The direction a response is most sensitive to

The three tolerances that do nothing took the whole second-derivative matrix of a ladder’s magnitude at a ripple peak, diagonalised it, and read the list of numbers that came back: two eigenvalues of order one and three at $4\times10^{-9}$. That is the rank, it is the finding, and it is half of what a diagonalisation produces.

The other half is the eigenvectors, and the machinery has returned them since the day it was written. Nothing had read them. Every measurement in this anchor takes the eigenvalues and stops, so what the collection knew about a tolerance was how much the response moves and never along what — and along what is a statement about the ladder rather than about the arithmetic.

It matters for a second reason. That essay closed by naming its own limit: the matrix it diagonalised belongs to one frequency, and a filter specification is about a band. Reading the directions makes the difference measurable, because a direction at one frequency can be compared with a direction at another and an eigenvalue cannot.

What an eigenvector of a curvature matrix is #

Expand the response about the nominal design in the relative component errors $x_i = \delta p_i/p_i$. At a ripple peak the gradient vanishes, which is the result the tolerance that can only take away established, so the first surviving term is

$$\frac{\Delta |H|}{|H|} \approx \tfrac{1}{2}\, x^{\mathsf{T}} Q x$$

with $Q$ the relative curvature matrix. An eigenvector of $Q$ is a combination of component errors — every part moved by a stated fraction of one another — and its eigenvalue is the curvature along it. The eigenvector belonging to the largest eigenvalue is therefore the direction in which the response falls fastest for a given size of error, and its opposite is the direction in which a batch of components is least forgiving.

It is not a component. It is a recipe, and the recipe has a sign pattern in it.

Two routes to a direction, which is not the same check as two routes to a number #

The rung below computed its matrix twice — once from the adjoint identity, once from four full re-solves per pair of parameters — and required the two live eigenvalues to agree. That is a check on the eigenvalues and it says nothing about the eigenvectors, which is a different quantity and one that can be wrong while every eigenvalue is right: a matrix perturbed inside a near-degenerate pair moves its eigenvectors a great deal and its eigenvalues hardly at all.

So the check is made on the direction. The two routes share no adjoint solve, no transposed system and no derivative identity, and their leading directions come back the same to $7.6\times10^{-7}$ radians at fifth order, $9.9\times10^{-7}$ at seventh and $2.2\times10^{-6}$ at ninth. That is the calibration everything below is quoted against, and it is worth having before anything surprising is claimed.

The first departure: no direction serves two peaks #

The two directions above overlap by 0.0603. Two unit vectors at that inner product are 86.5 degrees apart, which is very nearly orthogonal, and the same measurement at seventh order gives a largest overlap of 0.2747 across its three peaks and at ninth order 0.3226 across its four.

The consequence is a practical one and it undoes the most natural reading of the rung below. That essay found the two live directions and said, correctly, that the part to tighten is not a part but a combination. If the combination were the same everywhere in the passband, buying it would be a design decision with one answer. It is not the same. The combination worth tightening at the lowest ripple peak is almost exactly the combination that does nothing at the highest, and the reverse.

Reading the two of them component by component says how differently. At 554.9 Hz the direction is $(0.385, 0.497, -0.459, 0.497, 0.385)$: the two end inductors and the two capacitors move together and the middle inductor moves against them, and no single part carries more than a quarter of the combination. At 897.9 Hz it is $(-0.061, 0.458, 0.758, 0.458, -0.061)$: the middle inductor now carries more than half of it and the two end inductors have very nearly dropped out. So the two peaks do not merely disagree about the recipe — they disagree about which part is worth the most, and the part the lower peak wants moved against the rest is the part the upper peak wants moved with it. A component engineer given one of those two answers and asked to buy against the passband has been given the wrong half of the problem.

The symmetry survives, and that is worth saying because it is the one thing that does not change. Every leading direction, at every peak and every order tried, is symmetric under reversing the ladder — the property the reading that does not care which way round it is measures directly on the network, appearing again in the statistics of its tolerances. Symmetry constrains the direction to a subspace; it does not pick a direction out of it, and the passband picks a different one at every peak.

The second departure: the band edge is where the sensitivity is #

Reading the eigenvalue that belongs to each of those directions gives a number nobody had asked for. It is not constant across the passband and it is not nearly constant.

