The noise a true-RMS meter reads low

The average a square root pulls low took a true-RMS converter apart. It squares its input, averages the square through a one-pole filter, and takes the root — and because the root is concave, the mean of the root of anything that ripples is below the root of its mean. The reading is low whenever the averaged square has any ripple on it, by an eighth of that ripple’s power, and never high.

On a sine the ripple is a single line at twice the signal’s frequency, reduced by the averager’s gain there, and the bias falls as the square of that gain. With a 100-millisecond averager a sine is one per cent low below 1.86 hertz and a few parts per million low a decade above. For the waveforms that essay drew — sines, triangles, square waves, rectifier currents — the bias is a problem for slow signals and for nothing else.

That essay ended by naming the input that breaks the reassurance. Noise is not periodic, its square has no lines to be far above, and a true-RMS meter is the standard instrument for measuring it. The arithmetic suggested a bias that falls only as the first power of the averaging, and suggested a size — a part in a few thousand for a kilohertz of noise through a hundred milliseconds. This essay measures it.

The noise bandwidth that sets this bias is the same quantity the noise field measures from the other side. The bandwidth noise sees is the width a filter presents to white noise, which is exactly the $B$ in what follows; the floor a resistor sets is the noise a meter of this kind is usually asked to read; and what a meter multiplies by is the cheaper meter whose form-factor error a true-RMS converter was bought to avoid. The concave root that biases the reading is the curvature argument of how small is small signal, applied to a square root rather than an exponential.

The sine, for comparison #

The sine’s result is the baseline, and its shape is the thing to hold on to. Two curves fall with frequency: the reading’s ripple, as the first power, and its mean’s bias, as the second, because the bias is an eighth of the ripple’s square. Far above the averager’s corner the bias is negligible long before the ripple is. A meter that has settled its display on a sine of a few hertz is, to every practical purpose, unbiased.

The sine’s mechanism is worth seeing once more before the noise replaces it. At two hertz the averaged square is right on average and ripples by a large fraction; its root is below one on average by an eighth of that ripple’s power. Every part of that picture carries over to noise except one: the ripple on a sine’s averaged square is a single line that a slower averager removes quadratically, and the fluctuation on noise’s averaged square is a continuum reaching down to zero frequency that a slower averager removes only in proportion.

Noise, measured #

The input here is Gaussian noise of unit power, band-limited either to a band from zero to B or to a band of the same width B centred at three times B, and drawn from a seeded generator so that every number can be regenerated. Each record is 2²¹ samples at eight times the bandwidth. The converter is the same explicit one — square, one-pole average of time constant τ, root — started on the record’s own mean square and allowed twelve time constants to settle before any reading is kept.

The truth each reading is compared against is the record’s own root-mean-square over the stretch the readings cover, not the unit power the generator was asked for; a finite record’s power is itself a random number, and comparing against the nominal would add that randomness to the bias. And because the bias is small beside the scatter of a single reading, it is estimated with a control variate: the reading minus the first two terms of its own series about the truth, whose mean is the bias and whose scatter is the square of the ripple’s rather than the ripple’s. The standard error comes from batch means over twenty time constants.

The noise reading is low at every averaging time drawn, as the concave root requires. At a product of bandwidth and averaging time of ten it is 0.600 per cent low, with a standard error of 53 parts per million. At a hundred it is 672 parts per million low. At one — an averager whose corner is comparable with the noise’s own bandwidth — it is 4.55 per cent low.

Beside it the sine, read by the same converter at the same product of frequency and averaging time, is 3.96 parts per million low at ten and 0.0396 at a hundred. At ten the noise reading is fifteen hundred times further off than a sine’s; at a hundred, seventeen thousand times. The two fall with averaging at different rates, and that is the whole finding in one figure: the sine’s bias falls as the square of $f\tau$ and the noise’s as the first power of $B\tau$.

