Group delay from S21

In effect you could have problem with 2•Π phase wrapping when calculating dφ from S21. However usually (a part particular systems) you could assume GD is a continuous function then it will be possible unwrap the phase especially in case of small dω.
Moreover the error in the GD calculation will only occours when the phase is wrapped (i.e.: one ore more single points), since we are interested in dφ and not in φ

albbg said:

In effect you could have problem with 2•Π phase wrapping when calculating dφ from S21. However usually (a part particular systems) you could assume GD is a continuous function then it will be possible unwrap the phase especially in case of small dω.
Moreover the error in the GD calculation will only occours when the phase is wrapped (i.e.: one ore more single points), since we are interested in dφ and not in φ

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I worked for LeCroy before, the one for transient recorders and digital scopes. It is not the dφ that we care, it's the distortion of the waveform that we cannot cancel out. That is time domain, not frequency domain. We did series parallel A to D conversion, I can tell you, we cannot compensate with poles or zeros. I fixed a lot of problems by ONLY changing the delay lines to bigger, lower loss delay lines. We used to use about 10 to 20' of RG174, it never work no matter how we compensate it. Changing to RG-58 fixed everything.

It is one thing to talk theory and definitions, it's another thing to make it work.

I don't understand very much this long discussion.

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I agree. The term definitions are clear, there are effectively no open points.

I worked for LeCroy before, the one for transient recorders and digital scopes. It is not the dφ that we care, it's the distortion of the waveform that we cannot cancel out. That is time domain, not frequency domain. We did series parallel A to D conversion, I can tell you, we cannot compensate with poles or zeros. I fixed a lot of problems by ONLY changing the delay lines to bigger, lower loss delay lines. We used to use about 10 to 20' of RG174, it never work no matter how we compensate it. Changing to RG-58 fixed everything.

It is one thing to talk theory and definitions, it's another thing to make it work.

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This is an interesting topic, but not related to group delay, I think. In so far group delay theory don't help with transmission line distortions. There's a certain problem that popular lossy line models as used in most SPICE variants don't represent frequency dependend cable losses correctly. Aplac is the only exception I'm aware of. You can basically model it with simulators that support laplace (s-domain) descriptions, by using a sqrt(s) proportional attenuation, but the performance is restricted by FFT window length.

The waveform distortions by lossy transmission lines are caused by skin effect losses and primarly appear as frequency dependend attenuation. RC ladder compensation networks can partly correct the waveform distortions, they have been extensively used in the age of analog oscilloscopes (long before LeCroy company started to make it) and are still used today as cable equalizers for analog video and measurement technique.

P.S.: I appended an example of a lossy line simulation in LTSpice, using laplace description.

See also https://www.edaboard.com/threads/265991/

It is my believe that variation of velocity of propagation is the cause of the problem. Actually there is a part in "Field and Wave Electromagnetics" by David K Cheng that talked about distortionless transmission line. Of cause, there will be no variation in velocity in lossless tx lines, but there is no lossless line. The book gave for distortionless tx line where:

\[\frac {R}{L} = \frac {G}{C}\]

\[\delta=\alpha +{j}\beta=\sqrt{(R+{j}\omega{L})(\frac {RC}{L}+j\omega{C})}=\sqrt{\frac{L}{C}}(R+{j}\omega{L})\]

\[\Rightarrow\; \beta=\omega\sqrt{LC}\]

Where \[u_p=\frac {\omega}{\beta}=\frac {1}{\sqrt{LC}\] is constant with different frequency. This is in P440 to p443 in the book.

It is my experience that it is not the attenuation or the phase shift alone that is the problem for the distortion of the tx line. We cannot compensate with gain or phase shift. This is not something that you can see in the scope display. We do ADC in series parallel configuration where we sum the digitized signal with the delayed analog and look at the difference. There is no RC or high pass amplification that come close to compensate the distortion.

BTW, I don't think skin effect cause phase distortion as skin depth \[\delta_c=\frac {1}{\sqrt{\pi{f}\mu\sigma}}\]

albbg said:

I don't understand very much this long discussion.
.............
.............
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.

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Hi albbg,
as you don`t understand this "long discussion" - may I ask you what is "the signal", which you compare with f1 in your above cited sentence?
Do you mean simply f2 or something else?
(perhaps the meaning of the parameter GD is not as clear as it seems?).

LvW

I hold my statement, that the discussed waveform distortion isn't primarly related to group delay variation. Group delay is nevertheless affected. How do you conclude that skin effect attenuation can't cause group delay variation? The basic sqrt(s) proportional attenuation in fact does.

