So long as there was not a specified arrival time, then yes.
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What is group delay?
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HaveMeterWillTravel
Senior Member
If all frequencies arrive at the same time, then that is generally considered propagation time. For audio only, that may have no significant effects. However; if combined with video then it may be profound.
Depends upon whether it was past the host's bedtime.
Holmz
Major Contributor
How profound is the delay a 3’ cable versus a 6’ cable?
It might be profound in a GPS timing, or a CERN accelerometer… but does the TT and listener really care?
Holmz
Major Contributor
It seems easier to imagine the family in a line, and signifying all the “Doh Ray Me” simultaneously (in a burst).
However the father (capt Von Trapp) is further away and then Christy, and the kids from tallest to shortest are closer to you.
So while the sound started simultaneously, it arrived as “Doh Ray Me…”
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Phase delay and group delay are derived from phase. Phase falls out of the steady-state frequency domain approach we prefer over working directly in the time domain, going with phasors and the corresponding phasor phase, which is what is being plotted in a frequency response. Phase delay and group delay thus generally represent "steady-state delays", and so you generally have systems like lumped circuits that have no true delay (there is an immediate output for ANY input), but generally non-zero and frequency-dependent phase delays and group delays, that can also be negative, while the circuit itself is represented by a causal transfer function. Phase delay and group delay are apparent delays; a 'best bet' of sorts of the equivalent true delay that could have caused this apparent delay under steady-state conditions, but there are infinitely many systems that COULD have given you this phase delay/group delay at this particular frequency, but they would all give you different transient responses, even for that one frequency in question.
For example, a polarity flip has a 180 degree phase shift across all frequencies, so if you choose a particular frequency, and plot input vs output for a system that flips the sign of the input, but only look at the steady-state response, your phase delay will be half a period, but this from a system that has no inherent delay/true delay/transport delay (where would it come from, you have just flipped two wires essentially). If you don't know any better, you might now think that this apparent time delay is constant across all frequencies ("Phase is just time delay". No, it is not.), as this steady-state phase delay COULD have been from a true time delay, if you didn't know any better, but in fact the phase delay related to the polarity flip varies with frequency and for example goes to infinity at low frequencies, while at the same time (no pun intended) always having an immediate (but opposite) output for any non-zero input. If at low frequencies, you had only a true time delay at your disposal and no polarity flip, and wanted to delay half a period, you would indeed need a larger and larger true time delay to achieve this, and that is exactly what the phase delay tells you. But that does not mean that there IS this delay in the system, it is just an apparent delay. If steady-state is all that matters, then of course you can rightfully say that this is your delay of interest, but for impulse and step responses, it is crucial to operate with the actual phase and its resulting delays.
Only in the special case of a true time delay will phase delay (and group delay) take on that delay value, and the differential delay (phase delay minus group delay) will be zero, indicating a true delay (barring a polarity flip that obscure this). But, phase is not just a time delay, and so for the full information for a single frequency (transient and steady-state), you need more information than just the magnitude and phase response at that single frequency, as otherwise you only know the (apparent) phase delay for the steady-state condition, but not what happens transiently (if that is a word).
Same thing with group delay. You might find that a modulated signal looks to be delayed with you group delay giving you the offset value, but there likely was an immediate output from the system, so no "actual" delay, just apparent. Of course, the system could have a DSP delay, or transport delay from wave propagation, but the lumped systems that we use for representing transducers cannot have such true delays, as will be discussed later.
If these steady-state delays are negative at some frequencies, it just means that it will 'look' as if the output is negatively delayed once everything has settled to steady-state, but we know that all physical systems are causal, so at the most there can be an immediate output, but not an output before an input. Now, if you allow non-causal system parts in your total system, some or all of the negative delay of course could come from here, but let us just stick with physical systems here.
A general system has a minimum phase part transfer function and an excess phase part transfer function. The excess phase part, in general, has two parts; an allpass part (zeros in right hand side/plane, RHS/RHP) and a 'true delay' phase part with a linear phase. Both of these excess phase parts have constant magnitude. The latter has to have the linear phase to infinite frequency to be a true delay, but in practice, it will be difficult to check this, and so a true delay can be difficult to separate out, as you might approximate a true delay rather well with an allpass function.
