A frequency that wanders

A jitter that grows with the look found that a free-running oscillator has no jitter until a record length is named. Each of its periods is independent, so its edges walk, and the error left after fitting a capture’s best delay and frequency grows as the square root of the capture — ten decibels a decade of length, on a phase-noise curve falling twenty decibels a decade of offset. The loop a reference allows then locked that oscillator to a noisy reference and found the best loop at the ratio of the two noises.

Both essays used one kind of oscillator noise, and a real oscillator has two. Close to the carrier its phase-noise curve steepens from twenty decibels a decade to thirty, at an offset its data sheet may call the flicker corner. The cause is in the active device: its own low-frequency noise falls as one over frequency, the oscillator converts some of it into a wandering of its frequency, and a frequency that wanders as 1/f puts into the phase — the frequency summed — a spectrum falling as $1/f^3$. The walk’s periods were independent. A flicker clock’s are not: a period that ran long is likely to be followed by another that runs long, because the frequency that made it long has not had time to change.

The walk essay’s list of open questions put it in one line: a timing error growing as the record itself, which the best-line fit removes less of. This essay measures how much less, says where the change from one growth to the other happens, and finds that a locking loop has to be of a particular kind to stop it.

A clock whose frequency has a 1/f in it #

The oscillator is the walk essay’s — 160 MHz, a 0.1 picosecond spread in every period, sampling a 10.01 MHz tone — with a flicker of its frequency added. The flicker is built as a sum of first-order processes with corners three to a decade, each of unit variance; their sum has a spectral density within a fraction of a decibel of C/f from a quarter of the capture’s own lowest frequency up to half the clock rate. Its level is set by the flicker corner: the offset at which its contribution to the timing spectrum equals the walk’s. That is the number a phase-noise plot shows as the knee between thirty and twenty decibels a decade, and the hero figure puts it at 100 kHz.

The curves are exact expectations rather than averages. The timing error is a running sum of increments whose autocovariance is known — a spike for the walk, a sum of geometric decays for the flicker — so the expected mean square left after removing any polynomial can be computed directly: a reverse running sum, then one recursion per corner, for any capture length. Twenty-four seeded clocks at each of four lengths are the second route, and they agree to within the scatter that two dozen captures allow, which for this noise is wide; more on that below.

Over 1,024 samples the flicker costs 1.6 dB: 84.1 against the walk’s 85.7. Over 65,536 it costs 14.7: 52.9 against 67.6. Beyond a few thousand samples the flicker clock’s ratio falls 19.5 dB a decade of capture — twice the walk’s slope — and in the limit exactly twenty.

Why twenty decibels a decade, and where it starts #

A walk’s residual grows as the square root of the capture because its timing error at a point is the sum of independent periods, and a fitted line can take only its average slope. A flicker clock’s residual grows as the capture itself. The reason is self-similarity: flicker frequency noise looks the same at every time scale, so the frequency’s wander over a capture twice as long is the same shape, and the timing error it integrates to is twice as large. Removing the best line takes the capture’s average frequency, and what is left is the frequency’s wander within the capture, which by that same scaling is a fixed fraction of the whole. The exact expectation confirms it: the residual about the line, divided by the square of the capture, is the same to a part in a hundred from 256 samples to 4,096.

So there are two growths and they cross. The walk’s mean square grows as n and the flicker’s as $n^2$, so they are equal at one capture length, and the hero figure marks it: 2,277 samples, 14.2 microseconds, at a 100 kHz corner. Below it the walk dominates and the walk essay’s rules apply; above it the flicker dominates and they do not.

The crossing has a universal form. Move the slider and it moves inversely with the corner: 22,778 samples at 10 kHz, 7,592 at 30, 758 at 300. In every case the capture at which the flicker’s share equals the walk’s lasts 1.42 periods of the corner frequency. That is a rule a converter designer can use without a phase-noise plot’s fine print: read off the oscillator’s flicker corner, invert it, multiply by 1.4, and captures longer than that are flicker-limited.

Telling the two apart from a timing record #

A phase-noise plot shows the corner directly, but a converter designer often has only timing records — edge times captured by a time-interval analyser — and the question is how to find the corner in those. The statistic that does it is the one frequency metrologists use for exactly this purpose: the Allan deviation, the spread of the difference between the average frequencies of two adjacent stretches of a given length. For the walk, whose periods are independent, averaging the frequency over a stretch four times longer halves its wander, and the Allan deviation falls as the square root of the stretch. For flicker frequency noise it does not fall at all: a stretch four times longer has a frequency that wanders just as much, because the wander is the same at every scale.

