The floor a converter sets

The noise field opened with a sentence this essay is going to repeat with one word changed. It said: a resistor’s noise is 4kTR volts squared per hertz, independent of everything about the resistor except its resistance and its temperature — which is what makes it a floor rather than a property of a component, because a different resistor of the same value at the same temperature produces the same number.

A converter has a floor of exactly that shape. Its step is q, its error over a step is uniform, and the root mean square of a uniform distribution over a step is q/√12 — a number containing no signal, no bandwidth and nothing about the circuit. It is the second floor this collection has drawn, and the interesting thing about it is not its value but what happens when the two are put on one axis.

Two floors, and why they can be compared at all #

Putting these two quantities on one axis needs a word of justification, because they are arrived at by completely different routes and it would be easy to compare them wrongly.

The Johnson floor is a density integrated over a bandwidth. Its value depends on how much of the spectrum is being looked at, and this collection has been careful about that since the noise field opened: the bandwidth in question is an equivalent noise bandwidth, not a −3 dB point, and using the latter understates a single pole’s answer by 21%. For a 1 kΩ source over 100 kHz the number is 1.266 µV rms.

The quantiser’s floor is not a density and has no bandwidth in it at all. It is q/√12, full stop. What gives it a bandwidth is the sampling: the error’s power is spread across the band from zero to half the sample rate, so its density is (q²/12)/(fₛ/2) and depends on the clock. That is the fact behind oversampling, and it is the subject of a later essay in this field.

For this comparison the total is the right quantity, because the question is whether the converter’s whole error is larger or smaller than the whole noise arriving with the signal. So both are totals, both are in volts rms, and both refer to the same 2 V full-scale span.

Beside 1.266 µV, a sixteen-bit converter is seven times noisier than the resistor in front of it and a twenty-four-bit one is thirty-seven times quieter. The crossing is at 18.80 bits, which exists only as an arithmetic quantity — nobody makes an 18.8-bit converter — and is the more useful number for it, because what it says is that eighteen bits are worth having and twenty are not.

What the crossing depends on #

The crossing moves, and what moves it is the circuit rather than the converter.

Two decades of source resistance are worth about three and a third bits, and the arithmetic is the same one the noise field measured directly: the density goes as √R, so a factor of a hundred in resistance is a factor of ten in voltage, which is 3.32 bits. The gate checks that relation to a part in 10¹⁶ rather than quoting it, because a quantity that “goes as the square root” is exactly the kind of claim that survives being slightly wrong.

The bandwidth matters the same way and for the same reason, and it is the term a designer usually has some control over. Halving the bandwidth is worth half a bit; the anti-alias filter that the previous essay priced in clock rate is also setting this number, since the noise reaching the converter passes through it too.

Effective bits, and the thing it is usually not #

The industry’s name for the combined answer is the effective number of bits, and it is obtained by inverting the formula the next essay measures: take the total measured signal-to-noise ratio, subtract 1.76 dB and divide by 6.02.

That is a perfectly good definition and it hides the distinction this essay exists to draw. A part with a stated resolution of sixteen bits, fed from a 1 kΩ source in 100 kHz, has an effective resolution set by the sum of the two floors — 1.266 µV of thermal and 8.81 µV of quantisation, adding in quadrature to 8.900 µV, which is 15.99 effective bits. The thermal contribution costs a hundredth of a bit and is invisible.

The same part fed from a 50 kΩ source in 10 kHz gives 15.93 bits, for a different reason: the resistor’s noise is 2.830 µV and the quadrature sum is 9.253 µV. Still nearly invisible.

A twenty-two-bit part in the second circuit gives 17.64 effective bits, not 22, and the four and a third bits that went missing were never the converter’s. That is the case the crossing identifies and the effective-bits figure conceals: one number reports the answer, and two numbers say which of them to spend money on.

What cooling is worth, in bits #

The Johnson floor carries a temperature in it and the quantiser’s does not, which makes the crossing one of the few boundaries on this site that a designer can move without changing a component value.

