Two loops, and the mismatch between them

The rung below this one marched single noise-shaping loops of second, third and fourth order and measured the input amplitude at which each stops working — an amplitude that falls with the order, so that the fourth-order loop has a runaway at seven tenths of full scale and the first has none anywhere. That is why nobody builds a fourth-order single loop, and it is a genuine trade rather than an engineering failure: the shaping and the stability come from the same integrators.

The standard answer to it is not a better single loop. It is two of them.

The arrangement #

A first-order modulator produces an output that is its input plus its quantisation error shaped once. Feed that error — which is available, being the difference between what the comparator said and what was presented to it — to a second modulator. Differentiate the second one’s output digitally and add.

The first stage’s error appears twice with opposite signs and cancels. What is left is the second stage’s error, shaped by the sum of the two orders. Every loop in the arrangement is first order and unconditionally stable, and the cascade behaves as though it were second order.

Marched at an oversampling ratio of 64, that is 72.5 decibels against the first stage’s own 47.5, and across the four ratios drawn the cascade gains 14.9 decibels per octave of oversampling against a single first-order loop’s nine. Fifteen decibels an octave is second-order shaping, measured on a pair of first-order loops rather than assumed from a transfer function.

Nothing in the march reads a state a real converter could not: the first stage’s error is taken as y − u at the comparator, which is a quantity a switched-capacitor implementation has as a charge.

Where the cancellation actually happens #

The cancellation is between an analogue path and a digital one, and that is the whole of this rung.

The digital differentiator is exact. It is a subtraction of two numbers, it has no tolerance, and it does not drift.

The analogue path is not. The first stage’s error reaches the second stage through a capacitor ratio, an amplifier with finite gain and a settling that is not complete, so the gain it arrives with is the designed one times some error. What does not cancel leaks through with the first stage’s shaping rather than the cascade’s — one order lower — and below some mismatch that leakage is the entire noise floor.

So the arrangement’s advantage is a claim about the silicon rather than about the architecture, and the point of this essay is what the claim costs.

The curve has the shape the argument predicts: flat while the shaped second-stage noise dominates, then falling as the leaked first-stage noise takes over, with the transition where the two are equal in power.

Three decibels are gone at 5.29 per cent at an oversampling ratio of 64. That is a capacitor ratio — achievable, and not free, and not something the digital side can measure or correct.

The sign is not a convention #

The arrangement is usually drawn with a wave of the hand — feed the error forward, differentiate, add — and the hand-wave hides a place where it is easy to be exactly wrong.

Which sign the error arrives with depends on the integrator form. The delaying integrator marched here gives y₁ = x − (1 − z⁻¹)e₁, with a minus, so the second stage must be fed +e₁ and its differentiated output added for the two terms to subtract. Fed −e₁ and added, the two contributions reinforce, and the cascade’s in-band noise is worse than its own first stage’s.

That is not a subtle failure and it is not an obvious one either. The output still looks like a sigma-delta stream, the spectrum still slopes upward, and the arrangement still gains with oversampling — it simply gains at first order with twice the noise. The only thing that catches it is computing the first stage’s own figure alongside the cascade’s and requiring the second to beat the first, which is why that assertion is the first one in the site’s gate for this figure and why it is written as a margin rather than as a comparison of two curves.

The general form of the lesson is the one the two-routes habit exists for: a quantity that is a difference between two paths needs the two paths’ own answers kept, because the difference alone cannot say which of them moved.

Every octave of performance is an octave of matching #

The tolerance falls as the −1.11 power of the oversampling ratio — one over it, within the fit. At 32 the two paths may differ by 11.6 per cent; at 64 by 5.3; at 128 by 2.1; at 256 by 1.1.

The reason is a comparison of two powers. In-band, first-order shaped noise goes as (π/OSR)³ and second-order as (π/OSR)⁵, so the leakage term δ²·(π/OSR)³ equals the shaped term (π/OSR)⁵ when δ ∝ 1/OSR. Every octave of oversampling buys fifteen decibels and demands twice the matching, and the two requirements arrive together.

