Circuit Response
every model has an edge, and the figure carries it
Concept

Correlation time — where it appears

How long a fluctuating quantity remains related to its own past, beyond which its samples behave as independent draws. It is what decides whether averaging helps: a white sequence has none and a pink one has as much as the record is long.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Averaging a white sequence, and averaging a pink one. computed by solving, not by drawing. Both sequences are the same seeded white stream, one of them put through the 1/f network. Averaged in non-overlapping blocks, the white one's spread falls as n to the -0.510 ± 0.006 across five seeds — the √N law — and the pink one's as n to the -0.087 ± 0.013, which is very nearly not at all. A thousand-sample average buys a factor of 36.9 on the first and 2.1 on the second.

The corner where averaging stops working

Average N samples and the noise falls by √N. That is a statement about independent samples, and flicker noise's samples are not independent — its correlation extends over every time scale, which is what a spectrum with no bottom means. Measured on the same seeded stream filtered and not: the white sequence falls as the −0.510 power of the block length and the pink one as the −0.087 power, so a thousand-sample average buys a factor of 36.9 on one and 2.1 on the other.

noise · Device noise The amplitude at which a converter's floor becomes distortion. computed by solving, not by drawing. The quantisation error is transformed and the power in harmonics of the input is measured directly, at the bins those harmonics occupy. Undithered, the share rises from 0.29% at 230 levels to 76.5% at 1.3: the error has stopped being spread and has become a deterministic staircase locked to the signal. The sweep is drawn against the levels the input uses rather than against volts because it is then the same sweep for every converter — a twelve-bit part and a sixteen-bit part agree here to 0.0e+0, so what decides the character of the error is how many levels the signal crosses and not how many the converter owns. With 1 least significant bit of dither the share stays under 0.34% at every amplitude, for 3.03 dB of signal to noise. Adding noise to a converter's input improves what comes out of it, which is true and sounds like it should not be — and the amount is one whole step, not "some": a quarter of a step leaves 52.7% and a half leaves 25.6%, because a dither smaller than a step cannot make the quantiser cross one.

When a floor stops being a floor

Quantisation error is treated as noise and behaves like noise while the input crosses many levels. As the amplitude falls it stops: measured at the bins its harmonics occupy, the share of the error's power sitting in harmonics of the input rises from 0.29% at 230 levels to 76% at 1.3. The floor has not moved. It has become a distortion product locked to the signal, and the fix is to add noise on purpose — one whole least significant bit, and no more.

digital · Quantisation Subtracting removes kT/C entirely and doubles the amplifier — worth 31× at a megahertz and a loss above 60 MHz. computed by solving, not by drawing. The noise on one sample of a switched-capacitor stage, and on the difference of two samples taken a settled interval apart, against clock frequency. The reset level is the same number in both samples and cancels exactly; the amplifier's own noise is two independent samples and its variance doubles, measured at 2.000 against the 2 the correlation predicts. At a megahertz that is 63.8 µV down to 11.53 — 31 times in power. The two curves cross at 60.2 MHz, which is where the amplifier's own noise equals kT/C, and above it the subtraction costs more than it removes.

The sample that is subtracted

Three earlier measurements found floors that no gain moves and no filter reaches, because both arrive as numbers already sampled. One of them can be subtracted: the reset level a capacitor holds is the same number in two consecutive samples and cancels exactly. What it costs is that the amplifier's own noise is not — two samples of it are independent, so its variance doubles. That is thirty times better at a megahertz, a loss above sixty, and the crossing is the one an earlier essay computed for a different question.

noise · kT/C noise With 1 RC of tracking, sixteen samples average like 7.8 independent ones. computed by solving, not by drawing. The rms noise on the average of N consecutive held samples of a 1 pF capacitor, against N, for track times of 0.25, 1, 3 and 10 RC between samples, with the 1/√N law for independent samples dotted. A sample shares exp(−Tₜ/RC) of its value with the one before, so at 1 RC the correlation is 0.368 and sixteen samples are worth 7.81 independent ones; 1,024 are worth 473.6. The closed form is checked against a seeded march of the held sequence.

The average that is not N times quieter

Averaging sixteen samples of a 1 pF capacitor is supposed to take its 63.3 µV of kT/C noise down by four. With one RC of tracking between samples it takes it down by 2.8: the switch has not been closed long enough for the capacitor to forget the last sample, each shares 37 per cent of its value with the one before, and sixteen samples are worth 7.8 independent ones. At a quarter of an RC they are worth 2.6. Sampling faster never helps past a ceiling: however quickly the samples come, an average gathers at most one independent sample per 2RC of time — the continuous average's own limit. And a sampler that settles its signal to 8 bits has already waited long enough for its noise.

noise · kT/C noise

Named alongside it

The objects these essays reach for when they reach for this one.

AveragingkT/C noiseNoise bandwidthBilinear transformConvergence orderDesign tradeoffDitherEffective bitsJohnson noiseModel rangeMonte carloNoise figure

All concepts