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

Phase linearity — where it appears

How closely a network's phase falls in proportion to frequency, which decides whether the components of a signal arrive together or spread out. It is what a Bessel family buys and what a Chebyshev spends, and neither is visible in a magnitude response at all.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Group delay across the passband, at order 5. The Bessel filter's delay varies 0.1% below 0.8 of the corner; the Chebyshev's peaks near the band edge and is several times its low-frequency value. Every family here has the same half-power frequency, so this is a difference in behaviour rather than in scaling.

Flat magnitude, unflat delay

A filter that passes every frequency in its band at the right amplitude and the wrong time has not passed the signal. Group delay is the measurement that says so, it is absent from the classical comparison, and it varies by fifty per cent across the passband of the two families everybody uses.

filters · Group delay Two reasons the frequency is not 1/2πRC. The measured oscillation frequency sits below the network's own zero-phase frequency, and by two separate amounts. The amplifier's share falls as 1/ρ — the product of shift and ratio is constant to 6.7% over two decades — and the limiter's share is flat at 0.7713%. They are equal at ρ = 578, and above that a faster amplifier moves the frequency by nothing that matters.

The frequency that is not the formula

The Wien network's zero-phase frequency is one over two pi RC to every digit the arithmetic has. The circuit does not run there. With a perfect amplifier it runs 0.771 per cent low, because the limiter's harmonics are part of the waveform whose period is being measured; with a real one it runs lower still, by an amount inversely proportional to the gain-bandwidth product. The two are equal at a ratio of 578, and above that a faster amplifier buys nothing.

applied · Oscillator 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 The droop a zero-order hold imposes at 48 kHz. computed by solving, not by drawing. Holding each sample for a clock period convolves the output with a rectangle, so the spectrum is multiplied by a sinc: -0.143 dB down at a tenth of the sample rate, -0.912 at a quarter and -3.922 at half — which is exactly 20 log(2/π) and contains no design decision at all. The dots are the amplitude of the fundamental read out of the transform of the staircase itself, agreeing with the closed form to 0.008%. There is also half a sample of delay, 10.417 µs here, which is the reason a held reconstruction is not a droopy copy of the signal but a droopy copy that has moved.

The staircase on the way out

A converter does not emit impulses. It holds each sample for a whole clock period, which is a convolution with a rectangle and therefore a multiplication by a sinc — 0.14 dB down at a tenth of the sample rate, 0.91 at a quarter, and 3.92 at half, which is exactly 20 log(2/π). Nobody chose that droop and it is nearly eight times the half-decibel ripple of a Chebyshev passband. Measured on the transform of the staircase itself, it agrees with the closed form to 0.008%.

digital · Reconstruction A 1 kHz tone burst through a band-pass of Q = 10, centred on its resonance. computed by solving, not by drawing, marched twice — sine and cosine carriers — so the output's envelope is the modulus of the two, exactly. The input envelope (pale) peaks at 240 ms; the output's (dark) at 243.171 ms, 3.171 ms late. The group delay at resonance is 2Q/ω₀ = 3.183 ms. The phase there is zero, so the carrier itself is not delayed at all.

The envelope that leaves early

A band-pass's group delay at its centre is the delay of an envelope, not of a carrier. Marched twice — on a sine and a cosine carrier, so the envelope is their modulus exactly — a 1 kHz burst through a band-pass of Q = 10 comes out 3.171 ms late against 2Q/ω₀ = 3.183, while the carrier's zero crossings come out within 6.4 µs of where they went in. Through a notch, whose group delay at its centre is −3.810 ms, the envelope peaks 3.746 ms before the input's. Neither is a paradox, and both hold only for bursts long against the inverse bandwidth: 60.6 per cent wrong at 0.6 of it, 0.7 per cent at 20.

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.

Group delayDesign tradeoffFilter familiesModel rangeAll-passBesselButterworthChebyshevDelay equalisationGain–bandwidth productIntegrationLimit cycle

All concepts