A minimum phase transfer function can thus not have any true delay, so we will know a priori that any input will give an immediate output, although it may have some linear distortion (an indicator of this will be how much the phase delay differs from the group delay). We assume our systems to be causal and stable, and so we can find a causal and stable inverse of our minimum phase part. The allpass part will add to the linear distortion but the time delay part will not. The problem comes when we try to invert the excess phase part, as either we end up with an unstable inversion system from the allpass system requiring poles in the RHS to cancel out the zeros, or we have problems with causality, as the inversion system must be anticipatory to counter the true delay in the system.
The minimum phase transfer function on its own may be all that there is to model some relevant system, such as many (lumped) transducer models. It is, however, not quite accurate to say that a system(!) is minimum phase, as for example you might have a transfer function from the displacement at some point on the membrane or surround to input voltage that seemingly is described via a minimum phase (mp) transfer function, but move to another spatial point, and it is no longer mp. When people say that a loudspeaker driver is minimum phase, they are typically talking about the pressure at some distance-to-input voltage being described with a minimum phase transfer function, although at higher frequencies, this may break down, as the spatial 3D aspect somewhat collides with the notion of 0D transfer functions (due to modal aspects, not having a clear distance from the single microphone point to the various points distributed across the membrane for these small wavelengths, temperature changes affect sound speed and this is more critical at higher frequencies, and so on). But this minimum phase aspect generally goes out the window when looking at a complete multi-driver loudspeaker for which the crossover alone is likely non-minimum phase (allpass pressure response is what you are aiming for, and this is by definition non-mp), and when summing mp transfer functions you are not ensured a resulting mp transfer function.
Both the mp and the allpass transfer functions are (typically) described, or can be approximated, via so-called rational transfer functions with polynomials in numerator and denominator, and these rational functions cannot completely describe a pure time delay, although they can approximate such a true delay up to a certain frequency, with a finite number of poles and zeros. This again is very useful to know, as many physical systems are given via a lumped models, and you know then that there will be no true delay, as such circuits can be described via rational transfer functions.
No need to think about waves, and possible dispersion (in acoustics we typically work with a constant sound speed, so that makes it simpler), until these signal processing aspects are well understood.
For example, a polarity flip has a 180 degree phase shift across all frequencies, so if you choose a particular frequency, and plot input vs output for a system that flips the sign of the input, but only look at the steady-state response, your phase delay will be half a period, but this from a system that has no inherent delay/true delay/transport delay (where would it come from, you have just flipped two wires essentially). If you don't know any better, you might now think that this apparent time delay is constant across all frequencies ("Phase is just time delay". No, it is not.), as this steady-state phase delay COULD have been from a true time delay, if you didn't know any better, but in fact the phase delay related to the polarity flip varies with frequency and for example goes to infinity at low frequencies, while at the same time (no pun intended) always having an immediate (but opposite) output for any non-zero input. If at low frequencies, you had only a true time delay at your disposal and no polarity flip, and wanted to delay half a period, you would indeed need a larger and larger true time delay to achieve this, and that is exactly what the phase delay tells you. But that does not mean that there IS this delay in the system, it is just an apparent delay. If steady-state is all that matters, then of course you can rightfully say that this is your delay of interest, but for impulse and step responses, it is crucial to operate with the actual phase and its resulting delays.
Only in the special case of a true time delay will phase delay (and group delay) take on that delay value, and the differential delay (phase delay minus group delay) will be zero, indicating a true delay (barring a polarity flip that obscure this). But, phase is not just a time delay, and so for the full information for a single frequency (transient and steady-state), you need more information than just the magnitude and phase response at that single frequency, as otherwise you only know the (apparent) phase delay for the steady-state condition, but not what happens transiently (if that is a word).
