narrowband modulated signals.
So why do we even speak of group delay?
Group delay actually is always group delay no matter what the bandwidth.
They say narrowband because of the alteration to the signal that occurs if group delay changes over the bandwidth of the signal.
If you consider ever narrower bands of the spectrum group delay will vary less until you reach a point where there is no signifigant change in group delay over the entire band.
For that reason the narrower the bandwidth of a signal the less it will be altered when passing through a filter.
The tone burst in the above examples has a narrow bandwidth and passes through the AP filter almost unchanged exept for phase.
If you were to send a wideband signal through our 16th order LR AP filter with it's volatile group delay curve parts of that signals spectrum would be scattered over a range of delay times.
No one delay time could match that signal up with what comes out of the filter because what comes out the filter would no longer be the same thing (apple in orange out).
That is why they say group delay applies to narrow band signals, it's because one delay time could not explain the arrival of a wide band signal passing through a filter with a volatile group delay curve.
However no other type of delay specification could either.
In audio however we are not using group delay to calculate a single delay time that tells us exactly when a signal will arrive.
Instead we are using group delay to calculate the range of times over which a signals spectra will be scattered so we can quantify to what extent the system will alter that signal, which means the above considerations don't apply.
Group delay really is the one that gives the actual delay.
The above example predicting the arrival time of the tone burst shows it.
Let me explain further, this post isn't nearly long enough (I know because AC's hard drives aren't full yet so how could it be)...
Energy at a single frequency can never change amplitude or phase, ever!
Likewise time has no meaning whatsoever at a single frequency.
For any change in a signal to occur their must be energy at more than a single frequency.
This is where the 'group' in group delay comes from.
No information can be sent without a group of frequencies present, ever!
With just two frequencies present you can have a modulated sinusiod.
It will have events occuring at specific points in time.
Like when the minimum level is reached (null) and the maximum (peak envelope).
If you introduce a phase differential between the two frequency componets one up in phase and the other down in phase the phase will remain the same for the modulated sinusoid.
But now the the time where it's event's occur is changed.
The amount of time shift is simply the negative of the difference in phase divided by the difference in frequency or group delay=-d(phase(periods))/d(frequency(hz)).
Adjusting the phase of each frequency componet the same amount in the same direction has no effect upon the signal except that the phase has been changed.
The null will still occur at the same time and so will the peak envelope (all events actually).
The more frequencies present in the signal the more complicated the signal can be.
Real signals like "Oops I Did it Again" by Brittney Spears will have countless frequency componets throughout the spectrum.
Rearranging these componets phase will alter the signals event timing (every drum beat, cymbal crash and keyboard note) as given by the group delay equation.
The variation in group delay time over the spectrum indicates exactly how that signals spectra are scattered in time.
Phase delay simply tells what amount of delay time that would be required to produce the same phase shift at a particular frequency.
This delay time is not necessarily the amount of delay the signal will experience like would be arrived at with group delay it is only an amount of delay time that will give the same phase shift at a particular frequency.
As was shown above a filter with non-linear phase will cuase phase shift that does not directly correlate with it's delay time in every part of the spectrum.
Daryl
