The energy that arrives first

The phase the magnitude already knows showed that for one class of network the phase is not a separate measurement: fed only the magnitudes of a solved lead, Bode’s integral returns its phase to two hundredths of a degree. The phase a decibel buys turned the same relation into an exchange rate, ninety degree-decades of phase for every twenty decibels of gain change. Both were statements about frequency, and both ended at the same boundary: a network with an all-pass in it has the same magnitude and a different phase, and nothing that reads only the magnitude can tell the two apart.

The class has a name that none of that explains. A minimum-phase network is called that because, among all networks sharing its magnitude, it lags least. A lag is a delay, and a delay is a statement about time. So the class should have a face in the time domain that no frequency plot shows, and it does: the minimum-phase network delivers its energy first.

Two networks of one magnitude #

The recovery figure’s pair is the place to start, because it is the pair that cannot be told apart by magnitude.

That pair has a difficulty for a question about time: a lead passes its input straight through at high frequency, so its impulse response begins with an impulse, and an impulse has no finite energy to deliver on any schedule. The pair this page measures fixes that by putting a pole in front. A buffered resistor and capacitor at 1 kHz, then a buffered section with a zero at 3 kHz and a pole at 10 kHz, gives a low-pass that falls away at the top and passes one at direct current:

$$H(s) = \frac{1 + s/\omega_z}{(1 + s/\omega_1)(1 + s/\omega_2)}.$$

The mirror is the same network with the recovery figure’s all-pass at 3 kHz behind it. The all-pass’s pole lands exactly on the zero and cancels it, and its own zero — at +3 kHz, in the right half-plane — takes its place, so the mirror is $(1 - s/\omega_z)$ over the same denominator. Every factor has the same magnitude on the imaginary axis as before, and the two networks’ magnitudes agree at every frequency sampled to 4.4 × 10⁻¹⁶.

The same energy, by a theorem and by a solve #

Parseval’s theorem says the energy of a signal is the same whether it is added up in time or in frequency: the integral of $h(t)^2$ over time equals the integral of $|H(j\omega)|^2$ over frequency. Two networks with equal magnitudes therefore have impulse responses of equal energy, before either response has been computed.

Computed, they have. From the residues of each transfer function — the impulse response written as a sum of decaying exponentials, and the energy up to any instant as a double sum over pairs of them in closed form — both totals are 6029.319. The integral of $|H|^2$ over ten decades of frequency, which never forms a residue, gives 6029.305, two parts in a million away. The magnitude has decided how much energy each network’s impulse response carries, to seven figures, and has nothing to say about when.

It is worth pausing on what else that settles. The bandwidth noise sees is an integral of $|H|^2$, and so is every noise voltage built on it. A network and its mirror pass exactly the same white-noise power, and no measurement of noise can tell a zero on the left from a zero on the right.

When it arrives #

The two curves start at zero and end at one, and between them the minimum-phase curve is above the mirror’s at every instant drawn. Half the energy has arrived 5.61 times sooner. By twenty microseconds the minimum-phase network has delivered two thirds of everything it will ever deliver and its mirror a third; by a hundred microseconds nine tenths against two thirds.

That ordering is a theorem, due to Robinson for sampled sequences and holding for continuous responses as well: among all causal responses sharing a magnitude, the minimum-phase one has the largest partial energy at every time. What the figure adds is its size. The theorem is an inequality and says nothing about by how much; here the same total arrives five and a half times later, on a network whose magnitude plot is indistinguishable.

The mirror’s curve also has an instant at which it delivers nothing at all. Its step response turns round at 20.69 microseconds, and the slope of a step response is the impulse response, so at that instant the mirror’s impulse response is exactly zero and its energy curve is momentarily flat. The minimum-phase network has no such instant: its impulse response never changes sign, which is the same statement as its step response never turning back.

