The ceiling is not at the output

A floor and a ceiling put a floor and a ceiling on one amplitude axis and called the distance between them the dynamic range. Both ends were computed rather than quoted, which was the point of the essay, and both were computed at the same place: the input of one stage, which is also where its output is measured.

One stage has one place. A realised filter has ten or fifteen, and only one of them has a connector on it.

Every internal node has to stay inside the supply too. That sentence is obvious as soon as it is written down and it is not in the arithmetic above, because the arithmetic above has nowhere to put it. What follows is what happens when it is put in: the ceiling turns out to belong to a node the signal passes through and the floor to belong to the output, the two are measured on different parts of the circuit, and the difference between the range a filter has and the range an instrument reports runs to twenty-two decibels.

What is being solved, and where each end comes from #

The floor is the per-resistor budget: each resistor’s node split, a source of √(4kTR) put in series with it, the network re-solved, and the contributions integrated and summed in quadrature at the output. That end is unchanged and it is genuinely an output quantity — noise is what comes out.

The ceiling is not. The network is solved at every frequency on the same grid, every node’s magnitude is recorded, and each node keeps the largest value it ever reaches anywhere in the sweep. Relative to the input, so that one is the line a reader can find: a node above it clips before the input does, and on a ±15 V supply a node at 4.5 times the input means the input may be driven to 3.3 V and no further.

The usable range is then a ratio of two numbers taken from two different nodes, which is an unusual enough object to be worth stating plainly. Nothing in it is a property of a terminal pair.

The node that binds is the capacitor node of the highest-Q section, and the mechanism is the one where the Q comes from is about. A second-order section is flat at its own resonance at the output and its interior is not: the series arm and the shunt arm carry Q times the drive between them, and the section’s job is to have that cancel by the time it reaches its terminal. Cancellation at the terminal is not cancellation inside.

What the nodes are #

The realisation is worth naming once, because the node labels are otherwise arbitrary and the argument is about which of them the reader would have thought to probe.

Each conjugate pole pair becomes a resistor, an inductor and a capacitor in series from the previous section’s output to ground, with the response taken across the capacitor and handed to an ideal follower. So a section contributes three interior nodes: the junction of the resistor and the inductor, the junction of the inductor and the capacitor, and the follower’s own output. The second of those is the one across which the section’s response is defined, and it is the one that carries the resonant rise, because at resonance the reactive arm’s two halves are large and opposite and the resistor is what limits them.

On the fifth-order Chebyshev the three sections have quality factors of 4.545, 1.178 and — for the lone real pole — none at all, at natural frequencies of 960.8 Hz, 651.9 Hz and 342.1 Hz. The resistances go the other way: 364 Ω, 2.07 kΩ, 4.65 kΩ, because a section’s resistance is √(L/C)/Q and the highest-Q section has the smallest of them.

That inversion is the whole reason both ends have to be computed. The section with the smallest resistor contributes least to the floor and sets the ceiling; the section with the largest resistor contributes most to the floor and never comes near the ceiling. A quantity read at the output sees the sum of the first effect and none of the second.

The band that sets it #

A ceiling that is met at one frequency is not the same object as a ceiling that is met everywhere, and the sweep says which this is.

F0b reaches its 4.537 at 962 Hz, which is its own section’s natural frequency of 960.8 and not the filter’s corner of 1000. And it is the largest node in the network over only 4.2 per cent of the frequency grid — some fifteen points out of three hundred and sixty across six decades, a quarter of a decade wide. Everywhere else in the sweep the largest signal in the circuit is at the input or at the output.

So the ceiling above is the ceiling for a signal that contains that quarter-decade. A filter driven by a signal whose spectrum stops below 800 Hz never meets it, and its usable range really is the number an instrument at the output reports. That is a genuine qualification rather than a hedge, and it is the kind this collection is obliged to state: the boundary here is a band as well as a level.

