The levels the loop cannot shape

A one-bit noise-shaping loop has two problems, and the essays before this one measured both. The loop that is worse at full scale found that a one-bit loop of second order does best at six tenths of full scale and that higher orders run away at amplitudes that fall with the order. A floor, or five tones found that its in-band error, measured against what the loop’s own transfer predicts, gathers into lines rather than spreading into a floor. Both are statements about a quantiser with two levels, and the remedy the field reaches for in both cases is the same: give the quantiser more levels.

When a floor stops being a floor showed why that should help with the second problem. A quantiser’s error is white only when the signal visits many levels, and the share of the error in harmonics of the input climbs from a third of a per cent at 230 levels to three quarters at 1.3; one bit sits at the far end of that curve by construction. The same loop with a quantiser of sixteen levels sits much further along it.

What a multi-bit quantiser costs is not in the quantiser. It is in the levels it feeds back. In a one-bit loop those levels are two voltages, and two voltages define a straight line whatever their values. In a three-bit loop they are eight, and eight voltages are a straight line only if they are made exactly right. This essay measures what the extra bits buy and then what an error in the levels takes back, and finds that the second is decided by a number the loop cannot touch.

The loop, and how it is measured #

The loop is the second-order one in its textbook form: two integrators in a chain, the quantiser’s output fed back into both with weights one and two, so that its noise transfer is exactly $(1 - z^{-1})^2$ and its signal transfer exactly a delay of two samples. The weights come from requiring that, and the same requirement gives one, three and three for a third-order loop and one, four, six and four for a fourth: the binomial coefficients with the first set to one. This is not the loop the earlier essays marched, which scaled its integrators by a half to suit a one-bit quantiser. The full $(1 - z^{-1})^N$ is chosen here because it is the same transfer for a quantiser of any number of levels, so the comparison between bit counts is between quantisers and not between loops.

A quantiser of $b$ bits has $2^b$ levels spread evenly over plus and minus full scale and picks the one nearest its input. The output is the level it picked. What is fed back is that level plus its own error, which is what a converter in the feedback path does when its resistors or capacitors are not exactly in proportion. The input is a sine inside the band at a fixed fraction of the band’s width, the record is run for a quarter of its length before it is kept, and the error is the output minus the input delayed by the loop’s order.

The error is measured through a squared Hann window, and the window was the first thing the measurement had to get right. The shaped error rises forty decibels a decade towards half the sample rate, and a record whose two ends do not meet leaks that loud out-of-band noise into the quiet band through its window’s sidelobes. Through a rectangular window the ratio of this loop scattered by twenty decibels from one amplitude to the next, because each record’s ends happened to meet better or worse. A plain Hann window removed the scatter at these ratios and was not enough at the highest, where the band sits 150 decibels down; squared, its sidelobes fall thirty decibels an octave and the measurement is steady. The same leakage reads a third-order one-bit loop of the earlier kind six decibels pessimistic at a ratio of 256, which is worth knowing before any of those numbers is quoted to the decibel.

Six decibels a bit, and the amplitude back #

At half scale each bit beyond the second adds between six and seven and a half decibels: 82.5, 90.3, 96.8 and 102.0 for two to five bits. That is the uniform quantiser’s rule — each bit halves the step and quarters the error power — carried through a shaping loop unchanged, because the loop shapes whatever error it is given and a smaller error shaped the same way is smaller by the same factor.

The first extra bit is worth more, 13.1 decibels, because a one-bit quantiser does not have a step in the uniform sense. Its error is the distance from the integrator’s output to plus or minus one, and the integrator’s output wanders over a range that bears no fixed relation to the gap between the two levels. A two-bit quantiser is the first whose error is bounded by half a step, so going from one bit to two buys both a smaller step and an error that behaves.

The amplitude is the second return. With one bit the ratio peaks at 0.6 of full scale and falls beyond it, which is the edge the earlier essay measured. With three bits it peaks at 0.8, with four at 0.85 and with five at the top of the sweep. A one-bit loop at large amplitude is driven by an input close to the levels it has, so its shaped error is a large fraction of the distance between them; a fine quantiser always has a level within half a small step, and the shaped error never pushes it past its last one until the input is almost at full scale.

