Circuit Response
every model has an edge, and the figure carries it
Concept

Filter families — where it appears

The named sets of pole positions — Butterworth, Chebyshev, Bessel, elliptic — each of which spends the same order differently. What separates them is measured here on the built networks rather than tabulated: ripple, skirt steepness, delay variation and ringing.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

A single pole, and the brick wall that passes the same noise. computed by solving, not by drawing and integrated over 267 frequencies. The equivalent noise bandwidth is 1.5706 times the −3 dB point, and π/2 is 1.5708. A noise voltage computed with the corner frequency instead is 20.2% low. The area under the curve and the area of the rectangle are the same number.

The bandwidth noise sees

A single pole passes π/2 times as much noise power as a brick wall at its own corner frequency, so a noise voltage computed with the −3 dB point is twenty-one per cent low. Measured by integrating the solved response rather than taken from the table it usually comes from, the ratio is 1.5706 and π/2 is 1.5708. A five-pole Chebyshev's is 0.964 — less than one.

noise · Noise bandwidth The sample rate each anti-alias filter demands for 80 dB. computed by solving, not by drawing. A 20 kHz passband, and each filter must be 80 dB down by the frequency that folds back into it. The required rate follows, and it is a property of the filter rather than of the converter: Bessel 3.53× Nyquist, Butterworth 2.08×, Chebyshev 1.53×. The elliptic design at a selectivity of 0.8 is refused: its equiripple stopband has a floor at -74.1 dB, which is above the requirement at every frequency, so no sample rate satisfies it. At a selectivity of 0.5 the same order needs 1.34×. The floor is the selectivity's, not the order's.

What the filter in front costs

The filter that keeps a converter honest is normally chosen for its skirt. Measured against one requirement — eighty decibels down by the frequency that folds back into a 20 kHz band — the choice is not a decibel or two of skirt but a factor in the clock: Bessel demands 3.53 times Nyquist, Butterworth 2.08, Chebyshev 1.53. And one design is refused outright, because an elliptic stopband is a floor rather than a slope and no sample rate reaches past a floor.

digital · Anti-alias A fifth-order chebyshev delay, and 1 all-pass section in front of it. computed by solving, not by drawing at 60 frequencies across the passband. The delay varies by 891 µs from end to end; 1 all-pass section, with pole pair found by search rather than taken from a table, brings that to 498 µs — a factor of 1.79. The price is that everything is later: 722 µs more at direct current, which is more than the 392 µs of variation removed.

Flat delay, bought with more delay

An all-pass section has a magnitude of one at every frequency — measured here on a solved network as 1 to within 9 × 10⁻¹⁶ over six decades — which makes it the only thing that can change a filter's delay without touching its magnitude response. It flattens by adding. A fifth-order Chebyshev's 891 microseconds of delay variation comes down to 498, and everything leaves 722 microseconds later than it did.

filters · Group delay A 4 kHz Butterworth, mapped to 48 kHz two ways. computed by solving, not by drawing. The upper panel is the digital response on the unit circle; the lower one is where each analogue frequency lands. The bilinear transform compresses the axis as it approaches half the sample rate — -3.11% at a tenth of the rate and -15.2% at a quarter — so a design mapped straight through is -3.432 dB at its own corner instead of -3.010. Pre-warping puts that one frequency back exactly and no other: above it the pre-warped curve is the further of the two from the analogue prototype. It is a choice of where to be right, not a correction.

The corner that moved

The bilinear transform has to fit an infinite frequency axis onto a circle, so something must be compressed, and what is compressed is everything near half the sample rate. A 4 kHz Butterworth mapped to a 48 kHz clock is 3.432 dB down at its own corner instead of 3.010, and at a quarter of the sample rate the axis is 15.2% out. Pre-warping puts one frequency back exactly and no other — it is a choice of where to be right, not a correction.

digital · Digital realisation Every node of a Chebyshev 5, and the one that clips first. computed by solving, not by drawing. The largest signal each node of the realised network ever carries, over the whole frequency sweep, relative to the input. The output reaches 1.000 times the input and F0b reaches 4.537, so on a ±15 V supply the input can be driven to 3.31 V rather than 15.00 before something clips — and the thing that clips is not the output. With the floor at 249.7 nV that is 142.44 dB of range against the 155.57 dB an instrument on the output would report, a difference of 13.14 dB that no measurement at the output can see.

