The two mirrors that change places

The energy that arrives first built a low-pass with one real zero and set it beside its mirror, the same network with the zero moved across the axis into the right half-plane. The two had one magnitude to the last bit of a double, and so one impulse-response energy, and the minimum-phase network delivered that energy first at every instant: half of it by 11.27 microseconds, where the mirror took 63.23. That essay ended on the case it had not measured. A network with two zeros has not one mirror but three, and the ordering it found says only which of the four comes first.

This page measures the other three, and the answer is not a ranking.

Four networks from one magnitude #

The network is a low-pass with poles at 1, 10 and 100 kHz and two real zeros, at 3 kHz and 30 kHz, one between each pair of poles. It is built from buffered sections so that nothing loads anything else: a resistor and capacitor for the first pole, then two leads, each with its direct-current loss restored, carrying one zero and the next pole apiece. Three poles and two zeros make it strictly proper, so its impulse response starts at zero and carries a finite energy.

Moving a zero to the other side of the axis is done the same way it was done before, with a first-order all-pass behind the network at that zero’s frequency. The all-pass’s pole lands exactly on the zero and cancels it, and its own zero, in the right half-plane, takes its place. With two zeros there are four choices — both left, the lower one mirrored, the higher one mirrored, both mirrored — and all four have numerators $(1 \pm s/\omega_1)(1 \pm s/\omega_2)$ over one denominator. Each factor has the same magnitude on the imaginary axis whatever its sign, so all four networks share one magnitude, to 6.7 × 10⁻¹⁶ at every frequency compared, and one impulse-response energy, 9261.008.

The level the phase cannot know found that feeding the phase of any of the three mirrors into Bode’s companion integral returns a magnitude that is not theirs, falling forty decibels a decade faster above each all-pass. That is the frequency-domain face of the difference between them. This page is about the time-domain face.

The first and the last #

The two outer curves behave as the single-zero pair did. The network with both zeros on the left is above every other curve at every instant drawn, which is Robinson’s theorem: among all causal responses of one magnitude, the minimum-phase one has the most energy delivered by every time. The network with both zeros on the right is below every other curve at every instant drawn, by at least the last bits of a double. Half the energy arrives by 2.13 microseconds on the left and 11.2 on the right, and nine tenths by 50.2 and 167.

Those two facts together make a tempting picture, in which each mirrored zero adds some fixed quantity of delay and the four networks line up by how much delay they have been given. The two middle curves refuse it.

The two that are not ranked #

Mirror the lower zero alone and half the energy arrives by 2.80 microseconds, barely later than the minimum-phase network. Mirror the higher zero alone and it arrives by 10.1, nearly as late as with both mirrored. By that measure, the higher zero is the one whose side matters.

Ask about nine tenths of the energy instead and the order reverses. Mirroring the lower zero delays the nine-tenths point to 156 microseconds; mirroring the higher delays it only to 60.9. By that measure, the lower zero is the one whose side matters, and by a factor of two and a half.

Both statements are true of the same two networks, because their energy curves cross. Before 12.64 microseconds the network with its lower zero mirrored has delivered more; after it, less. There is no answer to the question of which mirror is the more delayed until a fraction of the energy is named, and no fraction is more natural than another. The ordering of the four networks is two chains that share their ends — first, then either middle network, then last — and not a line.

The shapes of the two middle curves say why each wins where it does. Mirroring the lower zero leaves the fast part of the response almost untouched: its curve follows the minimum-phase network’s up to a little over half, then stalls on a plateau while the slow remainder waits on the 1 kHz pole. Mirroring the higher zero does the opposite: the curve stalls early, near a quarter, and then catches up with the minimum-phase network’s tail. A zero among the fast poles decides the fast part of the energy, and a zero among the slow poles decides the slow part, and which of those a half-energy time measures depends on where the half falls.

The instant they swap #

The crossing has an exact explanation, and it is short enough to give in full.

