VNA signal source SNR importance for S-parameter measurements?

[matthuszagh]

To what extent does the SNR of a VNA signal source matter for S-parameter measurements? To make this a bit more concrete, imagine I have a signal source with an SNR of 100dB and then degrade the SNR to 80dB. Would this affect the S-parameter results and, if so, how and why? This question is purely out of curiosity.

My expectation is that this metric isn't super important. In a super-heterdyne system such as in a typical VNA, noise far from the carrier should be strongly attenuated by the bandpass IF filters and therefore shouldn't have a large effect on the measured response. On the other hand, I would expect close-in phase noise to have a measurable effect since this could pass through the filters and effectively reduce the frequency selectivity of the measurement. Is this logic correct? Anything I'm not considering?

[jwet]

Your intuition is likely correct but the calculations get hairy.  S parameter measurements have all kinds of interacting error sources and uncertainties.  The biggest error source is the return loss of your test set and directional couplers.  A good test set might have a return loss of 60 db which sets a fundamental limit on everything else.  Google "mismatch uncertainty" or get a good microwave reference text- lots has been written about the topic.  Be prepared to do a lot of algebra with equations with upside L's in the- Gamma, a small joke.  HP used to have nomograms and little slide rules that would calculate this kind of stuff.  I think the difference between 80 and 100 dBc will be well below the other errors and not a real concern.

[paul@yahrprobert.com]

What comes to mind is situations where you're trying to measure items that are far away from 50 ohms, such as the impedance of an end-fed halfwave antenna where the resistive part is thousands of ohms. Then the VNA is trying to measure the slight difference from a perfect reflection.  Then your calibration method and your noise levels all have to be very good.

[jwet]

True- VNA/Reflection measurements have limits imposed by the coupler and the sensitivity and noise.  They aren't great for SWR's much over 10.  There are other techniques like RF I/V and others.  HP has a kind of extended app note about measuring impedance- it might be called a handbook.  Its basically a sales tool that goes through all the options pretty well.

[matthuszagh]

Keysight has an app note called "Ultra-Low Impedance Measurements Using 2-Port Measurements" that discusses accurate VNA measurements of very low impedances, which relates to the point you're making. The limiting factor here though is receiver dynamic range (measurement down to 0.1ohm requires about 100dB dynamic range), not source noise.

[Marsupilami]

To what extent does the SNR of a VNA signal source matter for S-parameter measurements?

If you think about the ideal VNA it doesn't matter at all because you're measuring the ratio of your source and something multiplied by the source, thus it gets cancelled together with any noise content in it.
In a weird thought experiment you would get perfectly fine results using only noise as the source. (Which in practice obviously doesn't happen, but modulated signals can be used and those are similar in terms of the spectrum spread.)

In real life the nonlinearities, IF filter bw, path differences etc. will make this not strictly true but still source noise is the least of evils you have to face among VNA uncertainty contributors.
Receiver noise on the other hand, as you saw has a direct impact because it is independently added to your test and reference receiver measurements (completely uncorrelated) so the math won't help at all.
If you look at the raw receiver measurements, using the VNA as a quasi spectrum analyzer, without doing the ratio math, you would see a direct noise contribution from the source.

HTH.

[rf-messkopf]

If you think about the ideal VNA it doesn't matter at all because you're measuring the ratio of your source and something multiplied by the source, thus it gets cancelled together with any noise content in it.

This assumes that the DUT does not have a significant delay (think of a length of cable). If it does, the noise at the reference and test receivers becomes de-correlated, and no longer cancels completely.

[Marsupilami]

This assumes that the DUT does not have a significant delay (think of a length of cable). If it does, the noise at the reference and test receivers becomes de-correlated, and no longer cancels completely.

You're right, but if I'm not mistake that delay has to be significant compared to the time constant of the IF filter, in other words the phase (and magnitude) flatness of the DUT over the IFBW that matters.

E.g. something that's 10 wavelengths long at 1GHz ( ~3m ) will have a phase difference of 0.036° over a 10kHz IFBW.
I did some scribble math for it out just out of curiosity. In this case assuming unity noise power density, the total integrated power error magnitude is -64dB. This is only the noise. Getting OP's example of 80dB SNR in the IFBW this means that the source noise error contribution due to the DUT delay is -144dB.
But I was cherry picking parameters and my math isn't rock solid either. I can imagine there are legit measurement cases where it is a problem.

