What is group delay?

[thorvat]

However the upper plot shows the phase, which is more easy to see, and maybe hear, than the group delay.

M8, there is some serious misunderstanding here - phase and group delay are part of the same time domain phenomena which cannot be separated in how you see it, or hear it.

 

[Holmz]

M8, there is some serious misunderstanding here - phase and group delay are part of the same time domain phenomena which cannot be separated in how you see it, or hear it.

While true in a strict sense, maybe I should give some perspective.

• I associate GD and time domain performance more with correlations and impulse signals like percussion instruments.
• Phase performance, and the cross spectral phase measurement… I loop in with more of the steady state, or continuous wave signals.

Visually;
It gets almost absurd to consider the FFT for impulsive signals, unless it is mainly as a computational time saver for doing correlations.
We can look at the spectra of an impulse and it is not nearly as clear as seeing the impulse in the time domain.


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?

 

[Rottmannash]

One moment, please...

Thanks. That was very informative.

 

[witwald]

You keep on saying “at a given frequency” - which is incorrect. At a given frequency the “d(frequency in radians)“ would be zero.

The slope of the phase response curve as a function of frequency is still present at any given frequency. For example, just because one is at a particular point on a path walking up or down a hill doesn't mean that the slope at that point is zero.

A group delay is a [time or phase] delay _between_ various frequencies — either traveled through media or processed by, eg, a filter (while yes, it can be a function of frequency too).

Frequency point by frequency point, the group delay curve represents the time delay between the input signal and the output signal at a particular frequency, under steady state conditions. By reading the group delay at two different frequencies, it then becomes possible to determine the relative delay between those two different frequencies.

 

[witwald]

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

The system's impulse response should be sharp, as in the limit the system will be able to pass all signals within its bandwidth without alteration. That's a relatively realisable goal, an example being DC coupled amplifiers. With loudspeakers it's much more difficult, as they have low-frequency and high-frequency roll-offs and multiple drivers that cause spreading of the impulse response.

 

[pseudoid]

...group delay is rate of change of the phase...

... then, what becomes of 'phase delay' (leading or lagging, in absolute numbers)?

"Audio Group-Delay" is an FFT computation, based on the impulse response (real-time); it merely represents change in phase (not necessarily the "RATE of change") of the response as a direct function of frequency. This meaning of Group-Delay is at the "component/hardware level" (re: audio reproduction), and primarily evaluates phase-response distortion products that can/may/do (said to) impact transient response and image (sound stage) coloration.
I guess we can all agree to call it "Audio Group-Delay" but it should NOT be confused with its meaning and definition in "RF Group-Delay", born of the communication systems disciplines. Similar to everyone agreeing about the difference between "classical current flow" and "electron flow".

What is Group Delay and Why We Measure It (9minutes) looks at it from a different perspective. Roger Gibboni defines it as the difference in the phase of the signal versus the frequency (although he equates phase with time as being the same phenomenon)
Don’t ask why the audio engineers could NOT come up with a more descriptive name.
I dunno! Maybe something like 'Transient (or Dynamic) Phase Response"…

 

[witwald]

Group delay usually refers to a constant/linear d(phi)/d(f) -- in other words, phase is a straight line when plotted on a linear frequency axis.

Group delay generally varies with frequency in many systems that are of interest to us. For example, analog low-pass and high-pass filters, as used in crossover networks, have nonlinear group delay near their cut-off frequencies. So group delay is far from being constant in many systems, although there are of course regions where it will be approximately constant too.

More complex phase relationships, sometimes called variable group delay, are fairly common in digital filters and other DSP, cause the time delay to vary by frequency. d(phi)/d(f) is no longer a linear function. It can be an arbitrarily complex, non-linear expression.

If we implement a Butterworth low-pass or high-pass filter in the digital domain, it will have the same group delay characteristics as the analog version of the filter. Of course, due to DSP latency, there will also be a time delay added to the signal because of the time it takes to perform the computations and then output the results through the DAC.

There is a well-known class of digital filters that have a constant group delay, owing to their linear-phase characteristics. These are known as Parks–McClellan linear-phase FIR digital filters, and procedures for their design were published back in 1973. The relevant paper, A computer program for designing optimum FIR linear phase digital filters, describes their characteristics quite nicely.

Here's what a non-linear phase (variable group delay) for one DAC looks like (blue). Red/pink/salmon line is what it looks like after phase is linearized. Since the slope of the red line is 0 (it's horizontal) there is no time delay.

That was an interesting plot, which I hadn't seen before. Thanks for sharing. It was interesting to see that the delta phase in the passband was within ±4 degrees. That's not a lot of variation. At 4kHz, a 4-degree phase difference corresponds to a spatial shift of 0.85 mm.