The two middle numerators are $(1 - s/\omega_1)(1 + s/\omega_2)$ and $(1 + s/\omega_1)(1 - s/\omega_2)$. Multiplied out, both are $1 \mp c\,s - s^2/P$, with $c = 1/\omega_1 - 1/\omega_2$ and $P = \omega_1\omega_2$: they differ only in the sign of the middle term. Writing $g$ for the impulse response of the all-pole part and $g'$, $g''$ for its derivatives — each power of $s$ in a numerator becomes a derivative in time — the two impulse responses are $A \mp c\,g'$ with $A = g - g''/P$, and the difference of their squares is $-4c\,g'A$. Integrated from zero to $t$, both halves integrate exactly — $\int g' g = g^2/2$ and $\int g' g'' = g'^2/2$ — so the difference of the energies delivered is

$$E_{\text{low mirrored}}(t) - E_{\text{high mirrored}}(t) = -2c\left(g(t)^2 - \frac{g'(t)^2}{P}\right).$$

Two things follow. The factor $c$ stands outside, so the ratio of the zeros changes how far apart the two curves are and cannot change where they cross. And the crossing is where $g'/g = \sqrt{P}$: the instant at which the all-pole network’s impulse response is rising at a fractional rate equal to the geometric mean of the zeros, in radians a second. For zeros at 3 and 30 kHz that is 12.64 microseconds, found from the three poles alone, and it agrees with the crossing of the two solved energy curves.

The figure is that argument tested on solved networks rather than on the algebra. Three pairs of zeros with ratios of four, ten and twenty-five and a common product of 90 kHz² give three gap curves of different heights — the widest ratio makes the biggest difference between its two mirrors — crossing zero at one instant, agreeing to fifteen decimal places. A fourth pair with four times the product crosses earlier, at 8.199 microseconds, because the all-pole response rises through a larger fractional rate sooner.

Across a range of products spanning more than a decade, three ratios give one curve to fourteen decimal places. The curve falls as the product rises, more slowly than the inverse square root the very earliest instants would give. Just after an impulse, a response three poles deep rises as $t^2$, so its fractional rate is $2/t$ and the swap would come at $2/\sqrt{P}$; further along the response is no longer a pure power and its rate falls away more slowly. The start a step takes from infinity measured that early power law directly, from the initial-value theorem applied again and again, and the swap instant is where the power law hands over to the poles.

A second pair, further apart #

The crossing’s position depends on the product, and the rest of the picture depends on where the zeros sit among the poles. Moving the lower zero down to 1.2 kHz, close to the slowest pole, changes the picture’s proportions without changing its shape.

With the lower zero almost on the slowest pole, mirroring it barely moves the half-energy time — 1.20 microseconds against 1.17 — and moves the nine-tenths time by a factor of nine, from 9.41 to 84.0. The zero shapes almost nothing fast and almost everything slow. Mirroring the higher zero now ties the half-energy time exactly with mirroring both, at 6.52 microseconds, and parts from it only in the tail. The swap has moved to 16.3 microseconds, because the product has fallen from 90 to 36 kHz².

The energy has grown more than fourfold, to 42,750.88, which is the magnitude doing what a zero low in the band does: lifting the gain across every decade above it until the next pole brings it back. All four networks carry the new total, because all four still share one magnitude.

Steps that count the zeros on the right #

The earlier essay’s second reading was the step response, and its finding was that a mirrored zero makes the step start the wrong way. With two zeros that finding needs restating.

The two single mirrors start the wrong way, as the single-zero mirror did: down first, to −0.1777 with the lower zero mirrored and to −0.0576 with the higher, then up to one. The network with both zeros mirrored starts the right way. It rises to 0.0545 by two microseconds, turns, falls through zero to −0.1743, and only then recovers.

The direction a step sets off in is decided by the highest power of $s$ in the numerator, because just after the step that term is the one the response is made of. For two zeros that coefficient is $\pm 1/P$, with the sign $(-1)^k$ for $k$ zeros on the right: one mirrored zero makes it negative and the step goes down; two make it positive again and the step goes up. So a step that starts in the right direction is not evidence that a network is minimum phase. The earlier essay’s test — a step that starts the wrong way proves a zero on the right — is still sound, and its converse was never claimed; here is the case that shows why it could not be.

What does count the zeros is how often the step crosses its starting level. For each real zero $z$ on the right, the step response $y(t)$ satisfies $\int_0^\infty y(t)e^{-zt}\,dt = H(z)/z = 0$, so $y$ must change sign at least once to make that integral vanish; with two such zeros it must change sign at least twice, and the network with both mirrored does exactly that. The single mirrors cross once each.

And the depths do not follow the energy. The network with both zeros mirrored is last at every instant in the energy figure, yet its undershoot, −0.1743, is shallower than the lower zero’s alone, −0.1777. Deepest undershoot and latest energy are two different orderings of the same four networks, and the early rise the single mirror does not have is where the difference shows.