The density is 4kTR, so the voltage goes as √T and the crossing moves by half a bit for every factor of two in absolute temperature. From 290 K to 77 K — room temperature to liquid nitrogen — is a factor of 3.77, which is 0.96 of a bit; to 4 K is a factor of 72.5, which is 3.09 bits. A twenty-two-bit converter that is wasting four bits at room temperature is wasting one at 4 K, and the part has not changed.

That is worth stating because it is the one term in the arithmetic that is a property of the environment rather than of the circuit or the converter, and because it is easy to over-claim from. Cooling the source resistor does not cool the amplifier in front of the converter, and it does nothing at all to the quantiser. What it moves is one of two floors, and only the smaller one.

The semiconductor field measured the other side of the same coefficient: a diode’s drop falls 1.828 mV per kelvin, and every decade of bias current moves that coefficient by exactly $V_T \ln 10 / T$. A converter’s reference is a semiconductor object of the same kind, so a part specified at 25 °C carries a drift that is a property of its bandgap rather than of its bit count — which is a fifth limit, and the reason this site keeps insisting that a number quoted without its conditions is not a number.

Where the floor is not the floor #

Two boundaries sit above this one in every real converter, and both have been drawn elsewhere on this site or are drawn later in this field. Neither is the quantisation floor and both are routinely attributed to it.

The half-step overload at full scale. A mid-tread converter’s largest code is one step below full scale, so a sinusoid driven to exactly full scale clips — 327 of 8192 samples at eight bits — and loses 0.83 dB. That is an overload, not a quantisation error, and the next essay separates them by measuring both.

Aperture jitter. The sampling instant is not exactly the intended instant, and the resulting error is the signal’s slope times the timing error, so it grows with input frequency at exactly twenty decibels a decade and has nothing to do with the number of bits.

Put together with the crossing above, a converter has four limits and only one of them is in the bit count: the step size, the source resistance’s thermal noise, the front end’s aperture, and — for a signal that does not use the whole span — the amplitude at which the error stops being a floor at all, which is the essay after next.

The span between the two, which is what a system actually has #

A floor is only half of a range, and this collection has already drawn the other half once. The noise field’s dynamic range essay measured a single amplifier stage in a 10 kHz measurement and found 66.28 dB between its Johnson floor and its one-per-cent distortion ceiling, with the observation that every technique which raises the ceiling leaves the floor where it is.

A converter’s version of that span is quoted rather than measured, and the quotation is 6.02N + 1.76 dB: 49.9 dB at eight bits, 74.0 at twelve, 98.1 at sixteen. Set beside the analogue stage’s 66.28 dB, the numbers say something that is easy to get backwards. A twelve-bit converter already has more span than the amplifier in front of it. Everything above twelve bits in that signal chain is being spent on a range the analogue half does not deliver.

The comparison has two honest complications and both are worth naming.

The two spans are not measured the same way. The analogue ceiling is a distortion figure — the amplitude at which the harmonic content reaches a stated fraction — and the converter’s is an overload, a hard clip at a code. One degrades and the other stops, which is the same distinction the first essay in this field drew about the sample rate, arriving now in amplitude.

A converter’s floor is not a distortion-free floor. The claim 6.02N + 1.76 treats the error as noise, and the essay after next measures the amplitude at which that treatment stops being true — at which point the converter has a distortion ceiling of its own, at the bottom of its range rather than at the top, which no analogue stage has.

So the useful reading is not “a sixteen-bit converter has 98 dB”. It is that a signal chain has one span, the smallest of several, and the number a converter contributes to it is only the top line of a data sheet when nothing else in the circuit is measured.

The same sentence the instruments field already made #

This essay’s conclusion has been reached once before on this site, about a different instrument, and the repetition is the point rather than an accident.

The instruments field argues that a probe is part of the circuit: a reading is two solves, the circuit and the circuit-with-the-instrument, and the quantity drawn is the difference. A 1× probe on a 2 kΩ source makes the reading one per cent wrong at 6.8 kHz, and no property of the probe alone states that number — it needs the source impedance.

A converter is the same object one level further along. Its resolution is a property of the pair rather than of the part, its useful bit count needs the source resistance and the bandwidth before it means anything, and a data sheet quoting twenty-four bits has told a reader as much about what will be measured as a probe’s tip capacitance has told them about a rise time.