So the Allan deviation of this clock falls as one over the square root of the stretch for short stretches and flattens at a floor for long ones, and the knee sits at a stretch of the same order as the corner period — the time-domain face of the 1.42 found above. The flat floor is the most useful single number a flicker clock’s data sheet can carry: it is the fractional frequency error that no amount of averaging removes, and multiplying it by a capture’s length gives, to within the fit’s constant, the timing error that capture will see. A data sheet that gives a period jitter and no Allan floor has described the walk and left the flicker to be discovered.

The same corner, seen as a spectrum #

The walk essay’s spectrum had one slope. Add the flicker and the skirt bends: above the corner it is the walk, twenty decibels a decade; below it, the flicker, thirty.

The model and eight seeded clocks agree across four decades of offset — −131.6 against −132.2 dBc/Hz at 78 kHz, −152.9 against −153.2 at 625 kHz — and part company by two decibels only at the lowest offsets the record holds, where a periodogram has very few cycles to average. Measured between a thirtieth and a third of the corner, the flicker’s own slope is 29.7 dB a decade; well above it the total falls 20.3.

The thirty-decibel slope is what makes the growth with capture length steeper. The power in a skirt between an offset and the carrier diverges as the offset falls — for the walk’s skirt as one over the offset, for the flicker’s as one over its square — and the lowest offset a capture can see is one over its length. A capture four times longer reaches four times lower: the walk’s error gains a factor of four in power, the flicker’s sixteen.

A fit takes a fraction, not a slope #

The natural repair is a richer fit. A line removes a frequency offset; a parabola would remove a frequency that drifted steadily through the capture, which is what a wandering frequency looks like over a short enough stretch. A cubic would take more again. The walk essay’s open question was whether a quadratic term would make a flicker clock usable over long captures.

One capture shows what the fit is up against. The walk wanders a few picoseconds and leaves 2.38 ps rms about its best line. The flicker clock bends: its frequency is different at the end of the capture from the beginning, and in between it changes more than once. A line leaves 9.2 picoseconds, a parabola 6.8 — it absorbs the average drift and nothing finer, and the curvature that remains is the frequency’s wander on every shorter time scale.

The exact expectation turns that one capture into a rule. Against flicker, removing a line rather than the mean buys 7.63 dB; a parabola on top of the line buys 3.88; a cubic on top of that, 2.61. Against the walk the same three terms buy 3.98, 1.92 and 1.30. And each gain is the same at every capture length, from 256 samples to 16,384, to a few hundredths of a decibel. That is self-similarity again: a fit removes a fixed fraction of a process that looks the same at every scale, so it shifts the curve down by a constant and leaves its slope alone.

A quadratic term does not make a flicker clock usable over long captures. It is worth 3.88 dB — two-thirds of a bit — at every length, which is something, and it is the whole of what it is worth. Over 65,536 samples the flicker clock is at 52.9 dB with a line removed and 56.7 with a parabola, against the walk’s 67.6; no polynomial of modest degree closes a fourteen-decibel gap that is still growing at twenty decibels a decade. Forty-eight seeded flicker clocks of 1,024 edges give 3.34 dB for the parabola, within the scatter that a noise dominated by its slowest components leaves even after four dozen captures.

That scatter is itself a property worth stating. A flicker clock’s residual over one capture is dominated by the corners slowest compared with the capture, which the capture holds only a cycle or two of, so one capture’s residual can sit several decibels from the expectation in either direction. Twenty-four seeds put the mean square within about a decibel and a half. A walk’s residual over one capture is much more consistent, because it is built from many independent periods; the flicker clock’s jitter is not just larger over long captures but less predictable from one capture to the next.

What a locking loop does, and what kind it must be #

A locked clock does not walk. The loop a reference allows modelled the lock as a first-order pull towards the reference, and against the walk that was enough: the error stopped growing once the record was several loop time constants long. Against flicker it is not.

With the walk alone the first-order loop holds 87.1 dB at every length. With the flicker it gives 85.17 dB over 1,024 samples, 84.28 over 4,096, 83.55, 82.92 and 82.36 over 262,144 — still falling, and falling by the same amount for every fourfold longer record. Each fourfold step adds the same variance, which is a logarithm. A first-order loop’s error transfer has one zero at zero frequency, which turns the walk’s $1/f^2$ into something flat and integrable, but turns the flicker’s $1/f^3$ into $1/f$, whose integral diverges logarithmically as the lowest frequency in the record falls. The loop has slowed the flicker’s growth from twenty decibels a decade to a creep of about one decibel per decade of record — and has not stopped it.

A loop with an integrator in it has two zeros at zero frequency, because it removes a ramp in phase as well as an offset, and that turns the flicker’s $1/f^3$ into $f$, which is integrable with room to spare. The same time constant with an integrator and a damping of 0.707 holds 87.5 dB at every length, 87.53 over 1,024 samples and 87.45 over 262,144. Six seeded clocks of 16,384 edges agree with both integrals to a third of a per cent.

