The ratio that does not walk to one

The bandwidth noise sees established the quantity: the width of the brick wall that would pass the same noise power as a given response, measured by integrating the solved network’s own squared magnitude rather than taken from the table it usually comes from. For a single pole it is 1.5706 times the −3 dB point and π/2 is 1.5708.

That essay also carried a sentence about what happens next, and the sentence is the one every account of the subject carries: the steeper the skirt, the closer the noise bandwidth gets to the −3 dB point, so the ratio walks down from π/2 towards one as poles are added. It was measured over five orders of one family and it was true there.

This page measures it over four families and eight orders, which is thirty-two realised networks and thirty-two integrals. One family does what the sentence says. The other three do three different other things, and one of them does not have the quantity at all.

What the ratio is for, and how much of the answer it is #

The quantity only matters because a noise voltage is √(density × bandwidth), and the density half is settled. The floor a resistor sets computes 4kTR exactly and checks it two ways that share no arithmetic; nothing about a resistor’s density depends on what is in front of it. The bandwidth half is where the whole of the design-dependence lives.

Because the voltage is the square root of the product, a ratio error of x is an error of √x in volts, which is the reason a factor of two in this quantity is worth arguing about and a factor of a few per cent is not. Two of the departures below are worth arguing about.

Using the corner frequency instead of the ratio — the mistake the rung below is named for — is 20.2 per cent low on a single pole and 0.32 per cent on an eighth-order Butterworth, so it very nearly repairs itself as the order rises. On a seventh-order Chebyshev with three decibels of ripple it is 17.9 per cent high instead, and on the eighth-order one 16.0 per cent low: the same mistake, on two filters one order apart, in opposite directions and both times by a sixth.

And the two adjacent orders differ from each other by √1.968 = 1.403 in volts. A design that moved from seven poles to eight to buy three decibels of stopband would find its noise floor forty per cent higher for a reason no part of the specification mentions.

What each entry is #

Two numbers, from the same network, on the same solve.

The noise bandwidth is ∫|H(f)|² df from zero upwards, normalised to the passband gain, taken on a logarithmic grid with the trapezoidal rule in linear frequency — the grid logarithmic because the response is, the rule linear because the integral is, and getting that pairing wrong is a silent factor of ln 10. The −3 dB point is found by bisection on the same solved response rather than taken from the design, because a realised eighth-order network is not exactly the polynomial it was derived from and the ratio is a ratio of two things that must be measured on the same object.

Both are properties of the built network. Every filter here is filterOf’s realisation: conjugate pole pairs become series-resonant sections with the output across the capacitor, a lone real pole becomes a resistor and a capacitor, transmission zeros become a capacitor across the inductor, and ideal followers separate the sections. It is the same machinery three families, one corner compares on the response and the resistor the noise comes from takes a noise budget of.

The integral’s upper limit is four decades above the corner. That is a decision rather than a detail, and the last third of this page is about what it hides.

The calibration, which is four filters that are one filter #

At order one there is no family. A Butterworth, a Chebyshev, a Bessel and an elliptic of order one are all a single real pole, because there is nothing about a first-order response for a family to differ in — no ripple to place, no delay to flatten, no zero to put anywhere.

So all four must return the same number, and the number must be π/2. They return 1.5706 and they agree with each other to better than two parts in a thousand.

That is worth more than it looks. Four independent pole-placement routines, four realisations, four bisections for the corner and four integrals, arriving at one closed-form constant, is a check on every part of the chain at once — and it is the only place in the table where an answer is known in advance. Everything below is quoted against it.

The table #

Butterworth’s column is the sentence everybody carries, and it is a good column: monotone, converging on one from above, and by order eight within two thirds of a per cent of a brick wall. A Butterworth of order eight really does pass almost exactly the noise its corner frequency suggests.

None of the other three columns is a version of that column with a different constant in it.

Bessel, which is least in the middle #

Bessel’s ratios run 1.5706, 1.1536, 1.0736, 1.0463, 1.0385, 1.0386, 1.0411, 1.0441 — down to order five and then up again, and still rising at 1.0468 and 1.0490 at orders nine and ten. A steeper Bessel passes more noise than a shallower one, past the fifth order, for the same nominal corner.