A fifth-order ladder is 12.7 times more sensitive at its upper ripple peak than at its lower one. A ninth-order ladder is 136 times more sensitive at its top peak than at its bottom one, and the ratio grows with the order because the peaks crowd towards the band edge as the order rises.

That is an eigenvalue, so it is owed a measurement of the network. Moving a ninth-order ladder one per cent along each peak’s own most sensitive direction and re-reading the response there gives 2.51 parts in ten thousand at the lowest peak and 377.5 at the highest. The ratio is 150 rather than 136, and the difference is the honest part: at the top peak a one per cent move is large enough that the cubic term is no longer negligible, and the quadratic form under-predicts the measured fall by 11.2 per cent. Moved a tenth of a per cent instead, the same prediction is right to 0.024 per cent at the lowest peak and 1.6 at the highest.

Why it rises towards the band edge is visible in where the peaks are rather than in any expression. A ninth-order ladder’s ripple peaks sit at 335.9, 631.3, 850.6 and 967.2 Hz against a half-power frequency of one kilohertz, so their spacing falls from 295 Hz to 219 to 117: they crowd towards the edge, and the top one sits inside the region where the response has begun to turn over. A reflection zero with the skirt immediately above it is held in place by less of the response than one sitting in the middle of a flat passband, and the curvature is what that costs. That is an observation about four measured frequencies and three measured curvatures rather than a derivation, and it is quoted as one.

This changes what a tolerance analysis at a stationary point is worth. The stationary-point argument invites a measurement at a ripple peak, because they are all equivalent in the theory that produces them — every one sits at exactly half the source, and the halving is the structural constant the two resistors a ladder was designed between is about. They are not equivalent in their curvature, and an analysis done at the first peak understates the last one by two orders of magnitude.

Rank two, whatever the order #

The rank does not move.

A ninth-order ladder has nine reactances, nine tolerances and a curvature of rank two at every one of its ripple peaks. Seven of its nine directions do nothing there.

The count does not grow with the order because it is not a property of the order. A ripple peak is a reflection zero sitting on the imaginary axis; a point on the imaginary axis is two real numbers — where it is along the axis, and how far a perturbation has taken it off — and $n$ component errors map onto two numbers with an $(n-2)$-dimensional kernel however large $n$ is. The null space grows while the live subspace stays where it is, so the fraction of a tolerance budget that a ripple peak can see falls as $2/n$: three fifths of it is spent on nothing at fifth order and seven ninths at ninth.

That is a statement about a point in the band, and it is the statement the rung below could make. The band is the interesting object.

What the whole band cannot see #

Each peak is blind to $n-2$ directions and each peak is blind to a different $n-2$ directions. What is invisible to the passband as a whole is the intersection, and there is a clean way to compute it: sum the projector onto each peak’s two live directions over every peak in the band, and diagonalise the sum. Its kernel is what no peak sees.

Exactly one combination survives, at every order, and the arithmetic of why is the simplest thing in the essay. An odd-order ladder of $n$ reactances has $(n-1)/2$ interior ripple peaks. Each contributes two directions. That accounts for $n-1$ of the $n$, and one is left over — a coincidence of counting that is the same at fifth, seventh and ninth order because it is the same counting.

The direction the arithmetic leaves over is, at fifth order and at ninth,

$$(0.587,\; -0.264,\; 0.413,\; -0.264,\; 0.587)$$

$$\begin{aligned}&(0.473,\; -0.235,\; 0.317,\; -0.270,\; 0.307,\\ &\quad -0.270,\; 0.317,\; -0.235,\; 0.473)\end{aligned}$$

and the signs alternate along the ladder, which is the signature of a change in impedance level rather than in frequency: the inductive reactances rise while the capacitive ones fall, so every time constant is preserved and every impedance is not. It is not a pure scaling, because the components are not equal — a pure scaling would put $1/\sqrt{n}$ on every one — and that is the difference between this direction and the one the same filter a thousand times larger is about, which moves the terminations too.

The one thing a maximum cannot report #

Perturbing along that direction and looking at what actually changed identifies it outright.

The direction is the ripple depth. A Chebyshev ladder is fixed by four things — its order, its cutoff, its ripple and its impedance level — and its reflection zeros are placed by the first two alone. The ripple factor decides how deep the response dips between those zeros and how far down the stopband sits, and it moves neither zero. So there is a one-parameter family of ladders with identical reflection zeros and different ripple, every member of which passes through exactly half the source at exactly the same frequencies, and no measurement made at those frequencies can tell one member from another.