It is also order-dependent in a way that runs against intuition. At order three the binding node leads over 51.2 per cent of the sweep; at five it is 4.2 per cent and at six 2.5. A higher-order filter has a taller internal peak that is met over a narrower band, so the ceiling gets worse and rarer at the same time — which is exactly the shape of a limit that gets found in the field rather than on the bench.

The family that has none of it #

The claim is not that every filter does this. It is that nothing about a reading at the output says whether this one does, and the cleanest evidence for that is a family where the reading is right.

The reason is that a Bessel’s sections have no Q worth the name. Its poles sit far from the imaginary axis by construction — that is what flat delay, bought with more delay is buying — so no section has an interior resonance to peak at. It is the same property, read on a different quantity, that makes a Bessel’s skirt gentle and its noise total the highest of the three families.

So the three families rank one way on selectivity, the other way on internal headroom, and a third way on their noise. Three families, one corner compares them at the same −3 dB point and what a steep skirt costs prices the selectivity in overshoot and delay. This is a third price on the same trade and it is the largest of the three.

The node moves #

If the binding node were always the same one, the whole finding would collapse into a correction factor: measure at the output, subtract a family-and-order table, and carry on. It is not always the same one.

At orders three through six the binding node is F0b, in the section with the highest Q. At seven it is F1b, in the section with the second highest, because the first section’s own attenuation before its peak has by then grown enough to matter and the second section is driven by a signal that has already been shaped. Which node binds is therefore a property of the whole chain rather than of the worst section, and it is exactly the kind of thing that has to be solved rather than reasoned about.

Fifteen filters, and the reading that does not move #

The two ends behave completely differently across a family of designs, and putting them on one axis is what makes the point.

A quantity that barely moves across fifteen genuinely different designs is not measuring the thing those designs differ in. That is the whole objection to the output reading, and it is worth separating from the objection that it is optimistic. Being optimistic by a known amount would be tolerable; being nearly constant while the truth moves by twenty-two decibels means the reading contains almost no information about the quantity it is named after.

The floor is what it is measuring, and the floors are alike for a reason that the resistor the noise comes from makes explicit: these filters are all normalised to the same corner and built at the same capacitance, so their resistances land in the same decade and their noise bandwidths are all within a few per cent of the corner. Three hundred nanovolts, give or take, fifteen times.

The ceilings are not alike, and they are not alike in a way that no summary statistic recovers. Bessel’s is at the output at every order. Butterworth’s rises smoothly with order — 1.155, 1.413, 1.701, 2.000 and 2.301 times the input at orders three to seven — which is nearly a straight line in the order and would tempt anyone into a rule. Chebyshev’s rises far faster and not smoothly: 1.784, 2.974, 4.537, 6.530, 13.257, where the last step is a doubling rather than the forty per cent the four before it average. A table of five numbers with one of them twice what the trend predicts is not a table anyone should extrapolate, and the reason for the outlier is the node moving rather than the peak growing.

The two ends move in opposite directions #

The most useful consequence is a design decision that comes out backwards if only one end is computed.

The rung below this one found that writing a cascade’s sections in a different order changes its noise and not its response, because a section’s noise is filtered by everything after it. The same reordering changes the ceiling, and in the other direction: putting the high-Q section first buries its noise under the rest of the chain and puts its resonant rise where the signal is still at full size.

Three decibels is not a large number and the direction is the whole content of it. A design that computed the floor, found the quieter arrangement and stopped there has chosen the narrower circuit, and it has done so on a calculation that was correct about everything it measured.

The comparison grows with order and keeps its sign: 5.45 dB at four, 3.08 at five, 7.34 at six and 7.82 at seven, every one of them in favour of the arrangement with more noise in it. It is not monotone, because the two ends are made of different things and neither moves smoothly, and that is itself worth noticing — a trade whose sign is stable and whose size is not is a trade that has to be computed per design rather than settled once.

What moves which end #

The two ends do not respond to the same design moves, and knowing which moves which is most of what the pair of readings is for.

