What the filter in front costs

The two essays before this one established that half the sample rate is a boundary with no gradient, and that past it a converter’s output is a faithful record of something that was not there. The practical response is to remove the offending content before it arrives, with a filter — and the filter is an ordinary analogue circuit, made of the same components, from the same four families, that this site has already computed from their definitions and measured on the networks that result.

What is new is the currency. Everywhere else on this site a filter choice is paid for in ripple, in delay variation or in ringing. Here it is paid for in clock rate, and the spread between families is a factor of two and a third rather than a decibel or two.

Stating the requirement so that it has an answer #

“An anti-alias filter must remove everything above half the sample rate” is not a specification, because no filter does that and the previous essay explained why: a signal with nothing above a frequency is a signal that never started. What can be specified is a level.

The requirement used here is the one a converter’s own data sheet implies. A passband ends at $f_\mathrm{pass}$. Any input at or above $f_\mathrm{s} - f_\mathrm{pass}$ folds into that passband, so the filter must be down to the stopband level by $f_\mathrm{s} - f_\mathrm{pass}$, and everything above that frequency must stay down. Given a filter, the smallest sample rate satisfying it follows:

$$f_\mathrm{s} \ge f_\mathrm{stop} + f_\mathrm{pass}$$

where $f_\mathrm{stop}$ is where the filter reaches the level and stays there. That last clause carries more weight than it looks, and the next section is about it.

The stopband level is a choice, and the natural one is the converter’s own quantisation floor: there is no point removing an interfering tone to eighty decibels if the converter’s own arithmetic sits at seventy-four. That ties this figure’s slider to the bit count of the part, which is where the design actually closes — a twelve-bit converter asks for about seventy-four decibels and a sixteen-bit one for about ninety-eight, and the filter’s demand on the clock moves with it.

Why the stopband edge is scanned before it is bisected #

Finding $f_\mathrm{stop}$ looks like a bisection: walk up in frequency until the response goes below the target. Written that way it is wrong, and wrong in a way that produces a confident, plausible, three-times-too-low answer.

An elliptic response does not fall monotonically. It has transmission zeros in the stopband, comes back up between them, and holds an equiripple level thereafter. A bisection asking “is the response below the target here” finds the first notch and reports it as the stopband edge — a frequency at which the filter is momentarily deep and generally is not.

So oversamplingFor scans first. It sweeps the response over twelve decades on a logarithmic grid, records the last frequency at which the response is still above the target, and only then bisects in the neighbourhood of it. Everything past that point is below the target, which is what the requirement actually asks for.

This is the same shape of error as one the filters field already recorded: a maximally flat response is flat to rounding over its first decade, so scanning it for local maxima with a greater-than-or-equal test finds thirteen “ripple peaks” in a Butterworth passband, every one a floating-point artefact, and then measures each of them. In both cases the machinery works and answers a slightly different question from the one intended.

What the families cost, in clock #

With the requirement stated and the edge found honestly, the answer is a table, and it is the reason this essay exists.

Bessel’s flat delay costs a clock 2.3 times the Chebyshev’s, which is 2.3 times the power in every digital stage downstream and 2.3 times the throughput in whatever consumes the result. Put beside the delay figure the filters field already measured — Chebyshev’s group delay varies 49% over the band against Bessel’s 0.06% — the trade is now legible in two currencies at once, and neither is a matter of taste.

The gate checks the ordering rather than the numbers, at every attenuation the slider offers, because the ordering is the claim and the numbers are the illustration: Bessel demands more than Butterworth, which demands more than Chebyshev, at forty decibels and at a hundred. It also checks that the demand falls with order for a fixed family — 5.50×, 2.82×, 2.08× and 1.76× for Butterworth at orders four, six, eight and ten — so that the trade is legibly order against clock and not one against nothing.

The refusal, and what it is a refusal of #

The last row has no bar, and it is the most interesting row in the figure.

An elliptic filter’s stopband is equiripple: it comes back up between its transmission zeros, and the level it comes back up to is a floor. The filters field measured this directly and stated the consequence in the language of design — a section’s transmission zero and its stopband floor are one choice, and asking for a zero close to the passband is asking for a shallow floor. Here that consequence arrives as an outright refusal.