[rf-messkopf]

But I was cherry picking parameters and my math isn't rock solid either. I can imagine there are legit measurement cases where it is a problem.

On a second thought, it comes down to how the noise can be modeled, and how the VNA measures. Consider the power waves \(b_2\) and \(a_1\) the VNA actually measures, say, in a transmission measurement, where \(S_{21}=b_2/a_1\). Let's assume for the moment that we can model the noise contribution of the source by an additive stochastic process \(X(t)\).

Now there are several questions. For example, does the VNA do
$$
\frac{b_2+X(t+\tau)}{a_1+X(t)},
$$
where \(\tau\) is the time delay of the DUT, or is some averaging applied, say over a time interval of length \(T\), and if yes, does the VNA do
$$
\Bigl\langle\frac{b_2+X(t+\tau)}{a_1+X(t)}\Bigl\rangle_T,
$$
or
$$
\frac{\langle b_2+X(t+\tau)\rangle_T}{\langle a_1+X(t)\rangle_T},
$$
where \(\langle\cdot\rangle_T\) means the linear average over \(T\), i.e., \(\langle f(t)\rangle_T=\frac1T\int_0^Tf(s)\,\mathord{\rm d}s\). The measurements in the \(b_2\) and \(a_1\) channel could be taken at different times, which means that \(\tau\) would include this time offset as well.

Also, there is the vector error correction, which depends on more measurements than \(b_2\) and \(a_1\) alone, as was already remarked above.

And I'm not sure if the noise contribution at the power wave level can actually be modeled by an additive stochastic process, and which autocorrelation function or power spectral density it would have. In a real VNA the \(b_2\) and \(a_1\) are determined by IQ mixing, where the phase is determined from \(\mathop{\rm arctg}(q(t)/i(t))\), and the magnitude from \(\sqrt{i^2(t)+q^2(t)}\), from which the complex power wave amplitude is calculated. So it is not obvious what the power spectral density of \(X(t)\) would be. That would need a more detailed investigation.

The conclusion for now is that I'm not sure how much compensation of the source noise actually takes place in a real VNA and how source noise can be modeled. This seems to be more complicated than I thought at first.

[Marsupilami]

This would make an excellent community paper. We just have to derive it properly.

$$
\frac{b_2+X(t+\tau)}{a_1+X(t)},
$$

What I don't like about this is here a1,b2,X and tau are independent of each other.
X as the source noise is already included in a1 and tau is parameter of the operation that turns a1->b2. If it's linear then e^(iw*tau) multiplication takes care of the time shift. The useful signal component in a1 suffers the same time shift as the noise.

The mixer gives me pause though. I got super confused while writing it down. It needs more thought.
And you're right about the distribution too. In the previous post I integrated the time shift induced phase slope multiplied by a constant power density over the IFBW which is yet another approximation.

Are you fun at parties too?
 
p.s. I commend your use of properly formatted equations dear Sir. I have to ramp up my game.
p.s.2: Where's the man when we need him the most? I feel we should have a Bat-signal for such cases. Dr. Dunsmore. HALP PLS!? (Can you tag people here?)

[rf-messkopf]

This would make an excellent community paper. We just have to derive it properly.

Absolutely. Maybe it is possible to come up with a better estimator for power wave ratios that has a lower noise contribution. 

X as the source noise is already included in a1 and tau is parameter of the operation that turns a1->b2. If it's linear then e^(iw*tau) multiplication takes care of the time shift.

The thing is that \(\tau\) is not necessarily equal to the phase delay of \(S_{21}\) (when we talk about a transmission measurement as an example). If you extend \(\tau\) by a time \(\tau_1=2\pi/\omega\), then \(\mathord{\rm e}^{\mathord{\rm i}\omega(\tau+\tau_1)}=\mathord{\rm e}^{\mathord{\rm i}\omega\tau}\), i.e., the phase of \(S_{21}\) is not changed at that frequency, but a longer time delay has an influence on the noise and increases the de-correlation between the reference and measurement channels in the VNA. So you have to consider the actual delay separately from the phase of \(S_{21}\).

