The zeros that have no side

The energy that arrives first established the time-domain meaning of minimum phase: two networks with the same magnitude carry the same energy, by Parseval’s theorem, and the one with its zeros in the left half-plane delivers that energy first. The two mirrors that change places took it to two real zeros and four networks, found the all-left and all-right networks fixed at the ends of the ordering, and found the two between unranked. Both essays put their zeros on the real axis.

The zeros that circuits actually place deliberately are rarely there. A notch filter’s zeros sit on the imaginary axis, a pair of them at the frequency to be removed; an elliptic filter’s sit there too, one pair per stopband null; and a lightly damped bridge or twin-T puts a complex pair just off the axis. Such a pair can be mirrored just as a real zero can — its reflection in the imaginary axis has the same magnitude contribution at every frequency — and the question both earlier essays left is how much of the energy that mirror delays, and how that depends on how close the pair sits to the axis.

The answer turns out to have a simple law, and the law contains a limit worth stating first: a pair on the axis is its own mirror. A notch’s zeros have no side.

The pair and its mirror #

The network has three real poles, at 1, 10 and 100 kHz, and a numerator $s^2 + 2\zeta\omega_0 s + \omega_0^2$: a pair of zeros at $\omega_0$ with damping $\zeta$, at $-\zeta\omega_0 \pm j\omega_0\sqrt{1 - \zeta^2}$. The mirror replaces the numerator with $s^2 - 2\zeta\omega_0 s + \omega_0^2$, whose zeros are the same points reflected to the right. At every real frequency the two numerators have the same magnitude, $\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2}$, so the two networks share their magnitude exactly and differ by the factor

$$A(s) = \frac{s^2 - 2\zeta\omega_0 s + \omega_0^2}{s^2 + 2\zeta\omega_0 s + \omega_0^2},$$

a second-order all-pass. Its group delay is

$$\tau(\omega) = \frac{4\zeta\omega_0(\omega_0^2 + \omega^2)}{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega_0\omega)^2},$$

which is $4\zeta/\omega_0$ at direct current, rises to a peak of about $2/(\zeta\omega_0)$ near $\omega_0$, and falls away above.

The energy each network has delivered by time $t$ comes from its residues in closed form, the double sum the earlier essays used. For the hero figure the mirrored network was also marched in time — its impulse response integrated from the polynomial coefficients by fourth-order Runge–Kutta, and the square of the output summed — and the march agrees with the residues to six parts in ten million of the total at every instant checked. Both networks carry the same energy, as Parseval requires. The left network is ahead at every instant, as the minimum-phase theorem requires, and at a damping of a half with the pair at 30 kHz the mirror delays half the energy by 10.62 µs.

Linear in the damping #

The number 10.62 is not arbitrary. The pair sits at 30 kHz, well above the network’s 1 kHz band, so almost all of the energy is at frequencies where the all-pass’s delay is still its low-frequency value, $4\zeta/\omega_0 = 10.61$ µs. The mirror, seen from inside the band, is a delay line.

Swept over the damping, every measure of the delay follows $4\zeta/\omega_0$. At $\zeta = 0.01$ the half-energy time moves by 0.21 µs; at 0.1 by 2.12; at 1 by 21.2 — the law to the hundredth of a microsecond, and the same for nine-tenths of the energy and for the energy’s centroid. The delay is linear in the damping, and it goes to zero as the damping does.

That is the result in its plainest form. A lightly damped zero pair barely knows which side it is on. At $\zeta = 0.01$ the mirrored network is a minimum-phase network’s twin delayed by a fifth of a microsecond, on a response whose energy takes eighty microseconds to arrive by half. In the limit — a pair exactly on the imaginary axis, which is what a notch filter is designed to have — there is no mirror at all: the reflection of a point on the axis is the point itself. The class of minimum-phase networks is defined by where its zeros are, and for a notch’s zeros the definition has nothing to decide.

Above $\zeta = 1$ the pair is no longer complex. It becomes two real zeros, one moving down towards the band and one moving up, and the lower one eventually reaches the frequencies where the energy is. Then the law fails, as it should: at $\zeta = 1.5$ the half-energy delay is 31.7 µs against the law’s 31.8, but the centroid’s is 29.7, and the earlier essays’ two-real-zero behaviour takes over.

The centroid, read in either domain #

The law for a pair above the band is the low-frequency limit of something exact, and the exact statement is worth having because it covers the pair inside the band too.

