What a steep skirt costs

Every filter design begins with the same negotiation. Something has to be passed, something has to be rejected, the two are close together in frequency, and the question is what it costs to separate them. The classical answer names three families and describes them with adjectives. This essay replaces the adjectives with three measurements, taken on the same solved networks.

The one thing that does not vary #

Before the differences, the similarity, because it bounds everything else.

Far above its corner, every all-pole filter of order n falls at 20 n decibels per decade. Five poles, a hundred decibels per decade; seven poles, a hundred and forty. That is not a property of the family — it is arithmetic, following from the fact that a polynomial of degree n in the denominator eventually dominates and the magnitude falls as the nth power of frequency.

So a filter’s asymptotic slope is bought entirely with components, at a fixed and non-negotiable rate, and choosing a different family buys none of it. What a family choice affects is what happens in the octave or two either side of the corner, where nothing is asymptotic yet, and that is exactly the region where filters are actually specified.

The three measurements #

Passband deviation is the peak-to-trough range of the magnitude across the band below 0.8 of the corner, in decibels. It is measured rather than taken from the ripple specification because the two are not the same thing: a Chebyshev’s ripple is its specification, but the Butterworth and Bessel also deviate — by drooping — and a comparison that reports one family’s ripple against the others’ zero is comparing a measurement with a definition.

Attenuation at three times the corner stands in for the whole stopband. Three is chosen because it is inside the region where the families differ and outside the corner itself, and because a transition ratio of about three is what a real specification tends to demand.

Group-delay variation is the ratio of the slowest to the fastest group delay across the same passband, as a percentage. This is the quantity the classical table leaves out entirely, and it is the one that decides whether a filter can be used on anything with edges in it.

What comes out #

At order five, with half a decibel of ripple allowed to the Chebyshev:

The middle column is the reason anybody accepts ripple: sixteen decibels, a factor of six, for nothing but a different arrangement of the same five poles. The right-hand column is what it costs, and the cost is not the ripple.

At order five the Chebyshev’s delay varies by 49% across the passband and the Bessel’s by 0.06% — a ratio of about eight hundred. At order seven it is 69% against under 0.01%. Those are not close calls, and they are invisible in every magnitude plot ever drawn of these filters.

The column that behaves strangely #

Reading the middle column against the order produces the expected picture: everything improves, Chebyshev fastest.

Reading the right-hand column against the order does not. The Bessel’s delay variation falls steadily — 2.3% at order three, 0.06% at five, below 0.01% at seven — which is the family converging on its own design goal. The other two do not converge on anything. The Butterworth sits between 37% and 48% at every order from three to eight; the Chebyshev wanders between 49% and 84% without a trend.

That non-monotonicity is real and it is worth pausing on, because it is the kind of thing a table of adjectives cannot express. Adding poles to a Chebyshev does not steadily worsen its delay; it moves the peak of the delay curve around relative to the band edge, and whether the ratio across the band goes up or down depends on where the peak lands. The measurement at order five happens to be a local best.

The practical reading is that delay behaviour cannot be inferred from the order, and a design that is acceptable at order five may not be at order six. Which is a strong argument for measuring rather than extrapolating, and it is the reason this figure has a slider rather than a caption.

Ripple is not the cost #

The commonest way this trade is taught puts ripple on one side and steepness on the other: accept half a decibel of ripple, receive sixteen decibels of stopband. That framing survives the measurements above but only just, and it is misleading in two ways.

First, the ripple is smaller than the alternative’s droop. The Chebyshev’s 0.50 dB is its specification; the Butterworth’s 0.44 dB is a measurement of the same band. They are the same size. The Chebyshev’s deviation is distributed as several equal wiggles rather than one downward slope, which is arguably better rather than worse — it means the response is within half a decibel of its nominal everywhere rather than progressively worse toward the edge.

Second, and more importantly, ripple is a magnitude effect and is therefore the cheap part of the cost. Anything downstream that cares about half a decibel can usually be told about it. Delay variation of fifty per cent is a phase effect, it cannot be compensated without another filter, and it changes the shape of a waveform rather than its size.

