The ceiling a gain gives back

The ceiling is not at the output found that a filter’s dynamic range is a ceiling at one node over a floor at another. A fifth-order Chebyshev’s noisiest node is its output, and its largest signal is not: the first section’s capacitor node, F0b, carries 4.537 times the input at its own resonance, so on a ±15 V supply the filter clips at a third of a volt of input before anything at the output looks large. The range it actually has is 142.44 decibels; an instrument on the output would report 155.57. Thirteen decibels of range, one node nobody probes.

That essay left the obvious repair unmeasured, and the repair is a standard one. If one section’s node is too large, divide the signal in front of it and multiply it back afterwards, so that every node in the cascade peaks at the same level — the input’s — and the overall response is untouched. Filter-design texts call it equal-maxima scaling and recommend it as a matter of course. The question is how much of the thirteen decibels it gives back, and the answer is a good deal less than the recommendation implies.

Every node at the input’s level #

The cascade is the realisation of the earlier essays: three sections — a resonant pair of Q 4.54, a pair of Q 1.18 and a real pole — each a series resistor, inductor and capacitor with an ideal follower after it, from a one-kilohm-scale design at one kilohertz. Into it go four gains: one at the input and one in front of each later section, chosen so that the largest signal anywhere inside each section equals the input, and one at the output that restores the overall response.

Unscaled, the largest signal in each section is 4.537, 3.109 and 1.000 times the input. The second section is loud too — its input is the first section’s resonant output, and the node between its own resistor and inductor carries more than three times the input. The scaling gains that put every section at the input’s level are 0.220 at the input, 1.000 and 1.602 in front of the second and third sections, and 2.833 at the output. Their product is one, and the scaled cascade’s response at a kilohertz matches the unscaled one to the last digits.

The ceiling has moved exactly as intended: no node now carries more than the input. The floor has moved too, from 249.7 to 883.8 nanovolts, and the range — ceiling over floor — has gained 2.16 decibels of the 13.14 that the first section’s node was costing. Eleven decibels of the problem remain.

Whose noise the gains multiply #

The reason is in what the gains do to each section’s own noise, and the solve can answer section by section by carrying every resistor’s Johnson noise to the output through its own transfer function.

Each section’s noise is generated inside it and reaches the output through every gain that follows. The gains after the first section multiply to 4.537 — the factor by which its signal was divided at the input — so its noise rises from 75.0 to 340.2 nanovolts. The second section’s input is still divided by the same factor, because its own largest node needed no further division, and its noise rises by the same 4.537, 129.4 to 587.1. The third section’s signal was divided by 2.833, and its noise is multiplied by exactly that, 199.9 to 566.3.

So the trade is exact, and it is unfavourable whenever the loud section is early. Dividing the signal where it is largest divides it for every section until a gain restores it, and each of those sections then has its noise amplified by the gain that restores it. The ceiling falls by 4.537 at the one node that set it; the floor rises by a weighted mixture of 4.537 and 2.833 over every section. The net is the 2.16 decibels. The resistor the noise comes from found a resistor’s contribution set by everything after it and nothing before it; scaling puts a gain after every resistor in the cascade, and that is why it costs so much.

Fifteen filters, and the fraction given back #

The Chebyshev 5 is one filter. Across three families and five orders the same scaling recovers very different fractions of very different costs.

The five Bessel filters have no internal peak — their gentle poles put no node above the input — and sit at the origin with nothing to recover. The Butterworths have modest peaks, costing a decibel and a quarter at order three and seven decibels at order seven, and scaling recovers none of it at orders three and four and between one and two decibels at five to seven. The Chebyshevs are where the peaks are, and the recovery is erratic: nothing of 5.03 decibels at order three, half a decibel lost at orders four and six, 2.16 of 13.14 at five and 3.77 of 22.45 at seven.

