Flatness, and the two currencies it is bought in

Two earlier essays here have measured one multiplication. The staircase on the way out followed the hold’s sinc across the band and found a droop nobody chose — 3.92 dB at half the clock, which is 20 log(2/π) and contains no design decision at all. The nulls are where nothing is followed the same sinc above the band and found that its image rejection collapses to exactly zero at the top of it.

Nobody accepts the droop. It is corrected in every converter that is asked to be flat, and the correction is put in one of two places: in the digital filter before the converter, or in the analogue filter after it. Those are usually described as the same operation applied in two convenient locations.

They are not the same operation and they do not charge in the same currency. What is worse, the thing they are both being paid for turns out to be cheaper than it looks — because the droop costs nothing that anybody has ever measured it costing.

What the droop does not cost #

A droop is a multiplication, and a multiplication applied to a signal is applied to everything that arrives with the signal.

The quantisation noise a converter produces is added at the same place the samples are — it is a property of the number, not of the analogue output — so it goes through the same hold and is attenuated by the same sinc. At a tenth of the clock the signal is 0.14 dB down and so is the noise. At half the clock the signal is 3.92 dB down and so is the noise. The ratio between them does not move. The whole of six decibels a bit survives a hold untouched.

That is the first thing to establish about the droop and it is almost never said, presumably because a data sheet quoting 3.92 dB looks like a loss and a reader assumes it is a loss of something. It is a loss of flatness. Nothing else about the converter is worse for it — not the resolution the floor a converter sets measures, and not the aperture a picosecond, read as bits prices.

Which fixes what any correction can be worth. A correction cannot recover signal-to-noise ratio that was never lost, so whatever it does, it does to the response shape alone — and since it cannot buy anything but flatness, everything it costs is a net cost. That is a stronger way to frame the comparison than “which correction is cheaper”, and it is the reason both prices below are worth knowing exactly.

The digital price is exactly the disease #

Multiply the samples by the reciprocal of the sinc before they reach the converter and the band comes out flat. The correction is a digital filter, it can be made exact to whatever length of filter is thought worth it, and it costs no analogue parts.

Its price is headroom, and the arithmetic is one line. The reciprocal of the sinc is greater than one everywhere except at direct current, so a sample sequence that was already at full scale cannot be multiplied by it. Everything has to come down first, by the largest factor the correction will apply — which is the reciprocal of the sinc at the band edge, which is the droop.

So the flat level a digital correction reaches is the level of the droop’s worst point, not of its best. A band that stops at 0.4 of the clock droops 2.420 dB and flattens at 2.420 dB below what the uncorrected converter gave at direct current. A band that stops at 0.49 droops 3.751 dB and flattens 3.751 dB down. The price is the disease, exactly, at every band width, with no constant of proportionality to argue about.

And because the noise floor is fixed at the converter’s output while the signal has come down, this is the one place in the argument where a correction does cost signal-to-noise ratio. The droop cost none; removing it digitally costs precisely as much as the droop was. A designer who corrects digitally and then notices the dynamic range has fallen has not found a defect; they have found the bill.

The analogue price is nearly a constant #

Put the correction after the converter instead and it is an analogue filter: a shelf with a zero and a pole, fitted to sit on top of the reciprocal of the sinc across the band.

It fits well when the band is narrow. At 0.2 of the clock the residual across the band is 0.014 dB; at 0.3 it is 0.079; at 0.4 it is 0.231; at 0.45 it is 0.368 and at 0.49 it is 0.541. A single zero and pole cannot track $1/\mathrm{sinc}$ near half the clock, where the reciprocal is heading for a pole of its own, and the residual grows accordingly. That is worth knowing but it is not the interesting price.

The interesting price is that the shelf is still rising where the images are.

A shelf is a smooth thing. It rises from unity at direct current to some plateau above the band, and it has no way of knowing that the band stopped at 0.4 of the clock — nothing in its two components encodes a band edge. So the worst image, at $1-r$ of the clock, is multiplied by whatever the shelf is doing there, and what it is doing there is most of the way to its plateau.

The right-hand column is the whole essay. The droop varies by a factor of six and a half down that table; the price of correcting it in analogue varies by a factor of 1.4, and in the opposite direction. A shelf built to remove half a decibel of droop raises the worst image by nearly five, and a shelf built to remove nearly four decibels raises it by three and a third.

