A picosecond, read as bits

Every boundary this collection has drawn is a value of something with a unit on it, and until now the units have been three: a frequency, an amplitude, a size. The site’s own rule says so — no model is drawn without the frequency, amplitude or size at which it stops being true — and the noise field already forced one amendment to it, adding the direction as well as the quantity, because a floor bounds a model from below.

This essay forces the other kind of amendment. Its boundary is a duration, and the reason it can still be drawn on a frequency axis is the whole content of the argument.

Why a timing error becomes an amplitude error #

A converter’s sampling instant is decided by a clock edge, and no clock edge arrives at exactly the same phase every cycle. The displacement is small — picoseconds — and it is not obviously a problem, because a picosecond is nothing beside a sample period of tens of nanoseconds.

It is a problem because the converter does not measure time. It measures a voltage, and the voltage it gets is the one the signal had at the instant it actually sampled. So an error δ in the instant becomes an error in the value:

$$\Delta v = \frac{dv}{dt}\,\delta$$

and for a sinusoid of amplitude A and frequency f the slope is at most 2πfA. Divide the signal’s root mean square by the error’s and the amplitude cancels:

$$\mathrm{SNR} = -20\log_{10}\!\left(2\pi f \sigma_t\right)$$

The amplitude cancels, which is the first surprising thing, and it means the jitter’s cost is the same fraction of full scale at every level — a floor in the same sense the quantiser’s is, and unlike the quantiser’s it cannot be escaped by using more of the range.

The result contains no property of the converter except the jitter. Not the bit count, not the sample rate, not the input impedance. Only the timing error and the input frequency.

Measured, on jittered samples #

The closed form is one route and the site’s rule is that a closed form gets a second one. Here the second is direct: draw a seeded Gaussian displacement for every sample, evaluate the sinusoid at the displaced instants, and difference the result against the same sinusoid sampled exactly.

The two share the value of the jitter and nothing else — one is a sum over four thousand random draws, the other is a logarithm — and they agree to 0.13 dB across three decades. The small constant offset is the difference between the peak slope and the root-mean-square slope over a cycle, which the closed form’s factor of 2πf takes at the peak and the measurement averages.

The gate checks something sharper than the agreement, because the agreement alone could survive both routes being wrong in the same way. It checks that the penalty is exactly twenty decibels per decade of input frequency, at every jitter, to 1.4 × 10⁻¹⁴ dB. That exponent is what makes the effect a specification rather than a curve: it says the damage is first order in frequency, which is what the slope-times-δ mechanism predicts and what an error of any other origin would not give.

What it is worth in bits #

Converting the ratio to bits with the field’s own formula makes the consequence legible.

A sixteen-bit converter with ten picoseconds of aperture jitter is a sixteen-bit converter up to 199 kHz and a ten-bit converter at ten megahertz. Nothing about the part changed; the input frequency did.

That is why a resolution and a speed are not independent specifications, and why the two numbers on the front of a data sheet — bits and megasamples per second — cannot both be achieved on the same signal unless the aperture is stated. The arithmetic is unforgiving: holding sixteen bits at 10 MHz needs 0.2 picoseconds, which is a demanding clock by any standard and is the reason high-speed converters are specified in effective bits at a stated input frequency rather than in bits.

The two floors together #

The previous quantisation essays established a floor from the step size and a crossing with the source resistance’s Johnson noise. The jitter is a third floor and it is worth putting all three on one axis, because which of them dominates is a function of the input frequency and swaps within the range of an ordinary signal.

For a sixteen-bit converter on a 1 kΩ source in 100 kHz of bandwidth:

At 20 kHz the aperture is the smallest of the three and the converter’s own arithmetic is the limit. At 200 kHz the aperture has drawn level with it. At 2 MHz the aperture is twenty decibels worse than everything else in the circuit, and the converter’s bit count has stopped meaning anything.

That crossing is the practical content of this essay and it is the same shape as the crossing with Johnson noise — two quantities with different dependencies, meeting at a point that can be computed — with one difference that matters. The Johnson crossing moves with the source resistance, which a designer chooses once. The aperture crossing moves with the input frequency, which changes while the system is running.

So a converter’s effective resolution is not one number even for one circuit. It is a curve, and the figure above is that curve.