The step that starts the wrong way #

A zero on the right makes the network’s output the low-pass response less a scaled copy of its own derivative, where a zero on the left adds that copy. At the instant a step arrives the response of the low-pass has not moved but its derivative has, so for a short while the derivative is most of what comes out, with a minus sign in front of it. The mirror’s output therefore sets off downward, reaches −0.170 of its final value at 20.69 microseconds, and only then turns towards where it is going.

That is the behaviour a right-half-plane zero is known for in feedback design, where it is the reason a loop containing one cannot be made fast by adding gain: a controller that sees its output move the wrong way corrects in the wrong direction. The frequency a device sets for itself finds exactly such a zero inside an ordinary common-emitter stage, where the collector-base capacitance carries the input’s change straight to the output before the transistor has responded to it.

Nothing about the magnitude warns of it. The two step responses end at the same value, pass the same noise and have the same energy, and one of them spends its first thirty microseconds heading the wrong way.

Where the zero sits #

The size of both effects depends on where the zero is against the poles, and moving it across three decades shows which way.

The two faces of a right-half-plane zero move in opposite directions as the zero moves, and they are not one number.

With the zero far below both poles, the zero shapes almost everything about the response. Each network’s impulse response begins with a large spike from the derivative term — upward for the zero on the left, downward for its mirror — and the spike carries most of the energy for both, at nearly the same instant. Half the energy has arrived by 5.03 and 5.16 microseconds, three per cent apart. The step responses are another matter: the mirror’s goes to −2.69, nearly three times its final value in the wrong direction.

With the zero far above both poles, the zero hardly shapes anything. The mirror’s step undershoots by half a per cent, which a noisy measurement would miss, and yet it delivers half its energy sixteen per cent later than the network it mirrors. Between the poles, at 3 kHz, the delay was five and a half times.

So a low zero is mostly undershoot and a high zero mostly delay, and the place where one is small is not the place where the other is. A check that looks for a step going the wrong way finds the low zero and misses the high one; a measurement of when a response’s energy arrives finds both, but needs the minimum-phase twin to compare against, which is exactly what a measurement does not have.

At 300 Hz the curves nearly coincide for the first few microseconds and part only in the tail: the mirror has delivered 87 per cent by twenty microseconds against 93, and 94 per cent by a hundred against 98. The theorem’s ordering still holds at every instant — the minimum-phase curve is never below — and what the lower zero has changed is where in time the difference lives. Most of each response’s energy is in its opening spike, which the two networks share apart from its sign, and the part the zero’s side decides is the slower remainder that follows.

The totals are worth one more look. Fifty times the energy of the 3 kHz pair from a zero moved a decade is the magnitude doing exactly what one solve, read four ways says a zero does — the gain rises twenty decibels a decade above it until the next pole — and the residues and the frequency integral agree on the new total to six parts in a million.

With the zero a decade lower, the minimum-phase network is no longer well behaved either. Its step response rises to 2.74 times its final value before it comes back, and its mirror falls to −2.44 before it rises. Both excursions come from the same derivative term, with opposite signs, and they are nearly the same size. Minimum phase is not a promise of a gentle response: a zero below the poles makes a large overshoot on the left and a large undershoot on the right, and the only thing the side decides is the direction.

The direction is the whole of the difference for anything that acts on the output. What a steep skirt costs prices a filter’s selectivity in overshoot, and an overshoot is at least a movement towards where the output is going. An output that first travels 2.44 times its final distance the other way is a different kind of object for a comparator, a limiter or a controller, which is why a right-half-plane zero in a loop is treated as a limit on how fast the loop can be made rather than as one more pole to compensate. The arrow that goes past where it settles found an overshoot that no phasor predicts; this is an undershoot that no magnitude predicts, and the two are the same kind of omission.

What reads the difference, and what cannot #

Every measurement that is a statement about magnitude is blind to this page. A swept-sine gain measurement, a noise measurement, a spectrum analyser’s display, the transfer power of a stage: each is an integral or a sample of |H|, and each returns the same answer for the network and its mirror to the last bit. The phase the magnitude already knows made the strongest version of that point from the frequency side — its integral, fed the mirror’s magnitudes, returns the minimum-phase network’s phase and is wrong about the mirror by exactly the all-pass’s own phase — and a measurement that reads only magnitudes inherits the same blindness, however careful it is.

