The count under the density

Every earlier essay about shot noise here has used it as a density. The floor a current sets set 2qI beside a resistor’s 4kT/R and found them equal at fifty millivolts of drop. The two generators that are one current divided one collector current’s 2qI between a transistor’s voltage and current noise. In each, the density went into an integral over a noise bandwidth and came out as a root-mean-square current, and the root-mean-square current was then treated as the width of a Gaussian floor.

The first of those steps is exact. The second is not, and the difference only matters at small currents — which is where shot noise is most often the only noise there is. This page separates the two.

Why the density is exact #

A current of $I$ amperes is $I/q$ carriers a second. When the carriers cross a barrier independently — which is what a junction arranges and what a resistor does not — the number crossing in any interval is a Poisson count, and a Poisson count’s variance equals its mean exactly, whatever its size. Take the interval as the window whose noise bandwidth is $B$, which is $1/(2B)$ long. The mean count in it is

$$N = \frac{I}{2qB},$$

its variance is $N$, and $N$ carriers in $1/(2B)$ seconds is a current variance of $q^2 N \cdot (2B)^2 = 2qI\cdot B$. That is the density times the bandwidth, with no approximation anywhere. The signal-to-noise ratio of the current against its own shot noise is $N/\sqrt N = \sqrt N$ exactly, and nothing about the carriers’ size or number makes that any less true.

What the variance does not say is anything about the count’s shape. A Poisson count has a skewness of $1/\sqrt N$, a lowest value of zero, and a probability $N^k e^{-N}/k!$ at each integer $k$. A Gaussian of the same mean and variance has none of these: it is symmetric, it runs to minus infinity, and it is continuous. The two agree on every second-order quantity and disagree about everything else.

A picoampere, where the Gaussian looks right #

At a picoampere in ten kilohertz the two are hard to tell apart by eye. The count’s bars fill the Gaussian curve, the skewness is 0.057, and a seeded train of independent arrivals, counted in twenty thousand windows, gives a mean and a variance within their own sampling errors of 312.1. This is the regime every density-based calculation assumes, and at this current it is almost right.

Almost is in the tails. The count passes its mean plus five standard deviations with probability 7.84 × 10⁻⁷, and the Gaussian says 2.87 × 10⁻⁷. The count’s upper tail is heavier because the count is skewed — a positive skewness moves probability into the long side — and five standard deviations is far enough out for a skewness of 0.057 to matter by a factor of 2.7. A threshold detector set at five sigma on the strength of 2qI would see nearly three times the false triggers it was designed for.

Ten femtoamperes, where it does not #

A hundred times less current, and the Gaussian stops describing the count. There are 3.12 carriers a window on average, and the count is a handful of integers: zero, one, two, three and so on, with probabilities that a Gaussian smears across a continuous range reaching below zero. One window in twenty-three, 4.41 per cent, contains no carrier at all. The skewness is 0.566. The five-sigma probability is 1.03 × 10⁻⁴, which is 359 times the Gaussian’s.

The density is still exact. The variance of the seeded counts is 3.108 against 3.121, which is the density’s prediction to within the sampling. A noise calculation that stops at a root-mean-square current is still right here. It is only the moment the root-mean-square current is converted into a probability — how often a reading will pass some level — that the calculation fails, and it fails by a factor of hundreds.

A femtoampere, where most windows are empty #

At a femtoampere in ten kilohertz almost three windows in four, 73.2 per cent, contain no carrier. The current is not a noisy level at all. It is a sequence of isolated arrivals separated by silences, and its mean of 0.31 carriers a window describes a signal that is almost never at its mean. A Gaussian of the same variance, centred on 0.31 with a standard deviation of 0.56, puts more than a quarter of its probability below zero, where the count cannot go.

This is not a remote regime. The input currents of electrometer amplifiers, the dark currents of small photodiodes and the leakage of a guarded node are femtoamperes to picoamperes, and a ten-kilohertz bandwidth is modest. Where the trouble is at the input builds a photodiode amplifier; pointed at a faint enough source, the diode’s own photocurrent is a handful of carriers a window, and everything this page measures applies to it.

Tails, against current #

The comparison at three currents becomes a curve when the current is swept, and the probabilities are summed exactly from the Poisson distribution rather than sampled.

The upper tail is too heavy at every current drawn, and the excess falls slowly: 2.74 times at a picoampere, still more than ten per cent above the Gaussian until about 162 pA. The curve is jagged because the count is an integer: the threshold $N + 5\sqrt N$ moves continuously with the current, and every time it crosses an integer the tail probability drops by one term. The three-sigma curve is closer to one everywhere, since a skewness moves a distant tail far more than a near one.

