The line from Bessel to Butterworth

Three families, one corner computed Butterworth, Chebyshev and Bessel filters from their definitions, built each as a network and measured them at one order and one half-power frequency. Butterworth is flat, Chebyshev is steep, Bessel has good delay — and the essay’s point was that none of those is a number until it is measured: the step every comparison skips, the attenuation at a stated multiple of the corner, the variation of the delay across the band. The zeros that buy an order added the elliptic family, which buys selectivity with zeros rather than poles.

Both essays treat the families as a menu. A designer chooses one, then an order. This essay asks whether the menu is the right picture, and measures the answer: at a fixed order the families are not separate choices but points on one curve, the curve is a frontier on which every gain costs something, and the familiar names mark three places on it — two ends and a turning point — rather than three kinds of filter.

A line between two families #

A Bessel filter’s poles and a Butterworth filter’s poles of the same order can be paired off by rank — the real pole with the real pole, the lowest-Q pair with the lowest-Q pair, and so on — and each Bessel pole moved towards its Butterworth partner: its distance from the origin interpolated geometrically, its angle linearly, by a fraction $m$ running from 0 (Bessel) to 1 (Butterworth). That is the classical transitional Butterworth–Thomson construction. Each intermediate set is renormalised to the same one-kilohertz half-power point, built as a network of sections, and stepped.

At $m = 0$ the filter is the Bessel: its step overshoots 0.77 per cent, settles within one per cent in 0.68 periods of the corner frequency, and attenuates twice the corner by 14.1 decibels. At $m = 1$ it is the Butterworth: 12.78 per cent, 2.21 periods, 30.1 decibels. Between them every measured quantity moves monotonically: at $m = 0.5$, 4.53 per cent, 1.27 periods, 20.7 decibels; at $m = 0.75$, 7.99 per cent, 1.58 periods, 25.7 decibels. There is no step in any of them and no place where the filter becomes a different kind of thing. Bessel and Butterworth are the two ends of a line.

Three poles walking #

The line is easiest to see in the plane where the poles live.

From Bessel to Butterworth the three poles in the upper half-plane — one real, two complex — move outward from the origin and swing towards the imaginary axis together, the outermost pair furthest. Past Butterworth the Chebyshev ripple carries the same three on, onto an ellipse that narrows towards the axis. The quantity that matters for the step is the quality factor of the pair nearest the axis: 0.92 for the Bessel, 1.62 for the Butterworth, 5.56 for a one-decibel Chebyshev. Every move along the line is a rise in that Q, and a rise in Q is the whole of what selectivity costs in time: a higher-Q pair is a more selective resonance and a longer-ringing one, which the arrow that goes past where it settles measured for a single resonator.

One frontier at a fixed order #

Two numbers summarise what a designer trades: how strongly the filter rejects a frequency twice its corner, and how long its step takes to settle within one per cent. Plot every member of the family in those two numbers.

The transitional family runs from the Bessel, 14.1 decibels at 0.68 periods, to the Butterworth, 30.1 decibels at 2.21 periods, and the Chebyshevs carry on from 33.0 decibels at 2.29 periods for a ten-thousandth of a decibel of ripple to 47.0 decibels at 4.97 periods for one decibel. Along the whole curve every step that buys attenuation costs settling time, and no member is beaten on both numbers by another. That is the definition of a frontier, and it means that at a fixed order the choice of family is a choice of where on the frontier to sit — a single number, the exchange rate between rejection and time that the application can afford.

The exchange rate is not constant. From the Bessel to the halfway transitional member, 6.6 decibels of attenuation cost 0.59 periods of settling, eleven decibels a period; from the Butterworth to the one-decibel Chebyshev, 16.9 decibels cost 2.76 periods, about six a period. The frontier is steepest — the most attenuation per period — between the Bessel and the Butterworth, which is the region a designer who needs both a clean step and some selectivity will be in, and it is exactly the region the three-family menu has no names for. What a steep skirt costs measured the same trade as attenuation against order; this is the trade at a fixed order, and it is continuous.

Overshoot is not the price #

The usual time-domain number quoted beside a filter family is its step overshoot, and along the Chebyshevs it tells the wrong story.

