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# Group delay from S21

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#### 1a2s3d4f

##### Member level 3
Hi everyone,

I am confuse about the concept of group delay from S21. What different between group delay and phase delay. Can some one explain why we need to measure the group delay.

Thanks

In general, S21 is the measurement of the complex output/Input transfer function. Complex means that it represents both amplitude (that is gain or attenuation) and phase relationship between the input and the output signals.
The group delay is the variation of the phase due to the variation of the angular frequency (with the sign changed), that is:

GD = -dφ/dω

if we know S21 at two different frequencies, let say "f1" and "f2", since ω=2•Π•f, then:

GD ≈ -phase[S21(f1)-S21(f2)]/[2•Π•(f1-f2)]

I used "≈" because the exact definition requires the limit f2 --> f1

From the s2p parameters the phase of S21 can be known directly; if instead you have imaginary and real part of S21, then:

phase(S21)=arctg[Im(S21)/Re(S21)]

Points: 2

Points: 2

### Khashia

Points: 2
In general, S21 is the measurement of the complex output/Input transfer function. Complex means that it represents both amplitude (that is gain or attenuation) and phase relationship between the input and the output signals.
The group delay is the variation of the phase due to the variation of the angular frequency (with the sign changed), that is:

GD = -dφ/dω

if we know S21 at two different frequencies, let say "f1" and "f2", since ω=2•Π•f, then:

GD ≈ -phase[S21(f1)-S21(f2)]/[2•Π•(f1-f2)]

I used "≈" because the exact definition requires the limit f2 --> f1

From the s2p parameters the phase of S21 can be known directly; if instead you have imaginary and real part of S21, then:

phase(S21)=arctg[Im(S21)/Re(S21)]

Hi Albbq,

I would like to say thank you for pointing this out. I assume if the GD is too much, the BER will be impacted or signal will be loss.

BTW, what the different between propagation delay from GD.

Thanks

The group delay is the delay in the propagation of the information from input to output. Ideal linear phase systems exhibit a constant group delay over frequency; real systems have instead a non perfectly linear behaviour that results in a non-constant group-delay.
This group-delay variation affects the signal, and if too high (with respect to the type of modulation we are considering) can cause loss of performances.

you can have a huge group delay, like going from the earth to the moon, and the modulation will get thru just fine!

It is group delay unflatness (i.e. certain frequencies are delayed differently than other frequencies) when you have trouble.

tony_lth

### tony_lth

Points: 2
you can have a huge group delay, like going from the earth to the moon, and the modulation will get thru just fine!

I think, group delay has nothing to do with the time a signal needs to travel from A to B.
It is defined as the slope of the phase function of a two-port and applies to a small-band signal only (example: AM with a spectrum that is small if compared with the carrier frequency)

Let's consider a signal travelling into space; it will take a time, let say "tAB" to go from A to B.

We can analyse the behaviour at two frequency "f1" and "f2" --> T1=1/f1 and T2=1/f2.

The two phases, in radiant, will then be given by:

φ1=2•Π•tAB/T1=2•Π•tAB•f1
φ2=2•Π•tAB/T2=2•Π•tAB•f2

and applying the group delay definition (GD=-dφ/dω):

GD=-(2•Π•tAB•f2-2•Π•tAB•f1)/[2•Π•(f2-f1)]=-tAB (the sign only indicates the direction of propagation)

This means the group delay is the time required to travel from A to B.

Please note that the system is linear and GD is constant.

This means the group delay is the time required to travel from A to B.
Sound like an incorrect generalization at first sight. I presume you mean "the group delay of this specific system". For a system that is only causing delay time, group delay is equal to delay time. But there are other possible causes of group delay respectively phase dispersion dφ/dω than delay time.

tony_lth

### tony_lth

Points: 2
I presume you mean "the group delay of this specific system". For a system that is only causing delay time, group delay is equal to delay time.
Yes I was just answering the sentence of LvW that said "I think, group delay has nothing to do with the time a signal needs to travel from A to B."

