The selectivity that is not free

What a steep skirt costs replaced the adjectives in the filter table with measurements, and the currency it found was delay: the steepest of the three all-pole families distorts delay eight hundred times more than the gentlest. The zeros that buy an order then added a fourth family that does not fall because its denominator grows, and found the price of that in the same picture — a stopband that stops falling.

Both essays priced a trade with two quantities in it: how steep, and what it costs. The elliptic family does not have a trade with two quantities in it. It has three, and the third is not chosen by anybody.

A designer holds a passband ripple, a transition band — where the stopband is required to begin — and a stopband attenuation. An order fixes a relation between the three. Choose any two and the third is returned by the degree equation, which is a statement about elliptic integrals and not a design decision. The machinery for it has been in this collection since the elliptic family arrived, returning that third number on every call, and only one value of the second has ever been drawn.

What is being solved #

The elliptic approximation places its poles and its transmission zeros through the Jacobi function $\mathrm{cd}$, evaluated by descending Landen transformation. The quantity that decides everything is the ratio of the complete elliptic integral to its complement, and the degree equation says that the order multiplies it. Written through the nome $q = e^{-\pi K'/K}$ it is one line, $q_1 = q^{\,n}$, and the modulus $k_1$ recovered from $q_1$ then gives the stopband ripple factor $\varepsilon_s = \varepsilon_p/k_1$.

So the specification a reader would write down —

$$A_s = 10\log_{10}\!\left(1 + \varepsilon_s^2\right)$$

— has the passband ripple in it through $\varepsilon_p$, the order and the transition width through $k_1$, and nothing else at all. There is no term for the topology, no term for the components and no term a designer can lean on.

The shape of that figure is the whole argument in one picture, and the part worth staring at is the right-hand end. A stopband asked to begin at 1.05 times the corner — a transition band five per cent wide — is worth 21.3 decibels at order five and half a decibel of ripple. That is not a filter anybody would ship, and it is exactly the specification a person writes when the trade is presented as two quantities and the third is assumed to be free.

The trade is not a trade, it is an addition #

The five curves in that figure look parallel. They are parallel, exactly, and that is a stronger statement than it appears.

Because $\varepsilon_s = \varepsilon_p/k_1$ and $k_1$ depends only on the order and the transition width, the stopband in decibels is the ripple’s contribution plus the geometry’s, and the two do not interact. Relaxing the passband ripple from a half decibel to three buys 9.12 decibels of stopband, and it buys the same 9.12 whether the transition band is five per cent wide or a hundred. Relaxing from a tenth of a decibel to a whole one buys 10.46, everywhere.

That is a designer’s exchange rate and it is a number rather than a direction. A specification that is two decibels short of its stopband can be met by relaxing the ripple by an amount that is calculable in one line, without re-running anything — provided the stopband is deep enough for the identity to hold, and the figure says where that is. Below about ten decibels of stopband the two ways of writing the same quantity part company, because $10\log_{10}(1+\varepsilon_s^2)$ is only $20\log_{10}\varepsilon_s$ when $\varepsilon_s$ is large. The collection’s own machinery has that distinction written into its docstring — the two differ by 0.14 dB at fifteen decibels of stopband and by nothing at forty — and this figure is the first thing that makes the difference visible rather than merely recorded.

The order is the only integer in it #

The three quantities are continuous and the fourth thing in the relation is not, and that turns out to be where most of the slack in a real design is hiding.

In decibels the stopband is very nearly linear in the order. Differencing the degree equation’s own output across orders gives 11.9714 decibels per order with the stopband beginning at 1.25 times the corner, 15.5496 at 1.67 times and 17.4539 at twice. Those slopes are not fitted and they are not a rule of thumb; they are the ratio of the complete elliptic integral to its complement, because $k_1$ falls as $q^{\,n/2}$ and so its logarithm is linear in $n$ with slope $10\pi K'/(K\ln 10)$. Evaluated directly that expression gives 11.9716, 15.5496 and 17.4539 — five figures against a numerical difference of a machinery that computes $k_1$ through a nome, twenty terms of a theta series and a bisection, sharing only the elliptic integral itself.

So the order a specification needs is a real number, and it is almost never one. Half a decibel of ripple, a stopband beginning at 1.25 times the corner and forty decibels of rejection needs order 5.1101. Nobody builds a filter of order 5.1101. Rounded up to six, the same design returns 50.652 decibels — 10.65 decibels more than was asked for, which is not a margin anybody chose and is not free either, since it is a whole extra section.

