Using an SA and a computer in place of a VNA?

[Kalvin]

Yes, I did understand that. I was trying to provide counter-example so that one needs to have a known reference signal in order to be able to extract the phase information.

Now that I think about this, if the response of the DUT can be modeled using well known first and second order transfer functions and get a familiar Bode-plot response from the SA, one possibly could deduce the approximate phase response from that information.

[Kire Pûdsje]

As for the original question. Please check the prerequisites for the Kramers–Kronig and what this means for an arbitrary network.
In practice, it boils down to the point that if you know the network only consists of lumped elements, the phase response can be calculated from the amplitude response (well almost). However if the network also includes transmission lines (or just delay), it cannot be done. The problem lies in the fact that a transmission line has an infinite number of poles (and zeros) in the omega domain.

The consequence of problem is perfectly demonstrated by the question in one of the former posts.
How to measure phase of an ideal transmission line (with amplitude data only).

To be correct, phase can be extracted using amplitude only measurements, but then you need to perform something like a six-port measurement (not to be confuse with a analyzer with six ports).

[Bud]

Ther seem to be too many "if"s with this approach. There can hardly be many ideal components on a RF/microwave network where anything either a transmission line or a delay. "If" the proposed approach works on paper, real life implementation will be limited to specific use cases, plus not applicable to measuring active circuits. All of a sudden "bunch of expensive hardware" makes sense.

[rhb]

As for the original question. Please check the prerequisites for the Kramers–Kronig and what this means for an arbitrary network.
In practice, it boils down to the point that if you know the network only consists of lumped elements, the phase response can be calculated from the amplitude response (well almost). However if the network also includes transmission lines (or just delay), it cannot be done. The problem lies in the fact that a transmission line has an infinite number of poles (and zeros) in the omega domain.

The consequence of problem is perfectly demonstrated by the question in one of the former posts.
How to measure phase of an ideal transmission line (with amplitude data only).

I'm not sure I understand your assertions.

Causality requires that the imaginary part of the analytic trace be the Hilbert transform of the real part.
 
Kramers-Kronig describes the relationship between attenuation and dispersion. 

Causality is invoked in both cases, but they are not the same thing.  The analytic trace addresses the relationship between phase and amplitude.  Kramers-Kronig states that causality requires that frequency dependent attenuation implies dispersion. The medium must propagate different frequencies at different velocities.

The phase delay of an ideal transmission line is trivially measured via the amplitude response alone by placing a mismatch at each end of the transmission line.  This will result in reflections for waves traveling in both directions.  The resulting pole frequency provides the phase delay information for the transmission line,  This is typically done in the time domain (TDR), but is easily done in the frequency domain.

On further reflection, I think the biggest limitation to extracting the phase from the amplitude spectrum is the frequency span and sampling required.  A VNA can measure magnitude and phase at a single frequency.  Computing the phase from the amplitude spectrum depends upon redundancy from the measurement of many frequencies.

I've been given a couple of datasets with which to work, so I'm going to shut up until I've played with them.

[rfeecs]

The phase delay of an ideal transmission line is trivially measured via the amplitude response alone by placing a mismatch at each end of the transmission line.  This will result in reflections for waves traveling in both directions.  The resulting pole frequency provides the phase delay information for the transmission line,  This is typically done in the time domain (TDR), but is easily done in the frequency domain.

This is quite different from what you started out with:  that you can derive the phase from the amplitude response only.

If you are going to take multiple measurements with different mismatches, now you are going to a completely different method, like the 6 port relectometer:

To be correct, phase can be extracted using amplitude only measurements, but then you need to perform something like a six-port measurement (not to be confuse with a analyzer with six ports).

The six port reflectometer requires amplitude measurements only:
https://www.nist.gov/sites/default/files/documents/calibrations/mtt25-12.pdf

"A Microwave Network Analyzer Using Two 6-Port Reflectometers":
http://ieeexplore.ieee.org/document/1124353/

[Kire Pûdsje]

@rhb. It can be done theoretically, but as said the infinite number of poles is the problem.
It is like stating that 0*x will always result in 0. However this statement also breaks down when x approaches infinity.


Just thinking again about obtaining the phase response from the amplitude response, I realized, this is what is performed all the time when synthesizing filters from scratch.
We already performed this in the second year at the technical school.
Starting with a power transfer. (|s21|^2).
Using conservation of energy to calculate reflection (1-|s21|^2)
finding the Hurwitz polynomial.
Extraction poles/zeros or perform pole splitting, etc. to generate series/paralel L/C or LC

Again here we know that by defining the polynomial in the omega-domain, the network is by definition a network consisting of only lumped elements. (Or the theoretical UE unit element).
(To be fair, in the commensurate domain a polynomial can be found from which, a transmission line filter could be synthesized. This on the other hand cannot contain lumped elements).

