The loop a reference allows

A jitter that grows with the look sampled a 10 MHz tone with a free-running oscillator whose every period had a spread of 0.1 picoseconds, and found that the clock had no jitter until a length of time was named. Its edges walked: each was timed from the one before, so the error at an edge was the sum of every period’s error up to it, and the residual after fitting the best delay and frequency grew as the square root of the record — 80.3 dB over 4,096 samples, 68.2 over 65,536, still falling. A locking loop stopped it. Pulling each edge back towards a perfect reference with a time constant of a hundred periods held the ratio at 87.0 dB at every length; a thousand periods held it at 77.0.

The loop in that essay had a perfect reference, and so the conclusion it drew was one-sided: a wider loop was always better, because it corrected the walk sooner. No reference is perfect. A crystal divided down, a recovered clock, a distribution network — each delivers edges with timing errors of their own, and the phase detector sees those errors as surely as it sees the oscillator’s. Whatever the loop corrects towards, it also follows. So a wide loop that suppresses the oscillator’s walk passes the reference’s noise through to the clock, and a narrow loop that rejects the reference lets the oscillator wander. There is a best loop, and this essay finds it.

A loop that follows two things #

The model is the walk essay’s, with one term added. The clock’s timing error at edge $k$ is

$$e_k = a\,e_{k-1} + (1-a)\,r_k + w_k, \qquad a = e^{-1/\tau},$$

where $w_k$ is the oscillator’s fresh period draw, spread $\sigma_o$ = 0.1 ps, and $r_k$ is the reference’s timing error as the phase detector sees it at that edge, white with spread $\sigma_r$. The loop pulls the error towards the reference’s version of zero with a time constant of τ periods. With $\sigma_r = 0$ it is exactly the locked clock the walk essay drew.

Both noises enter a first-order filter with the same pole. The oscillator’s walk is high-passed — the loop removes its slow wander and leaves what changes faster than τ — and the reference is low-passed, because the loop averages it over about τ comparisons before acting on it. The stationary variance is the sum of the two,

$$\sigma_e^2 = \frac{\sigma_o^2}{1 - a^2} + \sigma_r^2\,\frac{1-a}{1+a} \;\approx\; \frac{\sigma_o^2\,\tau}{2} + \frac{\sigma_r^2}{2\tau},$$

and the hero figure draws both parts and their sum against the loop’s time constant for a 2 ps reference.

The oscillator’s share rises as $\sigma_o\sqrt{\tau/2}$ — the walk the earlier essay measured, stopped after about τ periods. The reference’s falls as $\sigma_r/\sqrt{2\tau}$, the error of an average over 2τ independent comparisons. Two power laws of opposite sign cross once, and their sum is least near the crossing: at

$$\tau^* = \frac{\sigma_r}{\sigma_o}, \qquad \sigma_e^* = \sqrt{\sigma_r\,\sigma_o}.$$

For the 2 ps reference that is twenty periods and 0.453 picoseconds, which on the 10 MHz tone is 90.9 dB. Seeded locked clocks of 65,536 periods each sit on the closed form at every loop tried: 1.000 ps at two periods, 0.656 at five, 0.452 at twenty, 0.649 at eighty, 1.19 at three hundred and 2.11 at a thousand.

The geometric mean, and what it forgives #

The second of those two expressions is the surprising one. The clock’s best jitter is not the larger of the two errors, nor their sum, but their geometric mean. A reference a hundred times noisier than another makes the best clock ten times noisier, not a hundred; one four times noisier makes it twice as noisy.

The slider runs the reference from half a picosecond to five. At 0.5 ps the best loop is five periods and the clock 0.235 ps, 96.6 dB; at 1 ps, ten periods and 0.324 ps, 93.8 dB; at 5 ps, fifty periods and 0.711 ps, 87.0 dB. A tenfold range of reference noise moves the clock by a factor of three, because the best loop moves with it: a noisier reference is answered by a narrower loop that averages it over more comparisons, and that narrower loop pays for itself in oscillator wander, at the square root.

The forgiveness has a limit, and the same expression states it. A fourteen-bit converter needs 86.0 dB on this tone, which is a clock of 0.79 ps; the best loop delivers that from an oscillator of 0.1 ps a period only if the reference’s error is below $0.79^2/0.1$ — about 6.3 picoseconds. A designer with a quieter oscillator can afford a noisier reference in exact proportion, and nothing else helps: no choice of loop gets a 10 ps reference and a 0.1 ps oscillator to fourteen bits.

A budget in a product #

The geometric mean turns a jitter budget into a statement about a product. The clock’s best error squared is $\sigma_r\sigma_o$, so a requirement on the clock is a requirement on the product of the reference’s error and the oscillator’s period spread, and either can be traded for the other at a fixed ratio. That is a different shape of budget from the one a converter designer usually writes, in which each source is allotted a share of the total and the shares are added in quadrature.

