The frequency a band moves at

A picosecond, read as bits gave aperture jitter its rule: a sinusoid of frequency $f$ sampled at instants wrong by $\sigma$ root-mean-square has a signal-to-error ratio of $-20\log(2\pi f\sigma)$, and ten picoseconds hold sixteen bits up to 199 kHz. A floor, or a line then showed that the same number of picoseconds can arrive as a floor or as a pair of lines, and it ended on a signal it had not measured: one that is itself broadband. For such a signal, it said, the distinction between the clocks disappears and the total is the whole story.

The total is then the question, and the rule in hand does not answer it. It was derived for one tone, and a signal filling a band has no single frequency to put in it. The usual practice is to use the top of the band, on the principle that the fastest component is the one the jitter hurts most. That is a bound and not an answer, and this essay measures how loose a bound it is: for a flat band it is 4.77 dB pessimistic, for the spectra most real signals have it is ten to thirty, and the quantity that replaces the top frequency is a mean over the whole band that anyone holding the signal’s spectrum can compute.

A band, sampled and measured #

The signal is forty-eight equal tones at random phases, spread evenly across a band from zero to its top frequency, each an exact number of cycles in a record of 16,384 samples so that nothing leaks and the signal’s power is exactly the sum of its tones’. It is sampled twice, once at the ideal instants and once at instants each displaced by a fresh draw from a seeded Gaussian of ten picoseconds, and the difference between the two sample sets is the error. The ratio of the signal’s power to the error’s is measured at seven top frequencies, from 100 kHz to 100 MHz.

At a top frequency of 1 MHz the measured ratio is 88.64 dB. A single tone at 1 MHz, sampled by the same clock, gives 84.04 dB, which is the rule. The band is better than its fastest component by 4.6 dB, and across the whole sweep the dots sit parallel to the tone’s line at the same distance: 20 dB a decade, the same slope, displaced upwards. The band’s top frequency can rise by a factor of 1.73 before the band is as badly off as a tone at its top.

That factor is $\sqrt 3$, and the line the dots lie on is the tone rule evaluated not at the top frequency but at 580 kHz, which is the band’s rms frequency — the square root of the power-weighted mean of the squares of its tones’ frequencies. For a flat band from zero to $f_{max}$ that is $f_{max}/\sqrt3$, and the measurement agrees with the rule at that frequency to 0.12 dB, which is the scatter of 16,384 random timing errors.

The error is the slope, sample by sample #

The reason is visible one sample at a time. A sample taken at $t + \delta$ instead of $t$ is wrong by $x(t+\delta) - x(t)$, and for a timing error of picoseconds against a signal that changes over microseconds that is the slope times the timing error, $x'(t)\,\delta$, to an accuracy that the square of the next term decides.

The figure computes both, for the first 160 samples of the band at 1 MHz: the error measured by sampling twice, and the slope at each instant multiplied by that instant’s draw. They agree to 7.4 parts in a hundred thousand of the largest error, and the shaded envelope — the magnitude of the slope — shows the error swelling where the waveform is steep and shrinking where it turns. Nothing about the band’s top frequency appears in any single sample’s error. What appears is the slope at that instant.

So the error’s power is the timing error’s variance times the slope’s mean square, $\sigma^2\,\overline{x'^2}$, and the slope’s mean square is a property of the signal’s spectrum. Each tone of amplitude $a$ at frequency $f$ contributes $a^2(2\pi f)^2/2$ to it, the tones’ cross terms average away over a record, and the ratio of signal power to error power is

$$\frac{P_{signal}}{P_{error}} = \frac{1}{(2\pi\sigma)^2\,\overline{f^2}},\qquad \overline{f^2} = \frac{\sum a_k^2 f_k^2}{\sum a_k^2},$$

which is the tone rule at the frequency $f_{rms} = \sqrt{\overline{f^2}}$. For a single tone that is the tone’s own frequency and nothing is lost. For anything else it is a weighted mean, and a mean is always below the maximum it is taken over.

Four spectra with the same top #

A flat band is the gentlest departure from a tone. Most signals worth sampling have far less power near their top than near their middle, and for those the rms frequency falls a long way.