Moreover, as already remarked, the VNA does not necessarily measure the reference (i.e., \(a_1\)) channel and the \(b_2\) channel simultaneously. If not, this time delay would have to be added to \(\tau\).

Also, the idea is that the \(a_1\) and \(b_2\) are only the useful signal component, and thus are considered noiseless. The total signal after the receiver and signal processing chain of course includes noise, thus \(a_1\) and \(b_2\) have to be modified. A simple noise model would be an additive stochastic process \(X(t)\) with certain properties (distribution, autocorrelation function, stationarity, etc.). This is like in communications engineering, where additive white Gaussian noise is used to model a radio channel: If \(s(t)\) is the useful signal, it becomes \(a\cdot s(t)+X(t)\) at the receiver input, where \(X(t)\) is a Gaussian white noise process. One would have to check whether additive noise is a good model in the case of power waves in a VNA.

And you're right about the distribution too. In the previous post I integrated the time shift induced phase slope multiplied by a constant power density over the IFBW which is yet another approximation.

Depending on the phase noise spectrum of the source and the IF bandwidth, this might actually be a good approximation.

Are you fun at parties too?

Not sure. 

p.s. I commend your use of properly formatted equations dear Sir. I have to ramp up my game.

When you know LATeX it is actually very simple.

[Marsupilami]

The thing is that \(\tau\) is not necessarily equal to the phase delay of \(S_{21}\) (when we talk about a transmission measurement as an example). If you extend \(\tau\) by a time \(\tau_1=2\pi/\omega\), then \(\mathord{\rm e}^{\mathord{\rm i}\omega(\tau+\tau_1)}=\mathord{\rm e}^{\mathord{\rm i}\omega\tau}\), i.e., the phase of \(S_{21}\) is not changed at that frequency, but a longer time delay has an influence on the noise and increases the de-correlation between the reference and measurement channels in the VNA. So you have to consider the actual delay separately from the phase of \(S_{21}\).

I agree. I deliberately did not write \(b_{2}=S_{21} \cdot a_{1}\)  , but if you keep the correct unwrapped phase I believe it will check out and still can be handled as multiplication.

Moreover, as already remarked, the VNA does not necessarily measure the reference (i.e., \(a_1\)) channel and the \(b_2\) channel simultaneously. If not, this time delay would have to be added to \(\tau\).

Now this I completely overlooked and you're absolutely right, this completely 100% invalidates my point. Bummer. 
It has to be a 4 receiver architecture for what I said to have a chance to work.
I'm getting more and more convinced by the minute that this is actually a legit concern, or at least I'm not sure why it isn't.

Also, the idea is that the \(a_1\) and \(b_2\) are only the useful signal component, and thus are considered noiseless. The total signal after the receiver and signal processing chain of course includes noise, thus \(a_1\) and \(b_2\) have to be modified. A simple noise model would be an additive stochastic process \(X(t)\) with certain properties (distribution, autocorrelation function, stationarity, etc.). This is like in communications engineering, where additive white Gaussian noise is used to model a radio channel: If \(s(t)\) is the useful signal, it becomes \(a\cdot s(t)+X(t)\) at the receiver input, where \(X(t)\) is a Gaussian white noise process. One would have to check whether additive noise is a good model in the case of power waves in a VNA.

That's fine, I just wanted to point out that the way how X contributes to \(a_1\) and \(b_2\) is different.
For the sake of simplicity let's consider the \(\tau < 1/2f\) so we don't have to deal with the phase wrapping.
Then keeping the wave quantities purely for the useful signal we can write the DUT input equals to \(a_1 + X\) while the output will be \(S_{21}\cdot a_1 + S_{21}\cdot X = b_2 + S_{21}\cdot X\)

(these equations are killing me, I'm editing this for the 5th time)

[Marsupilami]

I'm getting more and more convinced by the minute that this is actually a legit concern, or at least I'm not sure why it isn't.

What I'm thinking now is maybe it doesn't have to be correlated, it's enough if the total integrated power contribution is equal between the receiver readings.
I see more and more fault in my ways. I assumed the cancellation by the division, but the integration by the detector happens before the division so that's just not happening the way I imagined.

To what extent does the SNR of a VNA signal source matter for S-parameter measurements?

I'm sorry dude, at this point disregard everything I said earlier.