The even orders lose because of their output. A Chebyshev of even order with half a decibel of ripple has a passband that peaks 0.5 decibels above its direct-current gain, so its output itself carries 1.059 times the input; scaling every internal node to the input’s level does not touch that, the output becomes the ceiling at 1.059, and the floor has risen anyway. The odd orders end in a real pole whose largest node is the input’s own level, and their output does not ripple above the input, so the ceiling after scaling really is the input; whether that is the whole of the difference between odd and even, these figures do not separate. In no case does scaling give back more than about a quarter of what the peak was costing — the Butterworth 7’s 1.92 of 7.24 decibels is the best fraction, and the Chebyshev 7’s 3.77 of 22.45 the best absolute amount.

Ordering and scaling together #

The earlier essays found that the order of the sections matters too: putting the high-Q section first exposes its resonant rise to the full input, and putting it last lets the earlier sections’ attenuation shrink the signal it rises from. Ordering and scaling are both rearrangements that leave the response alone, and they can be combined.

Unscaled, the six orders span 3.08 decibels, from the design order’s 142.44 to 145.52 with the Q-1.18 pair first, the resonant pair second and the real pole last. Scaled, they span 2.60, and the best is 147.19 decibels in the same order. Ordering and scaling are nearly independent improvements: each gives about the same increment whether or not the other has been made, and together they recover 4.75 decibels of the 13.14 the design order was losing.

That is the most this realisation can give back by rearrangement. What remains — about eight decibels — is the price of a pole pair with a Q of 4.54 in a realisation that holds its resonant energy at one node. No gain distribution can remove it, because the resonance itself is what stores the energy, and the energy is a voltage on a capacitor.

Where each node peaks #

Equal maxima is easy to misread as “every node has the same waveform”, and it is worth seeing that it is not.

After scaling, the three section outputs peak at 1.000, 0.624 and 0.353 times the input, and the filter’s output — the same response as before — at 1.000. The second and third sections’ outputs stay below the input’s level because their largest nodes are not their outputs: in each, the node between the resistor and the inductor swings further than the capacitor the follower reads. The scaling has brought those internal nodes to the input’s level, and they peak at different frequencies, the first near its own resonance and the later ones across the passband.

A designer probing a scaled filter with an oscilloscope would therefore see three section outputs at three different levels and might conclude the scaling was wrong. It is right; the nodes that bind are not the ones with followers on them. That is the earlier essay’s point again, one level down: the ceiling is not at the output of the filter, and within a section it is not at the section’s output either.

The range read three ways #

Put the numbers on rails, which is how a designer meets them.

On ±15 volts the output reading says 155.57 decibels, the unscaled filter has 142.44 and the scaled one 144.60. All three move twenty decibels a decade with the rail, because the ceiling is proportional to it and the floor is not, and the gaps between them are the same on every rail: 13.14 decibels lost to the internal node, 2.16 recovered by scaling. A designer who needs another ten decibels will find them in a higher rail — a factor of three — much sooner than in any gain distribution.

A floor and a ceiling set out the range as one number between two boundaries; these figures say that both boundaries are properties of where the signal and the noise are inside the network, and that moving one moves the other.

The same trade in arithmetic #

Equal-maxima scaling is recommended as a matter of course in one place where it really is close to free, and the contrast explains why it is so expensive here.

A digital filter computed in fixed-point arithmetic has a ceiling at every node — the largest number the word can hold, beyond which the arithmetic overflows and wraps round, which is a far worse failure than clipping — and a floor at every node, the rounding error each multiplication adds. Which section goes first measured both for a cascade of biquads: the round-off floor running from 24 to 100 least significant bits across the orderings, the largest value any section reaches over eighteen decibels. Scaling there is mandatory, because an overflow is a gross error, and the scaling factors are powers of two that a shift implements without any arithmetic at all.

What makes it cheaper there is where the noise is added. A rounding error is added at a fixed absolute size — half a least significant bit — at the output of each multiplier, after the signal has been scaled; a section whose input has been divided by four has the same rounding noise as before and a quarter of the signal, so its own signal-to-noise ratio falls by exactly the factor the scaling took, and nothing earlier in the chain is affected. The analogue cascade above has the same structure and one important difference: its noise is generated by resistors at a fixed absolute size too, but the gain that restores the signal amplifies every section’s noise behind it, not only the section that needed dividing. In a digital filter the designer can restore the level inside the next section’s coefficients, after its own rounding; in this cascade there is no “inside” to put the gain.