The ceiling is not at the output

A fifth-order Chebyshev's noisiest node is its output and its largest signal is not. F0b carries 4.537 times the input, so on ±15 V the range is 142.44 decibels and not the 155.57 an instrument on the output reports. Across fifteen realised filters the output reading spans 3.97 dB and the range they actually have spans 22.34 — and one family of the three has no internal peaking at all, at any order.

noise · Dynamic range The noise bandwidth of four families at 8 orders, and the one that has none. computed by solving, not by drawing. Every one of the 32 entries is an integral of the realised network's own squared magnitude, divided by that network's own measured −3 dB point. At order one the four families are the same filter and return 1.5706, which is π/2 — the calibration the rest of the table is quoted against. Only Butterworth then does what the ratio is usually said to do: it falls at every order, to 1.0065 at 8. Bessel is least at order 5 (1.0385) and rises to 1.0441; Chebyshev alternates with parity, 0.9637 at five against 1.0686 at six; and an even-order elliptic has no noise bandwidth at all, because its stopband comes back up to a constant — its magnitude at the top of the range moves by 0.00 decades per decade of frequency, so the integral grows with whatever limit it is stopped at.

The ratio that does not walk to one

A single pole passes π/2 times as much noise as a brick wall at its corner, and every account of it says the ratio falls towards one as the skirt steepens. Over thirty-two realised filters only Butterworth does that. Bessel is least at order five, 1.0385, and rises again; Chebyshev alternates with parity and the two branches separate only above 0.1968 decibels of ripple; and an even-order elliptic has no noise bandwidth at all, its integral returning 380 or 38,005 depending on where it was stopped.

noise · Noise bandwidth An order-8 Butterworth: the whole sketch is 3.0103 dB out at the corner, and its 4 sections −5.85 to +8.17 dB. computed by solving, not by drawing. The error of the straight-line sketch against the solved response — for each buffered section of an order-8 Butterworth lowpass at 1.00 kHz, and for the whole cascade. At the corner the sections are out by −5.852 dB (Q = 0.5098), −4.418 dB (Q = 0.6013), −0.915 dB (Q = 0.9000), +8.175 dB (Q = 2.5629), which add to −3.0103 dB: the whole filter's error, the same 10 log 2 as a single pole. The whole sketch is never further out than that anywhere; the section with the highest quality factor is +8.343 dB out at 1.04 kHz. The quality factors multiply to 1/√2.

The corner error a filter hides in its sections

The straight lines of an eighth-order Butterworth filter are 3.0103 dB out at the corner and nowhere worse — the same as one pole, at every order. The lines of the four sections it is built from are out by −5.85, −4.42, −0.92 and +8.17 dB there, which must add to the whole because the sections multiply. The whole sketch never gets worse with order and the worst section's error grows as 20 log(n/π). And the section the sketch misrepresents most is the one whose frequency error moves the filter most: 0.312 dB for one per cent, against the 0.173 dB any section's stopband shift gives.

frequency · Asymptotic approximation The bandwidth a windowed density is divided by, against the one bin an analyser assumes. Computed from the weights, from the coefficients in closed form, and out of the bins by Parseval — three routes agreeing to a part in 10¹². A record of N samples weighted by a window and transformed gives bins whose noise bandwidth is N·Σw²/(Σw)² in units of the bin spacing: 1 exactly for an unwindowed record, 1.5000 for Hann — three halves, exactly, because (a₀² + ½a₁²)/a₀² is (0.25 + 0.125)/0.25 with no record length in it — 1.3628 for Hamming, 1.7268 for Blackman and 2.0044 for Blackman–Harris. A density divided by one bin instead is wrong by the figure beside each bar. The bandwidth is not ordered with the sidelobe: Hamming's first sidelobe is 11.2 dB below Hann's and it costs less bandwidth as well, so the received sequence of windows fails at its second entry. What Hann buys is the far skirt instead, and that is a different measurement.