[jjoonathan]

I don't like the "delayed noise" model.

1. Source noise would have to be so high that it survives reflection, coupling, and mixing while still landing above the noise floor of the receiver. This does not seem likely, especially at lossy microwave frequencies, unless you are using a PNAX signal generator module to generate I/Q modulated noise or something (at which point I'd argue that it's a noise-like signal, not noise). It seems much more likely that the source noise sinks beneath the noise floor of the receiver and therefore gets replaced with new, uncorrelated noise.

2. Mixing has the potential to mix several different bands of noise together, unless you carefully balance preamplification, preselection, and mixer loss. Spectrum analyzers take great care with this and VNAs generally don't: samplers used to be common, folding noise from dozens or hundreds of bands down to IF, and harmonic mixing is still common today. PNAX noise receivers are the only potential exception that springs to mind (potential because I'm not 100% sure this is a design objective of theirs), but the fact that they are optional sort of makes them the exception that proves the rule, doesn't it?

I would model the noise as uncorrelated. For S11 I would expect excess SNR, so it wouldn't matter except insofar as you could trade SNR for speed. Speed is always welcome! For S21 your SNR is going to determine your ability to inspect filter stopbands, but I suspect this is obvious enough that people are ignoring it rather than missing it.

Needless to say, Joel Dunsmore's book has a detailed examination of these subjects. I think I recall seeing an equation that turned receiver noise into trace noise.

[jjoonathan]

Here is an 8510 sampler LO. I think in this particular picture you are seeing a conduction angle designed to favor low bands, but still, imagine you have a signal at 25GHz. Imagine that it's 20MHz above one of those harmonics and is therefore headed for IF. You probably can't help but notice that there are a dozen+ other, higher amplitude LO harmonics, each with +-20MHz sidebands, all with noise, and they're all going to pile into the same IF.

Here is what that looks like for S21: look how high the noise floor is on a 50GHz measurement. That's because the poor samplers are operating at harmonic 100 or something so they have 99 bands of noise layered on top of the signal. S11 will have a similar problem, but even so, the rated residual directivity is 36dB. Compare to 35dB on a modern N5245B (tech specs, page 26)! My guess is that the modern instrument is actually slightly better and the extra dB went into better methodology, but still, my point is that a TON of additional noise on the 8510 didn't degrade the directivity spec.

That's impressive, but the 8510 is slow as balls. To the extent that more SNR => less time averaging => more speed, more SNR is better. The PNAX is wicked fast, and I'd obviously swap my 8510 for a PNAX in a heartbeat if my budget permitted 

[rf-messkopf]

I assumed the cancellation by the division, but the integration by the detector happens before the division so that's just not happening the way I imagined.

Yes, if this is the case (and I assume it is in any real VNA) the discussion is moot.

I don't like the "delayed noise" model.

If you wanted to take advantage of some source noise cancellation between the reference and measurement channels, you would have to include such a delay. Whether this is worthwhile in a realistic situation, as you rightfully question, is another problem.

Even though this may be an academic question, it is still interesting: Suppose you have two quantities affected by partially correlated noise, and you want to estimate their quotient. What is the most efficient estimator, and what would be the error?

Needless to say, Joel Dunsmore's book has a detailed examination of these subjects. I think I recall seeing an equation that turned receiver noise into trace noise.

Section 2.3.3 talks about receiver noise, and there is a formula to convert the noise floor to trace noise. There are also some remarks on the phase noise of the source, which, according to Section 2.3.3.1, can dominate trace noise at higher levels (i.e., away from the noise floor). There are however no explicit noise models discussed.

[jjoonathan]

Right, but the modeling choice to focus entirely on receiver noise (and LO phase noise) rather than LO SNR is pertinent.

I think we are all in violent agreement.

[coppercone2]

This assumes that the DUT does not have a significant delay (think of a length of cable). If it does, the noise at the reference and test receivers becomes de-correlated, and no longer cancels completely.

You're right, but if I'm not mistake that delay has to be significant compared to the time constant of the IF filter, in other words the phase (and magnitude) flatness of the DUT over the IFBW that matters.

