Implementing Kramers–Kronig causality check on real VNA s-params

[yefj]

Hello , Kramers–Kronig need infinite BW but I measure s-params from 0 to 70GHZ.
how do you reccomend to deal with the missing data?
Thanks.

 

[Baluncore]

To work perfectly, Kramers-Kronig requires a bandwidth from DC to daylight, but perfection is impossible, so you must settle for less.

The same is true in time domain reflectometry, TDR, where the incident step must be instant and the sampling rate infinite. For that you must buy a more expensive storage oscilloscope, say, with a BW of 100 GHz and a sampling rate of 240 Gss.

The advantage of time domain reflectometry, is that you do not need an IFFT or an integration, but TDR still has the bandwidth problem of Gibb's Effect, which breaks causality by appearing before the step.

 

[yefj]

Hello Baluncore,what is the proper way to implement Kramers-Kronig on my s-params which has only 10MHz-67GHz? for causality checking.

 

[Baluncore]

It will depend on the format of the data available.
Google AI to find papers and code.

 

[yefj]

Hello Baluncore, first I need to do the extrapolation .
In the article they say that they use the polinomial but they dont say how they got the polinomial coeffients.
Is there a manual that explains better the polinomial extrapolation methedfology in the Kramer -Kronig method?

 

[Baluncore]

Causality Verification Using Polynomial Periodic Continuations for Electrical Interconnects.
https://imapsjmep.org/article/39885...ontinuations-for-electrical-interconnects.pdf

 

[yefj]

Hello Baluncore ,I am trying to implement something simple with the kramer kronig method.
I have s-param file is there some manual which is more practical?
Thanks.

 

[Baluncore]

I am trying to implement something simple with the kramer kronig method.
I have s-param file is there some manual which is more practical?

There is nothing simple about the Kramers-Kronig method.

Do not start with real data, you will need to follow a step-wise process of software development using synthetic data.
1. First define a simple transmission line model, with loss and reflection coefficients.
2. Synthesise the TDR response expected, including attenuation and internal reflections.
5. Reverse the process to analyse the transmission line model from the TDR response. As you test and refine your code, make the transmission line model more complex each time, and check the result against your model.

Test the following two steps separately, then insert them into the above process sequence.
3. Compute the band limited S-parameters for the synthesised TDR over a fixed band.
4. Reverse the process by computing the TDR from the S-parameters.

Then you can apply the process to a real TDR response, and check that you get sensible results.

 

[yefj]

Hello Baluncore,The Kramer Kronig method uses hilbert transform .Why its not mentioned in the manual?

 

[Baluncore]

Hello Baluncore,The Kramer Kronig method uses hilbert transform .Why its not mentioned in the manual?

The way a process is defined mathematically, is not necessarily the way it is computed numerically.
"Multiplication by the sign function in the time domain corresponds to the Hilbert transform ( i.e. convolution by the Hilbert kernel 1/πω ) in the frequency domain."
https://en.wikipedia.org/wiki/Kramers–Kronig_relations

See: Understanding the Kramers-Kronig Relation Using A Pictorial Proof. White Paper.
By Colin Warwick, Signal Integrity Product Manager, Agilent EEsof EDA
https://hpwiki.mcguirescientificservices.com/_media/application_notes:5990-5266en.pdf

 

[yefj]

Hello baluncore ,I read the manuals especially this one which is closer to what i need.
regarding the steps you told me to follow ,how can I understand hilbert transfort causality from finite BW s-params.
After step 4 how do I use the Kramer kronig method?
Thanks.

https://hpwiki.mcguirescientificservices.com/_media/application_notes:5990-5266en.pdf

1. First define a simple transmission line model, with loss and reflection coefficients.
2. Synthesise the TDR response expected, including attenuation and internal reflections.
5. Reverse the process to analyse the transmission line model from the TDR response. As you test and refine your code, make the transmission line model more complex each time, and check the result against your model.

Test the following two steps separately, then insert them into the above process sequence.
3. Compute the band limited S-parameters for the synthesised TDR over a fixed band.
4. Reverse the process by computing the TDR from the S-parameters.

 

[Baluncore]

You tell me you want to visit the region of Kramers-Kronig, that is a long journey.

I read the manuals especially this one which is closer to what i need.
Regarding the steps you told me to follow ,how can I understand hilbert transfort causality from finite BW s-params.

Reading about a process, is not a substitute for actually building the numerical simulation. Reading a travel brochure, or an itinerary, is insufficient. You must actually take the journey, and write your own code.

After step 4 how do I use the Kramer kronig method?

That is a surprise you will discover, only when you reach that point in your exciting journey.
Your journey, will take you to your destination.

You have selected your destination, you must now take the first step.
What software will use to model the TDR of real transmission line networks?