That windfall is worth spending rather than banking, and the exchange rates above say exactly what it buys. Kept at order six and held to forty decibels, the stopband may instead be required to begin at 1.127 times the corner rather than 1.25 — bisected on the degree equation, and a fifth of the transition band given back. Or, keeping the transition band, the passband ripple may be tightened from half a decibel to 0.0454, which is an order of magnitude of flatness for nothing. Or the difference may be taken in settling time, which the last two figures of this essay price.

None of those three is visible from a design flow that computes an order, rounds it up and stops. What makes them visible is that the relation has three continuous quantities in it and one integer, so rounding the integer up always leaves something on the table, and which of the three it is returned in is a decision. The habit is the one the same filter a thousand times larger applies to impedance: a quantity that looks like a consequence is usually a free parameter that nobody noticed had been set.

The reproduction, which is also a warning #

Before the selectivity is swept it has to be established that a design at a selectivity nobody has drawn is still the filter it claims to be. The comparison the family arrived with is the instrument.

The design and the network agree to a tenth of a decibel, which is what makes every number after this attributable to the sweep rather than to the synthesis. And the crossover has moved: at the collection’s usual selectivity the all-pole family overtakes at about two corners, and asking for the steeper skirt has pushed it out to 2.9 while making the elliptic response eighteen decibels deeper in between. Both halves moved, in the same direction, for one change of one parameter.

The stopband is not a slope #

The word “attenuation” hides what the elliptic stopband actually is, and the third quantity is meaningless without it.

Twenty-eight decibels of stopband is bought and sold in that one picture, and neither the nulls nor the floor is where a reader of the specification would look. The nulls are extremely deep — they are genuine transmission zeros and what actually fills a null measured what it takes to fill one, which is loss in the arm and not component tolerance — and they are also narrow, so a signal three per cent away from one gets nothing from it. What a stopband requirement is actually about is the level between them, and that level is the number the degree equation returns.

There is a second reason the level and not the nulls is the right thing to specify, and it is about what a stopband is for. A null rejects one frequency; a stopband rejects a band, and what gets through a band is set by its worst point. So an elliptic design’s rejection of a broadband interferer is its floor, not its nulls, and the two differ here by more than sixty decibels. The only case in which the nulls are the answer is a single interfering tone whose frequency is known and stable — a mains harmonic, a local oscillator — and that is a notch problem rather than a filter problem, priced separately by what actually fills a null and by the inductor that is a capacitor, which is what decides whether a real arm can hold a null where it was put.

Where the floor is in the circuit #

The floor is not a property of the approximation that then has to be realised. It is a capacitance ratio, and it is visible in the netlist.

Each section that carries a transmission zero splits its capacitance in two. The part across the inductor makes the series arm an open circuit at $\omega_z = 1/\sqrt{LC_p}$, which is the zero; the total sets the pole pair at $\omega_0 = 1/\sqrt{L(C_p+C_2)}$, which is therefore always below it. Far above both, the inductor is out of the picture and the section is a capacitive divider passing $C_p/(C_p+C_2)$ — which is the same ratio, and is exactly $(\omega_0/\omega_z)^2$.

Two routes to one number, sharing the netlist and nothing else: a product of capacitance ratios read out of six element values, and a magnitude response solved at a frequency. They agree to 0.02 of a decibel. The half-decibel offset against the degree equation is the normalisation the family’s own machinery warns about — the equation refers its stopband to the top of the ripple band, and at an even order direct current is the bottom of it — and it is exactly the ripple, at every order and every selectivity checked.

The odd orders are the more interesting frame. At order five the two notch sections hold 18.42 decibels between them and the stopband measures 38.68, which is not a discrepancy: an odd-order design has an unpaired real pole, its section is still falling where the notch sections have flattened, and the stopband level is the two together. A thousand corners out that response is 85 decibels down and going, because there is nothing to hold it up. So an elliptic filter of odd order does not have a floor at all, and the sentence “the elliptic stopband comes back up” is true of the even orders and true only near the band edge of the odd ones. That is the sort of distinction the a ladder is not a cascade rung is about arriving from the other side: what a realisation does is not always what the approximation says.

The fourth quantity, which the degree equation does not mention #

Three quantities are in the specification. There is a fourth in the part, and no line of the degree equation contains it.

