A floor, or a line

A picosecond, read as bits established the arithmetic. A converter samples at t + δ instead of t and gets a value wrong by the signal’s slope times δ, so the damage is proportional to frequency and to nothing else about the converter: the signal-to-noise ratio is −20·log₁₀(2π·f·tⱼ), which is why a sixteen-bit part is a twelve-bit part at ten megahertz.

That number is a power. It is correct, it was checked against a measurement that shares only the value of tⱼ with it, and it does not say what the error looks like — because where the error lands in frequency is decided by where the jitter is in frequency, and a root-mean-square in picoseconds has thrown that away.

The totals agree, and the pictures do not #

The two clocks give total error powers of −87.43 dB and −87.55 dB relative to the signal, against the closed form’s −87.55. They agree with each other to 0.12 of a decibel and with the expression the rung below derived to within the same. Nothing is in dispute about the total.

The random clock spreads that total over two thousand bins, at −120.5 dB each. The modulated one puts it into two lines at the signal frequency plus and minus the modulation, at −91.5 dB. The worst single line differs by 19.4 decibels between two clocks that a specification cannot distinguish.

Which of those two numbers matters is not a matter of taste. A spectral mask, an adjacent-channel requirement and a spurious-free dynamic range are all written about the worst line. A signal-to-noise ratio and an effective number of bits are written about the total. A converter specified adequately for one can fail the other by twenty decibels with no change to its clock’s root-mean-square jitter.

The gap is the processing gain, and it is measurable #

The difference between a floor and a line is not a property of the clock; it is the processing gain of whatever transform is looking at it. Spreading a fixed power over N/2 bins puts 10·log₁₀(N/2) less in each of them, which for a 4,096-point record is 33.1 decibels.

The way to demonstrate that it is the processing gain rather than a coincidence is to change the record length and watch which quantity moves.

The floor falls 3.1 dB per doubling — it is a density, and a longer record has narrower bins. The line does not move at all: 3.8 dB of scatter across sixteen times the record length, which is the measurement’s own repeatability rather than a trend.

So the gap between the two clocks grows with the measurement, from 22 dB at 1,024 samples to 36 dB at 16,384. Two parts that measure the same on a short record are not the same part, and a longer look is what tells them apart.

What the two clocks do to a receiver #

The 19.4 decibels have a direct reading in the two applications that care most, and the readings point in opposite directions.

For a communications receiver, a strong unwanted signal near the wanted one is sampled with the same clock, and the clock’s jitter puts sidebands of the strong signal on top of the weak one. That is reciprocal mixing, it is proportional to the strong signal’s amplitude, and it is a line rather than a floor whenever the clock’s phase noise is a line — so the receiver’s blocking performance is set by the worst spur, not by the total. The modulated clock above is 19 dB worse at exactly this.

For an instrument measuring a spectrum, the same 19 dB appears as a spurious response that moves when the input frequency moves and stays a fixed offset away from it. That is the signature: a spurious line whose offset from the carrier is constant is a clock artefact, and one whose absolute frequency is constant is not.

For a sampling oscilloscope the same jitter appears as horizontal blur on a repetitive waveform, and there the total is what matters rather than the worst line — which is why an instrument’s clock is specified in picoseconds and a receiver’s is specified as a phase-noise curve. Two industries, two specifications, one physical quantity, and the reason for the difference is the one measured above.

Where a modulated clock comes from #

A clock whose jitter is a tone rather than a random sequence is not a contrivance. Three ordinary mechanisms produce one and all three are present in a real system.

A phase-locked loop — a feedback system with all the properties what is left at crossover describes — has a reference and a loop bandwidth, and reference feedthrough puts a tone at the comparison frequency onto its output — typically tens or hundreds of kilohertz, which is exactly the region modelled here. A switching regulator supplying the clock or the converter modulates the threshold crossings at its own switching frequency, a few hundred kilohertz — and what reaches the clock from that rail is the quantity what gets through from the rail measures. And crosstalk from a digital bus, of the kind the far end that cancels measures, puts a tone at whatever rate that bus runs.

