What is group delay?

Holmz

Major Contributor
I was trying to say, that an analogue crossover is minimum phase, whereas a digital one could be anything...

I seem to recall hearing that the 1st order XO is minimum phase, but I am not sure about the higher order ones?

retroflex

Member
Why is it meaningless? What if you want to know how the input signal is affected, even if it is a single frequency? The group delay curve will help inform, won't it?

Please provide an example that explains this, to help me.
No, if you want to see how a single frequency is affected you should look at the phase delay. If we take the filter from my earlier post (yesterday at 7:56 PM in my timezone) and feed it a single sine wave at 10kHz, we get the following response:

The delay between the input and output is the phase delay (~0.05ms), and not the group delay (~0.3ms). And this is not some trickery where the group delay happens to match up with the period of the sine wave, one can confirm this by looking at the phase response I posted of the filter.

I'm not sure how it's misleading? I just referred to a well-known class of FIR digital filters, not all classes of digital filters. It was but one example. Please provide some information on some design approaches other than that used by Parks–McClellan.
I think it's misleading because you sort of imply that it's tricky to make a linear phase FIR filter. It's really not, you can choose any coefficients as long as they're symmetrical. Here's the frequency response of a filter with coefficients [0.5, 1, 0.5] running at 48kHz:

It's perhaps not a very useful filter, but it does have constant group delay.

retroflex

Member
And to further elaborate on why the group delay is meaningless in this case, I've created two more all-pass filters with different Q, and hence different group delays at 10 kHz (the original one was Q=5):

These three filters all delay a 10kHz sine wave the same amount of time (just enough to flip it around), but they have wildly different group delays at 10kHz (the slope of the phase).

witwald

Senior Member
Forum Donor
I seem to recall hearing that the 1st order XO is minimum phase, but I am not sure about the higher order ones?
Yes, a pair of low-pass and high-pass first-order Butterworth filters sum to unity with zero phase shift at all frequencies. Hence, such a combination is minimum phase in nature, and is sometimes labelled as being "phase coherent".

However, the 3rd-order Butterworth low-pass/high-pass filters, have a non-zero phase response. They are all-pass crossovers, with phase shifting but a flat magnitude response. Hence, their summation is not minimum phase, even though the individual filters are. The same applies to Linkwitz–Riley filters, another filter topology that is commonly used in loudspeaker applications.

The special property of the first-order high-pass/low-pass Butterworth filter pairs is discussed by Small (1971), in his JAES paper, "Constant-Voltage Crossover Network Design".

ebslo

Senior Member
Forum Donor
The REW document on minimum phase gives a great example of a practical use for group delay in home audio. It shows how to use group delay (or better, excess group delay) to identify regions of a measured frequency response that are suitable for correction by equalization filters. Although it is written for room correction, it is also applicable to headphones if you have the measurements. It's a bit of a long read, but well worth it as it shows both what group delay is and why it's useful for this purpose.

OP
K

KTN46

Active Member
Thankyou all for your earnest replies and discussion. So based on what I have read, might I get you guys to look over my current understanding?

Phase: The phase of a signal can be understood as a "shift" in the soundwave from its source, measured as an angle between 0 and 360 degrees.

Phase Delay: Different frequencies can (through filters or speaker design?) exhibit different delays. This would be called Phase Delay.

Minimum Phase: I'm not sure what this means yet and how it plays into group delay.

Group Delay: This would be the derivative of the Phase Delay graph. I suppose it tells us how 'chaotic' a device's frequency response is respective to its surrounding frequencies?

I wonder why Phase Delay and Group Delay are discussed somewhat separately. i.e. can we not infer group delay from phase delay? What is the benefit in focusing on one over the other?

I also wonder about the relationship between Group Delay and Distortion. For audibility, is lower group delay preferred?

Why is it also that there is much higher group delay in the bass region, and what tends to cause spikes in group delay (i.e. in many headphone measurements)?

dualazmak

Major Contributor
Forum Donor
I assume, no matter how deeply we may discuss theoretical aspects of "group delay", it should be better to "objectively observe" the air sound by precision microphone and simple audio analysis software like Adobe Audition in time-amplitude mode as well as in time-spectrum mode where no black-box type signal manipulation should be included; this is why I would like to rely on my rather primitive DIY methods recently developed and shared here, here, here and here.