The four signals share a top frequency of 1 MHz and differ only in how their power is spread over the three decades below it. With power rising in proportion to frequency, the rms frequency is 737 kHz — the continuous band’s $f_{max}/\sqrt2$ — and the signal does 2.62 dB better than the rule. Flat, 602 kHz on this logarithmic grid and 4.37 dB better. Pink, with equal power in every octave, 282 kHz and 10.96 dB better. Power falling as the square of frequency, which is the shape of most audio programme above its midrange and of most slowly varying sensor outputs, has an rms frequency of 34 kHz — thirty times below its top — and does 29.30 dB better than a tone at its top would.

Every one of the four matches the tone rule at its own rms frequency, to two tenths of a decibel. The shapes are chosen to bracket what occurs, and the spread between them — 27 dB, four and a half bits — is the range of error a designer is choosing between when the jitter budget is written against the top frequency. The rule written that way is safe. For the falling spectrum it is safe by a factor of thirty in frequency, which is a factor of thirty in the clock’s allowed jitter.

The lower edge, and when a band is a tone #

The flat band’s advantage depends on where it starts. A band from zero has an rms frequency of $f_{max}/\sqrt3$. A band from $r$ to one of $f_{max}$ has a mean square frequency of $(1 + r + r^2)/3$ of $f_{max}^2$, so its advantage over the tone at its top is

$$10\log\frac{3}{1 + r + r^2}\ \text{dB}.$$

The measured points follow it across nine bands: 4.60 dB for a band from zero, 3.59 from a quarter, 2.47 from a half, 1.17 from three quarters and 0.21 from 97 per cent, each within the 0.3 dB that the finite draw of timing errors and the finite count of tones allow. The advantage halves by the time the band starts at half its top, and a band a few per cent wide behaves like the tone at its centre.

That is the case the tone rule was written for, and it covers more than a pure sinusoid. A narrowband signal sampled well above baseband — an intermediate frequency digitised directly, a carrier and its modulation — has its rms frequency at its centre, and for it the rule is right with the centre frequency in it. The rule’s error is confined to wideband baseband signals, where it is always on the safe side and usually by more than a designer would guess.

What the same picoseconds allow #

Read the other way, the rms frequency is a statement about what a converter’s clock will support.

At ten picoseconds, sixteen bits of jitter-limited resolution end at 199 kHz for a tone, 344 kHz for a flat band and 734 kHz for a pink band over three decades. At twelve bits the same clock carries a tone to 3.18 MHz, a flat band to 5.50 MHz and a pink band to 11.7 MHz; at one picosecond every figure is ten times higher. The flat band’s factor of $\sqrt3$ is the measured one; the pink band’s 3.69 is its rms frequency’s ratio to its top, which depends weakly on how many decades the band spans — it is the square root of twice the natural logarithm of that span, for a span much greater than one.

At a top frequency of 1 MHz with ten picoseconds, the tone holds 13.67 bits and the flat band 14.43. Three quarters of a bit is small next to what a pink or falling spectrum gains, and it is still the difference between a converter meeting a specification and not meeting it, for no change in the part.

A budget written three ways #

The rule is most often used backwards — to set the clock’s specification from a resolution — and there the difference compounds. Take an audio channel with a 20 kHz band that must hold 120 dB against jitter alone. Written against a tone at the top of the band, the rule allows $10^{-6}/(2\pi\times20\,\text{kHz})$ = 7.96 ps. Written against a flat band from zero it allows $\sqrt3$ times that, 13.8 ps. Written against pink content from 20 Hz to 20 kHz — equal power per octave, the usual stand-in for programme material — the rms frequency is 20 kHz divided by $\sqrt{2\ln 1000}$, 5.38 kHz, and the allowance is 29.6 ps. And if the content falls as the square of frequency above 20 Hz, its rms frequency is the geometric mean of the band’s edges, $\sqrt{20 \times 20{,}000}$ = 632 Hz, and the clock may carry 252 ps.

Those four figures span a factor of thirty-two for one channel and one target, and each is right for its signal. A clock specified at 7.96 ps for a channel that only ever carries speech has been bought thirty times too good. That may be the right purchase — a test tone at 20 kHz is what the channel will be certified with, and a tone is the worst case — but it should be made knowing that it is the certificate and not the programme that sets the price.

The error these clocks produce has a spectrum of its own, and it is worth knowing what shape it is. Each sample’s error is the slope times an independent timing draw, and a sequence multiplied by independent draws is white whatever the sequence was: the band is shaped, the error is not. So the jitter error from a pink programme lands as a flat floor across the whole Nyquist band, at the level the rms frequency sets — including the octaves where the programme itself is quiet. The ratio this essay measures is a total; per octave, the error is heaviest relative to the signal exactly where the signal is weakest, at the top of the band. That is the one place the tone rule’s pessimism is partly earned back. It is also the opposite of the behaviour when a floor stops being a floor found in quantisation error, which stops being white and locks to the signal as the signal shrinks: jitter error with independent draws stays white at any amplitude, because its randomness is in the clock and not in the signal’s relation to a grid.

