The sample before the sample

The capacitor that remembers put a dielectric’s slow polarisation in a sample-and-hold and measured what comes back. Hold ten volts long enough for every branch to follow, acquire zero through a switch, open the switch, and the held voltage creeps towards where it was, by a fraction of the step. It said the error is zero for a constant input and proportional to the change since the last acquisition, and it drew a conclusion that has a natural sound: that makes it signal-dependent, and a signal-dependent error appears as distortion rather than as a line in a specification. Ten seconds, and fifteen minutes then showed that the test which defines the part is really a count of decades, with the reading set by a ratio of two durations.

That conclusion can be tested in one line, and it does not hold. An error proportional to a step in the signal is proportional to the signal. The whole circuit is resistors and capacitors. So the map from one sample to the next is linear in every state it holds, and a linear map has no harmonics to give. The memory is real and it can be large, but what a dielectric does to a train of samples is a filter, with a long and very small response. That changes what it is, how to specify it, and whether anything can be done about it. Something can.

One period is one matrix #

The capacitor is the model of the earlier essays: a main capacitance with seven relaxation branches behind it, log-spaced in time constant from 10 ms to $10^4$ s, with equal capacitance a decade, and sized to read 0.2 per cent on the standard test. An acquisition forces the top node to the sample, so every branch relaxes towards it by a fraction $1 - e^{-T_a/\tau_k}$. A hold opens the switch and the node exchanges charge with the branches, which is a linear system. So one period, acquisition and hold together, takes the state of the branches and the sample into a new state and a reading, by fixed matrices. They only need computing once for a given acquisition and hold, by marching each basis vector through the hold.

The second route is the march of the netlist from the earlier essay, with the same part and timings. For the first reading after the step they agree within 3 per cent; the difference is that the march has a real switch resistance. Everything after that is the matrices, and they make a long record cheap: 4,096 samples cost a few matrix-vector products per sample.

Reading the same step for a long time #

The picture on the opening figure is the one the earlier essay measured, carried on past its first sample. With a 1 ms hold, the first reading after the input steps from 10 V to 0 is 109 ppm of the step. The tenth is 46.7. The hundredth is 4.5, and the four-hundredth is 0.904. That is a slow decline, and the reason for it is what an acquisition can and cannot do. A 100 µs acquisition reaches a branch of 100 s time constant by a millionth of the distance per sample. So the slow branches keep nearly all of their ten volts through hundreds of samples, and each hold they leak a little of it back.

The longer the hold, the larger the first reading, as the earlier essay’s law says. But look at the later readings on the slider: 9.64 ppm on the tenth at a 0.1 ms hold, 46.7 at 1 ms, 50.2 at 10 ms and 50.6 at 100 ms. The tenth reading stops depending on the hold. By then the amount of time that has passed is large against the fast branches, which have given up what they had. So a reading of a settled zero is limited at each stage by whichever branches have not yet given up, and on a long enough record the early hold’s advantage has gone. A converter using a shorter hold gets a better first reading, and it does not get a much better later one for very long.

The half that never arrives looked at another reading that does not get where it is going in the time available, and in both cases a time scale in the part far exceeds the time scale of the measurement.

The filter’s weights #

A linear system with one input is completely described by its response to a unit impulse, and that response is a set of weights.

With a 1 ms hold this reading contains 10.3 ppm of the last sample, 3.87 ppm of the one ten back and 0.0391 ppm of the one a hundred back. Those are very small numbers, but there are thousands of them and they all have the same sign, so all earlier samples together come to 109 ppm. That is, not by coincidence, the first reading after a step. A step from a long-held ten volts to zero has a first reading equal to the sum of all of those weights times ten volts, and that is what the first figure’s 109 ppm measured all along.

So the error the standard description called “a fraction of the step” is really the sum of a filter’s tail. A reader who wants a sharper statement has it: the reading is $y_n = h_0x_n + \sum_{k\ge 1}h_kx_{n-k}$, and a part with a stated absorption and a stated timing has a stated set of $h_k$. That is exactly how anyone would describe an audio delay effect or a converter’s memory, and its small size does not change its form. The tail falls off as the inverse of the lag at long lags, a direct consequence of the equal capacitance per decade of relaxation time: the three holds all converge to about 0.04 ppm a hundred back and 0.0004 a thousand back, in a way that has nothing to do with the hold. The dielectric’s spectrum of relaxation times is what sets that shape.