A Chebyshev’s overshoot rises from 13.79 per cent at a ten-thousandth of a decibel of ripple to 15.24 per cent at 0.05 decibels, and then falls: a one-decibel Chebyshev overshoots 10.19 per cent, less than the Butterworth’s 12.78. Read as a figure of merit, overshoot would say the one-decibel Chebyshev has the better step of the two. Its settling time says otherwise: it lengthens at every step of ripple, from 2.29 to 4.97 periods, more than twice the Butterworth’s.

The overshoot is the height of the first peak, and it is decided by how the response’s several ringing components add at that instant. With more ripple the highest-Q pair rings longer and at a lower amplitude relative to the others, and the first peak’s height can fall while the tail lengthens. What a designer pays for selectivity is the ringing — the time until the output is trustworthy — and the overshoot is a poor proxy for it. The cliff before the fastest settling found settling time a treacherous quantity for a single pole pair, with discontinuities in it; across a family it is still the honest one, because it is what the application waits for.

A specification picks an order, then a point #

The frontier moves with the order: more poles buy more attenuation at every settling time. Drawing the frontiers of several orders on one plane turns a specification into a reading.

Each order’s frontier lies to the right of the one before it: more attenuation for the same settling. A specification of 25 decibels at twice the corner, with the step within one per cent by 1.6 periods, falls between them. No third- or fourth-order filter of any family reaches it — the fourth-order Butterworth falls short even on attenuation, at 24.1 decibels, and the fourth-order Chebyshevs that clear 25 decibels take more than two periods to settle. The fifth order reaches it with the member three-quarters of the way from Bessel to Butterworth: 25.7 decibels at 1.58 periods. Neither named family at order five would do: the Bessel is too gentle and the Butterworth too slow.

That is the practical use of the line. The design procedure the frontiers support has two steps, in this order: find the lowest order whose frontier passes the specification, then find the point on that frontier that meets it with the most margin on the side the application cares about. The family name is whatever that point happens to be called, and quite often it has no name.

The same reading also says what an order is worth in this currency. At a settling time of 1.6 periods, the fourth-order frontier offers about 23 decibels at twice the corner and the fifth about 26: three decibels for a pole. At the Butterworth end the step from fourth to fifth order is six decibels. A pole buys less near the Bessel end than near the Butterworth end, because the Bessel’s poles are spread along a line in the plane rather than round a circle, and an extra one adds more delay than selectivity — the same reason the Bessel’s attenuation at twice the corner barely moves with order at all, from 12 decibels at third order to 14 at sixth.

Why three points have names and the rest do not #

If the families are points on a line, it is fair to ask why those points were named and the others were not. The answer is that each named family is the solution of an optimisation, and the optimisations are about properties of the response that are easy to state and hard to trade.

The Butterworth is the filter whose magnitude is maximally flat at zero frequency: every derivative of the magnitude that can be made zero there is zero. The Bessel, or Thomson, filter is the one whose group delay is maximally flat at zero frequency, which is what makes its step so clean — a delay constant across the band delays every component of the step equally, and the step arrives intact — the property flat delay bought with more delay found an all-pass section buying back for the other families, at a price in total delay. The Chebyshev is the filter whose passband error is spread evenly, as equal ripples, so that its attenuation outside the band is the greatest any all-pole filter of that order can have for that ripple. Each is the best possible filter by one criterion, which is why each has a closed form and a name, and why a designer reaching for one feels on solid ground.

The transitional members optimise nothing so simply stated. A filter three-quarters of the way from Bessel to Butterworth is neither maximally flat in delay nor in magnitude; it is a compromise, and compromises do not get closed forms. That is a historical reason for the menu and not a technical one. The figures above say that, for the two quantities most applications are specified by — rejection at a stated frequency and time to settle — the compromise is often exactly what is wanted, and that the named families are the right answer only when the specification happens to fall at one of their points or at an end of the line.

There is one more reason the ends are attractive, which is that they are robust in a way the middle is not quite. A Butterworth’s maximal flatness means a small error in its parts moves its magnitude only at second order near zero frequency; a Bessel’s means the same for its delay. A transitional member has no such stationary property, and its response moves at first order with its parts. In practice the difference is small next to an ordinary tolerance, but it is a difference, and it belongs in the choice.