A phase measurement sees the difference, if it is trusted over enough band to find phase that never comes back. A time measurement sees it immediately and without needing a reference. The step either starts in the direction it is going or it does not; the energy either arrives with the network’s fast poles or waits on the slow part of the response. Neither reading needs the twin to compare against, and that is the practical asymmetry: a frequency measurement can only show that a network is consistent with being minimum phase, while a step that goes the wrong way shows that it is not.

The limit of the time route is the one the sweep drew. A mirrored zero far above the poles moves the energy late and the step by half a per cent, so the step’s verdict is only as good as the resolution of the step, and the energy’s verdict needs the minimum-phase twin to compare with. Between them they cover the two ends of the zero’s range, and in the middle — a zero among the poles, as at 3 kHz — both are large.

What the all-pass adds, which is delay and nothing else #

The mirror is a network with an all-pass in it, and an all-pass is a pure delay element in every sense except the simplest: it passes every frequency at full strength and delays each by its group delay. Flat delay, bought with more delay uses all-pass sections deliberately to straighten a filter’s delay without touching its magnitude, and the price named there — that straightening delay always adds delay — is the same fact as this page’s, seen in frequency. A first-order all-pass’s group delay is largest at low frequency, where the energy of a low-pass’s impulse response mostly is, and so it moves the bulk of that energy later.

And a length of line is the limit of the same thing. The staircase in time measures a metre of cable as 4.83 nanoseconds of delay whose phase grows in proportion to frequency: an all-pass with infinitely many sections, changing no magnitude and delaying every frequency equally. A network is minimum phase exactly when it contains none of that, and the energy figure is the measurement of how much delay a single first-order section of it holds.

How the numbers were obtained #

Each transfer function is recovered from the network’s own nodal matrix by sampling its determinant on a circle, and its impulse response is the sum of exponentials over its poles with residues N(p)/D′(p). The energy delivered by each instant is the closed-form double sum over pairs of poles, and the total is checked against the integral of $|H|^2$ over ten decades of frequency, taken on 8,001 solves of each network. The two magnitudes are compared by solving both networks at a common set of frequencies. Each half-energy time is bisected on the closed form. The step responses come from the same residues and, independently, from marching each netlist by the trapezoidal rule; the lowest point of a step response is read from 601 evaluations of the closed form.

What it does not say #

It mirrors one zero. A network with k real zeros has $2^k$ networks sharing its magnitude, one for each choice of side, and the ordering this page measures says only that the all-left choice delivers first; how the others rank against each other is not measured here.

It uses strictly proper networks, because the question needs a finite energy. A network with a direct path, like the lead of the recovery figure, still has a minimum-phase member and a mirror, but its impulse response begins with an impulse and the comparison has to be made on the remainder.

And it treats a zero exactly on a pole as a case to step around. There the minimum-phase network loses an order and the mirror keeps it, and the residue expansion used here does not apply to the repeated pole the mirror then has.

Still open: the level the phase cannot know, the most-delayed mirror, and a measured step #

The magnitude from the phase. The integral in the first two essays runs from magnitude to phase; its companion runs the other way and returns the log-magnitude of a minimum-phase network from its phase alone, except for one additive constant no phase measurement contains. Measured on the lead, it would find the shape of the gain to the precision of the sweep and its level wrong by exactly that one number.

The mirror that delivers last. With several zeros, the network with every zero on the right is maximum phase, and it should deliver its energy last of all the networks sharing its magnitude. A network with two zeros has four mirrors; ranking their half-energy times and undershoots would say whether the ordering is the simple one or whether a zero’s position matters more than its side.

Deciding the class from a step. A step that starts the wrong way is sufficient evidence of a right-half-plane zero and not necessary evidence — a mirrored zero far above the poles undershoots by very little. How small an undershoot a noisy measured step can resolve, against where the zero sits, would turn this page’s sweep into a test with a detection limit.