A floor, or five tones

The sharpest sentence this field has produced about a one-bit modulator is that three quarters of its in-band error sits in five lines. One bit, and where the noise went records it and the loop that is worse at full scale builds on it: 74.6 per cent of the error in the five largest lines of the band, against the 1.953 per cent a white error spread over 256 lines would put in any five. A factor of thirty-eight, and the conclusion drawn from it is that the shaping law’s white-noise argument is false about this loop.

The conclusion is right. The factor is not, and the reason is in the comparison rather than in the measurement. A shaping loop exists to make its error not white. It multiplies the quantiser’s error by a transfer function that rises with frequency, so the largest lines in the band sit at the band edge whether there is a tone anywhere in the picture or not. Comparing against a flat spectrum counts the loop doing its job as evidence that the loop is misbehaving.

What is being measured, exactly #

A first-order modulator is marched forward one sample at a time with a one-bit quantiser inside the loop, driven by a coherent sinusoid a quarter of the way up its own band, over a record of 32,768 samples at an oversampling ratio of 64. The band holds 256 lines and the test tone sits in line 65, which puts the band edge at 3.94 times the input frequency.

The error is the output minus the input, sample by sample, and its power is evaluated line by line at the bins below the band edge. Two numbers come out. The five-line share is the fraction of that power sitting in the five largest lines, whatever they turn out to be. The flat share is 5 over 256 — 1.953 per cent — which is the share five lines of a spectrum with no structure in it would hold.

The second is a null hypothesis, and a null hypothesis is a model of what the measurement would say if nothing interesting were happening. That is where the trouble is: a flat error is not what would happen if nothing interesting were happening. It is what would happen if the loop were not there.

The null the loop itself prescribes #

The loop’s own noise transfer function is not a modelling choice. It follows from the difference equations that are actually marched: each integrator accumulates its own gain times the difference between the stage before it and the previous output, so writing each integrator state as a combination of the input and the output and solving for the output gives the transfer function from the quantiser’s error to the output directly.

The textbook form is $(1 - z^{-1})^L$ for $L$ integrators, which is the transfer function of a loop whose coefficients are all one. Only the first-order loop here has that; the higher orders use a fifth, because unity gains at second order put the first integrator at four times full scale, which is a real amplifier that has to swing it. Using the loop’s own equations rather than the textbook form costs a dozen lines of arithmetic and is the honest object, so it is what is used.

And it makes almost no difference, which is worth knowing rather than hiding. At third order the two expressions differ by 41.94 decibels at line 1 of the band, 41.95 at line 64 and 42.17 at line 256 — a level, not a tilt. Loop coefficients scale a noise transfer function very nearly without bending it, so the share the two forms put in five lines is 13.02 per cent against 12.87. The correction this essay is about is therefore not the choice of expression. It is the choice of null, and the textbook form would have produced it just as well.

Evaluated over the 255 lines of the band that do not hold the tone, that transfer function puts 5.74 per cent of its power in its five largest lines. Against 1.953. The null is nearly three times what it was taken to be, before any measurement has been made at all.

Why five lines of a shaped error are 5.74 per cent #

The number can be recovered by hand, which is the check that the transfer function is not doing anything surprising.

Below the band edge the frequencies are small — line 256 of 32,768 is a twentieth of a radian — so $|1 - e^{-j\omega}| = 2\sin(\omega/2)$ is very nearly $\omega$ itself, and a first-order loop’s line power goes as the square of the line number. The five largest lines are then lines 252 to 256, and their $h^2$ over the sum of every $h^2$ up to 256 is 5.739 per cent. The exact transfer function gives 5.74.

The same arithmetic with $h^4$ and $h^6$ gives 9.375 and 12.873 per cent for two and three integrators, against 9.49 and 13.02 exactly. So the null rises with the order for a reason that has nothing to do with tones: a steeper ramp puts more of its total in its top few lines. The last twenty lines of the band hold 21.6 per cent of a first-order null, 33.4 per cent of a second-order one and 43.3 of a third — before anything interesting has happened at all.