None of those is described by a root-mean-square figure and all of them are usually specified as one. The specification a system engineer needs is the clock’s phase-noise spectrum, which is the same information the two curves in the first figure carry.

What the closed form is entitled to #

It is worth being exact about the rung below rather than treating this one as a correction to it, because the earlier result is not wrong anywhere.

The expression −20·log₁₀(2πf·tⱼ) is a statement about total error power and it is exactly right for both clocks here. What it is not is a statement about a spectrum, and it was never claimed to be one. The mistake this rung guards against is a use rather than a derivation: taking a total-power figure and comparing it against a specification written about a single line.

The same distinction is made in the quantisation anchor. When a floor stops being a floor found a quantisation error that is not noise at all — for a signal coherent with the sample rate it is a small set of discrete lines, and the same total power appears as a spur rather than as a floor. Two anchors, two mechanisms, one structural point: a total is not a spectrum.

Integrating a phase-noise curve, and what is lost #

The bridge between the two specifications is an integral, and it is worth writing out because it is where the information disappears.

A clock’s phase noise is a spectral density in decibels relative to the carrier per hertz, plotted against offset from the carrier. The root-mean-square jitter is the integral of that density over some offset range, converted from radians to seconds. So the picosecond figure is derived from the spectrum — by an operation that maps a curve to a number, which is not invertible.

Two curves with the same integral are two clocks with the same quoted jitter, and the two in this essay are exactly that pair: one flat, one a delta function, same area. The specification is not under-determined by accident; it has been deliberately reduced.

The offset range of the integral is the second thing lost. A jitter figure quoted over 12 kHz to 20 MHz and one quoted over 10 Hz to 1 MHz are different numbers for the same clock, and the low-offset end matters for a receiver while the high-offset end matters for a sampler — so two correctly quoted figures for one part can differ by an order of magnitude.

Two routes, and what they share #

The measurements here are made by sampling a sinusoid at jittered times and transforming the result, which is a simulation of the experiment. The comparison quantity is a closed form derived from a first-order expansion of the sine.

Those two share the value of tⱼ and nothing else — no expansion, no transform, no assumption about the jitter’s distribution — which is what makes their agreement to a tenth of a decibel evidence. And they disagree exactly where they should: at large jitter the first-order expansion overstates the error, because a sine’s slope is not constant over a large timing excursion.

Both spectra scale identically with the jitter, which is the check that the difference between them is about the shape of the jitter and not about its size. Five times the picoseconds is fourteen decibels on the floor and fourteen on the line.

What a coherent sample set is doing here #

Every measurement in this essay uses a signal frequency chosen so that a whole number of cycles fits in the record, and that choice is doing work worth naming.

Without it the signal itself leaks across bins, and the leakage — tens of decibels down but spread over the whole spectrum — is indistinguishable from a jitter floor. That is the same hazard six decibels a bit has to navigate when it measures a quantiser’s own floor, and the same repair. The measurement would then be reporting the window function rather than the clock. Choosing a coherent frequency, with the cycle count coprime to the record length so that the samples do not repeat, removes the leakage exactly rather than approximately.

That is a measurement condition of the same kind the capacitance that is not one number finds in a bridge’s own test amplitude: not a detail of the apparatus, but part of what the number means.

The number a converter’s data sheet does print #

Aperture jitter appears on a converter’s data sheet as a picosecond figure and it is the converter’s own contribution — the uncertainty of its internal sampling instant relative to its clock input. That is a real quantity, it is usually a few hundred femtoseconds, and it is almost never the one that decides the answer, because the clock delivered to the pin has more jitter than the part adds.

