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

Pole-zero mirror — where it appears

The reflection of a pole or zero across the imaginary axis of the s-plane, from left half to right or back. A zero and its mirror give the same magnitude at every frequency and different phase, so mirroring changes only delay; the difference between the two is an all-pass.

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

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 complex pair of zeros at 30.0 kHz, damping 0.5, and its mirror: one magnitude, and half the energy 10.6 µs later — 4ζ/ω₀ is 10.6. computed by solving, not by drawing. A low-pass with poles at 1, 10 and 100 kHz and a pair of zeros at 30.0 kHz with damping ζ = 0.5, against the same network with the pair mirrored into the right half-plane; the fraction of the impulse response's energy delivered by each instant, from the residues, with the mirrored network also marched from its polynomials (dots, within 6e-7). Both carry the same energy. The left-half-plane network is ahead at every instant; half its energy arrives at 75.4 µs against 86.0, a delay of 10.62 µs, where the mirror's all-pass delays a low frequency by 4ζ/ω₀ = 10.61 µs.

The zeros that have no side

A complex pair of zeros can be mirrored into the right half-plane without changing a magnitude anywhere, and what the mirror adds is a second-order all-pass whose delay is 4ζ/ω₀ at low frequency. Above the band that is the whole story: with the pair at 30 kHz on a 1 kHz low-pass, half the energy arrives 0.21 µs later at ζ = 0.01, 2.12 µs at 0.1 and 21.2 µs at 1, linear in the damping — so the zeros of a notch, which sit on the axis, have no side worth choosing. Inside the band the all-pass's delay peaks at a millisecond exactly where the notch has removed the energy, and the energy's centroid moves by the delay averaged over |H|², which for fifteen pairs from 2 to 100 kHz matches a quadrature over frequency to five parts in a hundred thousand. And unlike one mirrored zero, a mirrored pair never starts a step the wrong way: it is only late.

frequency · Minimum-phase

Named alongside it

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

All-passGroup delayDelay equalisationExcess phaseFilter familiesMinimum-phasePhase linearityRight half-plane zeroStep responseVerification

All concepts