I am trying to understand group delay too.
This is from https://en.wikipedia.org/wiki/Group_delay_and_phase_delay

Group delay is a useful measure of time distortion, and is calculated by differentiating, with respect to frequency, the phase response versus frequency of the device under test (DUT). The group delay is a measure of the slope of the phase response at any given frequency. Variations in group delay cause signal distortion, just as deviations from linear phase cause distortion.
Units of Group delay as defined is Rad/(Rad/Sec)=seg. So it is consistent with the time to travel. If GD = -dφ/dω is constant it means that all frequencies will have the same time delay. It can be high or low but the same for all frequencies.
If GD = -dφ/dω is not constant then it means that different frequencies will have different delay times and then distortion results.
A condition for GD is that the phase response of the system is a linear function of the frequency. That is if we take the dφ/dω of a linear function we get a constant.

Yes I was just answering the sentence of LvW that said "I think, group delay has nothing to do with the time a signal needs to travel from A to B."
I agree to the viewpoint, that group delay has something to do with delay time (travel time, flight time, whatsoever...), because it's the same value for a class of systems. But it's a different quantity, in so far LvW is systematically right. He's referring to an important property of group delay. It's defined for systems where a delay time can be neither measured nor theoretically derived. It can be understood it as a generalization of delay time. It's mainly useful for the description of systems with frequency dependend transfer function, e.g. filters or band limited channels as mentioned by LvW. They generally expose a non-constant group delay, except for those that are designed purposeful with linear phase like FIR filters with symmetrical impulse response.

A condition for GD is that the phase response of the system is a linear function of the frequency. That is if we take the dφ/dω of a linear function we get a constant.
I presume you meaned to write "a condition for constant group delay".

albert22

### albert22

Points: 2
Yes I was just answering the sentence of LvW that said "I think, group delay has nothing to do with the time a signal needs to travel from A to B."

Albbg, sorry to say but in post#7 you have applied the group delay definition in an uncorrect way.
I repeat: Group delay has nothing to do with the time a signal needs to travel from A to B.

The parameter "group delay" must be applied to a "group" of frequencies only (e.g. a modulation signal), which are very close to each other if compared with the corresponding mean value.
As an illustrative example, take a carrier that is AM modulated with a signal that occupies a spectrum of only less than 1% of the carrier frequency. When such a compound signal "travels" through a 2-port (amplifier, filter,...) it suffers from phase distortions caused by the phase function of the 2-port. Only if the group delay is constant (linear phase function) all frequencies are delayed by the same amount with the result of zero phase distortions.
Thus, the meaning of the term group delay is NOT to describe the delay of a signal but to characterize the phase distortion of a compound signal carrying some kind of information.

- - - Updated - - -

Units of Group delay as defined is Rad/(Rad/Sec)=seg. So it is consistent with the time to travel.

Albert, just one comment: Does this sentence mean that "it is consistent" with the travelling time just because both parameters are measured in seconds?

albert22

### albert22

Points: 2
FvM Sorry I meant:
A condition for constant GD is that the phase response of the system is a linear function of the frequency. That is if we take the dφ/dω of a linear function we get a constant.
LvW
I incorrectly take into account a previous post that related group delay with delay or time to travel. Two or more posts went thru when I was writing so I dint see that I was trying to elaborate on a wrong assumption. Now I see that it has nothing to do with that. Thanks.
Can I think that the delay time of each sinusoidal component is related with the phase delay instead?

Still banging my head to understand the subject, I found this:
...the group delay. It is a function of the frequency ω and we colloquially say it is "'the time delay of
the amplitude envelope of a sinusoid at frequency ω".
here
https://www.dsprelated.com/blogimages/AndorBariska/NGD/ngdblog.pdf
is a discussion of negative GD which (for me) clears up my confusion regarding time delay.

tony_lth

### tony_lth

Points: 2
is a discussion of negative GD which (for me) clears up my confusion regarding time delay.
You can read it as a prove that group delay is a different quantity respectively has "nothing to do" with signal travel time.