So the honest statement of the trade is: steepness is paid for in delay, and the ripple is a distraction. That is not the usual framing, and it comes directly from the third column being measured instead of omitted.

Order against family #

Since both the order and the family buy stopband attenuation, a natural question is which is the better currency, and the measurements answer it.

Going from order five to order seven buys the Butterworth 19 dB at three times the corner, at a cost of two more poles — which in a passive realisation means two more components and in an active one means another section. Going from Butterworth to Chebyshev at order five buys 16 dB for no components at all.

So the family change is close to free and the order increase is not, which explains why Chebyshev designs are common in anything cost-sensitive. What the free option costs, again, is delay: the order increase leaves the delay variation roughly where it was, and the family change nearly doubles it in the worst cases.

There is also a practical limit on order that the measurements do not show. Each additional pole pair sits closer to the imaginary axis in the same family — the poles fan out along the circle — so a high-order filter contains sections with high quality factors, and a high-quality section is sensitive to component tolerance in exactly the way a low-quality one is not. A tenth-order Chebyshev on paper is a filter whose passband shape depends on components that do not exist to that accuracy.

Where the trade comes from #

The three columns are not independent quantities that happen to move together. They are three readings of where the poles are, and seeing that makes the trade feel inevitable rather than empirical.

A steep transition requires a pole close to the imaginary axis, because the response near a pole falls off at a rate set by how close the driving frequency comes to it, and a pole near the axis is approached closely. That is the middle column.

A pole close to the imaginary axis is a lightly damped resonance, and a lightly damped resonance stores energy for many cycles before releasing it. That is a long delay, concentrated near the pole’s own frequency, which is near the band edge — the right-hand column.

And a lightly damped resonance has a peak in its magnitude, which shows in the passband as ripple unless it is cancelled by the neighbouring sections — the left-hand column.

So all three columns are the same measurement of the same pole, expressed in three units, and no arrangement separates them. A designer can choose where on the curve to sit; nobody can leave it, because the thing being traded is one geometric quantity.

The only escape is to stop using poles alone. A filter with transmission zeros — an elliptic response — achieves a steeper transition without moving its poles as close to the axis, by placing nulls just above the band edge instead. That buys a little on the trade rather than escaping it: the zeros make the transition steeper for a given pole placement, and the poles are still where the delay comes from. It is the reason elliptic filters are steeper than Chebyshev at the same order and still worse in delay.

What the measurements do not cover #

Two things are outside the scope of these figures and it is worth naming them so the numbers are not over-read.

The realisation is idealised. The sections here are separated by ideal followers, so each one’s response is its own. A passive ladder — the usual realisation for radio-frequency work — has its sections loading one another by design, and its element values come from a different synthesis. The response is the same, because both realise the same poles, but the sensitivity to component values is not.

The components are ideal. Every capacitor here is a capacitance and every inductor an inductance, with no series resistance and no self-resonance. A real filter’s stopband does not descend for ever: at high enough frequency each capacitor becomes an inductor and each inductor a capacitor, and the stopband floor stops falling and can come back up. That boundary is the subject of a different field in this collection, and it is the reason a real filter is always measured over a much wider band than it was designed for.

The specification a filter is usually given #

The three measurements in this essay are the ones that can be taken from a response. A specification is usually written the other way round, and translating between them is where the order comes from.

A filter is normally specified by four numbers: a passband edge with a maximum loss allowed there, and a stopband edge with a minimum attenuation required there. The ratio of the two edges is the transition ratio, and it, together with the attenuation, determines the minimum order for each family. For a Butterworth the relation is explicit enough to invert directly; for a Chebyshev it involves an inverse hyperbolic cosine and gives a smaller answer for the same requirement.

The reason to mention it here is what it does to the third column. A designer choosing the minimum order that meets a magnitude specification will, by construction, choose the family with the sharpest transition — and will therefore land on the largest delay variation available. The specification process actively selects for the property it does not mention.

