The model that makes jitter measurable in minutes instead of days. Five assumptions, one deterministic jitter that is not the deterministic jitter, and a random jitter that quietly absorbs whatever else is smooth.
Deck 07 ended on an awkward fact: total jitter at a low error ratio can only be measured by counting errors, and counting enough of them takes minutes at $10^{-12}$ and days at $10^{-15}$. The dual-Dirac model exists to escape that, and it is worth being clear that escaping it is all it is for.
The components of jitter combine by convolution, so the distribution of crossing times has three regions. Near the crossing point it is dominated by the deterministic part. Far out in the tails it is dominated by the Gaussian random part. In between it is a mixture. If the deterministic distribution were known, the two could be deconvolved and the tails predicted exactly — but it is not known, and there is no practical way to deconvolve without it.
So the model supplies the simplest deterministic distribution that could possibly work: two Dirac delta functions, separated by an amount the fit determines, each convolved with a Gaussian of width $\sigma$. The tails of the result are two Gaussians displaced from each other, and that is something a short measurement can be fitted to and a long extrapolation can be run from:
$$\mathrm{TJ(BER)} \approx \mathrm{DJ}(\delta\delta) + 2\,Q_{\mathrm{BER}}\,\sigma.$$
It is a Gaussian approximation to the outer edges of the jitter distribution, displaced by a fitted amount. Everything in this deck follows from taking that description literally rather than treating the two fitted quantities as though they were the physical random and deterministic jitter.
The model is usually invoked without its premises. They are worth writing down, because four of them are reasonable and the fifth is routinely false.
The first two are industry-wide conventions rather than findings, and they are what makes estimating from low statistics possible at all. The third is sound. The fourth is the one this deck is mostly about, and it is not a claim about reality — it is a deliberate simplification whose consequences have to be tracked.
The fifth, stationarity, was the subject of a slide in deck 07 and is the one most obviously untrue of a real system: rails droop, clocks are deliberately modulated, temperatures drift, links retrain. A distribution that moves while you measure it is broader than any of its instantaneous forms, and no amount of fitting recovers that.
Plotting the error ratio against sampling position on a logarithmic axis gives the familiar bathtub, whose flanks are curved. Plotting $Q(\mathrm{BER})$ instead straightens them: a pure Gaussian tail becomes a straight line whose slope is $\sigma$ and whose intercept is where the Dirac impulse sits.
That is not merely cosmetic. It turns the dual-Dirac fit into a straight-line fit, which is why every instrument implements it this way; and it makes a non-Gaussian tail obvious by eye, because the line bends. A logarithmic axis hides exactly that.
The same information on the axis everyone uses. The horizontal opening at the target error ratio is the eye width the receiver has to work with, and everything a clock recovery circuit does is an attempt to sit in the middle of it.
Notice how the two contributions behave differently as the target tightens. The deterministic part shifts both walls inward by a fixed amount whatever the error ratio; the random part steepens them, and its contribution grows with Q. A link dominated by one responds quite differently to a tightened specification from a link dominated by the other.
This is the single most common confusion in jitter analysis, and it follows directly from taking the fourth assumption literally.
Real deterministic jitter never follows a two-impulse distribution — deck 07 showed what it actually looks like for this channel — so it is unreasonable to expect the deterministic jitter extracted from the dual-Dirac model to equal the actual peak-to-peak spread. They are different quantities and deserve different names:
A fitting parameter. It is the separation the two impulses would need in order for two displaced Gaussians to reproduce the observed tails. It can always be measured, on any instrument, and it is what standards specify. It is not a property of the signal alone.
The actual peak-to-peak spread of the deterministic distribution. It is a property of the signal, it is what a designer usually pictures, and it can only be measured in special cases — such as when the whole distribution can be computed, which is what the next slide does.
