The attenuation a flat delay cannot buy

The line from Bessel to Butterworth turned three textbook choices into points on one curve. Interpolating each Bessel pole toward its Butterworth partner moves every quantity smoothly, and at a fixed order and corner the members form a frontier: a fifth-order filter’s attenuation at twice the corner climbs from 14.1 decibels as a Bessel to 30.1 as a Butterworth while its 1 per cent settling stretches from 0.68 periods of the corner to 2.21, and the Chebyshevs carry the same line on. No member beats another on both.

It closed with the axis a pulse system cares about and a frequency-domain specification leaves out: the group delay’s flatness across the passband. Flat magnitude, unflat delay had already found the two families everybody uses varying their delay by fifty per cent across their passbands. The question left was whether flatness moves monotonically along the line as overshoot and settling do, and how it trades against attenuation.

It is monotone. The trade it makes with attenuation is not a trade along a frontier but a wall: a filter can be made as delay-flat as anyone likes by adding order, and no amount of order buys a delay-flat filter more than about fourteen decibels at twice its corner.

The delay, two ways #

The filters are the pole sets of the earlier essay: Bessel and Butterworth poles of order five, each pole’s radius interpolated geometrically and its angle linearly between partners of the same rank, every set renormalised so that its half-power point is one kilohertz, and Chebyshevs of stated ripple renormalised the same way. For an all-pole filter the group delay has a closed form, a sum over the poles: each pole at $p$ contributes $-\mathrm{Re}(p) / ((\mathrm{Re}(p))^2 + (\omega - \mathrm{Im}(p))^2)$, a bump centred at its own imaginary part and as wide as its distance from the axis.

The second route is the realised network. Each pole set is built as a cascade of sections, solved at each frequency, and its phase differenced at a hundred-thousandth of the frequency either side; the negative slope of the phase is the group delay. At a tenth, a half, nine tenths and one and a half times the corner, for every pole set in the figure, the two agree to two parts in ten thousand. That agreement matters more than usual here, because the quantity of interest is a spread of a fraction of a per cent across a band, and a delay computed one way and compared another way could manufacture a spread out of nothing.

Monotone, and lost fastest at the start #

The spread grows at every step: 0.40 per cent at the Bessel end, 59.6 at the Butterworth, 191 at a 1 decibel Chebyshev. So the answer to the first half of the question is yes — flatness is monotone along the line, as overshoot and settling are, and the three axes order the members identically. A member is never flatter and also faster-settling than its neighbour on the Bessel side.

What the line adds is how unevenly the flatness is lost. The first eighth of the way from Bessel to Butterworth takes the spread from 0.40 per cent to 2.25 — more than fivefold — while attenuation at twice the corner has moved only from 14.1 to about 15. A Bessel’s flatness is a delicate property, a cancellation of the poles’ bumps that the slightest movement of the poles undoes, and the first step away costs most of it.

Over the lower half of the passband the contrast is starker. There the Bessel’s delay varies by six parts in a million and the Butterworth’s by 12.4 per cent. The Bessel’s delay is flat to a very high order near zero frequency — that is how it is defined — and what spread it has is all near the corner.

How the spread grows with the band #

The spread is not one number for a filter; it is a function of the band it is measured over, and the function has a characteristic power. Between a quarter and a half of the passband the Bessel’s spread grows as the band to the power 9.8 and the Butterworth’s as the band to the power 2.3. A maximally flat delay of order five has its first non-zero deviation at the tenth power of frequency — five poles, ten derivatives of the delay set to zero — and the measured 9.8 is that order showing through. The Butterworth’s delay has a quadratic term and nothing cancels it.

The practical reading is that “Bessel filters have flat delay” is a statement about a band well inside the corner. Over a quarter of the passband it is perfectly true, seven parts in a billion. Over the whole passband it is 0.40 per cent, and to twice the corner, where a pulse’s harmonics still carry energy, it is 32 per cent. A pulse whose spectrum reaches the corner sees a Bessel’s delay vary, and it is the band the pulse occupies that decides which number applies.