That is the honest statement of what the architecture does. It does not escape the trade the rung below found. It moves it: out of the loop, where it was a stability constraint on the integrators and therefore a hard limit on the achievable order, and into the analogue path between stages, where it is a matching specification — and a matching specification is a thing a process can be bought to meet, while a stability limit is not.

Why five per cent is a good answer and not a comfortable one #

Five per cent sounds generous for a capacitor ratio, and integrated capacitors match to a fraction of a per cent, so the arrangement looks safe by an order of magnitude. Two things narrow that.

The first is that three decibels is a large amount to give away. A design specified at the matched figure with no margin has spent its whole budget; the mismatch that costs a quarter of a decibel is about a third of the three-decibel figure, and that is 1.7 per cent at an oversampling ratio of 64 and 0.4 at 256. Which of the two numbers a designer needs depends on whether the cascade’s advantage was being spent or banked.

The second is that δ is one number standing for several mechanisms. A capacitor ratio is the best-behaved of them. The amplifier’s finite gain contributes an error of order 1/A which is a few parts in ten thousand for a good stage and a per cent for a fast one; the settling contributes an error that depends on the signal and is therefore not a constant gain at all; and both drift with temperature and with supply. The five per cent is a budget for the sum, not for the capacitors.

And the direction of the failure matters. When the leakage dominates, what appears in the band is the first stage’s quantisation noise — which for a one-bit first stage is large, is only first-order shaped, and carries the idle tones a first-order loop is notorious for. So the failure mode is not a gently raised floor: it is the first stage’s tonal signature appearing in a converter whose architecture was chosen partly to avoid it.

What the leakage does to the spectrum #

The in-band number is a summary and the spectrum says more, because the leaked term and the shaped term have different shapes.

The shaped term rises at forty decibels a decade — second order — so almost all of it is at the top of the band and above it, which is what makes decimation cheap and what makes oversampling worth so much. The leaked term rises at twenty, so it is comparatively flat across the band, and it is larger at the bottom than the shaped term by a wide margin even at mismatches where the two are equal in total power.

Two consequences follow that the single number hides. A converter’s low-frequency performance degrades first as the mismatch grows, which for an instrumentation converter is the part that matters most and for an audio one the part that matters least. And the crossover between the two mechanisms is a frequency as well as a mismatch: at any given mismatch there is a frequency below which the leakage dominates and above which the shaping does, and that frequency moves up through the band as the mismatch grows until it leaves the top of it.

That is why a specification written on the total in-band ratio can be met by a converter whose low-frequency noise is dominated by leakage — the same shape of concealment the tonal-share measurement in the rung below was introduced to prevent, where an error that is a set of tones and an error that is a floor have the same total power and are not the same thing.

What this costs that the single loop does not #

The comparison is not free in the other direction either, and three costs are worth naming beside the matching.

Two modulators instead of one. Twice the analogue circuitry, twice the reference load, twice the clock energy, and a second comparator whose own offset is a second error source.

The output is not one bit. A single loop’s output is ±1 and drives a one-bit digital-to-analogue converter, which is inherently linear because it has two levels. The cascade’s combined output is the sum of a one-bit stream and a differentiated one, which takes several levels — so the decimation filter is wider and, on the digital-to-analogue side, the inherent linearity is gone and the levels have to match, which is the quantisation field’s own subject arriving as a matching requirement. That is the same problem this essay is about, arriving in a different place.

And the second stage sees a signal it was not designed for. Its input is the first stage’s quantisation error, which is a full-scale-ish, roughly white, one-bit-derived waveform rather than a band-limited signal. Scaling it to keep the second stage’s integrators inside their range is a design step with no counterpart in a single loop, and getting it wrong shows up as exactly the same mismatch — a gain error between the stages — which is why the scale factor and the matching are not independent problems.