At a selectivity of 0.8, an eighth-order elliptic design’s floor is at −74.1 dB. The requirement is −80 dB. There is no frequency at which the response satisfies it, so there is no sample rate that satisfies it either, and oversamplingFor returns a refusal rather than a large number. That is the site’s standing habit — a solver that never declines is untested — applied to a design question instead of to a matrix.

The refusal is specific and it is worth saying what it is not. It is not a claim that order cannot help. The degree equation deepens the floor with order at a fixed selectivity — 38.7 dB at order five, 50.7 at six, 74.6 at eight for one selectivity, and 110.8 dB at order twelve with a selectivity of 0.85 — so a deeper floor is always purchasable with more sections. A first version of the gate asserted that order could not rescue a shallow floor, which is simply false, and the check refused it.

What order buys and frequency does not is the distinction the refusal encodes. Measured on the response:

The all-pole design falls at 48.2 dB per octave, exactly eight times 6.02, so every doubling of the clock is worth another 48 dB and a faster clock is always an available answer. The elliptic design moves by 4.8 dB across three octaves — which is to say, not at all. Doubling the clock buys it nothing, quadrupling it buys it nothing, and the only two things that help are a deeper floor bought with order or a lower selectivity bought with transition width.

That is a boundary of exactly the kind this site is built to draw, and it is not a frequency. It is a level, and it is impassable by the usual remedy.

What the bit count asks for #

The slider on the figure is the attenuation, and leaving it as a free choice is a way of avoiding the question. The natural setting is the converter’s own floor: an interfering tone folded down to a level below the quantisation noise is a tone that has been removed as far as the converter can tell.

That makes the requirement a function of the part rather than of taste.

and the ratio the figure reports moves with it. Measured on the same eighth-order Butterworth design, an eight-bit system asks 1.53 times Nyquist, a twelve-bit one 1.95, a sixteen-bit one 2.55 and a twenty-bit one 3.39. Every four bits of resolution is worth roughly another 0.45 of a turn on the clock, for the same filter, because 24 dB of extra attenuation is half a decade of extra skirt at this order.

Two qualifications, and both are the kind this site prefers to state than to bury.

The converter’s floor is not the system’s floor. The essay on the floor a converter sets measures the crossing at which the source resistance’s Johnson noise overtakes the quantiser’s — 18.8 bits for a 1 kΩ source in 100 kHz — and past that crossing the requirement above is asking the filter to remove interference to below a level nothing else in the circuit reaches. There is no gain in that, and the honest requirement is the larger of the two floors.

A folded tone is not spread out. Quantisation noise is spread across the band; a folded interferer is a line. A line at the level of the total noise power is far more visible than the noise is, so the usual practice adds margin — ten decibels is common — and the figure’s slider is where a reader can put it.

The filter’s own contribution, which is not free either #

A steeper filter is more components, and every resistor in it is a noise source. That is not a rhetorical point on this site: the noise field measured the equivalent noise bandwidth of each family on the built networks and found the odd-order Chebyshev passing less noise than a brick wall at its own corner — 0.9637 at order five — while an even-order one passes more.

So the filter chosen to keep interference out also decides how much of the resistors’ own noise reaches the converter, and the ordering there is not the ordering above. Chebyshev demands the lowest clock of the all-pole families and admits the least noise at odd orders; Bessel demands the highest clock and its flat delay is the reason anybody pays it. The three currencies — clock rate, delay variation, noise bandwidth — do not rank the families the same way, which is why the choice cannot be reduced to a single figure of merit and why this site draws all three.

What happens with no filter at all #

The figure’s argument is about how much filter. It is worth spending a paragraph on none, because the failure is more specific than “noise gets in” and because one of its symptoms is routinely misread.

With no filter, everything the front end can see arrives folded. A converter’s input bandwidth is normally several times half its sample rate — deliberately, so that the sampling instant catches the signal rather than a smoothed version of it — so the folded band is not a narrow strip above the wanted one but everything up to the front end’s own corner, stacked. A part with 500 MHz of input bandwidth and a 100 MHz clock folds five bands’ worth of whatever is present onto the wanted one.