 

[witwald]

"Audio Group-Delay" is an FFT computation, based on the impulse response (real-time);

Group delay can be obtained from a plot of the phase response as a function of (linear) frequency. Of course, the FFT can be used to conveniently compute the magnitude and phase response of the system under test. However, there are such devices as transfer function analyzers that can compute the phase difference between an input and an output signal without resorting to an FFT.

 

[witwald]

Don’t ask why the audio engineers could NOT come up with a more descriptive name.
I dunno! Maybe something like 'Transient (or Dynamic) Phase Response"…

They didn't because group delay is a characteristic of the steady-state response of a system, after the transient response has died away.

 

[witwald]

I guess we can all agree to call it "Audio Group-Delay" but it should NOT be confused with its meaning and definition in "RF Group-Delay", born of the communication systems disciplines.

There is really no distinction between Audio and RF group delays. They are all plots with time on the vertical axis and frequency on the horizontal axis. The magnitudes of the frequencies and the times that are involved may be different, but the underlying equation is the same.

 

[witwald]

It gets almost absurd to consider the FFT for impulsive signals, unless it is mainly as a computational time saver for doing correlations.

The FFT is very useful when undertaking transfer function analysis of a system. For example, test signals based on maximum length sequences (MLS) can be used to compute the impulse response of the system being investigated, and subsequent FFT processing can provide the frequency response function (both magnitude and phase response).

We can look at the spectra of an impulse and it is not nearly as clear as seeing the impulse in the time domain.

But if the impulse produces a smooth bandpass frequency response function, or otherwise, we learn a lot about the system's behaviour.

 

[retroflex]

Frequency point by frequency point, the group delay curve represents the time delay between the input signal and the output signal at a particular frequency, under steady state conditions.

I'm sorry, but this is not really true. It's meaningless to talk about the group delay when the input is just a single frequency, the phase or phase delay is much more relevant then. But you're totally correct in that it's meaningful to talk about the group delay of a filter at a particular frequency, and your hill metaphor is very good. But one has to be careful to separate the two cases, a filter always has a certain group delay at a certain frequency, but the delay the filter puts on an input signal with a certain frequency is not necessarily related to its group delay.

For example, low-pass and high-pass filters all have nonlinear group delay near their cut-off frequencies.

This is not the case, a digital FIR low- or high-pass filter can have a constant group delay across every frequency.

There is a well-known class of digital filters that have a constant group delay, owing to their linear-phase characteristics. These are known as Parks–McClellan linear-phase FIR digital filters, and procedures for their design were published back in 1973.

This is true but somewhat misleading, I'd argue. All FIR filters with symmetrical coefficients have a constant group delay, not just ones designed with the Parks–McClellan algorithm.

 

[pkane]

Group delay generally varies with frequency in many systems that are of interest to us. For example, low-pass and high-pass filters all have nonlinear group delay near their cut-off frequencies. So group delay is far from being constant in many systems, although there are of course regions where it will be approximately constant too.

If we implement a Butterworth low-pass or high-pass filter in the digital domain, it will have the same group delay characteristics as the analog version of the filter. Of course, due to DSP latency, there will also be a time delay added to the signal because of the time it takes to perform the computations and then output the results through the DAC.

There is a well-known class of digital filters that have a constant group delay, owing to their linear-phase characteristics. These are known as Parks–McClellan linear-phase FIR digital filters, and procedures for their design were published back in 1973. The relevant paper, A computer program for designing optimum FIR linear phase digital filters, describes their characteristics quite nicely.

That was an interesting plot, which I hadn't seen before. Thanks for sharing. It was interesting to see that the delta phase in the passband was within ±4 degrees. That's not a lot of variation. At 4kHz, a 4-degree phase difference corresponds to a spatial shift of 0.85 mm.

Having implemented all these (linear, minimum phase and Butterworth) filters in an arbitrary filter generator for all of these from scratch, I am somewhat familiar with them and their characteristics

 

[witwald]

Thing is, it take us a really long time for our hearing apparatus to even recognize the presence of a 50 Hz tone;

I suppose that would depend on how our hearing system works. I'd be concerned how that tone is presented to the listener. 50 Hz tones may take time to build up to a steady state in a listening room, so is that part of the time taken to recognise the presence of that tone? If we move away from a continuous tone scenario, to say a kick drum with a 50Hz fundamental, how long does it take us to realise that the kick drum has been activated? I expect that when convolved with the room acoustics, that kick drum will have a few reflections that the listener may discern, either as separate entities or blended with the first arrival.