The bandwidth a bin is not

A density read off a transform is divided by a bandwidth, and the bandwidth belongs to the window rather than to the bin spacing — exactly three halves of a bin for a Hann window, so a density divided by one bin is 1.761 decibels high in power. Measured three ways that share only the weights. And the received sequence of windows fails at its second entry: Hamming's first sidelobe is 11.2 decibels below Hann's and it costs less bandwidth as well, with Hann collecting its debt at the fourth sidelobe and every one after it.

noise · Noise bandwidth An all-pole filter's stopband floor: -120 dB, from 10 fF nobody drew. computed by solving, not by drawing. A butterworth 8 realised at a 20 kHz passband, drawn twice: from its poles, which fall at 48 decibels an octave for ever, and from the same netlist with 10 femtofarads of stray capacitance from its input to its last shunt node — a fraction of a picofarad between two tracks. The second stops falling at -120.0 dB at 136 kHz, because the stray bypasses every pole in front of it and what is left is its ratio to the shunt capacitance it divides against. The poles alone predict -293 dB a decade further out, which is 173 decibels lower. So an all-pole design has a stopband floor exactly as an elliptic one does — -140 dB at 1 fF, -130 dB at 3 fF, -120 dB at 10 fF, -110 dB at 30 fF, -100 dB at 100 fF, which is 20.0 decibels per decade of stray — and above that frequency a faster sample rate buys nothing at all, which is the refusal recorded for the elliptic row and not for this one.

The floor every filter has

The elliptic design was refused because its equiripple stopband is a floor and no sample rate reaches past a floor, while an all-pole design falls at six decibels an octave per pole for ever — so a faster clock is always an answer for one and never for the other. On paper. Ten femtofarads of stray capacitance from a filter's input to its output, which is a fraction of a picofarad between two tracks, puts the eighth-order Butterworth's stopband at −120 decibels from 136 kilohertz onwards where its poles predict −293. Every all-pole design has a floor, and it is a layout rather than a design.

digital · Anti-alias The order is worth 0.68 bits and a factor of 29 in clock. computed by solving, not by drawing. A butterworth design at a 20 kHz passband, at each order from two to ten and at three impedance levels, with the sample rate it demands for eighty decibels and the resolution its own resistors' noise allows. The clock demand falls steeply — 50.50×, 11.27×, 5.50×, 3.65× and on down to 1.76×, a factor of 28.8 — and the bar widths carry it. The noise rises: 2.00 µV at order two against 3.22 µV at ten, which is 0.68 of a bit. So the order is cheap in the currency everybody worries about and expensive in nothing. What is expensive is the impedance level: every resistance goes as one over the capacitance and a noise voltage as the square root of a resistance, so a decade of capacitance is 1.66 bits — the same at every order, to 0.000 of a bit — and the three curves are ten, one and a tenth of a nanofarad a section. The ceiling on resolution is set by a choice nobody prices, and the order, which everybody argues about, moves it by two thirds of a bit.

Two thirds of a bit for a factor of twenty-eight

An anti-alias filter costed in clock rate gets steeply cheaper with order — fifty times Nyquist at the second and 1.76 at the tenth — and the obvious objection is noise, since the filter is in the signal path and every section adds resistors. Measured, the objection barely holds: from order two to order ten the clock demand falls by a factor of 28.7 and the resolution the filter's own noise allows falls by 0.69 of a bit. What does cost resolution is the impedance level, at five thirds of a bit per decade of capacitance, and nobody argues about that at all.

digital · Anti-alias The Chebyshev 5's sections, each node's peak before and after equal-maxima scaling. computed by solving, not by drawing. Unscaled (lighter), section by section the largest signal anywhere inside reaches 4.537×, 3.109×, 1.000× the input, so the ceiling is 4.537× at F0b. With a gain of 0.220 at the input, 1.000 and 1.602 between sections and 2.833 at the output, every section peaks at 1× (darker) and the response is unchanged. The floor rises from 249.7 to 883.8 nV, so the range gains 2.16 dB of the 13.14 dB the internal peak was costing.

The ceiling a gain gives back

A fifth-order Chebyshev loses 13.14 dB of its range because its first section's capacitor node carries 4.537 times the input. The textbook repair is to scale: divide the signal in front of the loud section, restore it later, so that every node peaks at the same level. Done exactly, with silent gains of 0.220, 1.000, 1.602 and 2.833 and the response unchanged to the last digit, it lowers the ceiling to the input and raises the floor from 249.7 to 883.8 nV — and gives back 2.16 dB of the 13.14. Every section between the division and the restoration has its noise multiplied by what the signal was divided by. On the even-order Chebyshevs scaling loses half a decibel; on the seventh-order it recovers 3.77 of 22.45. Reordering the sections and scaling them together gives 4.75 dB. The rest of the ceiling belongs to the high-Q section and no gain moves it.