The broadband noise of the source folds too, and it folds repeatedly: each image band contributes its own share of the same flat density, so the total noise in the band rises by the number of image bands the front end admits. That is the mechanism behind a result that reads as paradoxical the first time it is met — a converter whose measured noise floor falls when a modest filter is added in front of it, without anything about the converter changing. Nothing improved; several copies of the same noise stopped arriving.

It is the same arithmetic the noise field already carries in the other direction. The equivalent noise bandwidth of a response is the width of the brick wall passing the same noise power, and a converter with no anti-alias filter has an equivalent noise bandwidth set by its front end rather than by its clock — which is why the noise figure of a sampled system is one of the few places where the analogue bandwidth being too wide is the defect.

Where the essay stops #

Two things adjacent to this are deliberately not here.

Digital filtering after a fast converter is the modern answer to all of the above: sample at many times the rate needed, let a cheap analogue filter do the small job of keeping out the images near the high clock, and do the sharp filtering arithmetically before decimating. It changes the arithmetic in this essay entirely and it is a claim about an algorithm rather than about a circuit, which is a boundary this collection has stated and stays on this side of.

The reconstruction filter after the output has the same trade with the arrow reversed and is mentioned in the previous essay rather than measured here, because the measurement is the same measurement and repeating it would be a second figure making one argument.

What is here is one requirement, five designs, and a currency conversion nobody usually performs: a filter family chosen in decibels, priced in megahertz.

The realisation, which changes the answer again #

One thing the table above quietly assumes is that the filter built is the filter designed. The filters field has already measured how far that is from free: at the ripple peaks a one per cent component moves a buffered cascade by 0.162 dB and a doubly terminated ladder by 0.003 dB, and the exponents rather than the ratio are the claim — the cascade’s error grows as the 0.99 power of the tolerance and the ladder’s as the 2.00 power.

That result lands on this essay with some force, because the requirement here is a stopband level and a stopband level is exactly what component tolerance moves. An elliptic design whose floor is 6 dB above the requirement is refused by the figure; an elliptic design whose floor is 3 dB below the requirement on paper is refused by the components, some of the time, in a way nothing about the design says. The margin a real anti-alias filter carries is therefore not a design margin but a tolerance margin, and the two are different quantities with different arithmetic behind them.

Two further results from that field say that the realisation problem is worse than a tolerance and in one case has no solution at all.

The Q the amplifier decides builds an active section with an amplifier a hundred times its corner — the usual rule — and finds the quality factor two per cent high and the pole frequency two per cent low, the same number in both directions, with the number being the designed Q divided by the ratio. Carried into a fifth-order half-decibel Chebyshev that is 2.1 decibels of ripple on a design specified at half of one. This essay chose Chebyshev on the strength of a 1.53× clock requirement computed from an ideal response; a realised one with ripple four times the design has a passband edge somewhere else and a skirt starting somewhere else, and the clock factor moves with it. And the ripple that is a temperature adds that the two per cent is not even a constant: with a tail current a resistor sets, the same design runs 0.82 dB of ripple at −40 °C and 1.05 at +125, crossing a one-decibel specification at 89 °C.

The band that does not close is the sharper one, because it is not a tolerance at all. A cascade of active sections can be built at three decades of impedance level at second order, one decade at sixth, and none at all at eighth — the band shuts by half a decade per order, because fifty ohms of amplifier output resistance binds from one end and two picofarads of stray from the other, and every added section brings three more nodes and one more amplifier. Every high-order design in the table above is an eight- or ten-pole filter in the families that need one, so the design this essay prices in clock is, at those orders, one that no choice of component values realises. A doubly terminated ladder has only one of the two binding quantities and still has four decades at order nine, which makes the passive realisation the one that survives the requirement — at the cost of inductors, in a signal path where the filter’s whole purpose is to be cheaper than a faster clock.

Which is to say the clock rate this essay prices is a lower bound reached by a filter that exists only on paper. What a real system pays is that number plus whatever margin the components demand, and the second term is a property of the realisation rather than of the family — so the three currencies of the previous section are really four.