It is not distortion #

And now the check. Distortion produces frequencies that are not in the input, and a filter cannot.

A sine at 64 cycles in 8,192 samples comes out 28.1 ppm low in amplitude and with 0.0023° of phase. Its largest harmonic, read after the transient, is $1.5 \times 10^{-12}$ of the fundamental. That is the arithmetic’s own floor. And the output for 10 V is ten times the output for 1 V, sample by sample, to seven parts in $10^{15}$ of a volt. So there is no signal dependence in the sense that distortion means: a large signal has a proportionally large error, the same fraction of itself. This is exactly the sense in which a capacitor’s reactance is “signal-dependent”, and no one calls that distortion.

This needs saying because the earlier sentence is a tempting one and it has practical consequences. A distortion needs a linearisation to fix it, and it has no simple inverse. A designer who believes this is distortion will look for a better capacitor as the only cure. A filter has an inverse, which the next section measures. It also sets where to look when a real converter does show harmonics: a capacitor’s voltage coefficient, measured in the capacitance that is not one number, is a true nonlinearity, and it is what will actually produce them. Those two defects are different in kind, and a real capacitor has both, so a measured harmonic on a real part is that second effect. Separating them is a matter of checking whether the error is proportional to the amplitude, or to its square.

The response #

Being a filter, the memory has a frequency response, and it is the transform of the weights.

At a very low frequency there is almost no error, because each sample differs little from the last and the weighted sum of earlier samples is nearly the current sample. Up at a tenth of the sample rate the samples have decorrelated over the lags that matter, the shortfall has reached about 114 ppm, and it stays there. Four sines through the model agree with the transform to a few parts in a hundred thousand of amplitude, which is the check that the whole description is sound. That 114 ppm is close to the 109 ppm the opening figure gave for a step, since a signal that decorrelates fast looks like a step at every sample. And here, the weights at lag one, which are 10 ppm, can make the top of the band a little different from what the rest of the weights are doing. They are too small for that to show at this scale.

A designer can read a specification from this. A gain error, with a shelf at about a twentieth of the sample rate, is what a part with this absorption presents for a signal. An instrument that calibrates its gain on a signal at the top of its band has removed most of that shelf, and one that calibrates with a direct voltage has not. It is a small version of what a calibration at the carrier removes, since the reading’s shortfall is a function of frequency and a calibration is at one frequency.

A filter can be undone #

If the reading is a known weighted sum of earlier samples, then each earlier sample’s contribution can be subtracted from the current reading, and the earlier corrected values stand in for the true samples. It is a recursion and it costs one multiply per weight per sample, which is nothing on any modern processor.

On a signal of random steps every 64 samples, the uncorrected reading is 51.6 ppm of full scale in rms. Subtracting one earlier sample barely matters: 46.4. Twenty take it to 7.0, fifty to 1.61, two hundred to 0.302 and a thousand to 0.0169. Each extra sample is working on a slower branch. With about fifty, most of the error has gone, a factor of 32, or from 14.2 bits to 19.2.

The real limit on this is the second line. With the absorption known only to 20 per cent, which is a reasonable statement about a production capacitor at an unknown temperature, the corrected error falls to about 5.5 ppm by fifty samples and then stays there, however many more are subtracted. That is about what a correction 20 per cent out should leave: 0.2 times the uncorrected error would be 10 ppm, and the actual figure is about half that because the signal’s own spectrum helps. So a software correction buys about three and a half bits in this application, from a part known only to 20 per cent, for no cost in hardware. Better knowledge of the part buys more.

The temperature matters because the absorption has one, as ten seconds, and fifteen minutes measured, and the spread between parts is at least that large. A stored coefficient is an estimate. In principle a converter could measure its own coefficient at start-up, since a calibrated step and the readings that follow it are exactly the opening figure, and the weights can be read from it directly. That is the same trick the two generators with one cause looks for in another setting: an error with a known form can be measured once by something that knows the form.