The shares in quadrature are the right budget when the loop is fixed, and they make the reference look expensive: at a loop of a hundred periods the 2 ps reference contributes 0.141 ps, and halving it saves a fraction of a decibel. They are the wrong budget when the loop is free, because the loop is the cheapest component in the system to change. Halve the reference’s error, retune the loop to half the time constant, and the oscillator’s share falls too — the clock’s error falls by the square root of two, three decibels, from one change.

The product also says where the arithmetic stops working. Fourteen bits on a 50 MHz tone needs a clock of 0.159 picoseconds, so $\sigma_r\sigma_o \le 0.0254$ $ps^2$. With the 0.1 ps oscillator that asks for a reference of 0.25 ps and a loop of two and a half periods — a loop correcting nearly every edge, outside the range where the loop behaves as a smooth average, and outside what a phase detector can do. The product is satisfied on paper and the loop cannot be built. The honest reading is that the oscillator must improve: at 0.02 ps a period, the same product allows a reference of 1.27 ps and a loop of 63 periods, both ordinary. A clock budget written as a product shows immediately which of its two factors has run out; one written as a quadrature sum does not, because it has fixed the loop and hidden the trade.

None of this changes the walk essay’s warning about specifications. Every number above is a stationary jitter, an rms over a record long against the loop, and a data sheet’s period jitter — the 0.1 ps — is still the wrong figure to put in the tone rule. What the reference adds is a second datum the clock’s specification must carry: an oscillator’s period spread and a reference’s floor fix the best loop and the best clock together, and neither alone fixes either.

The figure above, from the walk essay, is the hero figure’s right-hand branch. Its loops of a hundred and a thousand periods are five and fifty times the 2 ps reference’s optimum, and the ceilings they reach — 87.0 and 77.0 dB — are what the oscillator’s share alone allows. With the reference’s error added they would sit slightly lower: at a hundred periods the reference contributes 0.141 ps against the oscillator’s 0.71, and the sum is 86.9 dB, a tenth below. It is the narrow loops that the reference governs, and those were the loops the perfect-reference picture recommended without limit.

The same rule, read off two spectra #

A loop designer rarely thinks in time constants. The usual picture is spectral: an oscillator’s phase noise falls with offset from the carrier, a reference’s sits flat, and the loop’s bandwidth decides which of the two the locked clock follows at each offset. The optimum has a statement in that picture too.

The free oscillator’s walk is a skirt falling twenty decibels a decade, and the 2 ps reference is a floor at −160.0 dBc/Hz on this tone. They cross at 1.27 MHz. The corner frequency of the best loop, $f_s/(2\pi\tau^*)$, is 1.27 MHz — the same number, because the condition that the two shares be equal in the time domain is the condition that the two spectra be equal at the loop’s corner. Set the loop’s bandwidth where the free oscillator’s phase noise meets the reference’s.

Read off the locked spectra, the rule’s logic is plain. Below its corner the loop makes the clock follow the reference, so the clock’s spectrum there is the reference’s floor; above the corner the loop does nothing and the clock is the free oscillator. The ideal clock would follow whichever of the two is lower at every offset, and a first-order loop can only switch once. Switching where they cross takes the lower envelope everywhere but near the corner, where the two add and the locked spectrum sits three decibels above either. A loop of two periods switches at 12.7 MHz and holds the clock on the reference’s floor across a decade where the oscillator was quieter; a loop of two hundred switches at 127 kHz and lets the oscillator’s skirt through across a decade where the reference was. Each of those is a decade of excess noise, and integrated, each costs the same.

That last observation is exact in the long-loop limit, and it is the next figure.

What missing it costs #

Write the time constant as a multiple x of the best one. The variance is then proportional to $x + 1/x$, and the loss in signal-to-error ratio is

$$10\log_{10}\frac{x + 1/x}{2}\ \text{dB},$$

the same at x and 1/x, and for any pair of noises.

A factor of two either way costs 0.97 dB; a factor of ten costs 7.0. The optimum is flat, as optima of a sum of reciprocal power laws always are, and that is useful: a loop’s time constant is set by component values with tolerances, by a charge pump’s current and a filter’s capacitor, and a 30 per cent error in it costs 0.15 dB. What matters is not being off by a factor of ten, which is what happens when a loop is designed for one reference and then fed another, or when a loop bandwidth is chosen for settling time and its noise is never revisited.

The exact curves for the three references fall on the universal one except at the far left of the 2 ps reference’s curve, where a tenth of its optimum is a loop of two periods. A loop that short is outside the square-root forms: the clock’s error there is the reference’s almost unfiltered, and the curve flattens. No practical loop is in that corner, since a loop that corrects every second edge has a bandwidth of a tenth of the clock frequency.