The mechanism is the one that makes a Bessel a Bessel. Its poles are placed to flatten the group delay, not to shape the magnitude, and as the order rises they spread along a locus that keeps the delay flat by giving the magnitude an increasingly gradual transition. The −3 dB point of a high-order Bessel therefore sits well inside a skirt that is still descending a long way past it, and the integral collects what is out there. Adding a pole steepens the far stopband, which contributes nothing, and softens the region just past the corner, which contributes almost everything.

What a steep skirt costs prices selectivity in overshoot and settling and flat delay, bought with more delay prices Bessel’s flatness in the delay it adds. This is a third price and it is one nobody would look for, because it is a cost that rises with the order in a quantity everything else says should fall.

Two more numbers put the size of it in context. Using Butterworth’s eighth-order ratio for a Bessel of the same order — which is what a table indexed by order rather than by family invites — is 1.85 per cent in volts, small enough to be ignored honestly. But the rise from Bessel’s own minimum at order five to its value at ten is only 0.50 per cent in volts, which means the finding here is a shape rather than a hazard: it matters because the direction is wrong, not because the number is large.

That distinction runs through the whole page. A departure that is wrong in size is a correction. A departure that is wrong in direction is a rule that has to be discarded, because it will be extrapolated, and every extrapolation of it will go the wrong way.

Chebyshev, which alternates #

The cause is where each parity starts. An odd-order Chebyshev leaves direct current at unity gain and dips into its ripple band; an even-order one starts at the bottom of the ripple and comes up to unity. The −3 dB point is measured from the passband, so the two parities put their corner in different places relative to the same shape, and the integral is divided by a different number.

The size of the effect is therefore the ripple, and the two branches only become disjoint above a ripple of 0.1968 dB, found by bisection on the pair that binds — the highest odd order against the lowest even one. Below that the branches interleave and the parity is invisible; above it a Chebyshev’s noise bandwidth cannot be read off its order at all without knowing whether the order is even.

That is the second thing on this page that is not a correction to Butterworth’s column but a different shape. And it is the one that most affects a design, because the zeros that buy an order and the selectivity that is not free both encourage exactly the move — one more pole, a little more ripple — that this figure prices in the opposite direction.

The elliptic, which has no ratio #

An elliptic filter places transmission zeros in the stopband, which is what buys its skirt: a fifth-order elliptic reaches a given attenuation at a frequency a seventh-order Butterworth needs. The zeros are what the zeros that buy an order is about, and the price named there is that the stopband comes back up between them.

For an odd order there are more poles than zeros, so the magnitude falls as 1/f at the top and the integral converges. For an even order there are exactly as many zeros as poles, the magnitude tends to a non-zero constant — 0.616 at order two, 0.0489 at four, 0.00311 at six, 0.000197 at eight — and ∫|H|² df over an infinite axis diverges. The quantity does not exist. What an integration routine returns for it is its own upper limit multiplied by that constant squared, and nothing else.

Convergence is a statement about the increments rather than about the value, which is what the figure measures: the share of the answer contributed by the last decade of the sweep. For the odd orders it is a few parts in a hundred thousand. For the even ones it is 90.0, 89.6 and 42.8 per cent. There is no third case, and there is no value of the integral that can be quoted with a qualification.

The one that looks convergent #

The most useful of those numbers is the smallest, because it is the one that would have got through.

The rate at which an even-order elliptic’s integral diverges is its own stopband floor squared, and that floor falls by roughly sixteen times for every two orders. So order two is unmistakable, order four is obvious, order six needs a sweep two decades wider than anyone would use, and order eight looks entirely convergent over any range this site’s solver can reach without becoming ill-conditioned.

A number that is stable to three figures over four decades of integration limit, and is nonetheless not a limit, is exactly the failure this collection was built to catch — a boundary is a model and a tolerance is about believing a number because it stopped moving, and here the thing that stopped moving is an artefact of where the sweep was cut off.