The two add in quadrature if they are independent, which they are, so a part with 0.2 ps of aperture jitter fed from a clock with 2 ps has 2.01 ps — and the converter’s specification has contributed one half of one per cent. Choosing a better converter on that number is choosing on the smaller of two terms.

What the converter’s own figure does have is a spectrum too — it is dominated by thermal noise on the internal clock buffer’s threshold crossing, so it is broadband — and that is why it is fairly quoted as a single number while the external clock’s is not.

What is not measured #

No amplitude noise on the clock. A comparator converting a clock’s amplitude noise into timing noise does so through the slope at the threshold, so a slow edge turns supply noise into jitter at a rate set by volts per second. That is the dominant mechanism in many systems and it makes the jitter’s spectrum a copy of the supply’s — which is a strong argument that the spectrum is the right object, and it is not modelled here.

No correlation between the jitter and the signal. Both clocks here are independent of the input. A clock derived from the same reference as the signal generator has jitter correlated with it, and the sidebands then fall at the same place every time and add coherently over records — which is a third behaviour, distinct from both of the two measured.

And no reconstruction. Everything above is about sampling. 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 jitter argument transfers unchanged and the processing gain does not, because a reconstruction’s output is not usually looked at through an FFT.

What would have to change to make the two clocks equivalent #

There is one circumstance in which the distinction above disappears, and identifying it says what the distinction depends on.

If the signal being sampled is itself broadband — noise-like, filling the band, with no line in it — then the sidebands the modulated clock produces are convolutions of a broadband signal with a line, which is a broadband result. The spectrum of the error then looks the same for both clocks, and the total, which was always equal, is the whole story.

So the difference is a property of the signal as much as of the clock. A single tone makes the distinction maximal; a modulated carrier makes it intermediate, with the sidebands becoming a copy of the modulation displaced by the jitter tone; and a noise-like signal removes it entirely.

That is worth carrying into a specification argument. A converter for a communications receiver is looking at a spectrum that has strong lines in it, so its clock’s spurs matter. A converter in a control loop is looking at something broadband, and for it the picosecond figure is sufficient — which is not an accident of convention but a consequence of the sentence above.

The specification that does not determine the answer #

Two clocks, one number, two outcomes twenty decibels apart. The number is not wrong and it is not useless — it settles the total error power exactly, which is what an effective-number-of-bits figure is computed from. It simply does not contain the information that decides whether a system passes.

The information that does is a spectrum, and the reason it is rarely quoted is that it is a curve rather than a number. This collection’s habit is to draw the curve and then say where the number that replaced it stops being true, and here it stops being true the moment anything cares about a single line rather than about a total.

That shape of complaint is not confined to clocks, and two of its other instances are worth naming because together they suggest where to look for the next one. The constant that is a window takes a diode’s ideality factor — quoted as a number, defined as a derivative — and finds it has a value at every current and no value anywhere: eight one-decade fits to one curve return factors from 1.23 to 1.98, two of them straight to a few parts in a thousand, so the residual gives no warning at all. And the coefficient that is about one reading takes a ceramic capacitor’s temperature coefficient and finds it a coefficient of one particular measurement, with one number unable to determine the two parameters the part has, so that two parts a bridge cannot tell apart differ by 1.80 at the voltage they are used at.

The common structure is a specification that is a projection. A jitter figure in picoseconds projects a spectrum onto its total; an ideality factor projects a curve onto a slope at an unstated current; a temperature coefficient projects a two-parameter model onto one number read under one condition. In every case the projection is exact — nothing is being approximated — and in every case two objects with the same projection behave differently. The tell is always the same: the specification has fewer numbers in it than the model it is a specification of.

And what fixes it is always the same as well, and is always more expensive. The edges that are lengths makes the general observation about a related class, and the remedy here is its analogue: state the condition alongside the number. A jitter figure with a spectrum beside it, an ideality factor with the current it was fitted at, a temperature coefficient with the bias it was measured under. None of those is a harder measurement than the one already being made. Each is a harder thing to print on one line.