The floor every filter has

What the filter in front costs prices an anti-alias filter in clock rate rather than in decibels, and the row of its table with no bar in it is the one that essay is proudest of. At a selectivity of 0.8 an eighth-order elliptic design’s stopband floor is at −74.1 dB, the requirement is −80, and there is no frequency at which the response satisfies it — so no sample rate does either, and the figure returns a refusal rather than a large number.

The essay is careful to say what the refusal is not. It is not a claim that order cannot help, because the degree equation deepens the floor with order. And the distinction it draws is between two kinds of design:

The all-pole design falls at 48.2 dB per octave, exactly eight times 6.02, so every doubling of the clock is worth another 48 dB and a faster clock is always an available answer. The elliptic design moves by 4.8 dB across three octaves — which is to say, not at all.

That sentence is true of a set of poles and false of a circuit.

Ten femtofarads, and what it is #

A hundredth of a picofarad is a small number and it is worth saying what physical thing it is, because the instinct is to dismiss it.

Two parallel tracks a millimetre apart on a board have a few tenths of a picofarad per centimetre between them. Two pins of a connector have a picofarad or two. The two ends of a small filter board, facing each other across it, have some tens of femtofarads. An amplifier’s own input-to-output capacitance, through its package, is tenths of a picofarad. Ten femtofarads is therefore not a careful layout’s stray; it is roughly the best a careful layout achieves, and a careless one has ten or a hundred times it.

Put it from the filter’s input to the node feeding its output buffer and the response has a floor at exactly

$$\frac{C_\mathrm{stray}}{C_\mathrm{shunt}}$$

because above every pole the shunt capacitance is the whole of what is at that node, and the stray is a divider against it. Ten femtofarads against ten nanofarads is a part in a million, which is a hundred and twenty decibels, and the arithmetic has no order in it and no frequency in it:

Twenty decibels per decade of stray, exactly, because the floor is a ratio of two capacitances. And the last column is the size of the discrepancy: at a hundred and thirty-six kilohertz times ten, the poles promise two hundred and ninety-three decibels and the circuit delivers a hundred and twenty.

Why it is a floor and not a slope #

The floor is flat — measured at three, ten, thirty and a hundred times the frequency it arrives at, the response varies by less than a decibel and a half. That is a stronger statement than a rising response would have been and it is worth saying why, because the first version of this measurement required a rise and was refused.

The intuition behind the rise is that a parasitic gets worse with frequency, and for many parasitics it does. The capacitor that is an inductor measures an impedance that follows $1/2\pi fC$ for four decades and then turns round and climbs, ninety-nine times what its capacitance predicts a decade above its own resonance — and applying that here, by putting a nanohenry in series with every shunt capacitor, does produce a floor. At minus one thousand and thirty-six decibels, which is arithmetic rather than physics: each section’s own series inductance is 6.3 millihenries at these element values, so a nanohenry against it is a part in six million, and eight of those in cascade is a floor no measurement will ever reach.

The stray from input to output is different in kind because it bypasses the whole filter rather than one section, and because both the stray and the shunt it divides against are capacitances. Their ratio has no frequency in it. So the response falls at forty-eight decibels an octave until it meets the ratio and then stops, exactly, for ever — which is the same shape as an equiripple stopband and is the reason this essay exists.

What this does to the requirement #

That essay’s requirement is that the filter be down to a stated level by the frequency that folds back, and the level is usually the converter’s own quantisation floor: about 74 decibels for a twelve-bit part, 98 for sixteen and 122 for twenty.

Read those three numbers against the table above.

Twelve bits asks for 74 decibels. Every stray in the table clears it, and the design is a design problem rather than a layout one. This is the case that essay’s arithmetic describes correctly.

Sixteen bits asks for 98. A hundred femtofarads of stray leaves a floor of a hundred decibels — two decibels of margin, before component tolerance — so the layout has become a term in the specification. The converter’s own floor is the one the floor a converter sets computes, and it crosses the source resistance’s thermal noise at 18.8 bits — so the range in which a filter’s stopband requirement is the binding one is the range this essay’s floor sits in. A board whose stray is a tenth of a picofarad meets a sixteen-bit requirement and one whose stray is two tenths does not, with the same schematic and the same components.

Twenty bits asks for 122. The floor at ten femtofarads is a hundred and twenty decibels, so a twenty-bit anti-alias requirement is not met by any layout in the table, at any order and at any sample rate. The filter that satisfies it on paper does not exist on a board, and the reason is not a tolerance — it is a capacitance nobody drew.