Same thing with group delay. You might find that a modulated signal looks to be delayed with you group delay giving you the offset value, but there likely was an immediate output from the system, so no "actual" delay, just apparent. Of course, the system could have a DSP delay, or transport delay from wave propagation, but the lumped systems that we use for representing transducers cannot have such true delays, as will be discussed later.
If these steady-state delays are negative at some frequencies, it just means that it will 'look' as if the output is negatively delayed once everything has settled to steady-state, but we know that all physical systems are causal, so at the most there can be an immediate output, but not an output before an input. Now, if you allow non-causal system parts in your total system, some or all of the negative delay of course could come from here, but let us just stick with physical systems here.
A general system has a minimum phase part transfer function and an excess phase part transfer function. The excess phase part, in general, has two parts; an allpass part (zeros in right hand side/plane, RHS/RHP) and a 'true delay' phase part with a linear phase. Both of these excess phase parts have constant magnitude. The latter has to have the linear phase to infinite frequency to be a true delay, but in practice, it will be difficult to check this, and so a true delay can be difficult to separate out, as you might approximate a true delay rather well with an allpass function.
A minimum phase transfer function can thus not have any true delay, so we will know a priori that any input will give an immediate output, although it may have some linear distortion (an indicator of this will be how much the phase delay differs from the group delay). We assume our systems to be causal and stable, and so we can find a causal and stable inverse of our minimum phase part. The allpass part will add to the linear distortion but the time delay part will not. The problem comes when we try to invert the excess phase part, as either we end up with an unstable inversion system from the allpass system requiring poles in the RHS to cancel out the zeros, or we have problems with causality, as the inversion system must be anticipatory to counter the true delay in the system.
The minimum phase transfer function on its own may be all that there is to model some relevant system, such as many (lumped) transducer models. It is, however, not quite accurate to say that a system(!) is minimum phase, as for example you might have a transfer function from the displacement at some point on the membrane or surround to input voltage that seemingly is described via a minimum phase (mp) transfer function, but move to another spatial point, and it is no longer mp. When people say that a loudspeaker driver is minimum phase, they are typically talking about the pressure at some distance-to-input voltage being described with a minimum phase transfer function, although at higher frequencies, this may break down, as the spatial 3D aspect somewhat collides with the notion of 0D transfer functions (due to modal aspects, not having a clear distance from the single microphone point to the various points distributed across the membrane for these small wavelengths, temperature changes affect sound speed and this is more critical at higher frequencies, and so on). But this minimum phase aspect generally goes out the window when looking at a complete multi-driver loudspeaker for which the crossover alone is likely non-minimum phase (allpass pressure response is what you are aiming for, and this is by definition non-mp), and when summing mp transfer functions you are not ensured a resulting mp transfer function.
Both the mp and the allpass transfer functions are (typically) described, or can be approximated, via so-called rational transfer functions with polynomials in numerator and denominator, and these rational functions cannot completely describe a pure time delay, although they can approximate such a true delay up to a certain frequency, with a finite number of poles and zeros. This again is very useful to know, as many physical systems are given via a lumped models, and you know then that there will be no true delay, as such circuits can be described via rational transfer functions.
No need to think about waves, and possible dispersion (in acoustics we typically work with a constant sound speed, so that makes it simpler), until these signal processing aspects are well understood.
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Also interesting to see how companies that really should get this, get it wrong in so many ways. For example, https://www.dpamicrophones.com/mic-university/technology/polarity-phase-and-delay/. This notion of 'phase is a function of time' (https://audiouniversityonline.com/polarity-vs-phase/) is just wrong. If anything, "Delays are a function of phase"(!). Delays having unit of time, does not mean that they are as 'fundamental' as time itself. The temporal aspects are encoded in the phase; phase delay gives you the steady-state delay for each sinusoidal, but to have the full picture for a sinusoidal starting at some time t=T0, you need the transfer function across the entire frequency range. For some intuition on that: Group delay cannot be known at a single frequency, if you only know the complex phasor for the system at that one frequency, as you cannot find the derivative.
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It would seem that the content of the typical general equation for a sinusoidal waveform, y = sin(ωt + ϕ), is being well and truly ignored by the notion that 'phase is a function of time'. The phase, ϕ, is a constant in that equation.