The record length is part of the measurement #

Every number here is read from a transform of a marched record, and the rung below already found what that can go wrong in: its third-order ladder reported 101.6 decibels at an oversampling ratio of 256 and 110.5 when the record was made four times longer, because the band is n/(2·OSR) lines wide and a fixed record measures a narrower band the faster it is clocked.

The same rule applies here and is applied the same way: the record is chosen so that at least 256 lines lie below the band edge, at every oversampling ratio on the sweep. Without it the comparison across ratios — which is where the fifteen decibels an octave comes from — would be measuring the record rather than the architecture, and the exponent would come out too high in exactly the way that looks like a good result.

The signal tone is placed on a coherent bin, odd and coprime with the record length, so that it lands in one line rather than smearing across three and so that the quantiser does not see a shorter record than it was given. And the noise is every line below the band edge except the one the signal is in. None of that is specific to the cascade; all of it is machinery the rung below built and this one inherits, which is the useful sense in which a rung is a rung.

What is not modelled #

The analogue integrators are exact. Their gain is infinite, their settling is complete, and their noise is absent. Every one of those is a contributor to δ in a real converter, and modelling them individually is what would turn the single mismatch parameter here into a budget with lines in it.

The quantisers are ideal comparators. No offset, no hysteresis, no metastability. A comparator’s offset in the first stage is a direct-current offset in the converter, which is harmless; in the second stage it is an offset on the error signal, which the differentiator removes; and its hysteresis is what decides the idle-tone pattern at small inputs, which is the quantity the rung below measures as the tonal share and which nothing here reproduces.

There is no decimation filter. The in-band ratio is read from a transform of the error over the band, which is where a converter’s result is defined and not where it is read. A real decimator has a finite stopband, so some of the shaped noise above the band comes back in — and for a cascade, whose noise rises steeply with frequency, the decimator’s requirement is stricter than for a single loop of the same in-band performance.

And the mismatch is a constant. It is a gain error, fixed over the record. A real one varies with the signal, because the settling error does, and a signal-dependent leakage is a distortion rather than a raised floor — which is a different measurement with a different figure of merit.

What the gate checks #

The matched cascade is asserted to be worth more than fifteen decibels over its own first stage, which is the claim that the cancellation happens at all and would fail if the digital combination had the wrong sign. That check has already earned itself: with the second stage fed −e₁ and its output added, the two leakages reinforce instead of cancelling, and the cascade measured 41.5 decibels where the first stage alone measured 47.5 — a cancellation running backwards, and visible only because the first stage’s own figure was computed beside it.

A large mismatch is asserted to take most of the advantage back, so a version whose leakage path was disconnected would fail rather than draw a flat line.

The three-decibel mismatch is asserted to lie between a tenth of a per cent and half — the range in which it is a capacitor ratio rather than a wish — and no ratio in the sweep is allowed to make either loop unstable, which is the architecture’s own selling point stated as a refusal.

The shaping law is asserted at fifteen decibels an octave to within one and a half, measured across four ratios rather than at one, and the tolerance’s own exponent is asserted at −1 to within a quarter. Both, because the first without the second would be consistent with a cascade that worked and whose requirement did not tighten.

Where the anchor stands after three rungs #

The first rung marched a first-order loop and found nine decibels an octave coming out of an arrangement that was told nothing about frequency.

The second raised the order, found fifteen and twenty-one decibels an octave, and found the price: an input amplitude at which each order stops working, falling with the order.

This one takes the standard escape from that price and measures what it substitutes. The escape is real — second-order shaping from first-order parts, with no amplitude limit anywhere in the sweep — and what it substitutes is a five-per-cent matching requirement that tightens by an octave for every octave of oversampling.

Which of the two is the better trade is not a question this collection answers, because it depends on a process rather than on a circuit. What it can say is that the two constraints are of a kind: one is a boundary in amplitude and the other is a boundary in matching, and the essays around this one keep finding that the second kind is the one that binds.