noise · Dynamic range Fifth-order filters between Bessel (m = 0) and Butterworth (m = 1), all at one corner: the step at m = 0.5. computed by solving, not by drawing: each pole set interpolated between the Bessel and Butterworth poles of the same rank, renormalised to a 1 kHz half-power point, built as a network and its step taken from the residues. m 0: 0.77% overshoot, 0.68 periods to 1%, 14.1 dB at 2 kHz; m 0.25: 2.19% overshoot, 0.94 periods to 1%, 16.6 dB at 2 kHz; m 0.5: 4.53% overshoot, 1.27 periods to 1%, 20.7 dB at 2 kHz; m 0.75: 7.99% overshoot, 1.58 periods to 1%, 25.7 dB at 2 kHz; m 1: 12.78% overshoot, 2.21 periods to 1%, 30.1 dB at 2 kHz. Every quantity moves monotonically from one end to the other: the families are the two ends of one line, not two choices.

The line from Bessel to Butterworth

Bessel, Butterworth and Chebyshev are taught as three choices, and measured at one order and one corner they are points on one line. Interpolate each Bessel pole towards its Butterworth partner and every quantity moves smoothly: a fifth-order filter's step overshoots 0.77% and settles to 1% in 0.68 periods of the corner as a Bessel, 4.53% and 1.27 periods halfway, 12.78% and 2.21 periods as a Butterworth, while its attenuation at twice the corner climbs from 14.1 to 30.1 dB. The Chebyshevs carry the same line on to 47.0 dB at 4.97 periods, and every step along it buys attenuation with settling time — a frontier, with no member beaten on both. Overshoot is not the price: past 0.05 dB of ripple a Chebyshev overshoots less while ringing longer. And a specification of 25 dB at twice the corner within 1.6 periods is met first at order five, by a member nobody names.

filters · Filter families A chebyshev low-pass of order 5: where its impulse response balances, and where its step crosses half. computed by solving, not by drawing, the step built from the network's poles and the impulse response its derivative. The impulse response (dark, scaled to its peak) balances at 0.7090 ms; the group delay at a thousandth of the corner is 0.7090 ms — the same number by a different route. The step (pale) crosses half its final value at 0.7238 ms, 2.1% after it.

The delay that is a centre of mass

A filter's group delay at zero frequency is not an approximation to when its step arrives; it is exactly where its impulse response balances. Across three families and orders one to seven the two agree to 5.7 parts in a million by routes that share only the netlist. The step's half-way point is a different thing: ln 2 of the delay for one pole, approaching it from below for a Bessel filter, and passing it — by up to 16.9 per cent — for Butterworth and Chebyshev filters whose impulse responses swing negative. Elmore's RC-tree delay is the same centre of mass, and a uniform ladder network crosses half at 0.757 of it.

filters · Group delay The group delay of fifth-order filters along the line from Bessel to Butterworth, and of a 0.1 dB Chebyshev, all with one half-power point. computed by solving, not by drawing: the delay as a sum over each pole set's poles, held to the slope of each realised network's own phase at four frequencies. At zero frequency the delay is 0.386, 0.413, 0.449, 0.488, 0.515, 0.633 periods of the corner; across the passband it varies by 0.40%, 5.7%, 16.4%, 34.1%, 59.6%, 122.7% of that. The Bessel is flat to the corner and beyond; every step toward Butterworth bends the delay up into a peak near the corner, and the Chebyshev's peak is the largest.

The attenuation a flat delay cannot buy

Along the fifth-order line from Bessel to Butterworth the group delay's spread across the passband grows at every step, from 0.40 per cent to 59.6, and on through the Chebyshevs to 191 — monotone, like overshoot and settling, and lost fastest in the first eighth of the way. The third axis turns the frontier into something sharper than a trade. More order flattens a Bessel's delay four to seven times an order and buys it no attenuation at all: 14.17 dB at twice the corner at order six is the most any Bessel reaches, and from there the attenuation falls toward the 12.04 dB of the Gaussian every delay-flat filter tends to. A flat delay and a steep skirt cannot both come from poles.

filters · Filter families

Named alongside it

The objects these essays reach for when they reach for this one.

ChebyshevDesign tradeoffEquivalent noise bandwidthStopband attenuationAnti-alias filterBesselBrick wall filterButterworthFilter orderGroup delayModel rangeOversampling

All concepts