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

Noise bandwidth — where it appears

The width of the ideal rectangular filter that would pass the same noise power as the network does, which is not its half-power point. For a single pole it is π/2 times that point, and it is what a density has to be multiplied by to give a root-mean-square voltage.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

kT/C, on a circuit whose resistance is a clock and moves by 1e+4. computed by solving, not by drawing. The noise a switched-capacitor low-pass leaves on its holding capacitor, against the ratio of the two capacitors, measured by marching a seeded sequence through the recursion the circuit obeys. It is 6.328 µV at every ratio drawn — kT/Ch, with the holding capacitor and nothing else in it — while the resistance the arrangement behaves as moves from 1.00 kΩ to 10.0 MΩ across the same axis. The two lines below it are the two sampling events that make it: what the input switch leaves on the switched capacitor, and what the sharing switch leaves behind when it opens. Neither is the answer and their sum is, exactly, because the pole's own bandwidth factor is the same expression.

The noise a clock does not make

A hundred-megohm resistor has a noise density of 1.27 microvolts per root hertz. A switched capacitor that behaves as a hundred megohms has none of it: the noise on the capacitor it charges is kT/C, with the holding capacitor in it and nothing else — not the clock, not the switched capacitor, not the on-resistance. And it is exact rather than asymptotic, at every capacitor ratio from a thousandth to ten.

noise · kT/C noise 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 The same three stages, in two orders. computed by solving, not by drawing by Friis's cascade. With the low-noise amplifier first the chain's noise figure is 3.31 dB; with the mixer first it is 12.01 dB. The gain is identical either way — 48.0 dB — so the 8.70 dB is bought with nothing but an ordering. Everything after the first stage contributes 26.0% of the total. The 2.0 dB of cable in front has gain below one, so it multiplies every later stage's contribution rather than dividing it.

The loss in front, counted twice

A decibel of cable before an amplifier costs a decibel of signal, which everybody expects, and a decibel of noise figure, which is a separate decibel arriving from a separate place. Measured across the slider it is exact: 1.31 dB of chain noise figure becomes 2.31 with one decibel of cable and 9.31 with eight, every time, while the 8.70 dB that stage ordering is worth does not move at all. A lossless reactive network in the same position costs nothing, because only the real part of an impedance is warm.

noise · Johnson noise What each factor of attenuation buys on a 2.0 kΩ source. computed by solving, not by drawing at 12 probe ratios: the one-per-cent frequency bisected on the node with and without the probe, against the frequency a tip capacitance alone would predict. A one-to-one probe reaches 6.79 kHz and a hundred-to-one 692 kHz. The first step, from 1× to 2×, multiplies the bandwidth by 2.03 for a factor of two in signal; the two routes differ by at most 1.8% across the sweep, and they differ at all only because the probe's 1.0 MΩ is already 0.20% of the reading before any frequency is applied.

The probe that takes a tenth

A ten-to-one probe buys an order of bandwidth for a tenth of the signal, and on a two-kilohm source the bandwidth is exact: 6.79 kHz becomes 69.2 kHz. The tenth of the signal is not a tenth of the signal-to-noise ratio. Solved resistor by resistor, the noise referred to the tip goes from 1.782 µV to 55.78 µV — a factor of 31.3 — because the divider that does the attenuating is nine megohms and a megohm, and √(n(n−1)kT/C) on the cable's own capacitance has no source resistance in it at all.

instruments · Probe loading A true-RMS converter reads noise low by 1/(16Bτ): 0.600% at Bτ = 10 and 672 ppm at 100, where a sine at fτ = 100 is 0.0396 ppm. Seeded, and measured. An explicit true-RMS converter — square, one-pole average of time constant τ, root — reading Gaussian noise of unit power, against the product of the noise's bandwidth and τ, from 1 to 100. Each point is 2²¹ samples at eight times the bandwidth, with the reading compared against the record's own root-mean-square and its standard error from batch means. Bτ = 1: 4.550% ± 150 ppm (band from zero), 4.081% (band of the same width about 3B). Bτ = 2: 2.598% ± 114 ppm (band from zero), 2.411% (band of the same width about 3B). Bτ = 5: 1.149% ± 75.4 ppm (band from zero), 1.098% (band of the same width about 3B). Bτ = 10: 0.600% ± 53.1 ppm (band from zero), 0.576% (band of the same width about 3B). Bτ = 20: 0.309% ± 39.1 ppm (band from zero), 0.295% (band of the same width about 3B). Bτ = 50: 0.127% ± 26.3 ppm (band from zero), 0.119% (band of the same width about 3B). Bτ = 100: 672 ppm ± 19 ppm (band from zero), 623 ppm (band of the same width about 3B). The dashed line is 1/(16Bτ), an eighth of the averaged square's variance, which the readings approach above Bτ = 10 and fall short of below it. A sine read by the same converter at the same product of frequency and τ is low by 3.96 ppm at 10 and 0.0396 ppm at 100 — the square of the noise's rate rather than its first power.

