A jitter that grows with the look

Every jitter in a picosecond, read as bits, a floor, or a line and the frequency a band moves at had a value. An edge was displaced from its ideal instant by a draw from a fixed distribution, the draws were independent or a sinusoid, and the root-mean-square displacement was a number that did not care how many edges were looked at. That is the right model of a sampling clock that is locked to something clean. It is not a model of an oscillator on its own.

A free-running oscillator does not know where its edges should be. It knows only where its last edge was, and it times the next period from there, with an error. So its timing errors are not independent draws: each is the previous one plus a fresh draw, and the sequence is a random walk. Measured period by period, the walk and the independent clock can be identical — the same spread in every period — and their timing errors are nothing alike. This essay samples the same tone with both and finds that the walk’s jitter has no value at all until a length of time is named, and that a locking loop is what gives it one.

Two clocks with the same period spread #

Both clocks run at 160 MHz and sample a tone at 10.01 MHz, placed on an exact bin of a 16,384-sample record. Both have a period spread of 0.1 ps: the interval between consecutive edges differs from its nominal 6.25 ns by a Gaussian of that size. In the first clock that is achieved by displacing every edge independently from its ideal instant by a draw of $0.1/\sqrt2$ ps, which makes the period the difference of two independent draws and gives it the stated spread; in the second, by timing each edge from the previous one, which makes the period a single draw and the edge’s position the running sum of every draw before it.

The sampled tone is then fitted by least squares — a sine and a cosine at its own frequency, which absorb its amplitude and any constant delay, and the same two multiplied by time, which absorb a small frequency offset to first order. What the fit cannot absorb is the error. That is what a spectrum analyser locked to the tone, or a receiver tracking it, would report, and it is the fair measure: a delay is not an error to anyone who does not know where time zero was, and a clock running a third of a part per million fast is a frequency error and not a jitter.

The independent clock gives 107.0 dB at every record length from 256 samples to 65,536 — 107.2, 107.0, 107.0, 107.1 and 107.0 over the five lengths averaged across eight seeded clocks — which is the tone rule, $-20\log(2\pi f\sigma)$, with its edges’ 0.071 ps in it. The walking clock gives 92.6 dB over 256 samples, 86.4 over 1,024, 80.3 over 4,096, 74.5 over 16,384 and 68.2 over 65,536. It falls 10.1 dB a decade of record, and it has not stopped falling where the measurement stops.

What a walk is #

The two clocks’ timing errors, drawn edge by edge, show the difference directly.

The independent clock’s errors fill a thin band about zero, 0.070 ps rms, and look the same at every edge. The walk wanders — up past eight picoseconds and back — to 4.06 ps rms over the record, fifty-eight times the independent clock’s, with the same spread in every single period. Nothing about any one period distinguishes them. What distinguishes them is memory: the walk’s error at edge $k$ contains every draw that came before it, so its variance is $k$ times the period spread’s, and its rms over a record of $n$ edges is the spread times $\sqrt{n/2}$.

Most of that wander is not an error in the sense that matters, and the fit removes it. A constant offset is a delay; a steady drift is a frequency offset; the best straight line through the walk takes both, 0.32 parts per million of frequency in the record drawn. What remains is the walk’s curvature about that line — the part that is neither a delay nor a frequency — and its rms is 1.31 ps in this record. For a walk of $n$ steps the expected residual about the mean is the spread times $\sqrt{n/6}$, and about the best line the spread times $\sqrt{n/15}$: the fit buys a factor of $\sqrt{7.5}$ over the raw figure and changes nothing about the growth. The residual still grows as the square root of the record, which is ten decibels a decade of length — the slope in the first figure.

Two independent routes arrive at those numbers. The first works on the timing sequence alone: the rms of the walk about its mean and about its best line. The second samples the tone, fits it by least squares with no knowledge of the timing sequence, and reads the ratio from what is left. They share the draws and nothing else, and at every record length the second agrees with the first read through the tone’s slope, to within six tenths of a decibel across eight seeds.

The spectrum, which is phase noise #

A walk in the time domain is a spectrum with no bottom in the frequency domain, and the sampled tone shows it.

With independent edges the error is a floor, −144.0 dB a bin at every offset — the white floor a floor, or a line measured, lower here because the jitter is smaller. With edges that walk, the tone grows a skirt that falls 19.8 dB a decade of offset, the twenty that a walk’s spectrum must have, from −96.9 dB at 97.7 kHz to −127.6 dB at 3.91 MHz. That skirt is what an oscillator’s data sheet calls phase noise, in its commonest regime, and the picture is the reason the data sheet gives it as a curve.