[niconiconi]

The phase delay of an ideal transmission line is trivially measured via the amplitude response alone by placing a mismatch at each end of the transmission line.  This will result in reflections for waves traveling in both directions.  The resulting pole frequency provides the phase delay information for the transmission line,  This is typically done in the time domain (TDR), but is easily done in the frequency domain.

This is quite different from what you started out with:  that you can derive the phase from the amplitude response only.

If you are going to take multiple measurements with different mismatches, now you are going to a completely different method, like the 6 port relectometer:

Recently I just learned the existence of Kramers-Kronig relations, and I started to question "why do we need a VNA at all, if we can use K-K", the search engine brought me here. So the answer seems to be that...

1. Yes, theoretically it's doable and practically it's also doable for lumped circuits.
2. But "a transmission line has an infinite number of poles (and zeros) in the omega domain" so it has limited practicality for RF circuit.

So it's a nothingburger, what a disappointment...

---

Coincidentally, I also knew a little about 6-port reflectometer, others may find the following discussion interesting.

In early 2022 I have successfully used multiple known mismatches and a non-linear equation solver to find the complex reflection coefficient of a signal generator, using scalar power measurement only. I used the method from a presentation by METAS, Switzerland's national metrology institute.

It turns out that this indirect measurement method has important application on measuring the output impedance of a signal generator with Automatic Level/Gain Control and active feedback, such as a leveled sinewave generator used for RF power and oscilloscopes calibration in a metrology lab. The output mismatch of the 0 dBm, 50 MHz reference output on an RF power meter can also be characterized this way for error/uncertainty assessment for metrology.

Since the level/gain of the generator is sampled by a power splitter or coupler then regulated by the feedback, theoretically its effective output impedance or reflection coefficient is only a property of the coupling device, and it's NOT a property of the amplifier itself. "Effective" VSWR can be as low as a perfect 1.00.

Because of the non-linear nature of feedback, you cannot use most conventional methods to find this output impedance from linear network analysis, such as sending a signal into the output (well, you actually can, a Fluke paper described this method, but the signal must be small enough to avoid disturbing the feedback). METAS proposed that a better and more reliable method is observing the change of forward power by attaching different characterized reflective loads to "reverse engineer" the complex output impedance by solving a system of non-linear equations. This is very similar to a classic 6-port VNA.

METAS paper:
http://resource.npl.co.uk/docs/networks/anamet/members_only/meetings/32/20091016_anamet32_furrer.pdf

Another NPL paper also used a 6-port like method to characterize a power splitter.
http://resource.npl.co.uk/docs/networks/electromagnetics/071129/rfmt_howes/miall.pdf

Basically, consider the simple case of a signal generator connected to an ideal, but deliberately mismatched power meter. You do multiple power measurements, each with a different mismatch. If you write down the equations that describe the power delivered into the power meter, then each RF power measurement defines a circle on the complex plane for all the possible reflection coefficients, and the complex impedance is the intersection point of these circles, which can be numerically approximated (well, in this particular case, each RF power measurement doesn't really define a single circle but set of infinitely many possible circles, but you get the point...)

Practically, a "deliberately mismatched power meter" is usually infeasible, nobody makes these power meters. I tried to use a stub to simulate it, but the measurement error is truly enormous and the result is unusable. I'm not sure what exactly went wrong, but it doesn't matter. I just remembered, it was because the power delivered to the meter approaches 0.

METAS's method is a further refinement of the method. Instead of using a mismatched power meter, it uses a directional coupler with mismatched loads attached, with a power meter to sample the forward power to the load. To simplify the math a bit, each time a mismatched load is attached, the coupler is fully characterized by a VNA as a 2-port network, not a 3-port.

I have successfully replicated this experiment. To verify the measurement, a signal generator is used with a large attenuator attached. So that the return loss is mostly determined by the attenuator, not the generator. This is the same method used by METAS for verification. The red trace is the solved return loss, the blue trace is the "true" (VNA-measured) return loss. As you can see, the quality of the data still has some problems, including a large outlier, but you get the idea. My leveled-sinewave generator project was suspended for another project but I eventually plan to return to this project and fully describe this method in an article.

[T3sl4co1l]

More specifically, K-K is solvable when the system is minimum-phase.  (I think?)  This is the case for a one-port, but a special case for multiple.

If you know that your network is minimum phase (e.g. an all-pole filter), it should be solvable that way.  But if it's not (zeroes, real delay), no such luck.

Clearly, an all-pass filter isn't something you can pick up on, scalar; if the amplitude response is truly flat (within measurement error), you'd certainly have no reason (or even mere inkling) to suspect it's any more complex than unity.

Tim