The delay that is a centre of mass found that a filter’s group delay at zero frequency is exactly the centroid of its impulse response. A related identity holds for the energy: the time at which a response’s energy is centred, $\int t\,h^2\,dt / \int h^2\,dt$, equals its group delay averaged over all frequencies with $|H(j\omega)|^2$ as the weight. Two networks with the same magnitude have the same weight, so the difference of their energy centroids is the difference of their group delays, averaged over the same weight — which here is the all-pass’s delay $\tau(\omega)$, averaged over the network’s own $|H|^2$.

Both sides were computed, by routes that share only the polynomial. The time side is closed form from the residues: the centroid’s numerator is $\sum a_ja_k/(p_j + p_k)^2$, the same double sum as the energy with the denominator squared. The frequency side is a quadrature of $\tau(\omega)|H|^2$ over six thousand points on a logarithmic grid from a tenth of a hertz to a gigahertz. For fifteen pairs spanning the network’s band and well beyond it, the two agree to five parts in a hundred thousand.

The identity says something the half-energy times could not. A delay counts in proportion to the energy it is applied to. A mirror whose all-pass delays only frequencies the network does not pass delays nothing a measurement of energy can see.

Inside the band, the delay sits in the notch #

That is exactly the situation when the pair sits inside the band and is lightly damped — which is what a notch is.

With the pair at 3 kHz and a damping of 0.1, the mirror’s group delay is 21.2 µs at direct current and peaks at 1,028 µs at 2.93 kHz — a delay fifty times its low-frequency value, at almost exactly the frequency where the zeros have cut the response to −24 dB. The large delay is applied where there is no energy to delay. Averaged over the energy, it is 0.35 µs: smaller even than the low-frequency value, because this network’s zeros lift its response between 3 and 10 kHz and put most of its energy at frequencies above the pair, where the all-pass’s delay has fallen away.

In time, that shows as a mirror that hardly moves the middle of the energy curve and moves its tail. With the pair at 3 kHz and $\zeta = 0.1$ the half-energy times of the two networks differ by under a hundredth of a microsecond, and the nine-tenths times by 0.15 µs: the last of the energy is the part near the notch, delayed by the all-pass’s peak. A notch with its zeros a little off the axis, mirrored, is a notch that rings a little later after a transient — the ring at the notch frequency is where the delay went.

Flat delay, bought with more delay used second-order all-pass sections deliberately, to flatten a filter’s group delay across its band, and found that the flatness cost total delay. The same section, appearing as the difference between a network and its mirror, is the cost of putting the zeros on the wrong side — and the cost is largest exactly where a section designed to equalise delay is least useful, in a stopband null.

A delay with a fixed area #

There is a way to see both cases — the pair above the band and the pair in the notch — as one fact about the all-pass, and it comes from counting its phase rather than its delay. A second-order all-pass turns its phase through exactly a full circle, 360 degrees, between zero frequency and infinity, whatever its damping. On the imaginary axis its numerator is the complex conjugate of its denominator, so its phase is minus twice the denominator’s, and the denominator’s phase turns through half a circle as the frequency passes the pair — the same half circle at any damping, only faster or slower. Group delay is the slope of that phase against frequency, so its integral over all frequencies is fixed:

$$\int_0^\infty \tau(\omega)\,d\omega = 2\pi$$

for every $\zeta$. The mirror always adds the same amount of delay, measured as an area under the delay curve. What the damping decides is where that area is put. A heavily damped pair spreads it across several decades, starting at $4\zeta/\omega_0$ at direct current; a lightly damped one concentrates it into a peak of height about $2/(\zeta\omega_0)$ and width about $2\zeta\omega_0$ around the pair’s frequency, and in the limit of zero damping it is a spike of area $2\pi$ at exactly the frequency the pair removes.

The phase a decibel buys found an area theorem of the same kind for minimum-phase networks: the area under the phase across decades is fixed by how far the gain moves, however the network is arranged. The all-pass’s version is that the area under its delay is fixed by how many zeros it carries. In both, a design choice moves a fixed quantity around rather than creating or removing it, and the interesting question is always whether it ends up where the signal is.

For a notch the answer is no, by construction. The pair’s lightly damped delay is concentrated exactly where the pair’s own magnitude has cut the energy, so the energy-weighted average, the quantity the centroid identity says the time domain sees, multiplies a tall narrow peak of delay by a deep narrow trough of energy. The product is small, and it goes to zero as the damping does. That is the precise sense in which a notch’s zeros have no side: not that the mirror adds no delay, but that it adds all of its delay at a frequency the network has removed.