That is not an argument for over-designing. It is an argument for the specification carrying a fourth requirement when the signal has edges in it, stated in the same terms as the others: a maximum delay variation across the passband, in per cent or in milliseconds. Once that is written down, the order and family fall out of four numbers instead of three, and the answer is often a higher order of a gentler family rather than the minimum order of a sharp one.

Component tolerance, which bounds the order from above #

There is a ceiling on order that none of the measurements above shows, and it is worth stating because it is what stops the middle column from being a free lunch.

Every additional pole pair in a family sits closer to the imaginary axis than the last, so its quality factor is higher. A section with a high quality factor is a sharp resonance, and a sharp resonance is sensitive to the components that set it: the fractional error in a section’s natural frequency is about half the fractional error in the product of its inductance and capacitance, and the error in its quality factor is the full fractional error in its resistance.

At order five with half a decibel of ripple, the highest-quality section has a quality factor near nine, and one per cent components move its peak by enough to be visible in the passband. At order nine it is above thirty, and components good to a tenth of a per cent are needed to get the designed response rather than an approximation to it.

So the practical ceiling on filter order is set not by the mathematics but by what can be bought, and it is lower for the sharp families than for the gentle ones — a Bessel of order nine is easy and a Chebyshev of order nine is a precision instrument. That is a third axis on the trade, it is invisible in every response plot, and it is the reason high-order filters are usually built digitally, where the coefficients are exact by construction.

The habit this figure encodes #

The figure at the top of this essay is nine measurements arranged as bars, and there is a reason it is not a table.

A table invites the reader to compare within a row and makes comparison across rows difficult, because the three quantities are in three units — decibels, decibels and per cent — and no reader holds three scales at once. Drawn as three panels of bars, each with its own scale and each labelled with what “better” means, the shape of the trade is visible without arithmetic: the Chebyshev’s bar is longest in the middle panel and longest in the bottom one, and that is the whole design decision.

What the bars also carry, and a table would not, is the slider. Every one of the nine numbers is a function of the order, several of them do not move monotonically, and a reader who suspects that a comparison has been staged at a flattering order can check by moving the handle. Every position on it was generated by the same code with the same assertions at build time, so the check is a real one.

What the three currencies are worth once they are all measured #

The delay column on this page is a variation across a passband, and two later measurements say what that variation is and what it costs to remove. Flat magnitude, unflat delay is the underlying quantity, absent from the classical comparison entirely and varying by fifty per cent across the passband of the two families everybody uses. And flat delay, bought with more delay is the exchange rate for buying it back: a fifth-order Chebyshev’s 891 microseconds of variation coming down to 498 with everything leaving 722 microseconds later than it did, on all-pass sections whose magnitude is one to $9\times10^{-16}$ over six decades. So the steepness this essay prices in delay can be un-priced, at a rate, and the rate is unfavourable — the latency added exceeds the variation removed.

The other direction is the one where a family beats all three of these on their own terms. The zeros that buy an order reaches forty decibels at twice the corner with two poles where the steepest all-pole family here needs five, by putting zeros in the stopband instead of adding poles — and charges in a currency none of the three columns above contains: a stopband that stops falling, measured at −38.7 dB and overtaken by an ordinary Chebyshev two corners out. A filter whose stopband is a floor rather than a slope is a different kind of object from anything on this page, and the comparison here cannot be extended to it by adding a fourth bar.

And the whole comparison is between designs rather than between circuits. A ladder is not a cascade is the reminder that the same response built two ways has passband errors whose exponents differ — 0.99 against 2.00 in the tolerance — so a family chosen on the bars above can lose the margin it won to the realisation before anything is soldered.

Where the trade is paid #

An order buys attenuation at a known rate and pays for it in three currencies the field measures separately. Flat magnitude, unflat delay is the delay flatness, and Flat delay, bought with more delay is what buying it back costs. The band that does not close and The floor that outlives the arithmetic are the ceiling on the order itself — a structure that cannot be realised past order nine is not a structure with a steeper skirt available. Three families, one corner is the other axis of the same decision, and A ladder is not a cascade is the realisation that decides what a one per cent component does to any of it.