Why the rule was written for a tone #

None of this contradicts the floor a converter sets or any converter data sheet, which state an effective number of bits measured with a single full-scale sinusoid. That measurement is the right one to specify a part by, because it is reproducible and it is the worst case at a given frequency: of all signals with power at or below $f$, a tone at $f$ has the largest mean square slope for its power. The rule is correct as a bound on the part.

What it is not is a prediction for a signal. A system whose content is known — a microphone channel, a vibration sensor, a video line — has a spectrum, and the jitter it will actually suffer is computed from that spectrum’s second moment in one line. The quantisation formula carries the same hidden signal: its 1.76 dB is $10\log(3/2)$, the ratio of a full-scale sine’s power to that of a full-scale ramp, and six decibels a bit, and the half step blamed on it measured it to 0.15 dB one step inside full scale — with a sine. There the property of the test signal is its shape; here it is its single frequency.

There is a second reason the tone rule persists: a sine at full scale is the largest signal a converter can take without clipping, and a band-filling signal is not. The forty-eight tones drawn here have a crest factor of 3.39 against a sine’s 1.41, so they must be backed off by 7.6 dB below a full-scale sine’s rms to fit between the rails. That costs 7.6 dB of the signal-to-quantisation ratio, measured against a fixed floor. It costs nothing of the signal-to-jitter ratio, which scales with the signal — the error is proportional to the slope, and the slope to the amplitude. The two effects are independent and both have to be counted, and the jitter half is the one this essay measures.

Two routes, and what they share #

The measured ratios come from sampling a sum of sinusoids twice and differencing: no expression for the error is used. The predicted ratios come from the tones’ frequencies and amplitudes alone, through the rms frequency, with no samples. The two share the list of tones and the value of the jitter, and nothing else. The per-sample check shares the tones and the draws of the timing error, and it confirms that the first-order picture — slope times timing error — is the whole of the effect at picoseconds, to seven parts in a hundred thousand.

Every band’s tones sit on exact bins of the 16,384-sample record, so the signal’s measured power equals the sum of its tones’ powers to rounding, and the slope’s mean square equals the sum of its tones’ contributions to rounding. What varies between runs is the draw of the timing errors, which moves a measured ratio by about a tenth of a decibel. The seeded generator makes every run of every figure identical.

What it leaves out #

Correlation between the jitter and the signal. The timing errors here are independent of each other and of the signal. A clock whose jitter is correlated with the signal — the same reference shared by the source and the sampler — puts the error where the signal is, and the mean over the record no longer separates the two factors.

A jitter that is not independent from sample to sample. Every timing error here is a fresh draw. A clock whose errors are related to the previous edge’s — a free-running oscillator, whose timing wanders — has a jitter whose size depends on how long the record is, and the mean square slope is then multiplied by a quantity that grows with the look. The walk the core sees measured exactly that kind of accumulation in a relaxation oscillator’s timing, where it walks at nanoseconds per root period.

The reconstruction side. A converter going the other way has the same mechanism at its output, and the staircase on the way out is where that path’s own errors are measured. The rms-frequency argument transfers unchanged.

Still open: the wandering clock, the correlated one, and the band that is not stationary #

A clock whose errors accumulate. Everything here assumes a jitter with a value — an rms that does not depend on how long it is measured over. A free-running oscillator’s timing is a walk, its rms grows with the square root of the record, and the rule then needs a record length as well as a frequency and a signal. Measuring what a walk does to the same sampled tone, and what a locking loop does to the walk, is the next thing to draw.

A signal and a clock from one reference. When the jitter is a function of the signal rather than independent of it, the error’s power is no longer the product of two means, and whether the cross term helps or hurts depends on the phase relation — which a floor, or a line found decides everything for a modulated clock.

Content that changes. A signal’s spectrum over one record may not be its spectrum over the next — speech, a transient, a burst — and the rms frequency is then a quantity that moves from record to record. The jitter floor moves with it, and a converter whose budget was written against the long-term spectrum is exposed during exactly the loud, bright moments the budget averaged away.