The relationship between them is an inequality rather than an equality, and it runs in one direction:
$$\mathrm{DJ}(\delta\delta) \le \mathrm{DJ}(\mathrm{p\text{-}p}).$$
The model's deterministic jitter is the smaller number. Quoting it as though it were the peak-to-peak value therefore understates the real spread — which matters because the two are routinely compared across instruments, across vendors and against specifications that do not always say which they mean.
The inequality can be tested rather than asserted, because for this channel both quantities are computable. The deterministic distribution comes from enumerating the bit patterns against the equalised pulse response — the equaliser matters, because the sampler sits after it — and the model is then fitted to the tails of the resulting error-ratio curve exactly as an instrument would fit it.
The previous slide showed the fitted random jitter coming out larger than the random jitter actually present. That is not a defect of the fit; it is what a fit to the tails must do when part of the width is deterministic but smooth.
The mistake is expensive in exactly one direction. Random jitter is multiplied by $2Q$ — fourteen at $10^{-12}$ — and deterministic jitter is not, so attributing deterministic width to the Gaussian inflates the extrapolated total. Whether that is conservative or dangerous depends on whether you are the one specifying or the one complying.
There are two ways to extract the random part, and they fail differently. Fitting a Gaussian expression to the measured distribution is what most instruments do, and it is what this slide models — it absorbs smooth deterministic width. Measuring the timing-noise spectrum and identifying the broadband floor with $\sigma$ separates them better, because a deterministic contribution is not spectrally flat. Instruments differ substantially in which they use, and therefore in the answer they give for the same signal.
The model's purpose is extrapolation, so its most important failure mode is a component that the measurement cannot reach. Suppose the random jitter is not one Gaussian but two — a narrow one that is always present and a wide one that appears rarely, which is what an intermittent aggressor or a marginal supply looks like.
At the error ratio an instrument can reach, the rare component contributes almost nothing and the fit is indistinguishable from a clean one. At $10^{-12}$ it dominates.
This is the quantitative form of the warning in deck 07 about bounded uncorrelated jitter. A rare wide component is precisely what crosstalk from a bursty neighbour produces, and it is precisely what the acronyms have no name for.
Every specification, every datasheet and every correlation argument should state whether it means the model's fitted parameter or the real peak-to-peak spread. They differ by tens of per cent on a real channel, and the model's is always the smaller.
If the random jitter the fit reports drifts as the range is extended further down, something smooth and deterministic is being absorbed into it. A stable value across a widening range is the evidence that the separation is real.
On a Q-scale a clean Gaussian tail is straight. Curvature is not measurement noise; it is the signature of a second component, and it is visible long before it dominates.
Run the link with neighbours quiet and again with them driven. The difference is the bounded uncorrelated contribution, obtained directly rather than inferred from a model that has no category for it.
| Quantity | Meaning | Worth remembering |
|---|---|---|
| Dual-Dirac total jitter | $\mathrm{DJ}(\delta\delta)+2Q\sigma$ | A Gaussian approximation to the outer edges, displaced |
| DJ($\delta\delta$) | A fitting parameter | Always measurable; not a property of the signal alone |
| DJ(p–p) | The real peak-to-peak spread | Larger; measurable only in special cases |
| The inequality | $\mathrm{DJ}(\delta\delta)\le\mathrm{DJ(p\text{-}p)}$ | The commonest confusion in the field |
| Q-scale | $Q$ on the vertical axis | A Gaussian tail is a straight line; curvature is evidence |
| Fitted $\sigma$ | Absorbs smooth deterministic width | Inflated whenever DJ resembles a Gaussian |
| The five assumptions | Separable, Gaussian, bounded, two impulses, stationary | The last is routinely false |
| Extrapolation | Safe only if the tail really is one Gaussian | A rare wide component is invisible where it is measured |
Deck 8 of eleven in Signal Integrity & High-Speed Digital Design. Every figure on this page is computed by si_models/deck08 and embedded as data; nothing is typed in by hand.
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