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Sure. It is a function of frequency in general, but constant in time, and so a fixed value for each frequency.
always learning
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audiofooled
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Even in, as you say, special case conditions, it would only be steady state information for that single measurement location, hardly anything more?
In rooms I suspect things get further complicated taking into account wave propagation, net displacement, mode 0, gradient?
Here's polarity flip for a range of low frequencies:
A Broad Discussion of Speakers with Major Audio Luminaries
The field of audio is small. “String of temporary opinions” sadly largely covers it, like palaeoanthropology from where it comes. There are exceptions, e.g. directivity of loudspeakers, the early Nordic standards on FR in reproduction, momentary hearing thresholds etc. However, we rely so much...
www.audiosciencereview.com
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The link looks to be about a polarity flip only on one channel out of several, so of course the total output becomes affected by this particular change of 180 degrees phase shift at all frequencies on that particular channel.Even in, as you say, special case conditions, it would only be steady state information for that single measurement location, hardly anything more?
In rooms I suspect things get further complicated taking into account wave propagation, net displacement, mode 0, gradient?
Here's polarity flip for a range of low frequencies:
A Broad Discussion of Speakers with Major Audio Luminaries
The field of audio is small. “String of temporary opinions” sadly largely covers it, like palaeoanthropology from where it comes. There are exceptions, e.g. directivity of loudspeakers, the early Nordic standards on FR in reproduction, momentary hearing thresholds etc. However, we rely so much...
Regarding steady-state information in locations, you are now involving waves into a signal processing topic, so now you will have magnitude and phase aspects coming from the transfer function from sources to individual microphone points, so modes and distances come into play. But it still follows the signal processing basics, as the total transfer function is a sequence of transfer function (per point, as you say), looking steady-state (and typically zero initial conditions), and so phases can be added trivially for a total phase. As long as linearity is assumed, you can find steady-state transfer functions, and from there calculate the response for non-sinusoidal input signals, with transient and steady-state behavior included.
audiofooled
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This was about polarity flip in one channel of a stereo bass setup, so sum of the response of two channels measured separately (in room, at a 3,5m distance), where one would potentially have opposite polarity.
What is interesting to me is the way that phase delay and group delay (on a per frequency basis) are behaving, but the differential delay (phase delay minus group delay) and pressure tends to be very close to zero for this low frequency range.
I just thought at the time it would be interesting for the discussion of stereo bass and the perception of AE (Auditory Envelopment), where decorrelated signals are at play. My point in general was that audibility is largely affected by the time domain performance.
Anyway, I hope I didn't get this wrong and you are of course welcome to correct me.
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This was about polarity flip in one channel of a stereo bass setup, so sum of the response of two channels measured separately (in room, at a 3,5m distance), where one would potentially have opposite polarity.
What is interesting to me is the way that phase delay and group delay (on a per frequency basis) are behaving, but the differential delay (phase delay minus group delay) and pressure tends to be very close to zero for this low frequency range.
I just thought at the time it would be interesting for the discussion of stereo bass and the perception of AE (Auditory Envelopment), where decorrelated signals are at play. My point in general was that audibility is largely affected by the time domain performance.
Anyway, I hope I didn't get this wrong and you are of course welcome to correct me.
I dont' think you are wrong. Low frequency phase is probably problematic, since many filter functions will look as linear phase from DC to 'some low frequency', so it is difficult to distinguish between a pure time delay and a more traditional phase shift. I saw a video from Matt Poes recently (cannot find it now, https://www.youtube.com/@PoesAcoustics/videos), where he talked about a multisub setup that seemingly had a nice total phase, but one sub was delayed a full cycle (*) and it was not that notacable. Once it was cleared up, the total phase was even better, though. Flipping the polarity of one sub will not change how it will immediate output sound, but it will change the phase delay, and so once you start looking both at transient and steady-state behavior, you need to keep your cool ;-)
(*) Btw. you will see explanations about bass ports and how the sound is delayed a full cycle, and this is not correct in a pure time delay sense, but I will make an article about that at some point.