A curve falling twenty decibels a decade has an integral that depends on where it starts. Its power between an offset and infinity goes as one over the offset, so the total diverges as the lower limit goes to zero — which is the spectral statement of a timing error that grows without bound. Every quoted rms jitter derived from a phase-noise curve is therefore an integral over a stated band, and the lower limit of the band is the length of the look: an offset of $1/T$ corresponds to a record of length $T$. A jitter figure integrated from 12 kHz and one integrated from 1 kHz describe the same oscillator and differ by the square root of twelve.

One oscillator, six jitter figures #

The consequence for specifications is that the same clock has several honest jitter figures, and they do not agree.

The period jitter — the spread of one period — is 0.100 ps, and it is the only figure that is a property of the oscillator alone. The cycle-to-cycle jitter, the spread of the difference between consecutive periods, is 0.142: $\sqrt2$ times the period jitter, since consecutive periods of a walk are independent draws. Those two are measured in one or two periods and do not depend on anything else.

The other four each carry a length. The accumulated jitter over a thousand cycles — how far an edge has wandered from where a perfect clock would have put it a thousand periods later — is 3.25 ps, against $\sqrt{1000}\times0.1$ = 3.16. The rms over a 4,096-edge record is 4.66 ps against $0.1\sqrt{4097/2}$ = 4.53; about the record’s mean, 2.87 against 2.61; and about its best line, 1.58 against 1.65. The last three are within the scatter sixteen walks allow, which is wide because a walk’s rms over one record is itself a random quantity that varies a great deal from one walk to the next.

From the smallest to the largest is a factor of 46, for one clock, with every figure correct. A data sheet that prints “jitter: 0.1 ps” for this oscillator has told the truth about its periods and nothing about its sampling, and a converter designer who uses it in the tone rule has computed 104.0 dB for a clock that delivers 80.3 dB over a four-thousand-sample capture. The difference is not a margin of safety being eroded. It is the difference between two quantities with the same unit.

The same distinction has appeared in these essays in a different circuit. The decision taken where the ramp is slowest found a relaxation oscillator’s consecutive periods sharing a draw, so that its jitter accumulates at 3.771 per root period rather than at the per-period 3.399, and the walk the core sees carried that walk into a transformer’s flux. The walk measured here is the simplest member of that family, with independent periods and no sharing, and the sampling problem it poses is the same one.

What a locking loop does #

A clock used for sampling is almost never free-running. It is locked, by a phase-locked loop, to a reference that is itself stable over long times — and a lock is exactly a mechanism that stops a walk.

The loop is modelled by its effect: each edge’s error is the previous error shrunk by a factor $e^{-1/\tau}$ before the fresh draw is added, so a wander decays with a time constant of $\tau$ periods instead of persisting. Over records short against $\tau$ the locked clock is indistinguishable from the free one — the loop has not had time to act. Over records long against it the error stops growing, at a stationary spread of $\sigma/\sqrt{1 - e^{-2/\tau}}$, which is about $\sigma\sqrt{\tau/2}$.

With a loop time constant of 100 periods the ratio falls from 93.2 dB at 256 samples to 89.4, 87.6 and 87.3, and stops at 87.0 dB over 65,536 — the ceiling set by a stationary spread of 0.71 ps. With a time constant of 1,000 periods it falls further, to 77.6 dB, against a ceiling of 77.0 set by 2.24 ps. The free walk over the same record is at 68.2 dB and still going.

So the loop is what gives the jitter a value, and its bandwidth is what sets the value. A narrow loop — a long time constant — lets the oscillator wander for longer before correcting it, and the ceiling is lower; a wide loop corrects quickly and the ceiling is high, until the loop starts passing the reference’s own noise, which is a different problem. The oscillator’s walk contributes jitter only from offsets above the loop’s bandwidth: in the spectrum, the skirt stops at the loop’s corner and flattens below it. The integration band a data sheet quotes, when it quotes one, is usually chosen to start near that corner for exactly this reason.

What it costs a converter, in bits #

Put a fourteen-bit converter behind the free-running clock and the arithmetic becomes a capture-length rule. Fourteen bits is 86.0 dB. The walk’s ratio after the fit follows $104.0 - 10\log(n/15)$ dB for this tone and this period spread — the tone rule with the period jitter in it, less the walk’s growth — and that reaches 86.0 dB at $n \approx 940$ samples, 5.9 µs at 160 MHz. Captures shorter than that see the converter’s own resolution; longer ones see the clock. Over 4,096 samples the clock limits the system to 80.3 dB, 13.05 bits, and over 65,536 to 68.2 dB, 11.04 bits — three bits lost to a clock whose every period is as good as the part promised.