Why the first power #

The reason is where the square’s fluctuations live. A sine’s square is a constant plus one line at twice its frequency, and the averager is far above nothing but that line. Noise’s square has fluctuations at every frequency from zero upwards. For Gaussian noise of unit power band-limited to a width $B$, the fluctuation of its square has a spectral density, near zero frequency, of $1/B$ — the self-convolution of a flat band — and no averager, however slow, is far above zero frequency. A one-pole averager of time constant τ passes a noise bandwidth of $1/(2\tau)$ of that density, so the averaged square has a variance of $1/(2B\tau)$, and the reading is low by an eighth of it:

$$\text{bias} \approx -\frac{1}{16\,B\tau}.$$

The dashed line in the figure is that expression. The measured readings approach it as $B\tau$ grows: 0.600 per cent against 0.625 at ten, 0.309 against 0.3125 at twenty, 672 parts per million against 625 at a hundred, where the standard error is 19 parts per million and the gap is inside three of them. The figure checks the agreement above $B\tau = 10$ to five standard errors and six per cent of the prediction. Below ten the measured bias is smaller than the expression says — 4.55 per cent against 6.25 at one — because the expansion behind it assumes the averaged square’s ripple is small, and at $B\tau = 1$ a single reading scatters by a third of its value.

That is the number the previous essay reached for and could not measure. A kilohertz of noise through a hundred-millisecond averager is $B\tau = 100$: the meter reads it about 650 parts per million low, a part in fifteen hundred. The prediction said a part in a few thousand.

A band about a centre reads the same #

The expression has the bandwidth in it and nothing else about the noise, and that is worth checking, because the most common noise measurement is of a narrow band some distance from zero — the output of a filter, a spectrum analyser’s resolution bandwidth, a band-limited receiver.

The figure’s second set of points is noise of the same width $B$ centred at three times $B$. At $B\tau = 10$ it reads 0.576 per cent low against the band from zero’s 0.600; at a hundred, 623 parts per million against 672. The figure checks the two agree, to five combined standard errors and eight per cent, from $B\tau = 5$ upwards. The reason is the same self-convolution: a band of width $B$ at any centre folds onto zero frequency in its square with the same density, $1/B$, as a band from zero. The centre frequency moves where the square’s other fluctuations lie — near twice the centre — and a slow averager removes those entirely.

Below $B\tau = 5$ the two part company, 4.08 per cent against 4.55 at one, because an averager that fast also passes some of the fluctuations that differ between the two shapes. For a meter the practical reading is simple: a narrow band is as hard to measure as a low one of the same width, and a narrow band anywhere is hard. A ten-hertz resolution bandwidth read through a hundred milliseconds is $B\tau = 1$, and a true-RMS reading of it is four or five per cent low.

One reading, and the mean of many #

The bias is not what a user of the meter sees first.

A single reading of noise scatters about its own mean by half the square root of the averaged square’s variance, which is $1/\sqrt{8B\tau}$: at $B\tau = 10$, 11.05 per cent, eighteen times the bias; at a hundred, 3.62 per cent, fifty-four times it. The figure checks the measured scatter against the expression to six per cent. Nobody looking at one reading of noise, or at a display that wanders by eleven per cent, will notice that its centre is half a per cent low.

The two fall at different rates — the scatter as the square root of $B\tau$, the bias as its first power — and that difference decides when the bias matters. Averaging independent readings reduces their scatter as the square root of how many there are and leaves the bias exactly where it was. The mean of about 339 independent readings at $B\tau = 10$ scatters by as much as it is biased; at a hundred, 2,898. Beyond that the mean of the readings converges, confidently and to many figures, on the wrong value.

That is the uncomfortable property this measurement adds to the previous essay’s. For a sine, taking more readings was never the problem, because the bias was negligible. For noise, the practice that makes a measurement look better — logging a converter’s output for a long time and averaging it — makes the bias the whole of the error. The corner where averaging stops working found that averaging stops reducing the scatter of flicker noise; here averaging keeps reducing the scatter and stops reducing the error.

Where the bias is large enough to matter #

The expression makes it easy to sort measurements into ones where the bias is negligible and ones where it is not, and the sorting is not the one intuition suggests.

A wideband measurement is safe almost regardless of the meter. An audio amplifier’s output noise read across twenty kilohertz with a one-second averager is $B\tau = 20{,}000$, and the reading is about three parts per million low — far inside any meter’s accuracy. So is a broadband receiver’s noise floor read through a hundred milliseconds. The bias belongs to narrow bands, and a narrow band is precisely what a careful noise measurement produces: a filter is put in front of the meter to say where the noise is, and the narrower the filter, the more the meter’s own averaging has to do.