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Here are some figures from an upcoming article:
Group delay illustration. Input is two sinusoidals close in frequency, so their sum form an envelope, and the output has an envelope also. While there is an apparent positive delay seen steady-state between the two envelopes, there is an immediate output at t=0, since the system in question was a rational transfer function. One needs to calculate the full (transient and steady-state) response to see that phase delay and group delay are generally steady-state quantities only, which likely goes against common intuition about delays.
A positive phase delay can also be illustrated for some system without any pure delay. An immediate output, since the system in question is described via a rational transfer function (which can at the most approximate a true delay), but an 'apparent'/phase delay when viewed steady-state. I have shown the decay of the transient response also. In principle, this will never die out, but if we wait a couple of periods, we can assume the same conditions as looking across infinite time, which is what is what the frequency domain approach requires.
Once it is understood that these delays are generally steady-state/apparent delays, we see why there is no problem with for example a negative delay, here illustrated with a negative phase delay. There is an immediate output, no issues with causality, but no transient delay either. It just 'looks' as if the output response is negatively delayed, but only once steady-state conditions have been achieved:
And can we stop questioning how a polarity flip (sign change) is indeed a 180 degree phase shift across all frequencies, as if this would somehow indicate a pure time delay? Yes, it might steady-state(!) look like a time-delay, but infinitely many other systems can give the same steady-state responses, when only looking at this particular frequency. Drawing steady-state input and output sinusoids (at a single frequency) and concluding anything from there will not cut it. You need to calculate the full responses to see how the transient responses differ. The polarity flip will have a phase delay, but this is an apparent delay, that did not come from a pure time delay system. This phase delay is frequency-dependent, going towards infinity at low frequencies, as for long periods, it will steady-state "look like" an extreme delay was needed to cause this response.
Even when looking at a singular "frequency" (with a starting time, so not really a frequency, but at least a semi-infinite sinusoidal), one needs information across all frequencies(!) to know anything more than the steady-state response.
The article will soon be submitted to audioXpress, so it should be out in the beginning of 2026.
Group delay illustration. Input is two sinusoidals close in frequency, so their sum form an envelope, and the output has an envelope also. While there is an apparent positive delay seen steady-state between the two envelopes, there is an immediate output at t=0, since the system in question was a rational transfer function. One needs to calculate the full (transient and steady-state) response to see that phase delay and group delay are generally steady-state quantities only, which likely goes against common intuition about delays.
A positive phase delay can also be illustrated for some system without any pure delay. An immediate output, since the system in question is described via a rational transfer function (which can at the most approximate a true delay), but an 'apparent'/phase delay when viewed steady-state. I have shown the decay of the transient response also. In principle, this will never die out, but if we wait a couple of periods, we can assume the same conditions as looking across infinite time, which is what is what the frequency domain approach requires.
Once it is understood that these delays are generally steady-state/apparent delays, we see why there is no problem with for example a negative delay, here illustrated with a negative phase delay. There is an immediate output, no issues with causality, but no transient delay either. It just 'looks' as if the output response is negatively delayed, but only once steady-state conditions have been achieved:
And can we stop questioning how a polarity flip (sign change) is indeed a 180 degree phase shift across all frequencies, as if this would somehow indicate a pure time delay? Yes, it might steady-state(!) look like a time-delay, but infinitely many other systems can give the same steady-state responses, when only looking at this particular frequency. Drawing steady-state input and output sinusoids (at a single frequency) and concluding anything from there will not cut it. You need to calculate the full responses to see how the transient responses differ. The polarity flip will have a phase delay, but this is an apparent delay, that did not come from a pure time delay system. This phase delay is frequency-dependent, going towards infinity at low frequencies, as for long periods, it will steady-state "look like" an extreme delay was needed to cause this response.
Even when looking at a singular "frequency" (with a starting time, so not really a frequency, but at least a semi-infinite sinusoidal), one needs information across all frequencies(!) to know anything more than the steady-state response.
The article will soon be submitted to audioXpress, so it should be out in the beginning of 2026.
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