The orders that need bits #

The amplitude return has a sharper form at higher order. A loop with the full $(1 - z^{-1})^N$ has a noise gain of $2^N$ at half the sample rate — four for second order, eight for third, sixteen for fourth — and every one of those multiples of the quantiser’s error appears at its input. One bit cannot absorb that.

Second order holds to the top of the sweep on any quantiser, which is the reason second-order one-bit loops are the workhorse they are. Third order with this transfer holds nothing at all on one or two bits: the states run past ten times full scale within the record at every amplitude from 0.05 up. On three bits it holds to 0.55, on four to 0.8 and on six to 0.95. Fourth order needs four bits before it holds anything, and then only to a quarter of full scale.

The rule that comes out is roughly one bit of quantiser for each order of shaping, and every bit beyond that buys amplitude. It is a different answer to the problem the earlier essays found from the one two loops, and the mismatch between them measured, which kept every loop at low order and cascaded them instead. Both answers remove the amplitude edge of a single high-order one-bit loop, and both pay somewhere outside the loop. The cascade paid in the match between an analogue path and a digital one. The multi-bit loop pays in the match between its own levels.

Two levels are always a line #

The levels a quantiser feeds back are produced by a converter — a string of resistors, a bank of capacitors, a set of current sources — and none of them comes out exactly in proportion. The R–2R ladder is the commonest of those converters, and every level it makes is a ratio of resistors; the error that is a distribution measured why two devices made side by side still differ. Their mismatch is the input to this figure.

The one-bit curve is flat to a tenth of a decibel across three decades of level error, out to one per cent. That is the property that made one-bit loops dominant in the first place, and it is exact rather than approximate. Whatever two voltages the converter produces for its two codes, they lie on a straight line through the codes: the output is the input code times one number plus another. The loop forces the average of the fed-back voltage to follow the input, so a converter with a gain error and an offset makes the digital output a scaled and shifted copy of the input, which is a gain and an offset in the result and nothing else. Neither is noise and neither is distortion.

Eight levels do not lie on a line unless they are placed on one. Any error that is not a common gain or offset bends the converter’s transfer, and the loop, forcing the converter’s average to follow the input, forces the digital codes to follow the input through the inverse of that bend. The ratio falls twenty decibels for each decade of error — the error in the band grows in proportion to the error in the levels — and it falls below what one bit gives at errors that are small by any standard of component matching: 0.027 per cent for three bits is a match of about one part in 3,700, and 0.021 per cent for five bits about one in 4,800.

More bits make it slightly worse rather than better. Five bits are 11.7 decibels ahead of three with perfect levels and fall below one bit at a smaller error, because thirty-two levels each carrying its own error bend the transfer in more places, and the shaped error the extra bits saved is now a smaller quantity for the level error to exceed.

The error that oversampling does not reduce #

Shaping is what makes oversampling work: the loop pushes its error out of the band, and a higher ratio pushes the same error further. An error that enters where the input does is not pushed anywhere.

With perfect levels the in-band error falls fifteen decibels for every doubling of the ratio, the second-order law that one bit, and where the noise went first measured, now clean because a three-bit quantiser’s error is white. With a tenth of a per cent of level error it hardly falls at all: four tenths of a decibel an octave above a ratio of 64, and the curve is a horizontal line from there. Oversampling a loop with this converter in it buys nothing past a ratio of about thirty-two, and the sixty decibels the perfect loop gains between 16 and 256 are simply not available.

The reason is in where the error enters. The loop’s whole mechanism is a comparison between the input and the fed-back level, so an error in the fed-back level is indistinguishable from an error in the input. The loop passes it through with the signal transfer, which is a delay. Whatever part of it is noise is spread over the full band from zero to half the sample rate, and a higher ratio keeps a smaller share of it — three decibels an octave. Whatever part of it is a fixed function of the input stays exactly where it is.

Distortion, not noise #

The spectrum says which part dominates.

