# [SI-LIST] Re: Causality test

*From*: Chris Cheng <Chris.Cheng@xxxxxxxx>*To*: Yuriy Shlepnev <shlepnev@xxxxxxxxxxxxx>, 'Luciano Boglione' <l.boglione@xxxxxxxx>, "si-list@xxxxxxxxxxxxx" <si-list@xxxxxxxxxxxxx>*Date*: Sun, 22 Aug 2010 23:23:56 -0700

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 . 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**Follow-Ups**:**[SI-LIST] Re: Causality test***From:*Yuriy Shlepnev

**[SI-LIST] Re: Causality test***From:*Siming Pan

**References**:**[SI-LIST] Causality test***From:*Luciano Boglione

**[SI-LIST] Re: Causality test***From:*Yuriy Shlepnev

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