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Group Delay 101

DonH56

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For phase p and frequency f group delay GD is the negative of the derivative of the change in phase with respect to the change in frequency: GD = -dp/df. In general phase is a function of frequency p(f); that is, phase changes with frequency. That's the math behind it.

A change in phase is equivalent to a time shift, for example a 180-degree phase shift is like shifting the time by one-half cycle of the signal. The derivative is a fancy expression for the slope of a line, so it is a measure of phase linearity. A straight (linear) line has the form y = mx+b where every point x is multiplied by the slope m and added to offset b to produce a y value. For a straight line, m is constant (just a number, not a function of something else), and hopefully we remember this formula from school. Now replace m with GD so to get a straight line, that means the change in phase divided by the change in frequency must be constant, meaning group delay is a constant, and every frequency is delayed by the same amount of time. What goes in, comes out again, exactly as it was but just a little later in time.

Now, a pulse, or musical signal, includes many frequencies. If we send the signal through a component like an amplifier or speaker with constant group delay, then every frequency is delayed the same amount, and the output is just like the input except delayed in time. If the group delay is not constant, that means different frequencies have different delays through the component, so at the other end the signal will be "smeared" in time with different frequencies arriving at different times. Things like transient attacks from drums or instruments will not be as clean. There are various studies discussing just how far off the delay can be at different frequencies before we notice it, some referenced in the Wikipedia article mentioned previously (https://en.wikipedia.org/wiki/Group_delay_and_phase_delay).

A straight line of y = mx with m = 1 looks like the blue line below. If I make the slope (m) depend upon the input (x) we get the orange line, which is no longer perfectly straight. The next plot shows the error. This is what happens when group delay is not constant, the (phase) line is no longer straight.
1668966520819.png

1668966540322.png


Now try with some signals, in this case the first five frequencies in a square wave. The amplitudes decrease as frequency goes up and only odd harmonics are used (see https://www.audiosciencereview.com/.../composition-of-a-square-wave-important.1921/). Notice how all nine signals line up in the middle and again at the end.
1668966560378.png


If all the signals arrive at the same time, that is group delay is constant, then this is what you see when you add them all up:
1668966575373.png


A perfect square wave has frequencies out to infinity, and this is just five, so it does not have perfectly sharp (straight) edges and the top and bottom are not perfectly flat. It is perfectly symmetric, however, with edges and top and bottom the same across the entire period.

Now adjust the group delay so it is not constant but instead each successively higher frequency is shifted just a little bit more. Notice how the signals no longer align perfectly in the middle and at the end but are spread out a bit:
1668966595777.png


The output when we sum them all looks different and is no longer symmetric:
1668966610978.png


We can see the difference more clearly by showing them both on the same plot:
1668966638373.png


Hopefully this helps visualize how group delay can impact the signal, and why constant group delay is a typical design goal. - Don
 
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Doodski

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@DonH56 maybe a mention of the square waves being comprised of only odd order waveforms might help the reader for now and later.

EDIT: It's in the graph... :D
 
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DonH56

DonH56

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Thanks Don. Good to see you posting these fundamental articles. Promoted to home page.
Thanks Amir, life's been pretty crazy lately, not much time to post. This one fell out of a different thread and I decided to make sure I didn't lose it.
@DonH56 maybe a mention of the square waves being comprised of only odd order waveforms might help the reader for now and later.

EDIT: It's in the graph... :D
I forgot, sorry, added to the post. I did link to the earlier thread on building a square wave.
 

restorer-john

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@DonH56 Your example surely shows a negative group delay with harmonics arriving before the fundamental.
 
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deadwood83

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For phase p and frequency f group delay GD is the negative of the derivative of the change in phase with respect to the change in frequency: GD = -dp/df.
Thank you for posting it in a way I can understand. Spelling it out in words is one heck of a thing to read!

Thank you also for helping explain the REW tab!
 
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DonH56

DonH56

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@DonH56 Your example surely shows a negative group delay with harmonics arriving before the fundamental.
Yes, you are right (of course). I realized that after I set up the equations and noticed I left a delay in the fundamental when I was building the vectors. My fault; I usually do this with a time vector, but for this one I set it up in the phase domain. The concept is the same and I didn't have time to redo everything (had to get to rehearsal this afternoon). Besides, causality is for wimps. :) I figured it would get noticed, natch, one stupid sign change...
 

mARCELOCM

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what about the group delay from the speaker itself?
can you explain whats occurs with sealed and vented box?
 

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voodooless

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what about the group delay from the speaker itself?
can you explain whats occurs with sealed and vented box?
That’s purely the difference in frequency response. EQ (with minimum phase) or align them equally, and group delay will be the same as well.
 

mosco

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Even with the translator I understand. Thank you friend and thank you Amir for so much enjoyment.
 

MarcosCh

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As usual, thanks a lot for the article. I had read about group delay a few times (headphone reviews?) but had no idea what it meant.
A couple of questions.
Could someone explain what is the physical reason behind non constant group delay?
When not constant, does it always increase with frequency? In other words, is the second derivative always positive like in the examples?
Last one. Why do I have the impression I only saw group delay measurements in ASR for headphones? Are headphones particularly prone to non constant group delay?
Thanks again!
 

pma

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I have put this example into another thread, maybe it would help. Constant or non-constant group delay depends on phase response and this depends on amplitude response. Please see below amplitude response, phase response and group delay of the 5th order low pass filter. GD tells signal time delay in the pass band, in this case. It is the different kind of mathematical representation of the same thing.

LPF.png


FrPhGd.png
 

voodooless

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Constant or non-constant group delay depends on phase response and this depends on amplitude response.
The caveat here is that this is true for a minimum phase system, so basically, anything involving regular analog electronics or physical systems like loudspeakers. When using FIR filtering, just about anything goes.
 

Lunafag

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It's cool to look at but not really audible, even in extreme cases
And in the treble air itself will rotate the phase, Klippel says that a temperature change of only 2°C will result in a phase error of 180 degree at 10kHz in 5m measurement distance. Our ears aren't really built to hear that, if the shift is the same in both channels of course.
 

voodooless

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@DonH56, do you have audio files of your two example square-ish waves? Would be fun to see how different they sound?
 

fpitas

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Does rePhase make a difference or not?
It depends. Based on an admittedly "sighted" test, we could hear flattened phase at very low frequencies (below 200Hz) but not at the 800Hz LR4 crossover of a friend's speaker. The difference in any event was subtle. The bass was "tighter" to use that overworked term.
 
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