Six decibels a bit gives the exchange rate for both: the 13.14 decibels the Chebyshev 5’s internal node costs is 2.18 bits, and the 2.16 decibels scaling recovers is about a third of a bit. A designer choosing between a filter that loses two bits to its own resonance and a converter with two more bits than it needs has been told by these figures which of the two scaling can repair, and it is neither.

What eleven decibels is worth, and where else to find it #

The eleven decibels that scaling does not recover are best seen as a budget item rather than a defect. A filter that must pass a signal a third of a volt in amplitude without its first section clipping on fifteen-volt rails, and must stay above its own floor of a quarter of a microvolt, has the range it has; the question is only which part of the design pays.

There are four places to find decibels, and these essays have now measured each. A higher rail buys twenty decibels a decade, cleanly, at the cost of power and of amplifiers rated for it. Ordering buys three decibels on this filter by putting the loud section where the signal is already smaller. Scaling buys two, less with real gain stages. Impedance buys whatever current can be spent, by lowering the resistors’ noise in the sections whose noise the scaling multiplied. And the realisation itself — a ladder network that spreads its resonant energy across several nodes rather than holding it at one — is the only change that attacks the peak rather than redistributing it. The floor a converter sets made the same kind of argument for a converter: the resolution is a property of the circuit in front of it, and the place to spend is decided by which limit binds.

What the gains are, and what a real gain stage would add #

The scaled cascade is the unscaled netlist with four controlled sources inserted — one between the input and the first resistor, one between each follower and the next section’s resistor, one after the last follower — and every quantity is solved: peaks as the largest magnitude each node reaches over the sixty-points-a-decade budget grid, noise as each resistor’s Johnson density carried to the output by its own transfer and integrated over the same grid. The gains are chosen from the unscaled solve’s own peaks, and the scaled solve is then checked to put no internal node above the input and to reproduce the response.

The inserted gains are silent, as every amplifier in this field’s noise budgets is, and that flatters the scaling. A real gain of 2.833 is a non-inverting stage with two feedback resistors whose Johnson noise appears at its output multiplied by its gain, and an amplifier with a few nanovolts per root hertz of its own, which reaches the output multiplied by the stage’s noise gain rather than its signal gain — the arithmetic the gain added to buy a margin measured for a stage whose noise gain was raised on purpose. Both add to a floor that has already risen, and a divide-then-restore pair of stages pays for two such amplifiers to recover a ceiling a single node set. A gain of 0.220 is a resistive divider whose resistors add noise at the most sensitive point in the chain. Every one of the 2.16 decibels above is therefore an upper bound on what scaling buys, and on the even-order filters, where it bought nothing, real gain stages make it a loss.

The same filter a thousand times larger is the other knob, and a better one: scaling a section’s impedance down lowers its resistors’ noise without amplifying anyone’s, and it costs current rather than range. It applies section by section, which is exactly what the scaling here could not do — move one section’s floor without moving another’s.

Still open: noisy gains, impedance scaling per section, and a realisation that spreads the energy #

The gain stages with their own noise. A gain stage built from an amplifier and two resistors adds a floor proportional to its gain, and with that floor in the budget there is an optimum: some scaling that is less than equal-maxima, trading a little ceiling for a lot less floor. Its value for the Chebyshev 5, with amplifiers of a stated noise, is a one-parameter search on this netlist.

Impedance scaling as the companion knob. Lowering the impedance of the loud sections — the ones whose noise the scaling multiplies — by a factor costs current and lowers exactly the contributions that rose. Combining equal-maxima scaling with a per-section impedance scale chosen to cancel the floor’s rise would say how much of the remaining eight decibels a current budget can buy.

A realisation that does not hold the energy at one node. A leapfrog or doubly terminated ladder network realises the same Chebyshev with its resonant energy shared among several nodes, and its internal peaks are known to be smaller. Whether that realisation’s range at equal noise approaches the 155.57 decibels the output reading promised — and whether which section goes first has an analogue there — is the comparison between a cascade and a ladder network made on range rather than on sensitivity.