That is the structural difference between the two corrections and it is not a matter of degree. The digital price tracks the disease exactly; the analogue price is about four decibels whatever the disease is. Which means the analogue correction is at its worst where it looks most attractive — on a comfortably oversampled converter with a tiny droop, where somebody reaches for one passive shelf because the droop hardly seems worth a digital filter.

How well a zero and a pole can follow a reciprocal sinc #

The residual is worth a paragraph of its own, because it is a second edge and it is the kind this collection is built to draw.

$1/\mathrm{sinc}(\pi r)$ is not a shelf. It is unity at direct current, rises gently, and then goes to infinity at $r = 1$, where the sinc has its first null. A shelf built from one zero and one pole rises from unity to a finite plateau and stays there. The two curves agree over whatever interval the fit is taken across and must part company before the sinc’s null, so the quality of the fit is a statement about how much of the way to that null the band reaches.

A factor of two and a half in band edge costs a factor of forty in residual, which is the signature of fitting a finite thing to something with a pole in it. At a fifth of the clock the shelf is a correction; at forty-nine hundredths it is a gesture, and half a decibel of ripple has been introduced in the course of removing a 3.75 dB droop.

Compare what the digital route does with the same problem. A digital corrector is a sampled filter, and a sampled filter’s response is periodic in the clock — it has the same period as the thing it is correcting, so the reciprocal sinc and the corrector both live on the same circle and there is no pole to chase. A short symmetric filter of three or five taps tracks the reciprocal sinc to a few thousandths of a decibel across four fifths of the band, which is two orders better than anything a shelf does, with no components at all. The reason is not that digital filters are better; it is that the correction and the defect are the same kind of object.

That is the same distinction the corner that moved turns on from the other direction — an analogue prototype mapped onto a circle is not the filter that was designed, and the frequencies near half the clock are exactly where the mismatch lives.

Where that leaves the reconstruction filter #

The essay before this one measured what the hold gives the analogue filter: 25.6 dB of rejection at a twentieth of the clock, 5.38 at 0.35, 1.74 at 0.45, and nothing at half. Those numbers are a contribution to the filter’s budget and an analogue shelf takes most of the contribution back.

At a band edge of 0.3 the hold gives 7.36 dB and the shelf costs 4.23, so the net is 3.13 — the analogue correction has converted two-thirds of the hold’s free image rejection into flatness. At 0.2 the hold gives 12.3 dB and the shelf costs 4.70, leaving 7.6. At 0.45 the hold gives 1.74 and the shelf costs 3.58, so the net is negative: the corrected converter has a worst image above where an uncorrected one would have put it.

That last row is the one to carry. A wide-band converter with an analogue droop correction on it has worse image performance than one with no correction at all, and the analogue filter after it has to be a pole better to compensate — at whatever the band that closes with the order says a pole is worth across that transition, which costs a part, a tolerance and a group delay, and none of that appears anywhere near the discussion of the droop that started it.

The digital correction costs none of this. It multiplies the sampled spectrum, so it multiplies the images by the same factor it multiplies the band — but the images are copies of the corrected band and are attenuated afterwards by the hold at their own frequencies, exactly as before. The rejection figure does not move. What moves is only the level, and only by the droop.

The converters that have already chosen #

Two common architectures settle the question before a designer reaches it, and it is worth seeing why, because in both cases the reason is that the correction was free where it ended up.

A converter with an interpolating filter in front of it — which is nearly every audio converter and most instrumentation ones — already contains a digital filter whose whole job is to raise the sample rate. The droop correction is a modification of that filter’s coefficients and costs nothing at all: no extra taps, no extra arithmetic, no extra delay. The filter had to exist and its response had to be designed, so the reciprocal sinc goes into the design target beside everything else. That is why an oversampling converter’s data sheet quotes a passband flatness of a hundredth of a decibel and never mentions a hold at all.

A converter with no digital filter — a direct parallel converter driven from a processor, an arbitrary waveform generator writing samples at the output rate — has no such filter to modify, and adding one means adding a multiplier and a delay line to a data path that was chosen for its simplicity. There the shelf is the honest answer, and the price above is what it costs.