We are moving further from defining "Group Delay"... Which is NOT the delay between input and output of a Signal Processor. This particular is rather called processing delay.. for good reasons...

When measured by a device, the output signal contains both the processing delay as well as the inherent group delay of the system under test. The measurement cannot separate one from the other–it just measures the total behaviour. Hence, the group delay curve of the system itself will be vertically offset upwards by the amount of processing delay.

 

[witwald]

Most simple natural non linearities are minimum phase... including crossovers, RIAA EQ, speakers, phono cartridge cantilever and electrical resonances, and many others.

Do you actually mean "Most simple natural linear systems are minimum phase..."? All of the examples that you mentioned correspond to systems that are largely linear in their behaviour (e.g., they don't produce a lot of distortion). They are usually described by mathematical models based on the theory of linear systems.

 

[witwald]

A Speaker/Driver, and it's crossover (non digital!), are all minimum phase

A digital crossover can also be minimum phase, as we can easily create Butterworth or Linkwitz–Riley filters in the digital domain. If the DSP latency of the high-pass and low-pass sections of the digital crossover is properly accounted for, then we are left with the minimum-phase filtered responses.

 

[witwald]

I'm sorry, but this is not really true. It's meaningless to talk about the group delay when the input is just a single frequency,

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?

a filter always has a certain group delay at a certain frequency, but the delay the filter puts on an input signal with a certain frequency is not necessarily related to its group delay.

Please provide an example that explains this, to help me.

This is not the case, a digital FIR low- or high-pass filter can have a constant group delay across every frequency.

Thanks for pointing that out. Agreed. I've edited my post to be a little more specific and less confusing. I was thinking about passive analog filters at the time.

This is true but somewhat misleading, I'd argue. All FIR filters with symmetrical coefficients have a constant group delay, not just ones designed with the Parks–McClellan algorithm.

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 the one used by Parks and McClellan.

 

[Holmz]

The FFT is very useful when undertaking transfer function analysis of a system. For example, test signals based on maximum length sequences (MLS) can be used to compute the impulse response of the system being investigated, and subsequent FFT processing can provide the frequency response function (both magnitude and phase response).

You are sort of making the case for me ^here^ By saying the part in bold.

But if the impulse produces a smooth bandpass frequency response function, or otherwise, we learn a lot about the system's behaviour.

You can talk baout the frequency response till the cows come home, but when the system’s impulse response becomes a pure Dirac delta function, we know we have reached nirvana.

The FFT is very useful when undertaking transfer function analysis of a system. For example, test signals based on maximum length sequences (MLS) can be used to compute the impulse response of the system being investigated, and subsequent FFT processing can provide the frequency response function (both magnitude and phase response).

But if the impulse produces a smooth bandpass frequency response function, or otherwise, we learn a lot about the system's behaviour.

Thanks for helping my cause the part I bolted and underlined.
MLS are usually used for correlations, which is primarily explained as a time domain concept.
(And yes, I know that one can do a couple of FFTs, an operatiopna en and inverse FFT more computationally cheaply.)

... then, what becomes of 'phase delay' (leading or lagging, in absolute numbers)?

"Audio Group-Delay" is an FFT computation, based on the impulse response (real-time); it merely represents change in phase (not necessarily the "RATE of change") of the response as a direct function of frequency. This meaning of Group-Delay is at the "component/hardware level" (re: audio reproduction), and primarily evaluates phase-response distortion products that can/may/do (said to) impact transient response and image (sound stage) coloration.
I guess we can all agree to call it "Audio Group-Delay" but it should NOT be confused with its meaning and definition in "RF Group-Delay", born of the communication systems disciplines. Similar to everyone agreeing about the difference between "classical current flow" and "electron flow".
…

I am not being disagreable, but I cannot agree that the RF and Audio are any different.
(It is a mathematical concept to describe the variation of delay over frequency.)

Yeah - human multiplication is different from mathematical multiplication. But accounting math, statistics math, and stockroom maths are all the same, as they are maths.

Even a audio horn looks a lot like a waveguide horn. And the audio DSP math looks a lot like RF math. There are only a few ways to spell Fourier Transform… ”Fast”, ”Discrete”,… inverse, 2D, 3D.

 

[pseudoid]

There is really no distinction between Audio and RF group delays

I sweated the details as to the difference in two posts but you insisting a lack of distinction is way out of weight-class to 'argue'.

 

[dlaloum]

A digital crossover can also be minimum phase, as we can easily create Butterworth or Linkwitz–Riley filters in the digital domain. If the DSP latency of the high-pass and low-pass sections of the digital crossover is properly accounted for, then we are left with the minimum-phase filtered responses.

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