The frontier, with a third axis #

Plotted as attenuation at twice the corner against delay spread, the fifth-order line is a rising curve: every decibel of skirt is bought with flatness, 14.1 decibels at 0.40 per cent, 30.1 at 59.6, 47.0 at 191. That is the earlier essay’s frontier with a third axis added, and it says the same thing — no member beaten on both.

The Bessels of other orders are the surprise. Each is a different filter with its own attenuation and spread, and they do not lie along the line or anywhere near it. They lie almost vertically: order three spreads its delay by 6.51 per cent, order five by 0.40, order seven by a hundredth of a per cent and order ten by fifteen millionths — four to seven times flatter with each added pole — and their attenuation at twice the corner stays between 12.0 and 14.2 decibels throughout. Adding poles to a delay-flat filter buys flatness and nothing else.

The selectivity that is not free found the usual two-way drawing of the filter trade hiding a third quantity — where the stopband is asked to begin — that an order fixes along with the other two. Delay flatness is another quantity the drawing hides, and it is fixed differently. Attenuation and flatness trade along the line at a fixed order; order trades for flatness at a fixed attenuation along the Bessels; and nothing on either set trades order for attenuation once the delay is held flat. A designer who fixes the delay’s flatness has fixed which of the three quantities is left to spend, and it is not the skirt.

The Gaussian a flat delay tends to #

The Butterworth gains six decibels at twice its corner for every order, because each pole adds twenty decibels a decade and twice the corner is three tenths of a decade out. The Bessel does not gain at all past order six. It peaks at 14.17 decibels there and then falls: 13.68 at eight, 13.14 at ten, 12.96 at eleven.

The reason is the limit a Bessel filter is heading for. A maximally flat delay of increasing order approaches a linear phase over an ever wider band, and its magnitude approaches a Gaussian, $e^{-(f/f_c)^2 \ln\sqrt 2}$ when normalised to its half-power point. At twice that point a Gaussian is down by a factor of $e^{-4\ln\sqrt 2} = 1/4$, which is 12.04 decibels, and no amount of order changes it, because the limit has no order in it. Low-order Bessels are a little steeper than their Gaussian limit near the corner and rise toward a maximum; past the maximum each added pole moves the response closer to the Gaussian, and the attenuation at twice the corner falls toward 12.04.

That is the wall. A filter that is to be delay-flat across its band can be had at any flatness, and its attenuation at twice its corner is about fourteen decibels at the very best, whatever its order. Attenuation beyond that has to come from the skirt’s shape — which the delay forbids — or from somewhere other than poles.

The same limit, three times #

The Gaussian is turning up in this collection for the same reason each time. The delay a chain adds found a chain of one-pole stages tending to a normal edge by the central limit theorem, and the mean that makes a stopband found a cascade of means tending to a Gaussian response with a noise bandwidth falling as the root of the count. A Bessel filter is the third: the filter whose step has the least overshoot and whose delay is flattest is the one closest to a Gaussian, because a Gaussian is the one shape whose impulse response is as compact as its spectrum and has no ringing in either. In every case the Gaussian is what is left when a cascade has been made as well-behaved as possible, and in every case its skirt is the price.

Why a Bessel peaks at order six #

The rise and fall of the Bessel’s attenuation has a mechanism worth stating, because the peak is not an accident of normalisation. Every Bessel here is scaled so its half-power point is one kilohertz. At low order the poles sit far from the axis and the response falls slowly through the corner and then steepens, like any all-pole filter, so attenuation at twice the corner grows with order: 12.00 decibels at three, 13.41 at four, 14.06 at five. But a Bessel’s poles are placed to flatten its delay, not its magnitude, and as the order grows the poles are pushed into a shape whose magnitude is ever more nearly Gaussian across the corner region. A Gaussian falls more slowly beyond its half-power point than a Bessel of order six does, so past six the added poles are spent making the response more Gaussian rather than steeper, and the attenuation at twice the corner comes down.

The numbers settle the argument. At order eleven the Bessel is 0.92 decibels above its Gaussian limit at twice the corner; at order six it was 2.13 above it. The approach is slow, as approaches to a central limit are, and the delay that is a centre of mass is the reason it has to be an approach to the Gaussian rather than to something else: the delay at zero frequency is the impulse response’s centroid, and a delay that is flat over a widening band is an impulse response whose higher moments are being pinned, which is what a Gaussian is.