So the interesting case is the middle one: a converter running at two or three times the band, with a short filter available and a marginal reconstruction filter after it. That is where the two prices are genuinely comparable, where the band edge sits between 0.15 and 0.3 of the clock, and where the table above says the analogue shelf charges three to eight times what it buys. A designer in that position reaching for one passive network because the droop is “only a decibel” has made the most expensive of the three available choices, and has done so because the price is invoiced in a currency nobody was looking at.

What the two prices do not include #

That the shelf fitted here is the best analogue corrector there is. It is one zero and one pole, fitted by least squares in decibels across the band, which is what is actually built. More sections fit better in the band and are worse out of it, because a closer fit to a reciprocal sinc means a steeper rise, and the steeper rise is still rising at the image. The direction of the trade does not change with order; only the size of both numbers does.

That the digital correction is free of everything. It costs filter length and it costs the arithmetic that goes with it, which is the subject of the same filter, rounded twice rather than of this essay. What it does not cost is image rejection, and that is the comparison being made.

That the droop’s neutrality on signal-to-noise ratio survives every architecture. It holds for noise produced at or before the converter, which is quantisation noise and the reference’s. It does not hold for noise added after the hold — an output amplifier’s own noise sees no droop at all, so a converter whose floor is set by its output stage does lose ratio to the droop. The claim here is about the dominant case and is stated as such.

That either correction fixes the phase. The hold’s half-sample delay is untouched by both. It is exactly linear, so it costs no shape, but it is still there, and flat magnitude, unflat delay is the field where that distinction is the whole subject.

A shelf fitted, not quoted, and the level the digital route flattens to #

The shelf is fitted rather than quoted, by a two-parameter search in decibels over the band, and its residual is reported with every figure so that the flatness being claimed is measured rather than assumed.

The digital price is checked to be the droop identically, which is a statement about the flat level being the band edge’s, and would fail immediately if the correction were normalised at any other frequency.

The analogue price is checked across the whole slider as a ratio, because the claim is that it is nearly constant while the droop varies by a factor of six — a claim no single band width can support.

And the shelf is checked to do nothing at direct current, to within two hundredths of a decibel, which is where the droop does nothing either and is the one point the two curves must touch.

One defect, two bills, and neither of them the one expected #

The shape worth carrying out of this is that a correction’s price is not a property of the defect.

The droop is a single number — 20 log sinc(πr) — and the two ways of removing it charge quantities that are not even the same kind of thing. One charges output level, in proportion to the defect. The other charges image rejection, in an amount that is very nearly independent of the defect. A designer choosing between them is not choosing between a cheap fix and an expensive one; they are choosing which of two unrelated budgets to spend from, and the right answer depends entirely on which budget is tight.

For a converter with bits to spare and a marginal reconstruction filter, correct digitally. For one running at the edge of its dynamic range with a generous analogue filter after it, the shelf is the right choice — and it is the right choice for a reason that has nothing to do with the droop’s size.

What makes the comparison possible at all is that both prices are measurable in decibels on the same axis, which is the only reason the table above means anything. Two budgets quoted in different units would have left the choice to taste, and the habit of putting everything on one solved axis is what turns it into arithmetic.

Still open: the band edge as the free parameter, and the correction with a null in it #

Both prices are functions of $r$ and $r$ is a choice. Every number in this essay is a function of the band edge as a fraction of the clock, and a converter run faster than the signal needs moves $r$ down, which is the arrangement what the filter in front costs prices on the way in. The droop falls, the image separation rises, and the analogue shelf’s price falls too — but not at the same rate as the droop does, since the shelf’s price is nearly constant. Measuring the three against the oversampling ratio would say which of them oversampling is actually being bought for, and the answer is probably not the one in the marketing.

A corrector that knows where the band stopped. The shelf’s whole problem is that it is smooth and the band edge is not. A corrector with a transmission zero placed at the first image frequency would flatten the band and attenuate rather than boost at $1-r$ — which is an elliptic-shaped response rather than a shelf, costs two more components, and is exactly what a reconstruction filter with a deliberately shaped passband is. Whether the combined part is cheaper than a flat filter plus a separate shelf is a question about component count that the two prices here make answerable.

And what a return-to-zero converter does to both bills. A hold shorter than a clock period droops less and rejects less, so it needs less correction and has less rejection to give away. Whether the net is better or worse is not obvious from either number alone, because the two effects move the same direction for one correction and opposite directions for the other.