I have no problems to admit, that lossy transmission lines involve both frequency dependend phase and magnitude errors. I understand that we agree about the fact, that they can't be completely compensated by simple equalization networks. The feasibility of an approximate equalization depends on the required quality.

Classical cable equalizers are trying to reduce the waveform distortion empirically, ladder networks implement an approximation by nature, in so far there must be a residual error. I believe that it's difficult or even might be impossible to achieve a performance sufficient for the time delay of a cascaded ADC.

P.S.: I don't exactly understand why you say the classical lossy RLCG transmission line model is distortionless? Of course it involves waveform distortions. The problem is that the model hasn't to do with any real transmission line. It's an over-simpilfication without practical worth.

I am not expert in this, I just join in the discussion. I can only quote the book that talk about different velocity for different frequencies. Here is a link:

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

LvW said:

Hi albbg,
as you don`t understand this "long discussion" - may I ask you what is "the signal", which you compare with f1 in your above cited sentence?
Do you mean simply f2 or something else?
(perhaps the meaning of the parameter GD is not as clear as it seems?).

LvW

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With "signal" I was referring to a simple sinwave (CW). I think the GD definition is clear: it says how the signal spectrum is delayed, frequency by frequency. If GD is constant then the phase propagates as a ramp (with respect to the frequency), that is increases with frequency; this means the system is distortionless (I'm speaking about phase distortion). In fact if the system delays all the frequency with the same value (for instance due to a travelling over space), the phase delay will increase with frequency because you have to compare the fixed time delay with the period of the sinusoid that shorten as the frequency increases, so the same time delay will contain more period of sinusoid.
If the GD is constant, that is all the frequencies composing a wideband signal are delayd by a fixed time delay, the input signal (of course input to the system we are considering) will not be distorted since all the components will arrive at the output at the same time (i.e. after the constant time delay). If, instead, GD is not constant the output signal will results distorted since the components at different frequencies will reach the output at differnt instants.
I'm not sure I'm able to explain clearly enough what I'm trying to say.

Just to answer to Alan0354 I think he is right but I think also here we are speking about the theory of GD. Correct me if I'm wrong.

Hi albbg,

thanks for the long explanation - however, that was not my question. I know the meaning of a constant GD and the relation to the phase response.
I was referring to your post#19:
"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."

Here you consider just two frequencies f1 and f2 and, in addition, you speak about a "signal", which "will be more delayed at f1".
And because this was your explanation for the case of a negative group delay my question was simply if your term "signal" is identical to f1 or something else.
For my opinion, this does not explain the effect of a negative group delay. For my opinion, you have only compared two sinusoidal waves which have different phase delays.
Or did I misunderstand something?
LvW

Added later: Perhaps it is helpful to give my explanation for negative group delay (based on your example):

Suppose a system has a rising phase function within a certain frequency range. When two frequencies f1 and f2 (within this range) are connected to the system input (superimposed) the compound signal will appear at the system output with a negative delay (if compared with the input). That`s all. A more realistic example is an AM modulated signal - and that is the reason that some authors use the term "envelope delay" instead of "group delay".

I have difficulties to identify the exact controversial subject of the present discussion.

Presumed we have a channel characterized by a phase transfer function φ(ω). We can determine GD = -dφ/dω and may observe it's negative in a certain frequency interval (rising phase). I think, there's no doubt about the calculation method, apparently the controversy is about the physical meaning of negative group delay respectively how it can be visualized with real signals?

As a first remark, we must not necessarily assign a physical meaning to the group delay property to determine it, we can e.g. slowly sweep a sine generator and just measure φ(ω), without getting any qualitative insights about the system behaviour.

To "see" a group delay effect in time domain we must apply signals that have a structure in time and preferably have sufficient small bandwidth to assign it selectively to particular frequency range. The gauss shaped sine burst in the discussed negative group delay paper is a good example of such a signal.