A worked choice: the filter in front of a multiplexed converter #

A converter that samples several channels in turn through one multiplexer needs, in front of each channel, a filter that does two things: attenuate anything above half the sample rate enough that it does not alias, and settle after the multiplexer switches before the converter takes its sample. The first is a rejection at a stated frequency; the second is a settling time. The frontier is exactly the plane the requirement lives in.

Take a corner at a kilohertz, a requirement of 25 decibels at two kilohertz — the first alias that matters — and settling within one per cent in 1.6 periods of the corner, 1.6 milliseconds. That is the specification drawn as a dot on the last figure. The usual habit would be to reach for a Bessel because the application is about settling, find it gives only 14 decibels at twice the corner at any order, raise the corner to buy rejection, and lose bandwidth; or to reach for a Butterworth because the application is about rejection, find its fifth-order step takes 2.2 periods, and add a sixth pole. The frontier says the fifth-order transitional member three-quarters of the way along does both, with neither change: 25.7 decibels at 1.58 periods.

What the filter in front costs made the converter’s side of this argument — that the filter in front of a converter is chosen against a requirement, not a habit — and found the cost of a steep skirt in delay and in order. The frontier adds the settling time to that ledger, which is the cost a multiplexed system pays once per channel per sample and therefore the one that sets how many channels a converter can serve.

Where the line stops #

The transitional construction pairs Bessel and Butterworth poles by rank, and that pairing is only well defined while the two sets’ poles can be matched up one to one in order of angle. At order seven the matching crosses over — a Bessel pole of one rank sits nearer the Butterworth pole of the next — and the intermediate sets stop being a family, with attenuations that fall rather than rise partway along. The figures therefore stop at order six. A better construction, interpolating the polynomial’s coefficients or matching the poles by minimum total distance, would carry the line further, and the frontiers above would be expected to continue the pattern; this essay does not measure it.

The Chebyshev continuation is a second line, not the same one: it begins at the Butterworth and moves along a different direction in the pole plane, and the two meet at the Butterworth with a kink in the frontier’s slope. Whether there is a smoother family that passes through the Butterworth without the kink — and whether it would dominate either line near the join — is a question about what the frontier is when every pole placement is allowed rather than two one-parameter families.

How the filters were built and measured #

Every pole set is computed from its definition — Butterworth poles evenly spaced on a semicircle, Bessel poles as the roots of the reverse Bessel polynomial, Chebyshev poles on an ellipse set by the ripple — renormalised by bisection to a one-kilohertz half-power point, interpolated where the figure calls for it, and realised as a cascade of sections with ideal followers. The half-power point of every built network is bisected again on the solve and checked to three parts in a thousand. Steps are the residues of the realised network’s poles summed at three thousand instants over twelve periods; settling is the last crossing of the one-per-cent band, interpolated between samples; attenuation is the solved response at two kilohertz relative to its value at one hertz.

The filters are ideal: exact parts, ideal followers, no amplifier bandwidth, no noise. Flat magnitude, unflat delay is the other axis a family is chosen on, the delay’s flatness across the band, and it moves along the same line — flattest at the Bessel end — which is the reason the Bessel end exists at all.

Still open: the delay axis, a third parameter, and the frontier with zeros #

Delay flatness as the third axis. The frontier above has two numbers; a pulse system cares about a third, the variation of the group delay across the passband, which is smallest at the Bessel and grows along the line. Whether delay flatness is monotone along the transitional family as overshoot and settling are, and how it trades against the attenuation once the settling time is fixed, would turn the frontier into a surface.

A family with two parameters. The transitional family and the Chebyshev family are each one-parameter lines. Allowing the poles to move independently — each pair’s radius and angle free, subject to the half-power point — gives a much larger set, and the true frontier is its lower edge. Whether the transitional family is that edge between Bessel and Butterworth, or whether some unnamed placement settles faster for the same attenuation, is an optimisation over the pole plane that these two lines only bound.

Zeros on the line. An elliptic filter buys attenuation at a stated frequency with zeros, which the zeros that buy an order measured. A transmission zero placed just above twice the corner would move any member of this frontier to the right without moving its poles — but zeros change the step too, adding their own ringing. Where an elliptic member falls on this plane, and whether it extends the frontier or crosses it, is the comparison between buying rejection with poles and with zeros made at a fixed settling time.