With perfect levels the in-band error is the shaped floor and nothing else: forty decibels a decade, smooth from bottom to edge. With a tenth of a per cent of error in each level the bottom of the band has risen seventy decibels, and two lines stand out of it, at twice and three times the input’s frequency. A static error in the levels is a curved transfer, and a curved transfer applied to a sine makes harmonics — the mechanism what the limiter charges for measured in an oscillator’s diodes, arriving here from a converter’s resistors. A bend that is odd about the middle of the range makes odd harmonics and one that is even makes even ones; this draw of eight errors has more of the first, another draw moves the balance between the two, and the harmonics stay.

That changes what the multi-bit loop’s limit means. A designer reading the ratio figure might plan to average the level error away with a longer record or a slower output rate. The spectrum says the error is not noise to be averaged: it is distortion at fixed frequencies, as repeatable as the input, and no amount of filtering after the converter removes a harmonic that lies inside the band the filter is meant to pass.

Where the trade sits #

The two halves of this essay meet at a single number for each converter. A multi-bit loop is better than a one-bit loop only if its levels match better than a few hundredths of a per cent. Above that match, every advantage the extra bits bought in decibels is gone, and the advantage in amplitude and in the orders a loop can hold is all that remains — which is a real advantage, since a fourth-order loop that does not run at all on one bit is useless however linear its two levels are.

Matching of one part in four thousand is a demanding figure for any set of components made side by side, and it has to hold over temperature and time as well as on the day of manufacture. That is why practical multi-bit converters do not rely on matching. They rearrange which unit elements produce each level from one sample to the next, so that each element’s error is spent evenly over time, and the error that would have been a static bend becomes a noise that is itself shaped. The measurement here is the case without that rearrangement, and it fixes what the rearrangement has to achieve: turn an error that does not fall with oversampling into one that does.

What this leaves out #

The loop is ideal in every other respect. The integrators are exact, with infinite gain and no settling error; a real integrator leaks, and the leak puts a floor under the shaped error of its own. No dither is added, so the three-bit loop’s error is white because of its levels rather than because of noise — the distinction one step of dither per moment draws for a quantiser outside a loop. The quantiser’s thresholds carry no error, which is honest rather than generous: a threshold error inside the loop is shaped along with the quantisation error, so it matters far less than a level error, and that asymmetry is the whole point of the comparison. And the level errors are fixed and independent — a seeded draw from a normal distribution for each level — where a real converter’s errors are often correlated along the string, which bends the transfer smoothly and puts more of the error into the second and third harmonics than an independent draw does.

The stability figure uses one criterion, states within ten times full scale for each integrator over a record of 32,768 samples, at amplitudes in steps of 0.05. A loop held at 0.55 might run away at 0.56 in a longer record; the steps say where the edge is to within a twentieth of full scale and not more finely.

The number to keep #

A second-order loop with the full $(1 - z^{-1})^2$ transfer gains six or seven decibels for every bit beyond the second and its best amplitude rises from 0.6 to 0.95 of full scale; third and fourth order need about one bit of quantiser for each order of shaping before they hold at all. The fed-back levels enter where the input does and are passed unshaped, so a three-bit loop loses its lead over one bit when each level is out by 0.027 per cent, its error stops falling with oversampling — 0.4 dB an octave at 0.1 per cent — and the error arrives as harmonics. One bit is immune, exactly, because two levels are always a line.

Still open: the rearranged levels, the leaking integrator, and the thresholds #

Levels rearranged from sample to sample. A converter built from identical unit elements can choose which elements make each level, and rotating the choice turns a static bend into a noise whose spectrum the rotation sets. Whether a simple rotation shapes that noise by one order, and what level match it then needs to keep the three-bit loop’s 20 decibels over one bit, is the measurement that would turn this essay’s limit into a design.

Integrators with finite gain. Every integrator here is perfect. An integrator built round an amplifier of finite gain leaks, and a leak takes the shaped error’s zero away from zero frequency; how much gain a loop needs before the leak costs a decibel, and whether that depends on the oversampling ratio, is the companion question about the other analogue part of the loop.

The quantiser’s thresholds. An error in a threshold is inside the loop and is shaped with everything else, so it should matter far less than an error in a level. How much less, and at what threshold error the quantiser starts skipping codes and the shaping argument fails, would put a number on the asymmetry this essay assumes.