A single sine frequency hasn't a structure in time and doesn't allow to visualize group delay. A wide band pulse will be also affected by phase dispersion dφ/dω, but doesn't directly show the frequency selective effect. That's the reason why group delay can be best shown for small frequency "groups".

https://en.wikipedia.org/wiki/Group_delay_and_phase_delay
Wherein:
In signal processing, group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component. Phase delay is a similar measure of the time delay of the phase, instead of the delay of the amplitude envelope, of each sinusoidal component.
All frequency components of a signal are delayed when passed through a device such as an amplifier, a loudspeaker, or propagating through space or a medium, such as air. This signal delay will be different for the various frequencies unless the device has the property of being linear phase. (Linear phase and minimum phase are often confused. They are quite different.) The delay variation means that signals consisting of multiple frequency components will suffer distortion because these components are not delayed by the same amount of time at the output of the device. This changes the shape of the signal in addition to any constant delay or scale change. A sufficiently large delay variation can cause problems such as poor fidelity in audio or intersymbol interference (ISI) in the demodulation of digital information from an analog carrier signal. High speed modems use adaptive equalizers to compensate for non-constant group delay.


https://en.wikipedia.org/wiki/Heaviside_condition
Wherein:
A signal on a transmission line can become distorted even if the line constants, and the resulting transmission function, are all perfectly linear. This happens in two ways: firstly, the attenuation of the line can vary with frequency which results in a change to the shape of a pulse transmitted down the line. Secondly, and usually more problematically, distortion is caused by a frequency dependence on phase velocity of the transmitted signal frequency components. If different frequency components of the signal are transmitted at different velocities the signal becomes "smeared out" in space and time, a form of distortion called dispersion.

I don't think group delay has any implication on amplitude attenuation, mainly about velocity variation with frequency. i think ALBBG hit it right on the nail in post #2 in the first 3 lines already. From what I read in the above two Wikipedias, Group delay is mostly for time domain. You can look at it as \[\frac{\partial \varphi}{\partial \omega}\], but it is a lot more than that. S21 is purely frequency domain. I am no expert in all the detail, but seems like this is quite straight forward. Am I missing something?

I don't think group delay has any implication on amplitude attenuation, mainly about velocity variation with frequency. i think ALBBG hit it right on the nail

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I'm not sure what your post is targetting to. There's no doubt about the considerations as such. However, it hasn't been stated by anyone that group delay has an implication on amplitude attenuation.

In contrast, I mentioned that the waveform distortions observed with transmissions lines are not primarly caused by group delay variations rather than frequency dependend amplitude variations. It's clear however that both amplitude and phase variations are involved with lossy lines. A discussion if one or the other is the dominant effect for a specific waveform distortion would be somehow off-topic here. I already admitted, that both are present.

I think, I owe an additional comment on the RLGC lossy transmission line model. I said, it's inappropriate to model real transmission lines. This means, that it's unable to represent the real behaviour in wide band respectively time domain as long it's parameters are fixed. This doesn't exclude, that you can see some delay dispersion with this model as discussed in the Heaviside Wikipedia article. But it's far from a realistic transmission line.

FvM said:

In contrast, I mentioned that the waveform distortions observed with transmissions lines are not primarly caused by group delay variations rather than frequency dependend amplitude variations. It's clear however that both amplitude and phase variations are involved with lossy lines. A discussion if one or the other is the dominant effect for a specific waveform distortion would be somehow off-topic here. I already admitted, that both are present.

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In both article, they talking about variation of delay with frequency for group delay. From the articles, variation of delay ( velocity) cause the pulse ( comprises of a lot of frequencies) to spread, which is the major concern of transmitting information. I don't see "frequency dependent amplitude variation" as the major problem for transmission, you round off the pulse by slowing down the rise time like you round a square pulse. Group delay cause the components of the waveform to shift and "spread" the pulse and that's the distortion that cause the major problem with communication and it is group delay problem.

BTW. The variation of velocity for different frequency does not come from RLGC tx line representation, that is just from the article of distortionless transmission line. The velocity is calculate in post #15 that uses µ and ε to calculate velocity. It is the complex part that causes the variation in µ and ε with frequency and whereby causing the velocity variation as \[v\;=\;\frac {1}{\mu_c\epsilon_c}\]. That is time domain.

FvM said:

I have difficulties to identify the exact controversial subject of the present discussion.

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Not really "controversial" - but, for my opinion, it makes a considerable difference if the special case of a negative group delay is explained
* based on two single frequencies at the system output - as described by albbg in post#19 (Negative group delay simple means that, considering two frequency f1 and f2, with f2>f1 the signal will be more delayed at frequency f1. ), or
* based on the delay a compound signal suffers between system input and output.

That was the reason, I have asked albbg for clarification.

The variation of velocity for different frequency does not come from RLGC tx line representation, that is just from the article of distortionless transmission line. The velocity is calculate in post #15 that uses µ and ε to calculate velocity. It is the complex part that causes the variation in µ and ε.