Impedance scaling moves the floor and does not touch the ceiling — at all. Dividing one section’s resistance by s and multiplying its capacitance by the same s leaves the section’s transfer function identical, which is why the resistor the noise comes from can use it to buy noise down. It leaves the interior ratio identical too: the worst node of the fifth-order Chebyshev sits at 4.537145 times the input at scalings of one, four, sixteen and a hundred, to six figures, while the floor falls from 249.7 nV to 238.2. A ratio of two node voltages inside one section knows nothing about the impedance level the section is built at.

Section ordering moves both, in opposite directions, which is the trade the next section prices.

And the family moves both in the same direction, which is why the family choice dominates everything else here: a Bessel is quieter at the ceiling and louder at the floor than a Chebyshev of the same order, and the ceiling term is much the larger of the two.

That gives a clean order of operations. Choose the family for the response, take whichever ceiling it imposes, then choose the ordering on the difference of the two ends rather than on the floor, and only then spend impedance on the floor — because impedance is the one move that cannot make the ceiling worse.

Where else a ceiling belongs to another node #

This is not a fact about filters. It is a fact about any circuit whose signal passes through somewhere before it arrives, and the collection already has several.

The active ladder is the largest case and it has already been measured from one side. Eight amplifiers, and what they add finds an active realisation’s floor rising as the square root of its impedance scale and its worst internal swing rising as the scale itself, so the two ends close on each other and the range has a maximum somewhere in between. That is this page’s arithmetic applied to a realisation whose internal nodes carry the inductor’s current rather than the signal’s, and the swing there is a good deal worse than 4.5 times.

The probe is the smallest. The probe that takes a tenth divides by ten at the tip, so the instrument’s own front end sees a tenth of the signal and its floor is referred back through the same ten — a ceiling at one node and a floor at another, on a circuit with four components in it.

And a bridge has the property by construction: its arms carry the excitation and its output carries the difference, so the largest signal in it is never the one being measured. The rejection four resistors decide is about what the arms do to the difference, and the same four resistors decide how much headroom the arms have left.

What it does not say #

It does not say the range is 132.9 dB in any circuit anybody builds. Every amplifier in these realisations is a nullor: infinite gain, infinite bandwidth, no noise, and — the relevant one here — no output swing limit of its own beyond the rail it is being compared against. A real follower stops being linear well before its supply, which how small is small signal measures on the device and the constant that is a window collects as a band rather than a number, so the ceiling above is an upper bound and the ceiling below it is nearer.

Nor is the floor complete. It is resistors only, and the floor a circuit has measures what a real amplifier adds — 4 nV/√Hz in series with its input and 0.6 pA/√Hz across it — while eight amplifiers, and what they add puts them into a ladder and reads the total again. Both ends of every range on this page are therefore optimistic, and they are optimistic by different amounts, so the difference between the two readings is the durable part rather than either number.

The sweep is the last assumption and it is the ordinary one. A node’s worst value is the largest it reaches at any point on a logarithmic grid, so a peak narrower than a grid interval is under-read. The sections here have quality factors between 0.5 and 8 and their peaks are decades wide in comparison; a bank of high-Q resonators would need a grid with points where the circuit has them, which is the same discipline the noise integral needs and for the same reason.

The number worth carrying #

Twenty-two decibels, at a node with no connector on it. That is the worst of the fifteen, and the useful version of it is the spread rather than the worst case: the reading at the output moves by 3.97 dB across three families and five orders, and the thing it is named after moves by 22.34.

The second number to carry is smaller and does more work: 4.2 per cent of the sweep. The ceiling is met in a quarter-decade band around one section’s own resonance, so the difference between the two readings is a statement about a signal as much as about a circuit, and a filter whose input never contains that band really does have the range its output reports.

The habit is to ask where each end of a range was measured before believing the subtraction. A floor is an output quantity and a ceiling is not, and the two being quoted in the same decibels hides that they came from different parts of the circuit. On a filter the ceiling is usually at a capacitor node inside the highest-Q section; on a seventh-order one it moves to the second; and on a family with no internal resonance it is at the output after all, which is the case that makes the reading look trustworthy on the days it happens to be right.