What this says about choosing a capacitor #

The standard advice is correct and it stays correct: use a low-absorption dielectric. Polypropylene and polystyrene have absorption well below a tenth of a per cent, against the several per cent of a high-permittivity ceramic, and every number here scales with the absorption. A part ten times better has all these errors ten times smaller. That is usually the cheapest step and the first one to take. A filter correction is what comes after, and it opens a different option: a part chosen for some other virtue, with a modest absorption, can be used if its memory is characterised and subtracted. It also sets the size of what a better capacitor could still buy, since a small remaining absorption is easier to subtract than a large one.

It also matters what the circuit does with the reading. On a slowly varying signal, the opening figure’s large numbers do not appear at all, since each reading differs little from the last; the shelf in the response is what such a signal sees. A multiplexed converter, which goes from channel to channel with unrelated voltages, is the worst case. Its readings are all effectively steps, and the sample before the sample is another channel entirely. So what looks like crosstalk between channels in a multiplexed system, a small trace of one channel in the next that does not go away with more settling time, is often this. The current a capacitor adds up looks at a related memory of the current through the part, and the relevant test of a multiplexed front end is to read a grounded channel straight after a full-scale one.

What the matrix leaves out #

The switch here is ideal in acquisition, with the node forced exactly to the sample. A real switch has a resistance, so the acquisition’s own settling mixes into this, and the first sample’s real error includes a settling term; a first-order fix is to treat it as one more linear element in the same matrix. Charge injection from the switch is absent, which adds a fixed offset that a calibration can remove. The acquisition and hold are constant, where a real converter’s may depend on its code. And the model has equal capacitance per decade down to 10 ms, so at much faster holds it has no branches to speak of and the error falls too sharply. A real dielectric has relaxation at much shorter times, and the figures at a 100 µs hold are therefore lower bounds. Whether a real part’s weights at a one-sample lag are as small as these depends on that.

The linearity itself is a fact about this model. A real dielectric’s absorption has some dependence on the field, and it is not exactly proportional at a high field. So at very high fractions of rated voltage some of this could become a true distortion. On a film capacitor well below its rating it is linear to a very good approximation, and that is a claim worth checking on a particular part: that is where the amplitude test above is useful.

The number to keep #

A 0.2 per cent part holding for 1 ms reads a 10 V step 109 ppm wrong on the first sample, 46.7 on the tenth and 4.5 on the hundredth. The reading is a linear weighted sum of earlier samples, with weights of 10.3 ppm one back and 0.0391 ppm a hundred back, summing to the 109. A sine through it reads 28.1 ppm low with no harmonic above $10^{-12}$, so the earlier description of this as distortion was wrong. The response has a shelf of 114 ppm above a twentieth of the sample rate. And a filter can be undone: fifty earlier samples subtracted take 51.6 ppm to 1.61 ppm, and with the part known to 20 per cent the correction stops at 5.47, a gain of about three and a half bits for no hardware at all.

Still open: a start-up calibration, a temperature, and a multiplexer #

The part measuring itself. A calibrated step and the readings that follow it are the weights. A converter could run this at start-up, with noise on its readings. How many steps it takes to know the weights to a stated accuracy, given that the small ones lie well under one code, sets whether such a calibration can reach the 20 per cent figure here or do better.

The weights across temperature. Absorption has a temperature coefficient, and its relaxation times move as well, which in a log-time model is a shift along the lag axis rather than a scale of the weights. A correction with a single stored set would then leave a residue with a particular shape, and whether scaling alone captures it or a shift is needed decides how much of the correction survives a warm-up.

A multiplexer’s crosstalk. With several channels in rotation, a reading’s weights fall on samples of other channels, with the same numbers. A correction is then per channel, and the matrix is a block. How much of a multiplexed system’s channel-to-channel error is this rather than its switches is a question a single grounded channel can answer on a bench, and the model here says what to expect.