The lower tail does the opposite, and more sharply. A count cannot be negative, so below twenty-five carriers a window — where $N - 5\sqrt N$ is below zero — the probability of falling five standard deviations short is exactly zero. Above twenty-five it rises towards the Gaussian’s value from below. A threshold set below the mean to detect a drop in current is therefore safer than the Gaussian says, and one set above it to detect a rise is less safe, and neither error is visible in the density.

The general law is the central limit theorem, stated with its rate. A Poisson count is a sum of many independent contributions, and its distribution approaches a Gaussian as $N$ grows — but the approach is governed by the skewness, $1/\sqrt N$. The first correction to a Gaussian tail $z$ standard deviations out is the skewness times $(z^3 - 3z)/6$, which at five sigma is eighteen times the skewness. A five-sigma tail within ten per cent of the Gaussian’s therefore needs a skewness under about a two-hundredth: some forty thousand carriers a window, 128 pA in ten kilohertz, the same order as the 162 pA the exact sums give. At $N = 312$ it is nowhere near.

The boundary is a current per hertz #

Everything above was measured at one bandwidth, and the bandwidth is half of the story. The count in a window is $N = I/(2qB)$, so the same current read in a narrower bandwidth delivers more carriers a window and looks more Gaussian, and read in a wider one looks less. Halving the bandwidth doubles $N$ and divides the skewness by $\sqrt 2$. The condition for a Gaussian five-sigma tail — forty thousand carriers a window — is therefore not a current at all. It is a current per hertz of noise bandwidth:

$$\frac{I}{B} \gtrsim 2q \times 40\,000 \approx 1.3 \times 10^{-14}\ \text{A per Hz}.$$

In a megahertz that is thirteen nanoamperes; in a kilohertz, thirteen picoamperes; in ten hertz, a hundred and thirty femtoamperes. The ten femtoamperes of the second figure, read in a tenth of a hertz, would be comfortably Gaussian. So the familiar advice to average longer has a second meaning here. It lowers the noise, as it always does, by $\sqrt{N}$; and it also makes the noise Gaussian, by the same factor in the skewness, which is what lets a threshold be set from a root-mean-square value at all.

The converse is the case that matters for fast instruments. A photodetector read in a hundred megahertz to resolve a nanosecond pulse needs over a microampere before its dark current’s shot noise is Gaussian at five sigma. Below that, its false-trigger rate is set by the count, and the count is always worse in the direction a trigger looks.

What a counter does instead #

An instrument that counts carriers, or photons, instead of filtering a current never meets this problem, because it never pretends the count is Gaussian. It sets its threshold from the Poisson distribution directly: in a window where the dark count averages 0.31, as in the femtoampere figure, the probability of three or more counts is about four in a thousand and of five or more about two in a hundred thousand, and those numbers can be read straight off the bars. The Poisson tail is simpler than a Gaussian’s, not harder. It is a finite sum of terms each of which is written down in one line.

That is why counting takes over from current measurement at the smallest signals. The count is what the physics delivers, its statistics are exact at every size, and an instrument that reads it as a count inherits exactness that an instrument reading it as a current has to give up the moment it converts a variance into a probability.

The shape of the reading, not only the count #

A real instrument does not count carriers in a rectangular window. It filters the current, most simply with a single pole, and reads the filter’s output. A single pole whose noise bandwidth is also $B$ passes the same variance, $2qI\cdot B$, because variance is all a noise bandwidth is defined by. Its reading’s shape is another matter.

Campbell’s theorem gives every cumulant of a filtered Poisson stream: the $n$-th is the arrival rate times the integral of the $n$-th power of the filter’s response to one carrier. For a single pole of time constant $\tau$ the second is $qI/(2\tau)$, which is $2qI\cdot B$ with $B = 1/(4\tau)$, and the third is $q^2 I/(3\tau^2)$. The skewness they make is $(2\sqrt2/3)/\sqrt{I\tau/q}$, which in terms of the same $N$ is $(4/3)/\sqrt N$ — a third more than the rectangular window’s $1/\sqrt N$.

The reason is where the pole puts its weight. A rectangular window weights every carrier in it equally. An exponential weights the newest carriers most and the older ones progressively less, so the reading at any moment is dominated by fewer carriers than the noise bandwidth suggests, and fewer carriers means more skew. The seeded trains confirm both expressions at four currents, the counted ones against $1/\sqrt N$ and the filtered ones against $(4/3)/\sqrt N$, within their sampling errors.

So a noise bandwidth fixes a reading’s variance and does not fix its shape. The bandwidth noise sees defines the noise bandwidth by exactly the property that makes it blind here: it is the width of a rectangle with the same integral of $|H|^2$, and the integral of $|H|^2$ is the second cumulant alone. The third cumulant is an integral of the response’s cube, and two filters with one noise bandwidth have different ones.