The parameter "group delay" must be applied to a "group" of frequencies only (e.g. a modulation signal), which are very close to each other if compared with the corresponding mean value.
I agree with your consideration except for "must ... only". The group delay plot of a low-pass filter is an example how group delay can be determined for wide band signals as well. As previously mentioned, my viewpoint is the other way around. Group delay can be applied to small band signals, where an exact absolute signal delay or "travel time" can't be determined by nature of the signal, but isn't restricted to it. In the cases, where group delay is constant over frequency, it's effectively identical to a signal delay.

I thought group delay is about different time delay of different frequency components of a singal:

https://en.wikipedia.org/wiki/Group_delay_and_phase_delay

I thought group delay mostly used in transmission line and the cause is mainly dielectric loss where $\epsilon_c=\epsilon'+ {j}\epsilon''\;,\; \epsilon''=\frac{\sigma}{\omega\epsilon}$. Also the complex part of µ if applicable. It mainly affect the velocity of the propagation where $v=\frac{1}{\sqrt {\mu_c\epsilon_c}}$

S21 is general forward reflection coef.

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I agree with your consideration except for "must ... only". The group delay plot of a low-pass filter is an example how group delay can be determined for wide band signals as well.

Yes, I agree. My statement was not exact enough.
It is correct that the term "group delay" can be applied to wide-band signals also - however, the determination (calculation) of the group delay using the differential quotient of the phase function (negative slope) may be applied to small-band signals only.

In the cases, where group delay is constant over frequency, it's effectively identical to a signal delay.

Yes, also agreed - with one single restriction: This kind of signal delay must be caused by phase shifts only (and not by another type of delay: storage effects, travelling time of electromagnetic waves,..)

Yes, also agreed - with one single restriction: This kind of signal delay must be caused by phase shifts only (and not by another type of delay: storage effects, travelling time of electromagnetic waves,..)
At first sight, I don't see a reason why group delay can't be calculated e.g. for a delay line?

At first sight, I don't see a reason why group delay can't be calculated e.g. for a delay line?

For electrical delay line ( tx line), you should be able to calculate the group delay. As I posted in #15, the velocity change with frequency due to the imaginary part of both ε and µ. That changes the velocity of propagation in the tx line. That cause the waveform to distort.

I believe it is more than just phase shift even though you can look at it this way. But I don't think you can look at it as pole and zero as the phase shift can go over 360 degree. It is change of velocity of different frequency components of the signal.

I am not good in definition of S21 and group delay. In amplifiers/transistors that spec in s-parameter, I have not seen phase shift over 360 deg, I don't even see how you can represent s-parameter with phase shift over 360 as it wrap around. I don't see how you can use S21 for long transmission line as signal delay difference can be way over 360 degree phase shift. Can you represent delay of over 360 deg with s-parameter?

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I don't understand very much this long discussion.

The definition of group delay is clear: GD=dφ/dω, that is how the phase change as the frequency change. It doesn't imply the use complex waveforms; CW is enough. From S21 parameters taken with a network analyzer you can easily calculate the group delay just appling the definition:
GD≈{phase[S21(f2)]-phase[S21(f1)]}/[2•Π•(f2-f1)]

In case of simple linear delay, GD is constant. Since, in this case, the phase delay is a ramp with respect of the frequency then GD will be constant (the first derivative of a ramp is a constant) and represents the propagation delay.

Negative group delay simple means that, considering two frequency f1 and f2, with f2>f1 the signal will be more delayed at frequency f1. This could happen for instance in notch filters.

FvM

### FvM

Points: 2
Yes, from the Wikipedia, GD is related to $\frac {\partial \phi}{\partial \omega}$, Can S21 parameter work with phase over 360 degree as GD need to work with potential long delay shift?