Where S-parameters come from, the four ways they arrive damaged, and the uncomfortable demonstration that a model can match insertion loss to a fraction of a decibel and still get the eye wrong.
A vector network analyser sweeps a sinusoid and measures the ratio of what comes back and what goes through to what went in, in magnitude and phase, at each frequency. A time-domain reflectometer launches a step and records the reflection against time. These sound like different experiments and they are the same one: the two data sets are Fourier transforms of each other, and every modern instrument will show you either view of whichever it measured.
The frequency domain has the better dynamic range and the better defined calibration, so it is where characterisation is done. The time domain is where a human can see which feature is responsible, because position along the trace maps to time, which is why debugging is done there.
A network analyser measures only the frequencies it sweeps, so everything below the first point and above the last has to be supplied by assumption. Those assumptions are the subject of slides 06 and 07, and they do more damage than most measurement error.
The channel used throughout this deck is the one from SerDes_Equalisation, imported rather than re-fitted, so nothing that article quotes is disturbed by anything done here.
An instrument measures at its own connectors. Everything between there and the thing you care about — cables, adapters, launch structures, a length of trace on a test coupon — is in the measurement unless it is removed, and calibration is how the reference plane gets moved.
| Method | What it needs | Where it is used |
|---|---|---|
| SOLT (short, open, load, through) | Characterised standards, whose behaviour is known as a function of frequency | Coaxial work; entirely routine well past 20 GHz with good standards |
| TRL (through, reflect, line) | Only a through, some repeatable reflection, and a line of known length — the impedance of the line defines the reference | On-board and on-wafer, where good standards cannot be made; the reference impedance is the line's, not 50 Ω |
| 2x-thru / automatic fixture removal | A fixture built as two back-to-back copies of the launch | The usual method for board-level channels, because the fixture is easy to fabricate alongside the coupon |
It is sometimes said that open and short standards are hopeless at gigahertz frequencies and that this is why scattering parameters exist. Neither half is right. Characterised open, short and load standards are used routinely well into the tens of gigahertz, and TRL goes higher still. Scattering parameters exist because travelling waves are the natural variables of transmission-line theory — the natural bookkeeping for magnitude and phase from port to port — and matched terminations are a secondary convenience. The companion deck Matrix Methods in Network Parameters makes that argument properly.
De-embedding inverts a model of the fixture and applies the inverse to the measurement. It is exact if the model of the fixture is exact. The fixture, however, is fabricated to the same tolerances as everything else on the board, so its impedance is known to about ten per cent — and inverting a slightly wrong fixture leaves a residue behind that was never in the device under test.
Worse, the residue is a reflection, and a reflection is the one kind of error the later slides show to be most damaging, because it lands in the time domain wherever the fixture's electrical length puts it.
A passive network cannot deliver more power than it receives. For a reciprocal, symmetric two-port that reduces to a condition testable at every frequency:
$$|S_{11}|^2 + |S_{21}|^2 \le 1.$$
Measured data violates this routinely by small amounts, from noise, from calibration residue, and above all from de-embedding, which subtracts a model from a measurement and can easily subtract slightly too much. The consequence is not cosmetic: a time-domain simulator handed a channel that creates energy can oscillate, and a model that gains two per cent in amplitude reports an eye two per cent too open.
Enforcing passivity is a standard post-processing step and it is not free: it perturbs the data to make it obey the constraint, and where it perturbs it is a choice the algorithm makes rather than one the measurement supports. A data set that needs a large passivity correction should be re-measured rather than repaired.
A causal network produces no output before its input. Two tests follow from that, and they catch different failures.
The time-domain test transforms the data and looks for energy ahead of the first arrival. It is direct and interpretable. The minimum-phase test compares the measured phase against the phase the measured magnitude implies through the Kramers–Kronig relations — the same relations that deck 03 applied to a dielectric, here applied to a transfer function. It catches a failure the first test misses: a magnitude that has been edited without the dispersion that must accompany it.
An instrument sweeps to some upper frequency and stops. Three things then go wrong at once and it is worth separating them. Removing the high frequencies removes real signal content. Cutting abruptly multiplies the spectrum by a rectangle, whose transform rings. And the sampled spectrum makes the inverse transform periodic, so energy that should have gone off the end of the record wraps round to the beginning.
The usual guidance is to measure to at least three times the Nyquist frequency of the intended data rate, and five times for anything with sharp features such as a via stub resonance. The reason is not that there is much energy up there — there is not — but that the pulse response's shape near the cursor depends on the phase relationships across the whole band, and truncation disturbs them.
A vector network analyser cannot measure at direct current. Every S-parameter file therefore begins at some lowest swept frequency and the value at zero has to be supplied. That single missing point sets the baseline the entire time-domain response sits on, because it is the integral of the impulse response.
The correct treatment is to extrapolate the measured low-frequency behaviour down to zero, respecting the fact that a passive interconnect has a real, finite transmission at direct current set by its series resistance. Assuming zero — which some tools do by default, on the reasonable-sounding grounds that no data means no signal — removes the baseline and produces a response that droops.
Each defect is now applied to the same channel and the equaliser is redesigned against the damaged version, which is the honest comparison: a real receiver adapts to whatever it is given, so the question is not how much the waveform changed but how much margin survives adaptation.
Moving the reference plane by fifteen picoseconds looks alarming on a phase plot and changes nothing that matters: a pure delay is a perfectly causal operation, and shifting where a measurement begins is what de-embedding is for. A quality check that flagged it would be worse than useless, because it would train its users to ignore it.
Correlation between a model and a measurement is almost always judged on insertion loss, because insertion loss is one curve and it is easy to overlay. Here is why that is not sufficient.
Give the channel a small internal reflection — an echo, as an unnoticed impedance step anywhere along the line would produce. Its effect on the magnitude of the transfer function is second order in the reflection coefficient and hides inside the ripple everyone tolerates. Its effect on the pulse response is to put energy at a particular delay, and whether that matters depends entirely on whether the decision-feedback taps can reach it.
Correlate on the pulse response, or on something derived from it — the cursor, the residual intersymbol interference after a defined equaliser, the eye height, or the channel operating margin of deck 17. All of those are sensitive to where in time the energy lands, which is what the receiver is sensitive to. Insertion loss is a useful first check and a poor final one.
| Check | Test | What failure means |
|---|---|---|
| Passivity | $|S_{11}|^2+|S_{21}|^2\le 1$ | The model creates energy; simulators may oscillate |
| Reciprocity | $S_{21}=S_{12}$ | Measurement error, or a genuinely non-reciprocal device |
| Causality, time domain | Energy before the first arrival | Phase inconsistent with magnitude somewhere |
| Causality, minimum phase | Phase against the Hilbert transform of log-magnitude | Catches magnitude-only edits; baseline is nonzero for a channel with reflections |
| Band | Three to five times Nyquist | Truncation rings and distorts the pulse near the cursor |
| DC point | Extrapolate, do not assume | Assuming zero removes the baseline |
| Correlation | On the pulse response, not on insertion loss | Magnitude agreement does not imply time-domain agreement |
Deck 16 of eleven in Signal Integrity & High-Speed Digital Design. Every figure on this page is computed by si_models/deck16 and embedded as data; nothing is typed in by hand.
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