What a single pole’s readings look like #

At thirty femtoamperes, nine carriers a window, the pole’s readings have exactly the variance the density predicts, 0.999 of it. Their histogram leans: the most likely reading is a tenth of a standard deviation below the mean, the long tail runs upwards, and readings beyond three standard deviations above the mean occur 0.50 per cent of the time — nearly four times the Gaussian’s 0.135 per cent — while not one of twenty thousand fell three standard deviations below.

This is the picture an oscilloscope would show of a small photocurrent after a single-pole filter, and it is the picture a comparator’s threshold is set against. A reading that is usually a little below its mean and occasionally well above it is the signature of a current made of carriers, and a noise figure computed from the density contains none of it.

Where the Gaussian floor may be used #

The density is always right. The Gaussian reading of it is right when three conditions hold: the question asked is about the root-mean-square level, a signal-to-noise ratio, or anything else that is second order; or the carriers per window are many enough that the skewness is small against the threshold’s distance cubed; or the threshold is near the mean. For the questions most noise calculations ask — what is the floor, how much signal clears it — the first condition holds, and every essay built on 2qI is sound.

The questions that fail are the ones about rare events. How often a comparator trips on noise alone, how many false counts a photon detector records in the dark, how likely a sampled reading is to exceed a limit: each is a tail probability, each is set by the shape rather than the variance, and at picoamperes in kilohertz bandwidths the shape is not Gaussian. The noise a true-RMS meter reads low found a meter misreading noise because a square root sat inside an average; this is the same kind of error one moment later, a probability read off a variance that does not carry it.

And one current those earlier essays measured is safe by a wide margin. The base current of a bipolar stage, the current noise the two generators that are one current is built from, is a nanoampere at a microampere of collector current and a gain of a thousand: 312,000 carriers a window in 10 kHz, a skewness of 0.0018, and a Gaussian that is right to about four per cent at five sigma. The shape matters for the small currents a bipolar base does not carry.

How the numbers were obtained #

The count’s probabilities are summed from the Poisson distribution in logarithmic form, using a Lanczos approximation to the gamma function, away from the mean so that tails of 10⁻³⁰ are sums of small numbers. The Gaussian tails use a rational approximation to the complementary error function good to a part in ten million. The seeded arrival trains draw exponential gaps at the rate $I/q$ from a seeded generator: counted, the arrivals in each of 20,000 consecutive windows are tallied; filtered, each arrival adds $q/\tau$ to an output that decays with time constant $\tau = 1/(4B)$ between arrivals, run in for twelve time constants and then sampled every eight. Each seeded result is compared with its closed form inside a bound set by its own sampling error, including the excess kurtosis a small count carries.

What it leaves out #

It treats the carriers as independent. A resistor’s carriers are not — the lattice correlates them, which is why a resistor carrying a direct current has no shot noise at all — and a junction’s are independent only in the regime where their crossing times are short against the window. Carriers that arrive in bursts, as in an avalanche photodiode, have a variance larger than their mean and a different shape again.

It uses a single current with no other noise. A real reading adds the amplifier’s voltage noise and a resistor’s Johnson noise, both Gaussian, and a Gaussian added to a skewed count reduces the skewness of the sum. How much Gaussian noise it takes to make a femtoampere’s count look Gaussian is the question an instrument designer actually faces, and it is not measured here.

And the filters are the two simplest. A higher-order filter’s response has a different cube integral, and whether its reading’s skewness is closer to the rectangular window’s or further from it than a single pole’s is a calculation Campbell’s theorem makes easy and this page does not make.

Still open: a count with Gaussian noise added, a burst of carriers, and a filter of higher order #

A count read through a Gaussian floor. Adding the Johnson noise of a feedback resistor to the pole’s reading lowers the skewness as the ratio of the two variances, and a threshold’s false-trigger rate then moves back towards the Gaussian. The ratio of Gaussian to shot variance at which the five-sigma rate is within a stated factor of the Gaussian’s would put a number on when an amplifier’s own noise hides the carriers.

Carriers that do not arrive alone. An avalanche gain multiplies each carrier into a random number of them, which raises the variance by an excess noise factor and raises the skewness further. Measuring the tails of a multiplied count against its excess factor would say how much of the tail’s weight a gain stage adds that its noise figure does not report.

A filter of higher order. A second- or fourth-order filter of the same noise bandwidth has a response that rises and rings rather than jumping and decaying, and its cube integral may be smaller than a single pole’s. Computing the third cumulant for the filter families already measured here would say which reads a small current as the most nearly Gaussian, which is a filter property no noise bandwidth expresses.