Where the jitter comes from, which is not the converter #

One consequence of the amplitude cancelling is that the source of the timing error does not matter either, and that has a practical edge to it.

The jitter is the sum of the converter’s own aperture uncertainty and whatever the clock brings with it, and the second is usually the larger. A clock distributed over a board, through a buffer, across a connector, accumulates phase noise from every stage; a clock derived from a phase-locked loop carries the loop’s own noise shaped by its bandwidth. None of that is in the converter’s data sheet and all of it lands on the same axis.

Which means the number in the figure is a system quantity, in exactly the way the crossing with Johnson noise is: it needs the circuit around the part before it means anything. The field has now produced three of these — a resolution that needs the source resistance, an anti-alias requirement that needs the filter, and an aperture that needs the clock — and all three are numbers a part cannot state alone.

There is a fourth consequence worth naming, which is where the figure’s undersampling aside from the first essay lands. Sampling a 70 MHz signal at 400 kHz is possible and costs nothing in sample rate; it costs the aperture at 70 MHz rather than at 200 kHz, which by the table above is a loss of about fifty decibels. A technique that appears free in one currency is expensive in another, and the only way to see it is to draw the boundary in time beside the boundary in frequency.

Why the axis follows the jitter, and what that caught #

The figure’s frequency axis is not fixed. It spans a quarter of the sixteen-bit frequency to two thousand times it, so as the slider moves the jitter the axis moves with it — and the reason is a gate failure worth recording, because it is the third instance of one shape on this site.

The frequency at which jitter alone caps the resolution at sixteen bits runs from 9.93 MHz at a fifth of a picosecond to 19.9 kHz at a hundred: two and a half decades. A first version of the figure used a fixed axis of 80 kHz to 140 MHz, which covers the middle of that range and not the ends. So at both extremes of the slider the marked edge — the whole point of the drawing, and the number its caption quotes — was outside the axis and therefore not drawn.

Nothing about the figure looked wrong. The curve was there, the panel was there, the caption named a frequency, and the frequency was simply absent from the picture at two of the seven settings.

That is precisely the failure the scale phase built machinery for. edgeMark returns null when the value falls outside the plotted range, which is the right thing to draw and was a silent thing to do until droppedMarks was added and netcheck began rendering every generator at every slider value and requiring the list to be empty. It caught this one immediately.

The repair is a better axis rather than a quieter mark, and the figure now asserts that both marked resolutions fall on the axis at every setting. Which is the other half of the lesson: a gate that reports a missing mark is a gate that has found either a wrong axis or a wrong claim, and somebody has to decide which — but it can no longer be neither.

The site’s other boundary with a duration in it #

This collection has drawn one thing before whose independent variable was closer to a time than to a frequency, and comparing the two sharpens what is different here.

The transients field measured the amplitude at which a step becomes too big: a unity-gain follower with 1 MHz of gain-bandwidth and 0.5 V/µs of slew rate stops being linear above a 79.6 mV step. The mechanism is a rate — volts per second — and the boundary is stated as an amplitude, because the rate the circuit can deliver is finite and the amplitude the input asks for is what exceeds it.

The two mechanisms are the same derivative from opposite sides. There the circuit’s slope is limited and the signal’s is what exceeds it; here the signal’s slope is what does the damage, and the limitation is in the timing. So the slew boundary is an amplitude at a fixed frequency and the jitter boundary is a frequency at any amplitude, and each is the other’s complement.

Both are invisible to a small-signal model. A transfer function contains no rate limit and no sampling instant; the first has been the transients field’s point since it opened, and the second is this field’s.

What is measured and what is stated #

Two honest limits on the numbers above.

The jitter is Gaussian and uncorrelated between samples. That is the standard model and it is not what a real clock does: phase noise has a spectrum, usually rising towards the carrier, so successive sampling instants are correlated and the resulting error is not white. The consequence is that the total is right and the distribution in frequency is not — a real jittered converter produces skirts around the signal rather than a flat floor, which is more visible on a spectrum analyser and less visible in a total. Measuring that needs the noise field’s machinery applied to a phase rather than to a voltage, and it is not done here.