If we can implement precision time-scale zero marker in the input test signal tone, we can easily identify the played-back air sound time position in 0.1 msec accuracy "only using the recorded air sound", and we can also objectively see/observe the shape of the sine pulse in 0.1 msec time resolution so that we can also visually observe/analyze the possible distortion(s) (deviations from pure sine shape, as well as the aftershocks), if any, in the real air sound.

The objectively determined relative delays, if any, can be adjusted/compensated in 0.1 msec accuracy in case if you would use digital software crossover/EQ (DSP software) capable of group delay controls in the upstream of your audio system; and such delay adjustment and/or time alignment can be again precisely confirmed by air sound recording and analysis with Adobe Audition in time domain analysis as well as in wave shape matching and observation.

We can apply this simplified objective method both in high Fq zone, e.g. 1 kHz, for precision air sound "wave shape matching" in 0.1 msec accuracy, and also in low subwoofer-to-woofer Fq zone, e.g. 15 Hz - 100 Hz also in 0.1 msec - 1.0 msec accuracy.

thorvat

Senior Member
However, mathematically the system impulse response is usually sharpened by adjusting the phase versus frequency.

Rather than argueing that the phase is not able to be heard, we could ask if the system‘s impulse response should be sharp?

I find it hard to associate the word "sharp" with the looks of IR - is it really how you say it Down There?

Ok, let's get few things straight. Pretty much the only thing that introduces phase deviation in a passive speaker system is coming from the passive XO phase shift. Once you correct that phase shift with the FIR filter you will fix the looks of IR, but, as I already showed, the GD will remain pretty much the same.

You should be aware that, when EQ-ing the phase od the speaker, the target is not to flatten GD to zero but to flatten the excess phase graph to zero. Now, you could flatten the GD to zero as well, but that would introduce so much pre-ringing and that would, by all means be audible in a nasty way, while that typical rise in GD at low frequencies usually is not problematic at all.

Frgirard

Major Contributor
Thankyou all for your earnest replies and discussion. So based on what I have read, might I get you guys to look over my current understanding?

Phase: The phase of a signal can be understood as a "shift" in the soundwave from its source, measured as an angle between 0 and 360 degrees.

Phase Delay: Different frequencies can (through filters or speaker design?) exhibit different delays. This would be called Phase Delay.

Minimum Phase: I'm not sure what this means yet and how it plays into group delay.

Group Delay: This would be the derivative of the Phase Delay graph. I suppose it tells us how 'chaotic' a device's frequency response is respective to its surrounding frequencies?

I wonder why Phase Delay and Group Delay are discussed somewhat separately. i.e. can we not infer group delay from phase delay? What is the benefit in focusing on one over the other?

I also wonder about the relationship between Group Delay and Distortion. For audibility, is lower group delay preferred?

Why is it also that there is much higher group delay in the bass region, and what tends to cause spikes in group delay (i.e. in many headphone measurements)?
Phase : angular position at time T on the wavefront

Phase delay: A delay is a phase shift.

Minimal phase : a system where a change in amplitude corresponds to a change in phase

By Neumann
What is “group delay” and how does it affect sound quality?
In simple terms, group delay is the time it takes for an electrical input signal to become an acoustical output. It is frequency dependant and should ideally be zero seconds at all frequencies, but this is practically impossible.

Habu

Active Member
By Neumann
What is “group delay” and how does it affect sound quality?
In simple terms, group delay is the time it takes for an electrical input signal to become an acoustical output. It is frequency dependant and should ideally be zero seconds at all frequencies, but this is practically impossible.
Thank you Frgirar!

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Holmz

Major Contributor
I find it hard to associate the word "sharp" with the looks of IR - is it really how you say it Down There?

I can only speak for myself.

Not a lot of people to talk about group delay with.

(Sort of like the joke, ”what does an engineer use for birth control?”
Ans: “Their personalities.”)

Ok, let's get few things straight. Pretty much the only thing that introduces phase deviation in a passive speaker system is coming from the passive XO phase shift. Once you correct that phase shift with the FIR filter you will fix the looks of IR, but, as I already showed, the GD will remain pretty much the same.