That is the sentence the flat null cannot say. Five lines of 256 holding 1.953 per cent is a statement about a spectrum with no shape, and the whole apparatus under discussion exists to give the spectrum a shape.

The calibration, which comes before the correction #

A null computed from a linearisation of a hard nonlinearity is worth nothing until the loop has been made to produce it. The arrangement that does this already exists in the field and was originally a defect: driving the modulator with a large tone above its own band.

The loop that is worse at full scale records why that placement was wrong for measuring a ratio — the numerator was a signal the denominator’s band does not contain — and why it was kept anyway. A one-bit quantiser driven hard by an out-of-band input is dithered by it: the loop stops idling into patterns locked to anything in the band, and its in-band error becomes what the transfer function alone would give.

Two routes, and what they share is worth stating because that is what decides whether an agreement means anything. One is a marched nonlinear system — a comparator, an accumulator and a subtraction, 32,768 times, with no transfer function anywhere in it. The other is a complex expression evaluated at 256 frequencies, with no time in it at all. They share the three loop coefficients and nothing else, which is the same standard one step computed twice applies to a transient and what a network answers applies to a solve.

Agreeing to a part in five hundred per line, across two and a half decades of line number, is therefore a calibration of the null rather than a coincidence. The 5.74 per cent is measured.

The departure, and what it does and does not change #

Against the calibrated null the first-order loop’s in-band error is 17.4 times as concentrated as its own shaping at a fiftieth of full scale, falling to 8.9 times at nine tenths. Against the flat null the same two points read 51 and 26.

Two things follow and they point in different directions.

Within one loop, nothing changes shape. Both nulls are constants at fixed order, so the curve in the figure at the top is the same curve read against a different level. Every statement of the form “the error becomes less tonal as the input grows” survives untouched, and so does the mechanism behind it.

Across orders, the correction is not a constant. The shaped null is 5.74 per cent at first order, 9.49 at second and 13.02 at third, because each added integrator steepens the rise and pushes more of the error into the top few lines. So the ratio between the two nulls is 2.94, 4.86 and 6.67 — and a comparison of tonality across orders made against the flat null has a factor of two and a quarter drifting through it in the direction that flatters the higher orders’ improvement.

The concrete casualty is a sentence this field has been asserting: that three integrators do not finish the job, the error still being an order of magnitude from white. It is true as stated — 13.8 times the flat share — and it reads as though the loop had been left badly tonal. Against the shaping the same measurement is 2.07 times, which is a different report about the same object.

There is a second reading in that figure and it was not the reason for drawing it. The third-order curve does not fall with amplitude at all. It runs 39.2 per cent at a fiftieth of full scale, up to 52.0 at a fifth, down to 23.5 at three fifths, and then up again to 68.1 and 68.5 at eight and nine tenths. The rise at the top is the amplitude edge the loop that is worse at full scale located by watching the ratio collapse, and the ratio does collapse in the same place — 76.0 dB at seven tenths, 75.6 at eight tenths, 60.0 at nine. The concentration measure sees the same event. A loop being pushed past its useful input range does not merely make more error; it makes error that is concentrated in a few lines, which is what a quantiser spending long runs saturated produces.

What the five lines actually are #

The five-line share says how concentrated the error is. It does not say what the lines are, and the answer decides whether the number is about distortion or about the shaping.

The separation is exact rather than tolerant. A harmonic of a coherently sampled tone lands on an exact integer multiple of the tone’s own bin and nothing else does, so the power at multiples of bin 65 is harmonic distortion and everything else is not. With the tone a quarter of the way up the band only two harmonics fit below the edge — the second and the third — which bounds how much of this error could possibly be distortion at all.

So the answer to what a one-bit loop’s in-band error is made of is an amplitude rather than a category. At a fiftieth of full scale it is two tones, and they are harmonics of the input. At nine tenths of full scale it is a quarter distortion and three quarters shaped noise piling up at the band edge, and the five-line measure has stopped reading the tones and started reading the slope.

That is the same boundary when a floor stops being a floor measures on a bare quantiser, where the harmonic share of the error rises from 0.286 per cent at 230 levels to 76.5 at 1.3. The direction is the same — small signals make the error deterministic — and the device is not: a one-bit modulator has no levels to run out of, so an explanation phrased in levels crossed cannot be the explanation here.