With the lower zero near the slowest pole everything is larger — the lower zero’s undershoot reaches −0.5406 — and the structure is the same. Both-mirrored rises first, now to 0.1386, then falls to −0.5361, shallower again than the lower zero alone. The early rise has grown with the undershoot, since both come from the same two numerator terms and the lower zero now dominates both.

What a measurement of either kind can say #

Every one of these networks passes the same noise, because the bandwidth noise sees is an integral of the squared magnitude, and every one has the same energy. A measurement of gain alone cannot rank them, and the earlier essays on minimum phase found exactly that from three directions.

A measurement in time can, but only partly. The first and the last of the four are fixed at every instant, so a measured energy curve that lies above another at every instant is evidence about which has more of its zeros on the left. The middle two are not fixed, so two networks whose energy curves cross cannot be put in order by their zeros’ sides without knowing where the zeros are. And a step’s direction reports the parity of the zeros on the right, not their number, while its crossings of the starting level put a floor under their number.

The same distinction appears in feedback. The frequency a device sets for itself finds a right-half-plane zero inside one transistor stage, carried by the collector-base capacitance. Two such stages in cascade have two, and by the argument above the pair’s step can start in the right direction and still cross its starting level twice. A loop built around that pair would see its output go the right way first, which is exactly the case a designer looking for the wrong-way start would not flag.

Flat delay, bought with more delay adds all-pass sections on purpose, and each of them mirrors a zero into the right half-plane in this sense. What this page adds to that essay’s price — that straightening a delay always adds delay — is that the delay added is not one number either: two all-pass sections at different frequencies delay different parts of the energy, and which of two equalisers is the slower depends on which part of the response is being asked about. What a steep skirt costs prices selectivity in overshoot, and the overshoot there is measured on a step; here the step and the energy have been seen to disagree about ordering, so a price quoted in one of them is not a price in the other.

How the numbers were obtained #

Each network is solved by nodal analysis; its transfer function is recovered from the nodal matrix by sampling the determinant on a circle, and its impulse response is the sum of exponentials over its poles with residues from the recovered polynomials. The energy delivered by each instant is the closed-form double sum over pairs of poles, and the total is checked equal across the four networks; the four magnitudes are compared by solving each network at sixty frequencies. Half-energy and nine-tenths times are bisected on the closed form. The swap instant is found as a sign change of the difference of two energy curves on 801 logarithmically spaced instants and then bisected, and separately as the instant at which the all-pole impulse response’s fractional rate of rise equals the square root of the product of the zeros in radians a second, from its three poles alone. Step responses come from the same residues and independently from marching each network by the trapezoidal rule, and their lowest points are read from a uniform grid of 4001 instants that starts at zero.

What it leaves out #

It uses two real zeros. A complex pair of zeros moves across the axis together — mirroring one of a conjugate pair alone would give a network with complex coefficients — so a network with a zero pair has two members, not four, and the question of ordering inside it does not arise in the same way.

It places each zero between a pair of poles. A zero above the highest pole, or two zeros between the same two poles, changes which part of the energy each zero decides, and the closed form above still says where the middle two swap, but not whether the plateau-and-tail picture that explains which one wins where still holds.

And it keeps the relative degree at one. With four poles and two zeros the all-pole response rises as $t^3$ at first, the swap instant at large products would go as $3/\sqrt{P}$ rather than $2/\sqrt{P}$, and whether the middle two still cross exactly once is not shown.

Still open: three zeros, a complex pair, and the energy curve as a test #

Three zeros and eight networks. The all-left and all-right networks should still bound the others at every instant, and the six between form a lattice ordered by inclusion: which zeros are mirrored. Whether every pair that is not ordered by inclusion crosses exactly once, and whether each crossing still depends on a product of the zeros involved, would say whether the closed form here is a fact about two zeros or about any number.

A complex pair. A resonant zero pair in the left half-plane and its mirror in the right have one magnitude and very different steps, and the pair’s damping decides how much of the energy the mirror delays. Measuring the half-energy and nine-tenths times against the zero pair’s quality factor would extend these orderings to the zeros a notch filter and an elliptic filter actually have.

Ordering a network whose zeros are unknown. Given a measured energy curve and a candidate minimum-phase twin built from a measured magnitude, the gap between the two curves is a measurement of excess phase in time. Whether the gap’s shape — where it is largest, and whether it closes before the tail — identifies which zeros are on the right, as the gain gap identifies an all-pass’s frequency, is the test this page’s closed form suggests and does not run.