That is the boundary this essay adds to that essay’s, and it is the same shape as that essay’s own elliptic refusal wearing different clothes. There the floor was chosen, by picking a selectivity; here it is inherited, by building the thing. Both are levels, both are impassable by a faster clock, and only the first is in any design equation.

Two remedies and one that does not work #

A faster clock does not work, and that is the whole point. Above the frequency where the floor is reached the response does not improve, so raising the sample rate moves the fold-back frequency into a region where the filter has stopped attenuating. Every doubling is worth nought decibels rather than forty-eight.

A higher order does not work either, which is less obvious. The floor is $C_\mathrm{stray}/C_\mathrm{shunt}$ and neither term is a function of the order — adding sections adds poles in front of a divider that is already the limit. What a higher order changes is the frequency at which the floor is reached, and it moves it down: a steeper skirt meets the floor sooner. So more order brings the useless region closer to the passband.

Splitting the filter works. Two filters of half the order each, with a buffer and a physical gap between them, have two strays in cascade — so the floor is the product of two ratios rather than one, which is twice as many decibels. Twenty femtofarads twice is −108 dB per half and −216 dB together, against −114 dB for forty femtofarads once. That is the reason a high-attenuation filter is built as two boxes with a shielded wall between them, and the reason is arithmetic rather than tradition.

And shielding works, by making the stray smaller. A ground plane between the input and output tracks does not remove the capacitance; it terminates it, so the input’s stray charge goes to ground rather than to the output. That is the same mechanism where an open switch leaks to measures for a switch’s own feed-through, and the numbers are of the same size because the geometry is.

The measurement that would find it, and the one that would not #

A filter’s stopband is normally verified with a swept sinusoid, and there is a reason this floor survives that measurement on most benches.

To see a hundred and twenty decibels of attenuation an instrument needs a hundred and twenty decibels of dynamic range at the frequency in question, and it needs the signal path round the instrument to be quieter than that. A generator’s output and an analyser’s input, on the same bench, with a metre of cable each, have a stray between them of exactly the kind this essay is about — so an attempt to measure a −120 dB floor frequently measures the bench’s own feed-through instead, and reads a floor that is a property of the setup.

Three consequences, and the third is the useful one.

The measured floor is the worse of the two, so a bench with −100 dB of its own feed-through reports −100 dB for every filter, and a designer comparing two filters on it learns nothing about either.

The two are distinguishable by disconnecting the filter. Replace it with a short piece of wire and the reading is the filter’s; replace it with nothing at all — input and output both terminated, the filter absent — and the reading is the bench’s. That is the same two-measurement discipline the probe is part of the circuit sets out for a probe: measure the instrument’s own contribution by removing the thing being measured.

And the floor is easier to see in the time domain. A step into the filter’s input produces, at the output, an immediate spike through the stray followed by the filter’s own settling — and the spike’s height is the stray divided by the shunt capacitance, which is the floor. That measurement needs a hundred and twenty decibels of vertical range rather than of noise floor, which an oscilloscope has none of, but it needs only one instrument and no swept dynamic range at all. What it shows is a feature at $t=0$ that no transfer function with eight poles in it can produce, and an eight-pole filter’s step response has nothing at $t=0$ at all.

What the elliptic design’s floor has that this one does not #

The two floors are the same kind of limit and they differ in one respect that matters for design, so it is worth putting them side by side.

The zeros that buy an order establishes that an elliptic section’s transmission zero and its stopband floor are one choice: asking for a zero close to the passband is asking for a shallow floor. The floor is therefore a design parameter — computable before anything is built, tradeable against transition width, and deepened by adding order.

The stray’s floor is none of those. It is not in the design, it is not tradeable against anything in the design, it cannot be deepened by order, and it is not known until the board exists. What it can be traded against is geometry: a larger board, a shield, a split into two enclosures.

So the honest statement of that essay’s asymmetry, corrected, is not that all-pole designs have no floor. It is that an all-pole design’s floor is in a different document from its design — and the document it is in is the layout, which the schematic does not contain. That is these essays’ standing claim about schematics arriving in the place where a filter designer is least expecting it: not in the component values, which are exact, but in the two numbers the drawing has no notation for, which are the distance between the input and the output and what is between them.

Where this leaves that essay’s table #

That essay’s central table prices five eighth-order designs in clock rate for an eighty-decibel requirement: Bessel 3.53× Nyquist, Butterworth 2.08, Chebyshev 1.53, elliptic at a selectivity of 0.5 1.34×, and elliptic at 0.8 refused.