So the order of the loop is not a detail of its dynamics. Against a walk, a first-order loop and a second-order loop of the same time constant give nearly the same jitter, and the choice between them is about settling and tracking. Against flicker, the first-order loop leaves a jitter that has no value, in exactly the sense the walk essay meant — it depends on how long the record is — and the second-order loop gives it one. A clock specified with a single jitter figure and a flicker region close to its carrier is making a claim about its loop’s order that the specification does not state.

What it costs a converter #

The walk essay’s fourteen-bit converter needed 86.0 dB on this tone, and the walk alone held it for captures up to 953 samples. With a 100 kHz flicker corner the limit is 723 samples; with a 10 kHz corner, 916. At fourteen bits the flicker barely moves the boundary, because the boundary sits below the crossover, where the walk still dominates.

The damage is to long captures. Over 65,536 samples the walk alone gave 67.6 dB, 10.9 bits; with a 100 kHz corner the clock gives 52.9 dB, 8.5 bits, and with a parabola fitted 56.7 dB, 9.1 bits. Over 4,096 samples the corner costs 4.5 dB, 75.2 against 79.7. A spectrum analyser measuring a converter’s spurious-free range over long records, a radio receiver integrating a narrowband signal, a coherent sampler accumulating many periods of a waveform — all of them capture for much longer than 1.42 corner periods, and all of them see the flicker clock’s error growing at twenty decibels a decade past that.

Locked, the picture changes again, and what matters is the order of the loop. A first-order loop of a hundred periods leaves 82.4 dB over a quarter of a million samples and less beyond; a second-order loop of the same bandwidth leaves 87.5, which is fourteen bits with room. The difference is five decibels at this length and grows without limit at longer ones.

How the numbers were obtained #

The flicker is the sum of first-order processes with corners three to a decade from a quarter of the capture’s lowest frequency to half the clock rate, each started from its stationary distribution so that a short capture is not a transient, and scaled so that its timing spectrum meets the walk’s at the stated corner. The exact expectations use the increments’ autocovariance — the walk’s spike and the flicker’s sum of geometric decays — summed into the timing error’s covariance and projected onto an orthonormal polynomial basis, so a capture of 131,072 samples costs a few million operations. The locked clocks’ variances are integrals of the increments’ spectrum times each loop’s error transfer, on a logarithmic grid from a twentieth of the record’s lowest frequency to half the clock rate. The seeded clocks are the same construction run forward with a fixed seed and fitted by least squares.

What it leaves out #

The reference’s own flicker. A locked clock follows its reference below the loop’s corner, and a crystal reference has a flicker region too, usually far closer to the carrier. Below the loop’s corner the clock inherits the reference’s thirty-decibel slope, and over long enough captures it reappears whatever the oscillator does.

Frequency noise steeper than flicker. Below the flicker region many oscillators show a random walk of frequency, a phase-noise slope of forty decibels a decade from temperature and ageing. A first-order loop’s single zero leaves that as $1/f^2$ — a locked clock that walks like a free one — while the second-order loop’s two zeros flatten it, so the integrator that was optional against the walk and necessary against flicker is exactly sufficient here. A noise one order steeper again would need a third.

And everything is one tone. A band-filling signal replaces the tone’s frequency with the band’s rms frequency, as the frequency a band moves at found, and that correction multiplies the one here rather than interacting with it.

Still open: the capture shorter than the corner, a frequency that drifts, and the loop’s order as a budget #

Bursts against a flicker corner. A capture shorter than 1.42 corner periods sees mostly the walk, and the corner where averaging stops working found the same structure in amplitude: averaging pays until the flicker takes over and then stops. Whether the best capture length for a fixed total observation time — many short captures fitted separately, or one long one — has a closed form in the corner frequency is the time-domain version of that question.

A drift with a cause. Temperature changes an oscillator’s frequency smoothly and predictably, which is what a parabola fits. When the drift is a cause rather than a noise — a crystal warming after power-up — the fit’s quadratic term is worth much more than its 3.88 dB against flicker, and the question becomes how long after power-up a capture has to start before the residual is the noise’s rather than the warm-up’s.

The loop’s order as a design variable. A second-order loop stops the flicker at the price of a peak in its error transfer near the corner, which amplifies the reference’s noise there. Whether the best loop for a clock with both a flicker corner and a noisy reference is second-order at a narrower bandwidth or first-order at a wider one, and where the boundary between those lies in the two corners’ ratio, joins the reference essay’s geometric mean to this essay’s logarithm.