The first of those is not a small qualification, and the rung above this one measures how large it is. A floor, or a line takes two clocks with identical two-picosecond jitter, samples the same sinusoid with each, and finds the total error power the same to a tenth of a decibel — the closed form on this page, right for both — while one of them puts that error across two thousand bins at −120 dB each and the other puts it into two lines at −91.5. Twenty-eight and a half decibels of difference, with the same number of picoseconds on both specifications. The gap is the processing gain, and it grows with the record length, because a density falls as the record lengthens and a line does not.

So the sentence above — the total is right and the distribution in frequency is not — is exactly correct and understates its own consequence. A receiver does not care about the total; it cares whether the error sits under its wanted signal or beside it, and the two cases differ by whatever the record length is worth.

Nothing here separates the converter’s aperture from the clock’s. The measurement takes one number, and a system has at least two contributions adding in quadrature. Separating them on a bench is a real technique — measure with two clocks of known different quality — and this figure would be the tool for reading the result rather than a substitute for taking it.

Where the clock’s own share comes from is measured elsewhere in this collection, in the field that makes clocks. The frequency that is not the formula closes by naming phase noise as the quantity it does not reach: the cycle-to-cycle jitter of an oscillator’s zero crossings is a different measurement from the mean period, it comes from the noise in the loop rather than from any of the mechanisms that decide the frequency, and it needs a stochastic march rather than a deterministic one. And two thresholds because there is a floor measures the step that turns that noise into a timing error: a threshold with noise under it is crossed 22.7 times rather than once, and what fixes it is hysteresis measured in standard deviations of the noise rather than in volts. A clock edge is a threshold crossing, so the aperture figure this essay converts into bits has, at its origin, the same signal-to-noise ratio and the same slope that appear in this essay’s own arithmetic — once at the clock and once at the input.

What is here is the exponent, and the exponent is the durable part. Twenty decibels a decade, exactly, measured to fourteen decimal places against a closed form it shares only a jitter with; and a table that turns picoseconds into bits at a stated frequency, which is the conversion that makes the number on the data sheet mean something.

What the rule now has to say #

The site’s premise has been amended once already, and this essay is the second occasion.

It began as no model is drawn without the frequency at which it stops being true. The noise field made that inadequate: a floor bounds from below, so the rule became the frequency, amplitude or size and the direction was made explicit. What this field adds is a fourth quantity, and one that does not join the list comfortably.

A duration is not a frequency read backwards. A boundary at 199 kHz for sixteen bits is a consequence of a boundary at ten picoseconds, not a restatement of it: the same ten picoseconds gives 3.18 MHz at twelve bits and 159 MHz at four, so the frequency is a function of two things and the duration is the property of the part. Drawing the consequence and calling it the boundary would be the same mistake what the filter in front costs warns about — pricing a filter family in decibels when the currency is megahertz, where the choice between Bessel, Butterworth and Chebyshev turns out to be a factor of 3.53, 2.08 or 1.53 in the clock rather than a decibel or two of skirt.

So the honest statement of what this field measured is that a converter has boundaries in four units: a rate, a level, an amplitude and a duration, and only the first three can be put on one axis. The fourth is drawn as its consequence because that is what a reader can use, with the duration stated beside it.

That is a small amendment and it is worth making explicitly rather than absorbing. A collection whose whole discipline is naming the quantity at which a model gives out has an obligation to notice when it acquires a new quantity — and the alternative, quietly reporting a derived frequency as though it were the boundary, is exactly the class of error this site has caught in its own captions three times now.

The list has since grown once more, and by a quantity that is not a unit at all. The edges that are lengths gathers the boundaries whose axis is a distance — the 0.60 millimetres a gap’s field reaches into a window, the 200 microns between a track and its plane, the ten centimetres at which Kirchhoff’s laws are a degree out — and observes what separates them from every other boundary here: nobody chooses them at the schematic, they are set by whoever builds the thing, and they appear in no netlist. An aperture time sits oddly beside those and squarely with them on the one property that matters for a specification. It is not a design parameter either. A converter arrives with it, the circuit around it cannot improve it, and the only decision available is which part to buy and how fast to run the signal — which is why the table in this essay is drawn as bits against input frequency, with the picoseconds as a curve label rather than as an axis.