Eighty decibels is the number to read against this essay, and it is the favourable case. Ten femtofarads of stray gives a floor of a hundred and twenty decibels, forty below the requirement, so every row of that table stands — and the refused row is still refused, for its own reason.

Raise the requirement to the ninety-eight decibels a sixteen-bit converter implies and the table changes character. The floor at a hundred femtofarads is a hundred decibels, so an ordinary layout has two decibels of margin on a requirement that the design meets by a hundred. Which of the five families is chosen then matters much less than it did: all four reachable ones clear ninety-eight decibels comfortably on paper and all four are within a couple of decibels of the same floor on a board, so the clock-rate ordering the essay below computes — a factor of 2.3 between Bessel and Chebyshev — is being paid for a distinction the layout has erased.

That is worth stating as a design rule rather than as an irony. Beyond about a hundred decibels, the filter family stops being the variable and the construction becomes it. That essay’s three currencies — clock rate, delay flatness, noise bandwidth — are all properties of the poles, and they rank the families differently, which is why it draws all three. A fourth currency is a property of the box, and past a threshold it dominates all three.

The threshold is computable from one number a designer already has. Given a stray $C_s$ and a shunt capacitance $C$, the design is family-limited while the requirement is below $20\log_{10}(C/C_s)$ decibels and layout-limited above it. For ten nanofarads and ten femtofarads that is a hundred and twenty; for one nanofarad and a hundred femtofarads it is eighty, which is that essay’s own requirement — so a filter built at a tenth the impedance level on a careless board is layout-limited at the requirement that essay is about.

What this measurement leaves out #

One stray, in one place. A real filter has a stray from every node to every other, and the input-to-output one is only the largest term. The others produce floors of their own at higher levels, because they bypass fewer poles, and the total is their sum rather than the largest of them.

The buffers are ideal. The realisation puts an ideal buffer between sections, and a real amplifier has its own input-to-output capacitance and its own output impedance rising with frequency. Both add to the feed-through and neither is here. The band that does not close measures what an amplifier’s fifty ohms and two picofarads of stray do to the realisability of a cascade, and finds no eighth-order design at all at a single impedance level.

And the passband is unaffected, which is worth stating because it is why this is invisible. A part in a million of feed-through changes a passband response by a part in a million. Every measurement a filter is normally checked by — ripple, corner frequency, group delay, step response — is exactly right on a filter whose stopband is a hundred decibels short of its design. Three families, one corner measures every one of those on the same realised networks and none of them would notice.

Still open: the strays that are not one stray, and the split that doubles the decibels #

Every stray at once. The floor measured here is from one capacitance across the whole filter. Putting a stray between every pair of nodes — which is what a board has — gives a set of floors at different levels, and their sum is the real one. The arithmetic is a full capacitance matrix and the useful output is which pair dominates, because that is the pair a layout has to separate.

The split, measured rather than argued. The claim that two half-order filters with two strays give twice the decibels follows from the ratios multiplying and is not drawn. What it needs is the buffer between them to be real, since the whole benefit depends on the intermediate node being driven from a low impedance — and a buffer whose output impedance rises with frequency reintroduces a path.

And the floor of a passive ladder. Everything here is a buffered cascade, which has one stray path across it. A doubly terminated ladder of inductors and capacitors has no buffers, so its input and output are joined through those very impedances, and a stray across it is in parallel with a network rather than with a divider. Whether that gives a floor at the same level, or a different shape entirely, is a question the filters field’s own ladder-network tools could answer and this essay cannot.

What is checked #

The floor’s level is required to be the ratio of the two capacitances, twenty decibels per decade of stray across five values, to a decibel and a half. That is the closed form and it is what says the floor is a layout rather than a design.

The floor is required to be flat — three, ten, thirty and a hundred times the frequency it arrives at, within a decibel and a half of each other. The first version required a rise, on the reasoning that a parasitic worsens with frequency, and the refusal is the useful kind: it named the mechanism, which is that a ratio of two capacitances has no frequency in it.

The arrival frequency is bisected, not searched for as a minimum. The response is monotone down and then flat, so its minimum over a sweep is wherever the sweep stopped — which is how this first reported 60 gigahertz for a filter that stops improving at 136 kilohertz.

And the discrepancy is measured a decade past the arrival rather than at it, because at the arrival the two curves have only just parted, by definition. At the arrival they differ by thirteen decibels and a decade later by a hundred and seventy-three, and the second is the number a design is wrong by.