Locked with a time constant of a hundred periods, the same clock’s ceiling is 87.0 dB, 14.16 bits, and the converter keeps its fourteen bits at any record length. Locked with a time constant of a thousand, the ceiling is 77.0 dB, 12.50 bits: a slower loop, chosen perhaps to reject a noisier reference, has cost a bit and a half of a fourteen-bit converter at every capture longer than a few thousand samples. Neither number appears in the clock’s period jitter, and both follow from it once a record length and a loop are named.

Why the tone rule still works for locked clocks #

None of this overturns the tone rule for a well-locked sampling clock, which is most of them. Over any record much longer than the loop’s time constant, a locked clock has a jitter with a value, and that value is what goes into $-20\log(2\pi f\sigma)$. What the walk changes is which value. The number to use is the clock’s timing error integrated over the band from the loop’s corner upwards — or, equivalently, the rms timing error over a record, after the best delay and frequency have been removed, once the record is long against the loop. It is not the period jitter, which for the locked clock in the figure is 0.1 ps and gives 104.0 dB, when the clock actually delivers 87.0.

And for records shorter than the loop’s time constant — a burst capture, a single radar pulse, an oscilloscope’s single-shot acquisition — the loop has not acted, and the free walk’s figures apply. The same clock then has a better jitter over a short capture than over a long one: 92.6 dB over 256 samples against 87.0 over 65,536. Averaging longer to reduce noise, which the corner where averaging stops working found failing for flicker noise because its samples are not independent, fails here for the same reason in the time axis: the error is correlated across the record, and a longer record collects more of it rather than less.

How the numbers were obtained #

Each clock’s timing errors are generated edge by edge from a seeded Gaussian: independently for the white clock, cumulatively for the walk, and cumulatively with a decay factor $e^{-1/\tau}$ for the locked clocks. The tone is sampled at the 160 MHz edges displaced by those errors, and fitted by linear least squares on four functions — sine and cosine at the tone’s frequency, and each multiplied by time about the record’s centre — through the normal equations; the residual’s power over the tone’s is the ratio. The timing-domain route computes the same sequences’ rms about their mean and about their least-squares line, and reads a ratio through $-20\log(2\pi f\sigma)$. Each point is the mean over eight seeded clocks for the record-length figures, sixteen for the six jitter figures and twelve for the spectra. The spectra use a Hann window on the detrended sampled tone and average the power over the bins within a tenth of each plotted offset.

What it leaves out #

Flicker in the oscillator. A real oscillator’s phase noise has a region falling thirty decibels a decade close to the carrier, from flicker noise in its active device, as well as the twenty-decibel region modelled here. Its timing error grows faster than the square root of the record there, and the loop’s ceiling depends on it more strongly.

The reference’s own noise. The locked clocks are pulled towards a perfect reference. A real reference has its own walk, which the loop passes below its bandwidth, and the total is a sum of the reference’s jitter inside the loop band and the oscillator’s outside it.

The signal’s spectrum. Everything here samples a single tone, so the error is the tone’s slope times the residual timing error. For a band-filling signal the frequency a band moves at replaces the tone’s frequency with the band’s rms frequency, and the two corrections multiply.

Still open: the loop’s own noise, the flicker region, and the capture that is shorter than the loop #

The loop’s bandwidth as an optimum. A wide loop suppresses the oscillator’s walk and passes the reference’s noise; a narrow one does the opposite. With both sources modelled — a walking oscillator and a walking reference of different strengths — there is a loop time constant at which the sampled tone’s error is least, and it should be where the two clocks’ accumulated jitters are equal over one time constant. Measuring that optimum on the march would turn a loop-design rule of thumb into a number.

The thirty-decibel region. An oscillator whose phase noise falls thirty decibels a decade close in has a timing error growing as the record length itself rather than its square root, and the best-line fit removes less of it. How much less — whether a quadratic term in the fit would be needed to make such a clock usable over a long capture — is a question about the next term of the same least-squares fit.

Bursts. A capture shorter than the loop’s time constant sees the free walk, so a burst receiver’s clock is characterised by its period jitter and its walk alone, and a long-capture receiver’s by its loop. The same oscillator and loop, specified once, give two answers to two receivers, and neither data sheet figure is the one either receiver needs.