A short capture wants no loop at all #

Everything so far is about stationary jitter — what the clock’s error looks like over a record long enough to see every frequency the loop passes. The walk essay found that what a converter cares about is the error left after the record’s own delay and frequency have been removed, and over a short record the fit removes the slow part of the error. That changes the optimum, and in a direction that surprises.

The two parts of the locked clock’s error are first-order processes with the same pole. So the locked error is one first-order process, with innovation variance $\sigma_o^2 + (1-a)^2\sigma_r^2$, and the residual it leaves after a delay-and-frequency fit can be computed exactly for any record length — no seeds needed. Twenty-four seeded clocks sampling the tone over 4,096 samples agree with it to 0.07 dB.

Over long captures the best loop is τ* — 10, 20 and 50 periods for references of 1, 2 and 5 ps — whatever the length. As the capture shortens the best loop narrows: the slow wander a narrow loop leaves is exactly what the fit removes, so a narrow loop’s cost is partly refunded while its benefit, rejecting the reference, is not. And below a critical capture the refund wins outright. For a 2 ps reference, captures shorter than 140 samples are served better by the free oscillator than by any loop at all — 7.0 times τ*; for 1 ps, 78 samples, 7.8 τ*; for 5 ps, 316 samples, 6.3 τ*.

The transition is not gradual. Just above the critical capture the best loop is already 120 periods for the 2 ps reference, six times its long-capture optimum, and just below it there is none. The error as a function of the loop’s time constant has two minima — one at a finite loop, one at no loop — and the critical capture is where they trade places.

The figure shows why, in the units a converter designer uses. The free walk is 97.7 dB over 64 samples and falls ten decibels a decade; the 20-period loop is 95.4 dB over 64 samples and stops at 90.9. The two curves cross at 214 samples: a capture shorter than that sees less error from a clock with no loop than from the best loop for long captures. A burst receiver, a radar that samples one pulse, an oscilloscope’s single-shot acquisition — each is a short capture, and each is better served by a clock that has not been told about its reference.

That is the third question the walk essay left, answered with numbers rather than a direction. A clock’s loop bandwidth is a choice about the capture length as much as about the two noises, and the same oscillator and reference, locked once, give two receivers two answers. The loops four times too wide and four times too narrow lose the same 3.2 dB over a long capture, as the universal curve says; over 64 samples the narrow one is 8.5 dB better than the wide one — 97.5 against 89.0 — because the fit has refunded the wander it lets through.

What this loop is, and is not #

The loop is first-order. It corrects the clock’s phase in proportion to the error, with no memory of the frequency it has been correcting. Real phase-locked loops are almost always second-order — an integrator in the loop filter, so that a constant frequency offset leaves no phase error — and a second-order loop’s response has a peak near its corner whose height depends on its damping. The geometric-mean result survives in spirit, since it is a statement about two power laws crossing, but the constant in front of it and the cost curve both depend on the damping.

The reference’s error is white. That is the right model for a phase detector’s own noise and for a reference whose jitter is broadband, and the wrong one for a crystal oscillator, whose phase noise also falls with offset. A reference that walks has no floor for the oscillator’s skirt to meet; the optimum then depends on the ratio of the two walks’ rates, and over a long enough capture the reference’s walk dominates whatever the loop does.

And the oscillator’s walk is the only thing it does. Real oscillators have a region close to the carrier where the phase noise falls faster than twenty decibels a decade, from flicker noise in the active device. A loop’s corner usually sits above it, which is part of why the loop is there — but a narrow loop chosen to reject a noisy reference may put its corner inside that region, and then the oscillator’s share stops rising as the square root of τ.

Still open: the flicker region, a second-order loop, and a reference that walks #

The thirty-decibel region. Below its flicker corner an oscillator’s phase noise falls thirty decibels a decade, its timing error grows as the record length itself rather than its square root, and the best-line fit removes less of it. Whether a first-order loop can stop such an error at all — or only slow it, leaving a jitter that creeps up with the logarithm of the capture — is the question that decides whether a narrow loop is ever safe against a flicker-dominated oscillator.

A loop with an integrator. A second-order loop follows a ramp in phase without error, at the price of a response that peaks near its corner. The peak amplifies both the reference’s noise and the oscillator’s around the corner — the same trade the area that feedback only moves found for an amplifier’s sensitivity — so the optimum damping for jitter may be different from the optimum for settling, and the geometric-mean rule acquires a constant that depends on it.

A reference that walks too. With both clocks walking at different rates, the long-capture residual grows without limit whatever the loop, at the reference’s rate. The loop then decides only the short-capture behaviour and the crossover between the oscillator’s walk and the reference’s, and the question becomes the one the corner where averaging stops working asked of averaging: over what length is the slower walk worth following at all.