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

Butterworth — where it appears

The all-pole filter response that is as flat as possible in its passband, falling smoothly with no ripple at six decibels an octave for each pole. Its order alone sets how steep the fall is, so a stated rejection at a stated distance fixes the order it needs.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Three filter families at order 5, all with the same half-power point. At three times the corner the Chebyshev is -64.0 dB down, the Butterworth -47.7 dB and the Bessel -28.3 dB. The inset is the passband at forty times the vertical magnification, which is the only place the Chebyshev's half-decibel of ripple is visible at all.

Three families, one corner

Butterworth is flat, Chebyshev is steep, Bessel has good delay. None of those is a number, so the table they appear in cannot answer the question anybody has. Here each family's poles are computed from its definition, built as an actual network, and then measured — starting with the step every comparison skips.

filters · Filter families At 60 dB of rejection an eighth-order filter keeps 3 of the brick wall's 6 windows; a fourth-order one keeps 2. computed by solving, not by drawing. The sample rates at which a band from 100 to 120 MHz can be sampled without anything the filter in front passes at more than −60 dB folding onto the band, for a band with hard edges and for Butterworth bandpass filters of order 12, 8, 6 and 4 on the band's own edges; each row is labelled with the span, in megahertz, where the filter is still above −60 dB. A real filter moves each window's edges inward by half its transition band. The window from 120 to 200 MHz narrows to 124.4–193.2, 127.9–188.4, 132.8–182.4, 149.7–166.9 MHz as the order falls, and the lower windows close altogether. Every window is checked by folding every frequency the filter passes.

The windows a real filter leaves

A band from 100 to 120 MHz can be sampled at any clock in six disjoint windows — for a band with hard edges. Put a real bandpass filter in front and ask that whatever folds onto the band be 60 dB down, and every window's edges move inward by half the filter's transition band: an eighth-order Butterworth keeps three of the six, a fourth-order one two, and the window from 120 to 200 MHz shrinks from ±25 per cent of clock tolerance to ±19.1 per cent at order eight and ±5.4 per cent at order four. The windows depend on the rejection only through the rejection per order: 40 dB at order four, 60 at order six and 80 at order eight all leave the same ±15.7 per cent. So a filter's order is bought in clock tolerance, at one order per ten decibels.

digital · Anti-alias 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.

BesselChebyshevFilter familiesDesign tradeoffFilter orderGroup delayStep responseAliasingAnti-alias filterBandpass samplingClosed formNormalisation

All concepts