[SI-LIST] Re: Causality test
- From: Siming Pan <pansiming86@xxxxxxxxx>
- To: Chris Cheng <Chris.Cheng@xxxxxxxx>
- Date: Mon, 23 Aug 2010 09:07:46 -0700
Hi Chris,
I also did some studies regarding to causality checking based on K-K in
frequency domain. Actcually you can disregard the errors due to the upper
band limitations using "Generalized dispersion relations with subtractions"
(Piero Triverio, Stefano Grivet-Talocia, On checking causality of
bandlimited sampled frequency responses). This method also gives you a good
error band to evaluate the small error terms caused by truncation. However,
when directly implementing this method, one problem we met is when you
recheck the data that are enforced by replacing the imag part to the
calculated one, the results are still non-causal. But there are some methods
that can solve this problem.
Regards,
Siming Pan
2010/8/22 Chris Cheng <Chris.Cheng@xxxxxxxx>
> Yuri,
> I have a tough time imagining how to extend a band limited
> measurement/simulation above the Fmax using K-K.
> At least for the lower expansion I know what the DC imaginary part is :-D.
> Error also tends to blow up towards Fmax so I think I will lose the upper
> limit rather than extending it.
> Chris
>
> -----Original Message-----
> From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
> On Behalf Of Yuriy Shlepnev
> Sent: Sunday, August 22, 2010 12:15 PM
> To: 'Luciano Boglione'; si-list@xxxxxxxxxxxxx
> Subject: [SI-LIST] Re: Causality test
>
> Hi Luciano,
>
> In your best case scenario, when you know the S-parameters from DC to
> infinity continuously, you just have to verify the Kramers-Kronig (K-K)
> relations directly. Use real part of an S-parameter, compute imaginary
> through the K-K formula and compare it with the actual imaginary part. If
> S-parameters are described by formulas, the K-K relation can be verified
> analytically. Otherwise you have to introduce some metric to measure the
> deviation numerically - root-mean square error for instance. If this value
> is close to zero, S-parameters may be considered causal.
>
> Unfortunately, most of the time we have to deal with measured or computed
> discrete and band-limited S-parameters. We have discuss the problems
> arising
> from that at the tutorial at the recent DesignCon - available at
> http://www.simberian.com/TechnicalPresentations.php - see #2010_01.
>
> The simplest causality test that I know is the rotation test. Take
> S-parameter values at 3 consecutive frequency points in the complex plane,
> find center of the circle drawn through 3 points. The rotation with the
> increase of frequency should be mostly clockwise. Counterclockwise rotation
> is possible at the resonance frequencies. Ratio of clockwise rotation to
> the
> total rotation may be used as the causality measure. Non-physical and noisy
> S-parameters typically have this ratio close to zero or zero. If 100% of
> the
> rotation is clockwise, the S-parameters may be causal (not conclusive).
>
> Another way to verify the causality of the band-limited discrete data sets
> is to use causality boundaries introduced on the base of K-K relations
> introduced in:
> Milton, G.W., Eyre, D.J. and Mantese, J.V, Finite Frequency Range
> Kramers-Kronig Relations: Bounds on the Dispersion, Phys. Rev. Lett. 79,
> 1997, p. 3062-3064.
> Triverio, P. Grivet-Talocia S., Robust Causality Characterization via
> Generalized Dispersion Relations, IEEE Trans. on Adv. Packaging, N 3, 2008,
> p. 579-593.
>
> Note that with any band-limited data test you will not get a definite
> answer
> most of the time - there will be always some uncertainty. What matters is
> how you compute time-domain response of a system described with the
> discrete
> and band-limited S-parameters. Such computations based on convolution
> require interpolation and extrapolation of S-parameters. Different
> extrapolations will produce different time-domain responses with possible
> distortion of the original frequency-domain response. "Enforcement of
> causality" by computing real part from extended imaginary part also leads
> to
> hidden distortions of the original data set due to multiple possible
> extensions. Extraction of delay to make system look like delay-causal is
> also ambiguous for interconnects due to possibility of multiple dominant
> delays at high frequencies (as in Scott's example of direct communication
> between ports - we observe it quite often above 10 GHz). That all lead to
> substantial differences in the time-domain responses computed with
> different
> software on the base of the discrete band-limited S-parameters.
>
> Considering all that, the causality can be estimated as the RMS error of
> passive rational or any other causal circuit approximation of the original
> S-parameters. That is what is important for the analysis with S-parameters
> -
> passive, causal approximation from DC to infinity with the controlled
> accuracy over the originally provided frequency band. With such macro-model
> approximation you should expect exactly the same time and frequency-domain
> response computed with any circuit or system simulator. The delay-causality
> in such approximation may be valid only for signals with the harmonics
> below
> the maximal frequency in the original response.
>
> Note that passivity and causality cannot be discussed separately, because
> of
> passivity should be verified from DC to infinity to make sure that the
> response is causal and this is not possible without some kind of
> approximation. For instance, S-parameter models computed with some EM
> software may satisfy the simple passivity test eigenvals(S*S^H)<=1 over the
> provided frequency band, but cannot be extended to infinity (and sometime
> to
> DC) with causal functions without violation of passivity. Enforcement of
> passivity leads to larger RMS error of the approximation in the original
> frequency band. Larger approximation error points at some subtle causality
> problems in such cases.
>
> Best regards,
> Yuriy Shlepnev
> www.simberian.com
>
> -----Original Message-----
> From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
> On
> Behalf Of Luciano Boglione
> Sent: Saturday, August 21, 2010 4:06 PM
> To: si-list@xxxxxxxxxxxxx
> Subject: [SI-LIST] Causality test
>
> Hi folks,
>
> Is there a "test" that can be executed directly on a given S parameter set
> that can confirm causality of the network? Put yourself in the best
> possible
> conditions (e.g. off the top of my head, frequency ranges from DC to
> infinity continuously). I'm asking for something similar to a passivity
> test
> in the frequency domain.
>
> Thanks,
> Luciano
>
>
>
>
>
>
> .
>
>
> Sent via BlackBerry from T-Mobile
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--
Siming Pan
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