E.g. something that's 10 wavelengths long at 1GHz ( ~3m ) will have a phase difference of 0.036° over a 10kHz IFBW.
I did some scribble math for it out just out of curiosity. In this case assuming unity noise power density, the total integrated power error magnitude is -64dB. This is only the noise. Getting OP's example of 80dB SNR in the IFBW this means that the source noise error contribution due to the DUT delay is -144dB.
But I was cherry picking parameters and my math isn't rock solid either. I can imagine there are legit measurement cases where it is a problem.

Well, I did not get into the math yet (as budget permits for it to be useful) but VNA delay related measurements on lossy DUT are reputed to be particularly noisy/bad (to where the author of microwave101 'approves' the use of smoothing, which is normally regarded as marketing trickery, when it comes to group delay on what I assume might be long waterlogged cables or something). Phase shift/delay defiantly has a big impact on some measurements, if I recall because of noise.

[paul@yahrprobert.com]

Another situation where source noise could really mess you up is if you were analyzing quartz crystals or crystal filters which have their S parameters varying very rapidly vs frequency.
A caution I would also throw out there for you guys contemplating doing a lot of math is that the analyzer's calibration will have to be done using your noisy source.  If my boss told me to calculate the error due to that I'd start looking for a new job!

[Marsupilami]

1. Source noise would have to be so high that it survives reflection, coupling, and mixing while still landing above the noise floor of the receiver.

I don't see it guaranteed. I'm not saying it's not, I just don't know if it is and why. Here's one hypothesis:
One might assume that the same signal (more or less) is used as the stimulus and the LO for the mixers, coming from the same source.
In this case the noise contribution to the receiver readings via the LO is always going to be at least as much higher as the coupling factor of the couplers than the that of the noise from the stimulus or return. Since the LO is always present, the noise of it is already factored into the receiver noise thus the variable but smaller contribution from the stimulus/return signal is negligible.
I don't know if it's true though. Is the coupling factor high enough usually to make it so?

Also the ratio math still helps. Even if the noise in the response is not correlated to that of the stimulus, and it's delayed, and it's twisted a bit by the DUT transmission variation over the IFBW, it still will roughly be treated as the useful part of the stimulus, meaning that its integrated power over the IFBW will have a similar ratio arriving at the receivers as the useful component.
At least the difference between the stimulus noise ratio at the receivers and the stimulus CW ratio at the receivers is small.

[Marsupilami]

the analyzer's calibration will have to be done using your noisy source.

I feel the errors coming from a non-ideal calibration are easier to find in literature, so if we had the source noise contribution to the raw measurements figured out then to work up to the final corrected measurement uncertainty would be... well... slightly less extremely difficult.

If my boss told me to calculate the error due to that I'd start looking for a new job!

We all find fun at different places.

[virtualparticles]

Hello! Please see this paper and scroll to Transmission measurement uncertainty. The noise does matter.

https://coppermountaintech.com/introduction-to-the-metrology-of-vna-measurement/

[Joel_Dunsmore]

The details are covered in pages 109-111 of the second edition of www.tinyurl.com/JoelsMicrowaveBook but in general Source Noise affects trace noise when the noise decorrelates and when it rises above the noise floor of the receiver.  So we have two specs on trace noise, one which is "high-level trace noise" and depends on source power as the phase noise starts to affect the measurement, and on low level trace noise which depends on receiver noise floor. One figure of interest is Fig 2-34, reprinted here (note this was from before the new DDS, whose phase noise is so low it is never above the receiver noise floor):

[gf]

in general Source Noise affects trace noise when the noise decorrelates

In practice, is there actually a chance that it won't decorrelate between multiple receivers?

[Joel_Dunsmore]

[In practice, is there actually a chance that it won't decorrelate between multiple receivers?

In short, no.  But I had an instance where we were looking at trace noise across a frequency sweep and we found that when the raw data (as measured on uncorrected ADC results) was at +-90 degrees the traces noise was about 4 times what it was at 0 or 180 so there must have been some extra decorrelation when the two receivers were in quadrature, or thinking about it another way, there was some level of correlation. In fact, it is likely this is a result of very close in phase noise that one might expect to cancel in the normal ratio measurement.

In older systems, where the phase noise was significant in the source and LO (the signal driving the mixer) the loop dynamics were adjusted to minimize the phase noise at the IF offset frequency, but as we moved to better synthesizers we don't need to do that any more.