The overshoot does not move. Across a transition band narrowed by a factor of nineteen it stays between 13.6 and 15.2 per cent, and it is not even monotone. Anybody pricing the time-domain cost of selectivity by looking at the first peak — which is what an overshoot figure is — would conclude that selectivity is free in the time domain.

It is not. It is paid in the tail, where nobody photographs it.

A transition band nineteen times narrower costs 2.57 times the settling time to a tenth of a per cent, and 2.76 times to a hundredth. The one per cent figure moves less — 4.91 milliseconds to 9.68 — which is the same lesson as the overshoot: the tighter the tolerance the measurement is taken at, the more of the price is visible, because the price is entirely in how long the last small error takes to go away. This is the quantity the cliff before the fastest settling is about, met from the frequency domain rather than from the damping ratio.

The tail is one pole, and its residue is why the growth is not larger #

The mechanism is worth separating from the measurement, because the two numbers involved pull in opposite directions.

Narrowing the transition band moves the band-edge pole nearer the imaginary axis. Its quality factor goes from 5.28 to 18.03 across the sweep and its time constant from 1.737 milliseconds to 5.768 — a factor of 3.32. If that were the whole story the settling would grow by 3.32 as well. It grows by 2.57, and the missing factor is the pole’s own residue, which falls from 0.2027 to 0.0569 as the pole approaches the axis. A slow mode that starts smaller takes less time to become negligible, and the settling time is τ·ln(A/tol) with both terms moving.

That prediction uses one pole out of five and is checked against a full residue expansion over all of them. The worst disagreement across the six designs is 0.78 per cent at a hundredth of a per cent of settling, 3.32 per cent at a tenth, and 6.44 per cent at one — better the further into the tail the measurement is taken, which is the direction that says the approximation is the right one rather than a fit that happens to work. Early in the transient the other four modes have not died and one envelope is not the answer; late in it, there is nothing else left. That is the same discipline one step computed twice applies to a transient and where the behaviour is written down applies to a pole set — a second route that must agree, and a stated region where it does not.

For scale: an all-pole Chebyshev of the same order and the same ripple settles to a tenth of a per cent in 8.08 milliseconds and a Butterworth in 3.30. The widest elliptic design here is 9.04, close to the Chebyshev it is built on top of; the narrowest is 23.23, which is seven times the Butterworth. The time-domain price of the zeros themselves is small and the time-domain price of the selectivity is not. Anybody who has decided that a ripple is acceptable has already paid most of the first; what the second costs is set afterwards, by a parameter usually left at its default.

What it does not say #

It does not say the elliptic family is a poor trade. Order five at a transition band of 1.67 times holds 56.6 decibels, and the steepest all-pole family of that order needs to go out to 2.55 times to be that far down at all. The zeros do exactly what the zeros that buy an order measured them doing.

It does not say the transition width should be generous. Where the anti-alias filter in front of a converter sits, the transition band is decided by the sample rate and the signal band and there is nothing to negotiate — what the filter in front costs is the case where all three quantities are handed over at once. What this rung says is what the third one then is, and what the settling then is, both of which were previously found out by building the thing.

And it does not say the floor is what a real filter has. Everything above is an ideal realisation with ideal followers between the sections. A real notch has an arm with loss in it, and what actually fills a null measured that at twenty decibels per decade of arm resistance; a real cascade has amplifiers with finite gain-bandwidth, which the Q the amplifier decides prices at the designed Q over the ratio — and the band-edge section here has a Q of 18 at the narrowest design, so that section is where a real part would be found out first. The numbers above are the best case, and they are the right numbers for deciding whether the specification is worth writing.

The number worth carrying #

Three quantities, an order, and one line: at order five and half a decibel of ripple, 66.1 decibels at twice the corner, 38.7 at 1.25 times and 21.3 at 1.05. Relaxing the ripple to three decibels adds 9.12 wherever it is spent. Settling to a tenth of a per cent runs from 9.04 milliseconds to 23.23 across the same range, at a corner period of one.

The habit that goes with it is about what a free parameter is. The selectivity in this design sat at a default for as long as the family has been in this collection, and everything written about the family was written at that default — which made a two-quantity trade out of a three-quantity one and hid the fourth quantity entirely. A parameter with a default is a decision somebody made once, and the way to find out whether it was the right one is to sweep it and read what the other three do.