The noise a true-RMS meter reads low

A true-RMS converter reads a sine low by an amount that falls as the square of its frequency, and for anything but a slow sine that amount vanishes: 3.96 parts per million at ten times the averager's corner. Noise is not a sine. Its square fluctuates at every frequency down to zero, and the averager passes a share of that set by its own bandwidth against the noise's, so the reading is low by 1/(16Bτ) — the first power, not the second. Measured on seeded noise at Bτ = 10 it is 0.600 per cent low against a predicted 0.625; at 100, 672 parts per million. One reading scatters by eighteen times that, so no single reading shows the bias and the mean of a few hundred is nothing but bias.

power · Rms and average With the period 1.3% long, eight periods read a sine 0.599% wrong through a rectangle, 79.4 ppm through a triangle, 22.8 ppm through a Hann window. Integrated over each weighted window, the worst over every starting phase. The error in a sine's root-mean-square summed over N assumed periods 1.3% too long, against N, for four weights across the window. The rectangle is the plain sum: 0.640% at two periods and flat in N, δ/2 as the earlier measurement found. The triangle is flat in N too but 75 times smaller at eight periods. The Hann window falls with N, as the −2.67 power: 414 ppm at two periods, 0.293 ppm at thirty-two. The four-term Blackman–Harris is below 4.85 ppm at every count from two. A weight that falls to zero at the window's ends counts the extra sliver of period at almost nothing.

The window that forgives the period

A converter that sums a waveform's square over whole periods reads its RMS exactly, and one that sums over periods 1.3% too long reads a sine 0.6% wrong — δ/2, for any number of periods — because the extra sliver of period counts at full weight. Weight the window down to zero at its ends and the sliver barely counts. Over eight periods a triangular weight reads the same sine 79 ppm wrong and a Hann weight 23 ppm; and they are different kinds of cure. The triangle is the only one that changes the error's order, from first to second in δ, so it wins when the period is nearly right; the Hann weight keeps first order but falls with the number of periods, about as its inverse square, so it wins over long windows. Both cost nothing with the period exact — from two periods up. Over one, a Blackman–Harris weight reads even a sine 10% wrong. The price is noise: a Hann-weighted reading of noise scatters √1.5 as widely.

power · Rms and average 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 A 50 Ω source over 20.0 Hz–20.0 kHz: the best turns ratio is 10.7, giving 1.30 dB; the white optimum's 11.5 gives 1.31, and no transformer at all 13.2. computed by solving, not by drawing, the band integrals checked by quadrature. The drifting part of the essay before — 4 nV/√Hz and 0.6 pA/√Hz white, flicker corners at 10 Hz and 1 kHz — fed from a 50 Ω source through an ideal step-up transformer of ratio n, so it sees n²·50 Ω. The noise figure over 20.0 Hz to 20.0 kHz against n: least at n = √(√(E/I)/Rₛ) = 10.73, where the amplifier sees 5.76 kΩ, at 1.298 dB. The ratio that matches the white optimum, 11.55, gives 1.310 dB; the source connected directly, 13.23 dB. The slider is the band.

One turns ratio for a band

A 50 Ω source and an amplifier whose optimum is 6.67 kΩ want a step-up transformer, and the ratio that matches the white optimum, 11.5, is the one a design note computes. Over the audio band it is nearly right: 1.31 dB against 1.30 at the band's own best ratio, 10.7. Over 1 Hz to 100 Hz it is badly wrong — 9.2 dB where a ratio of 4.8 gives 5.4 — because the band's noise lives in its bottom decade, where the flicker has moved the optimum down. One ratio for a two-decade band costs at most 0.24 dB against a ratio that could follow the optimum. The transformer's winding resistance multiplies the whole noise factor by one plus the winding's resistance over the source's, not just its own share: a tenth of the source costs 0.41 dB where its own noise alone would cost 0.31. And 115 amplifiers in parallel are an ideal 10.7:1 transformer, noise for noise.

noise · Device noise

Named alongside it

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

Johnson noiseDesign tradeoffkT/C noiseModel rangeAveragingCorrelation timeDynamic rangeImpedance matchingMeasurement conditionNoise figureSeeded generatorSettling time

All concepts