What the table is checked against #

Thirty-two integrals is enough machinery to be wrong somewhere invisible, so it is worth saying what the checks are, since they are the reason the odd results above are reported as results rather than as suspicions.

The first is the order-one row already described: four families, one filter, one closed-form constant. It exercises the pole placement, the realisation, the corner bisection and the integral in one comparison, and nothing else in the table has an answer known in advance.

The second is that every ratio is above zero and every all-pole family’s magnitude at the top of the integration range is falling at its own order — eight decades per decade for an eighth-order Butterworth, and the same measurement that reports zero for an even-order elliptic. That is what separates a family whose integral has a limit from one whose integral does not, and it is measured on the response rather than argued from the pole count.

The third is the normalisation. Every filter is bisected to its own −3 dB point rather than trusted to arrive at the one it was designed for, because recovering a high-order network’s response from its polynomial is not the same as solving the network: a fifth-order cascade’s determinant has roots spread far enough that a sampled recovery quietly returns a different filter, and an early version of this machinery reported a corner of 657 Hz for a filter normalised to a kilohertz. Everything on this page is measured on the realised netlist, which is the same discipline what a network answers, and how the answer is checked sets out for the responses themselves.

What none of the three can check is the one thing the last two sections are about, which is whether the integral being computed exists. That question is not answered by any comparison of two routes, because both routes stop somewhere.

What it does not say #

It does not say the quantity is useless. For a Butterworth of any order it is a well-behaved number with a limit and a monotone approach to it, and Butterworth is what most anti-alias filters are. For an odd-order Chebyshev or elliptic it is equally well-defined, and the values below one are real: those filters genuinely pass less noise than a brick wall at their own corner, for the reason the bandwidth noise sees sets out.

It says the ratio is a property of a design rather than of an order, and that three of the four families in this collection break the rule of thumb in three unrelated ways. There is no correction factor, because there is no single direction: 0.720 for a seventh-order Chebyshev at three decibels of ripple, 1.0490 for a tenth-order Bessel, and no number at all for an even-order elliptic.

Two things it deliberately leaves alone. The integrals stop four decades above the corner, and a real filter’s response does not continue to the model’s infinity either — every capacitor becomes an inductor somewhere, which the capacitor that is an inductor measures, and every amplifier runs out of gain, so the physical stopband floor of an even-order elliptic is not a constant to infinity but a shape with its own edge. That does not rescue the quantity; it replaces a divergent integral with one whose value is set by a parasitic, which is a worse position rather than a better one. And the realisation’s own followers are ideal, so nothing here includes the gain roll-off that would change the tail on a real board.

What a filter designer does with it #

The practical form of all of this is short, and it is not “use a better table”.

Integrate the design. The whole of this page costs one solve of the realised network over a few hundred frequencies, which is a fraction of what the design itself cost, and it returns the number for that design rather than for its family. Every value quoted here came out of that and nothing came out of a table.

Check the tail before believing the integral. The one question that separates a number from an artefact is whether the magnitude at the top of the range is still falling, and it is a single evaluation. An all-pole response falls at its own order and always converges. A response with as many zeros as poles does not, and that is decided by counting rather than by looking at the answer.

And take the parity seriously if there is ripple. A Chebyshev’s noise bandwidth cannot be interpolated between adjacent orders at any ripple above a fifth of a decibel, because the two parities are on separate branches. That is the one result here that will change a design decision rather than a datasheet entry: the extra pole that buys stopband can cost forty per cent of the noise floor, and the two effects are quoted in different documents.

The number worth carrying #

Only one column in four. Butterworth walks from 1.5706 to 1.0065 and does what the sentence says; Bessel bottoms at 1.0385 and turns round; Chebyshev alternates by up to a factor of two at adjacent orders; the even-order elliptic has no value.

The habit that goes with it is narrower than the finding. A ratio that has been checked on one family and then stated as a property of order is the commonest way a correct measurement becomes a wrong rule, and the check that catches it is cheap: run the same integral on every design the collection can build, and look at whether the answer moves when the thing it should not depend on is changed. Here the thing that should not have mattered was where the integral stopped, and for four of the thirty-two entries it was the whole answer.