You should be aware that, when EQ-ing the phase od the speaker, the target is not to flatten GD to zero but to flatten the excess phase graph to zero. Now, you could flatten the GD to zero as well, but that would introduce so much pre-ringing and that would, by all means be audible in a nasty way, while that typical rise in GD at low frequencies usually is not problematic at all.

We are in pretty good agreement.

witwald

Senior Member
Forum Donor
Pretty much the only thing that introduces phase deviation in a passive speaker system is coming from the passive XO phase shift.
What about the natural response of the tweeter/midrange/woofer drive units? Being minimum-phase electrodynamic transducers, they also have some phase shifts of their own. Each of those drivers is a bandpass system, with its own low-pass and high-pass cut-off frequencies and associated roll-offs. The natural magnitude and phase response of each such driver needs to be accounted for when the crossover filtering is being designed.
You should be aware that, when EQ-ing the phase of the speaker, the target is not to flatten GD to zero but to flatten the excess phase graph to zero.
That will depend on what one is trying to achieve. Ideally, the multi-driver system would end up having a bandpass magnitude response function, completely flat in the passband. It would also be beneficial if it had a linear phase response (constant group delay). The excess phase is a concept related to comparing the difference between a minimum-phase system and one that isn't.

witwald

Senior Member
Forum Donor
You should be aware that, when EQ-ing the phase of the speaker, the target is not to flatten GD to zero but to flatten the excess phase graph to zero. Now, you could flatten the GD to zero as well, but that would introduce so much pre-ringing and that would, by all means be audible in a nasty way, while that typical rise in GD at low frequencies usually is not problematic at all.
It's not entirely proven that the pre-ringing on an FIR filter such as the one suggested here is audible, let alone in a nasty way. The filter impulse response will be convolved with the input signal, and the resultant signal passed through the analog loudspeaker system. A linear-phase FIR-corrected loudspeaker system will have the ability to reproduce a waveform that maintains all the relative phases of the various frequency components of the input signal. It will produce a more faithful rendition of the input signals, be they transient ones or more steady-state ones, or combinations thereof.

witwald

Senior Member
Forum Donor
Minimal phase : a system where a change in amplitude corresponds to a change in phase
In general, multi-way passive loudspeakers are not minimum-phase systems; they have an excess phase component as part of their phase response. It is possible to apply minimum-phase analog equalization to such loudspeakers, where the change in amplitude corresponds to a particular change in phase response.

thorvat

Senior Member
It's not entirely proven that the pre-ringing on an FIR filter such as the one suggested here is audible, let alone in a nasty way.

It has been proved, many times - you just need to put effort to dig for it. You can even find examples of sound files with audible preringing on this forum, you just have to search for it.

witwald

Senior Member
Forum Donor
It has been proved, many times - you just need to put effort to dig for it. You can even find examples of sound files with audible preringing on this forum, you just have to search for it.
@thorvat If it's not too much trouble, could you please point me to one or two posts that, in your opinion, provide good examples of the audibility of the preringing effect? Cheers.

witwald

Senior Member
Forum Donor
Why is it also that there is much higher group delay in the bass region, and what tends to cause spikes in group delay (i.e. in many headphone measurements)?
@KTN46 Can you point to a headphone measurement that has a spike in the group delay that you refer to?

thorvat

Senior Member
@thorvat If it's not too much trouble, could you please point me to one or two posts that, in your opinion, provide good examples of the audibility of the preringing effect? Cheers.

IIRC it was posted by @UliBru (Dr. Uli Brueggemann).

ernestcarl

Major Contributor
IIRC it was posted by @UliBru (Dr. Uli Brueggemann).

I remember this, but the example(s) were artificially contrived and highly exaggerated to highlight what pre-ringing "might" possibly sound like -- In real practical use this effect, if heard with very careful listening, would likely be much more subtle and difficult to recognize. It took me quite a long time to recognize pre-ringing myself -- only after using rather "extreme" levels of phase manipulation with rePhase.

thorvat

Senior Member
It took me quite a long time to recognize pre-ringing myself -- only after using rather "extreme" levels of phase manipulation with rePhase.

Not at all. All you need to induce quite audible pre-ringing is a single phase correction filter with a modest Q, say 5. What each of us considers "quite audoble" is a matter of discussion, of course.

As pre-ringing is indeed audible every serious EQ software has some kind of pre-ringing compensation, otherwise they would all flatten phase response to a horizontal line sitting at zero deg.

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