That is not a fact about the arithmetic and it is not a degeneracy. It is the definition of a doubly terminated ladder’s maxima arriving as a null space: the maxima are at one half because the network is matched there, and matching is a condition that says nothing whatever about the ripple.

The stopband number is the one to keep beside it. A move that is invisible at every maximum of the passband takes 2.32 per cent off the attenuation at twice the cutoff, which is not invisible anywhere and is the specification most filters are actually bought against.

Null to third order, and not to fourth #

A quantity measured at $10^{-10}$ is worth one paragraph of suspicion, for the reason the matrix that is ill, and the answer that is not exists to give: a small number produced by subtraction is what a cancellation looks like. The test is not a tighter tolerance, it is an exponent.

Four at the peaks and one everywhere else. The direction is null in the quadratic form and null in the cubic term above it, so doubling the move multiplies the departure by 14.2 rather than by four — and it is still $1.3\times10^{-7}$ at a move of 6.4 per cent, which is far outside any tolerance a real part is bought to. That is a null space rather than a small number, and the exponent is what says so. A quantity that merely happened to be small would grow as the square, like everything else in the matrix.

At the band edge and in the stopband the same direction is an ordinary first-order tolerance in the ordinary proportion. A combination that costs nothing where the specification is stationary costs full price everywhere the specification is not, and the two facts belong to one direction.

What an instrument that returns directions has to be able to do #

A direction is the easiest quantity in this collection to be wrong about without noticing. A wrong number looks wrong. A wrong vector of five numbers, normalised, symmetric under the ladder’s own reversal, looks exactly like an answer — and every downstream check a wrong one would face, that it is a unit vector and that it respects the network’s symmetry, a random vector passes.

The site’s other direction-valued machinery makes the point. A pole’s sensitivity is computed from the null vector of the nodal matrix at that pole, found by refining a starting vector against a solve; when that matrix came out exactly singular rather than singular to rounding, the solve declined it, the refinement could not take its first step, and the random starting vector was handed back unchanged. The sensitivity that followed was four orders of magnitude low, and nothing failed, because a random unit vector satisfies every assertion a real one does. The matrix is now shifted by one unit in the last place of its own scale before the vector is refined, and only after the unshifted solve has refused.

Which is why the two-route check above is on the direction and not on the eigenvalue, and why it is worth its four re-solves per pair. The equivalent habit is what every derivative, and the one that is zero applies to a gradient and the derivative of a root to a pole, and it is the third place in this anchor where checking that the answer is not an artefact of the arithmetic has been worth more than the answer.

What it does not say #

It does not say the null space is an economy. A batch of components does not vary along a chosen direction; it varies along all of them at once, and the two expensive directions at each peak are the ones a random spread lands in, which is what the tolerance that is not on any part is about from the other side. The kernel is a fact about the design, not a discount available to production.

It does not say a ripple specification is free. It says that the maxima cannot price it, which is a statement about where to measure rather than about what to buy — and a filter whose ripple is a specification has to be measured between its peaks, where the same combination is first order.

What it leaves a designer is narrower than the arithmetic and more useful than a caution. Three statements survive it. Measure at the top ripple peak, because every other one understates the tolerance the passband actually imposes, by up to two orders. Do not tighten a part: no eigenvector here is a component, the recipe changes between peaks, and buying a one per cent middle inductor buys against one peak and almost nothing against another. And price the ripple somewhere other than at a maximum, because the maxima are structurally incapable of reporting it.

And it is still one family. Every number here is a doubly terminated Chebyshev, because a Butterworth has no interior maxima at all: its passband is flat to rounding, so there is no stationary point away from direct current for any of this to be about. The comparison a ladder is not a cascade draws between realisations is the same limitation from the other direction, and it is why the argument here is visible in a rippled response and invisible in a maximally flat one.

The number worth carrying #

A ninth-order ladder’s passband has four ripple peaks, each blind to seven of its nine tolerances, each sensitive to a different combination from the others, and the curvature at the top one is 136 times the curvature at the bottom. Across the whole band exactly one combination is invisible, at every order, and it is the ripple.

The habit that goes with it is shorter. A stationary point is a good place to measure and a bad place to stop, because the properties that make it stationary are the properties it cannot report. When a measurement returns a null space, the next question is not how deep the null is but how wide — over how much of the band it holds — and the answer is usually that it holds at a point and nowhere near it.