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There's a direct relation between imaginary µ and ε components and the loss elements G and R in the RLGC model. The statement about the "distortionless" condition R/L = G/C implies, that the model shows distortion for other value pairs of R and G. In a usual coaxial cable G can be effectively ignored at frequencies below GHz, so the RLGC model already indicates waveform distortions because R/L = G/C is never met. But as said, for a realistic model, R and G must be made frequency dependend.

To close the circle to the initial question, group delay and attenuation variations of coaxial cables can be separately derived from a complex S21 measurement and you should be able to distinguish which parameter is more critical for your application. I know, that it's magnitude variation for a number of analog measurement problems, but of course there are other cases as well.

I agree that R and G is frequency dependent. R is dependent on skin effect and G definitely depends on dielectric loss. I never feel comfortable with the RLGC model, the L and C is really from definition of EM wave propagation.....From the wave equation. They just DEFINE L and C, not the other way around.

Yes, I did think a little about the distortionless tx line, I think it's more for brain tease than real life application......more like wishful thinking that you can somehow design a tx line so R/L=G/C!!! Back to LeCroy, at the time, even when I change the RG174 to RG58, I still had to have a few pole zero compensation to make it work. At least I can get there with compensation. There was no way to compensate with the RG174. There is no distortionless tx line OR lossless tx line!!!!

BTW, I am so glad to find this forum. I found this forum when I was looking for PCB layout package and learn the Eagle. But I am not exactly a pcb designer. Then I notice the antenna and EM section where that's my main interest. I am studying antenna design right now.

LvW said:

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)

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Yes, and a constant phase slope vs frequency is the same as saying a constant group delay. It is group delay unflatness (i.e. the group delay value changes at different frequencies) that screws up modern communications, especially complex modulations like 256 QAM, etc. If you had a perfectly flat phase shift increase vs frequency, as in say a length of 50 ohm line, it will not impede demodulation at all!

About negative GD, let's consider a simple op-amp based circuit: R feedback from output to inverting input and C from inverting input to ground.
Considering an ideal op-amp it's easy to calculate the transfer function as:

Vo/Vi=(1+jωRC)

The the input-output phase relation is given by:

φ(ω)=arctg(Im/Re) = arctg(ωRC)

From which we can derive the GD:

GD=-dφ(ω)/dω=-RC/(1+ω²R²C²)

Let's try a numerical example: R=1k, C=1n

φ(ω)=arctg(6e(-3)f)

and

GD = -1e(-6)/[1+4e(-11)f²]

taking for example f1=12kHz and f2=13kHz we will have

φ(12k)≈0.075 rad that is 994ns time delay
φ(13k)≈0.082 rad that is 998ns time delay

and

GDapprox=-[φ(13k)-φ(12k)]/(13k-12k)=-1034ns

applying instead the exact GD we have -994ns

P.S.: in my post #28 I wrote that phase increases, but I meant its absolute value increases while the sign is negative.

albbg said:

The group delay is the delay in the propagation of the information from input to output. .

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As is evident by the relationship; group delay is not the delay in propagation of information from input to output rather it is a "change" (more precisely change in phase) measured over frequency content of the signal. Thus when there is no change GD is zero no matter how much it takes for the information to propagate from input to output.

Hi albbg,

Of course, I agree to your calculation in post#38. However, I am afraid, there is something like a misunderstanding between us.
Let me explain in short:
*The original question was about the concept/meaning of the term group delay (post#1).
* In your post #4 and post#6 you spoke about the propagation delay (in space between A and B), which in this context is very „questionable“ (as prooved by other thread contributors).
* In your post#19 you have used - for my opinion - a very unclear description using two frequencies f1, f2 and, in addition, the term „signal“ (which gave reason to my question in post#25)
* In post#28 you gave us an explanation: „signal=sinewave“, which obviously is not correct in the discussed context.
* Now in post#38 you have presented an example - again using two discrete frequencies only.
_________________________
From all your contributions I had concluded that the PHYSICAL MEANING of the term „group delay“ was not quite clear to you (and remember: that was the original question!).
In your calculation, you did nothing else than to prove that the formula for this parameter gives a negative value in case of a rising phase.

But I think, the questioner would like to know: Which signal is delayed if compared with which other signal?
And this cannot be answered using only the phase properties (delay) of two discrete frequencies f1 and f2 at the output of a system.
To put it simple: You cannot compare the phase of two different frequencies!
You must compare the compound signal after superposition of (at least) two frequencies at two distinct points - namely at the input and at the output of the system under consideration
(as explained in my post#29 and#34).

LvW