The envelope’s view of the same spread #

A signal that is a modulated carrier inside the passband travels at the group delay of its carrier’s frequency, as the envelope that leaves early measured, and a spread of delay across the band is then a spread of arrival times across the carriers. For a fifth-order Butterworth the 59.6 per cent spread is about 0.31 periods of the corner frequency between an envelope near zero frequency and one near the corner; for the Bessel, 0.0015 periods. A system that sends several carriers through one filter and needs them to arrive together is asking for the left end of the line, and the frontier says what it costs in skirt.

Where the attenuation has to come from instead #

There are two ways past the wall, and both have been measured elsewhere. The zeros that buy an order found that transmission zeros give attenuation at a stated frequency without moving the poles, and a zero placed just above twice the corner would lift any member of the frontier to the right. What it does to the delay is less kind: a zero on the frequency axis adds no delay but changes the delay’s shape near itself, and a delay-flat filter with a zero beside its band is no longer flat there.

The other is to keep a steep filter and flatten its delay afterwards. Flat delay, bought with more delay found an all-pass section bringing a Chebyshev’s delay variation down from 891 microseconds to 498 at the cost of 722 microseconds of extra delay for everything. That buys attenuation and flatness together, which the poles alone cannot, and charges for it in total delay — the third quantity the frontier’s two axes did not count, and the one a real-time system cannot always give.

A worked choice #

A pulse system that must keep its delay within one per cent across its passband and reject a disturbance at twice its corner by twenty decibels has, from these figures, no all-pole answer at any order. The fifth-order line reaches twenty decibels just before halfway, where the spread is already fifteen per cent; the Bessels of any order sit at fourteen decibels. A one-per-cent spread is a Bessel of order five or above, and twenty decibels at twice the corner needs either a zero near twice the corner or an equaliser after a steeper filter.

If the specification is relaxed to the lower half of the passband — the pulse’s energy mostly well inside the corner — the Butterworth’s 12.4 per cent is still too much but a member a quarter of the way along the line, at 1.4 per cent over the lower half and 16.6 decibels, is close, and an eighth of the way along is within a per cent. The band the delay is specified over decides the answer more than the family does.

What the measurement leaves out #

The filters are all-pole and built at one order; the delay is the ideal pole-set’s, confirmed on networks of ideal sections. A real section’s amplifier adds a little phase of its own, which tilts the delay at the top of the band, and a Chebyshev’s passband ripple puts ripple into its delay that this spread treats as part of one number. And the spread is a peak-to-peak measure; a system that cares about the mean square deviation of delay, as a modem’s equaliser does, would read the same curves differently and find the Butterworth’s quadratic rise less damaging than its peak suggests.

The number to keep #

Along the fifth-order line the group delay’s spread across the passband grows at every step, 0.40 per cent for Bessel, 59.6 for Butterworth and 191 for a 1 dB Chebyshev, as attenuation at twice the corner goes from 14.1 to 30.1 to 47.0 dB. More order flattens a Bessel’s delay four to seven times a pole and buys it no attenuation: 14.17 dB at order six is the most any Bessel reaches, and they tend to the Gaussian’s 12.04. A flat delay and a steep skirt cannot both come from poles.

Still open: the zero beside a flat delay, the two-parameter family, and the delay a step reads #

A zero beside a delay-flat filter. A transmission zero just above twice the corner gives attenuation there that poles cannot. How much it bends a Bessel’s delay within its passband — whether a zero placed far enough out costs less flatness than the next step along the line does — is the comparison that would say whether a zero is the way past the wall.

A family with two parameters. The transitional line moves every pole together. Letting each pair’s radius and angle move freely, subject to the half-power point, might find placements flatter than the line for the same attenuation, below the curve this essay draws. Whether the transitional line is the true edge of that larger set is an optimisation over the pole plane.

The delay a step reads. A pulse passed through a filter is late by the delay and distorted by its spread, and the step’s overshoot and settling were the earlier essay’s axes. Whether the spread measured here predicts the pulse’s distortion better than the overshoot does — and over which band it must be measured to do so — is the time-domain test of the third axis.