Where the calibration stops, and what is not claimed because of it #

The out-of-band drive dithers a first-order loop by 41.1 decibels — 88.6 dB reported in band against 47.5 with the tone inside it. At second order the margin is 22.5 dB and at third it is 7.7.

That collapse matters because the margin is the calibration. A loop that is barely dithered by its own out-of-band input is still idling into patterns, so the spectrum it produces is not the transfer function’s spectrum, and the check cannot separate “the linearisation is wrong” from “the measurement still has tones in it”.

So the shaped nulls at second and third order are predictions, not measurements, and the numbers above that rest on them are marked accordingly. The first-order figure is a measurement, and it is the one the argument needs, because the flat null’s error is a factor of three at that order and grows from there rather than shrinking.

The record is part of the measurement #

One number in all of this is not a property of the loop and it is worth naming, because both nulls depend on it and one of them depends on it much more.

The band holds $n/2\,\mathrm{osr}$ lines, so at 32,768 samples and an oversampling ratio of 64 it holds 256. The flat null is 5 over that — halve the record and the flat null doubles to 3.9 per cent, with nothing about the modulator having changed. The shaped null moves too, because the top five of 128 lines is a larger share of a ramp than the top five of 256, but it moves less: the ramp’s total is dominated by its top end either way.

So the five-line share is a measurement condition rather than a property, in the sense the instruments field uses the phrase — a reading that belongs to the pair of the thing and the apparatus reading it. The bandwidth noise sees makes the same point about a noise floor, where the number quoted is meaningless until the bandwidth it was integrated over is stated. Here it is the line count, and this field has been quoting a share of five lines without it.

The right form of the statement therefore names the record: at 256 lines in the band, a first-order loop driven at half of full scale puts 74.6 per cent of its in-band error in its five largest lines, against 5.74 for its own shaping and 1.953 for a flat error. Two of those three numbers move if the record moves, and the ratio between the measurement and the shaped null is the most stable thing on the page.

What this does not say #

It does not rescue the white-noise argument. Seventeen times its own shaping is not a small excess, and at first order the error is very nearly all distortion at small inputs. The shaping law that predicts nine decibels an octave still rests on an assumption that is false about this loop, and one bit, and where the noise went is right that the measured 8.99 falls short of 9.03 for exactly that reason.

It does not change any signal-to-noise ratio. The total in-band error power is what it is; only the description of how it is distributed moves. Everything the field has measured about the shaping law, the amplitude edge and the cascade in two loops, and the mismatch between them is untouched.

And it does not make the flat share useless. It is the right null for a bare quantiser, where the error genuinely has no transfer function in front of it — which is why the same statistic works without correction in the quantisation field and needs correcting here. A null is a property of the arrangement rather than of the statistic, and the same number carried across an arrangement boundary is the failure, not the number.

What it opens #

A dither, which this loop does not have. A bare quantiser’s error is made white by adding one least significant bit of noise at its input, and the loop’s out-of-band drive is that trick performed by accident with a signal rather than deliberately with noise. Whether a small deliberate dither inside the loop moves the five-line share to the shaped null without spending the ratio is a measurement this field is one parameter away from making and has not made.

And a lower tone. Two harmonics in the band is a bound imposed by putting the test tone a quarter of the way up it. A tone a twelfth of the way up would put eleven harmonics below the edge and turn the harmonic share from a two-line statistic into a spectrum, which is what would be needed before the word distortion could be given a number rather than a direction.

The number worth carrying #

Five point seven four per cent, not one point nine five three — and 17.4 times rather than 51.

The habit that goes with it is about nulls rather than about loops. A statistic quoted as a multiple of a reference is only as good as the reference, and the reference is usually chosen when the statistic is written and then travels with it unexamined into arrangements it was never right for. The test is whether the null describes what the apparatus would produce with the interesting effect switched off. Here the interesting effect is a tone in the band, switching it off leaves a shaped spectrum rather than a flat one, and the difference between those two is a factor of three at first order and nearly seven at third.