What is different here, and worth the field having, is that the converter’s version of the sentence has a crossing in it. A probe’s loading gets steadily worse and there is no bit count at which it stops being the probe’s fault. A converter’s floor crosses the circuit’s, at a number that can be computed from two component values and a temperature, and on one side of that number the answer is “buy a better part” and on the other it is “the part is not the problem”.

The floor that has no source resistance in it #

The crossing measured here is a crossing between a quantiser’s floor and a source’s floor, and the second of those is computed as $4kTR$ over a stated bandwidth — which is the right calculation for a continuous-time front end and the wrong one for the front end most converters actually have.

A sampling converter charges a capacitor through a switch and then opens the switch, and the noise field has measured what that arrangement’s floor is. The total that has no resistor in it is the result: a larger resistor is noisier and makes a narrower filter, and the two dependences are exactly reciprocal, so five decades of resistance charging one picofarad give five decades of corner frequency, two and a half decades of density, and one total — 63.2762 microvolts at every one of them, which is the square root of $kT/C$ and contains no resistance at all. The noise a clock does not make sharpens it further: a switched capacitor behaving as a hundred megohms has none of the 1.27 µV/√Hz density that a hundred megohms of resistor would have, because what lands on the holding capacitor is $kT/C$ with the holding capacitor in it and nothing else — not the clock, not the switched capacitor, not the on-resistance, and exactly rather than asymptotically.

Which means the table above answers a slightly different question from the one a converter’s user has. Its horizontal axis is the source resistance, and for a sampled front end the source resistance cancels out of the floor entirely; what replaces it is the sampling capacitance, which is a property of the converter rather than of the circuit in front of it. The crossing still exists — a quantiser’s floor still halves with every bit while $\sqrt{kT/C}$ does not move — but it is a crossing between two properties of the part, and it moves with the input capacitance rather than with anything a designer puts in front of it.

Both answers are needed, and which one applies is decided by where the bandwidth is set. A front end whose filter is narrower than the sampling rate presents the sampler with a source whose noise has already been limited, and then the source’s $4kTR$ over that bandwidth is what arrives; a front end that lets the sampler see the full width of the source’s noise folds it, and the amplifier inside the sample measures the folding — the number of times white noise folds into the band is exactly the number of time constants the settling needs, so asking for two more bits of settling costs fifteen per cent more noise before anything else has changed.

The measurement this essay does not make #

One number has been used throughout and not measured: 1.266 µV for a 1 kΩ source in 100 kHz. It is johnson()'s answer, which is 4kTR integrated over a bandwidth stated as a number, and a reader who has followed the noise field will notice what is missing from it.

The bandwidth of a real front end is not a number, it is a response, and the bandwidth noise sees is where that is measured: a single pole passes π/2 times as much noise power as a brick wall at its own corner, so a noise voltage computed from the −3 dB point is twenty-one per cent low, and the ratio integrated on the solved response comes out 1.5706 against π/2’s 1.5708. A five-pole Chebyshev’s is 0.964, less than one — so the correction is not even reliably in one direction, and a front end designed for its skirt has a noise bandwidth that has to be computed rather than guessed. The honest version of this essay’s crossing therefore needs an equivalent noise bandwidth computed on the anti-alias filter that the filter essay has just chosen — which is available, and which would move every crossing in the table by a few tenths of a bit in a direction that depends on the family.

It is left as a stated bandwidth here for a reason that is about what the figure is for. The crossing’s existence and its dependence on the source resistance are the argument; the exact bit count is an illustration of it, and tying that illustration to a particular filter would make the figure about the filter. What the slider gives a reader instead is the range: two decades of source resistance move the crossing from 20.96 bits to 17.64, and a filter’s choice of noise bandwidth is worth a fraction of one of those three and a third bits.

The general form of this is worth saying once, since the field is new and will be extended. Every number here is a number about a pair — a converter and the circuit it is attached to — and the site has arrived at that conclusion from four directions now: a probe and its source, an instrument and its edge, an amplifier and its load, and now a converter and its resistor. What differs is only which of the pair is usually printed on the part.