[SI-LIST] Re: Causality test

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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