A step that is late, never wrong-way #

The energy that arrives first found one mirrored zero making a step start the wrong way — down to −0.170 of its final value before it turned round — and the two mirrors that change places found that the network with both of its real zeros mirrored did not start the wrong way at all. A complex pair is the same case.

The initial direction of a step is set by the sign of the first non-zero coefficient of the response’s expansion near $t = 0$, which is the ratio of the numerator’s and denominator’s leading coefficients. Mirroring one real zero flips the numerator’s leading coefficient and the step starts the wrong way. Mirroring a pair flips the sign of the middle coefficient only; the leading coefficient is untouched, so the step starts the right way. What is left is the delay: the mirrored step crosses half its final value 10.0 µs late, close to the 10.6 µs the all-pass gives at low frequency, and the two traces are never more than 5.2 per cent of the final value apart.

So the test the energy that arrives first left open — deciding a network’s class from a step — cannot detect a mirrored complex pair by looking for a wrong-way start, at any damping. It can detect it only as a delay, and a delay is indistinguishable from the network being slower, unless something independent says how fast it should be. That independent thing is the magnitude: the delay the magnitude implies, through the phase the phase the magnitude already knows recovers, compared with the delay the step shows. The gap between them is the all-pass, and its size is $4\zeta/\omega_0$.

Where the pair’s damping comes from #

The damping of a zero pair is not usually a design parameter in its own right; it is what a circuit’s losses leave. A twin-T notch with perfectly matched parts has its zeros on the axis; a mismatch moves them off it, into one half-plane or the other according to its sign. The null that is stationary in nothing measured how a notch’s depth depends on its components’ errors; the depth is the zeros’ distance from the axis, read from a different side. A notch whose null is 40 dB deep has its zeros at a damping of order a hundredth, and on this essay’s law a mirror at that damping costs a delay of about $0.04/\omega_0$ — a few microseconds for a notch at a kilohertz, against a response that takes milliseconds.

So whether a mis-trimmed notch’s zeros have drifted into the right half-plane is, for every purpose but one, irrelevant. The magnitude does not see it; the step does not start the wrong way; the delay is a fraction of a percent of the response’s own time scale. The one purpose is the one where the phase matters by itself — a notch inside a feedback loop, where a right-half-plane pair adds phase lag around the null frequency, 180 degrees at the null itself, and where the margin it eats is the all-pass’s phase, not its delay.

The law, and what it is the law of #

Mirroring a complex pair of zeros multiplies a network by a second-order all-pass, whose delay is $4\zeta/\omega_0$ at low frequency and peaks near $\omega_0$. With the pair above the band the mirror delays the energy by $4\zeta/\omega_0$: 0.21 µs at $\zeta = 0.01$, 21.2 at $\zeta = 1$, for a pair at 30 kHz. A pair on the axis has no mirror. In general the energy’s centroid moves by the all-pass’s delay averaged over $|H|^2$, which fifteen pairs confirm to five parts in a hundred thousand; inside the band the delay sits in the notch, where the energy is not, and averages to 0.35 µs for a 3 kHz pair at $\zeta = 0.1$ whose peak delay is a millisecond. A mirrored pair never starts a step the wrong way; it makes it late.

Still open: several pairs, a loop with a notch in it, and the pair recovered from a measurement #

Several pairs at once. An elliptic filter has a pair per stopband null. With several complex pairs, each mirror is an all-pass of its own and their delays add, but the ordering question of the two-mirror essay returns: which subsets of mirrored pairs are ranked against which, and whether the crossing instants depend only on products of the pairs’ frequencies as they did for two real zeros.

A notch inside a loop. In a feedback loop the relevant cost of a right-half-plane pair is phase, and phase is not averaged over energy: the margin is read at one frequency. Whether a notch placed in a loop to remove a resonance costs more margin with its zeros slightly to the right of the axis than slightly to the left, and by how much per unit of damping, is the question a loop designer would actually ask.

The damping from a measured response. Given a measured magnitude and a measured step, the delay gap between them is $4\zeta/\omega_0$ for a pair above the band. Whether that gap can be measured precisely enough to recover a pair’s damping — and its side — in the presence of noise, is the test with a detection limit that the level the phase cannot know began for a single all-pass.