A noise density measured with a one-hertz bandwidth and a one-second averager is $B\tau = 1$, and the reading is about four and a half per cent low — about 0.4 decibels, the kind of discrepancy that gets attributed to the filter’s bandwidth being slightly different from its label. A spectrum analyser that averages its displayed amplitude rather than its power, which is what an analyser’s video averaging does unless it is told otherwise, is the same measurement with a different name, and has a bias of the same kind; the well-known correction such instruments apply for averaging a logarithm is a larger instance of the same concavity, applied to a log rather than a root.

The averager’s shape #

Everything here uses a one-pole averager, because that is what an analogue true-RMS converter contains. The filter an average is found that a rectangular window of length $T$ has a noise bandwidth of $1/(2T)$ and that a one-pole of the same noise bandwidth takes 2.33 times as long to settle to one per cent. The bias depends on the averager only through the noise bandwidth it presents to the square’s fluctuations, so a rectangular average of the square with the same noise bandwidth has the same bias and reaches its reading 2.33 times sooner.

That does not change the conclusion about where to take the root — a rectangular window followed by a root is still biased — but it changes the cost of lengthening the average. A digital meter that averages the square over a sliding rectangle buys a given bias in less than half the time an analogue converter’s capacitor does, and the six seconds a ten-hertz band needed through a one-pole become about two and a half.

What a meter should do about it #

The bias is a known function of two numbers, so it can be removed by arithmetic: multiply the mean reading by $1 + 1/(16B\tau)$ when $B\tau$ is large. That needs the noise bandwidth, which is usually known — it is the filter in front of the meter — and the converter’s averaging time, which is usually printed. It does not work below $B\tau$ of about ten, where the expression overstates the bias.

The better remedy removes the root from inside the average. Averaging the square over the whole logging time and taking one root at the end reads the mean square exactly, whatever the ripple, because the square is linear in what it averages and only the root is concave. A meter that logs the averaged square rather than its root has no bias of this kind at all; one that logs its displayed reading and averages that has the whole of it. The filter an average is describes what the averager passes; this is a reason to take the root after it rather than before a second average.

Lengthening the averager also works, and costs time. Holding a noise reading within a tenth of a per cent needs $B\tau$ of about sixty, which for a ten-hertz band is six seconds of averaging — and a true-RMS converter’s reading falls more slowly than it rises, so a step down in the noise takes longer still to be read.

The explicit converter reaches within one per cent of a doubled value in 3.629 time constants and of a halved value in 5.006, so a six-second averager chosen to hold a ten-hertz band’s bias to a tenth of a per cent needs half a minute to follow a halving of the noise. That is the trade the bias forces on a narrow-band measurement: time to settle against error in the mean, with no setting that gives both.

What the measurement contains #

Gaussian noise. The factor of two in the square’s fluctuation density is a property of the Gaussian distribution; noise with a different distribution — a sum of a few sinusoids, impulsive interference, quantisation noise — has a different fluctuation density and a different constant in front of $1/(B\tau)$. The first power survives for any stationary noise whose square fluctuates at zero frequency; the sixteen does not.

Brick-wall bands. A real filter’s band has skirts, and the relevant width is its noise bandwidth rather than its nominal one. The self-convolution of a rounded band has a different density at zero frequency from a flat one of the same noise bandwidth — smaller, for the same power, since a flat band is the shape that concentrates its power most evenly — so the brick-wall bias is an upper bound on a real filter’s, and the width to put into the expression is the filter’s statistical bandwidth rather than its nominal one.

One seed a point. Each point is one seeded record, with a standard error from batches within it. The agreement with the expression is within the errors quoted; a different seed moves every point by about a standard error, and at $B\tau = 100$ that is three per cent of the bias.

Still open: the converter that knows the cycle, flicker noise, and the heater #

Summing over whole cycles. For periodic signals the bias can be removed entirely by averaging the square over an exact number of periods instead of through a filter. The cycle a converter has to know measures what that costs when the period is known only approximately, and finds a window error that does not fall with the number of cycles.

Flicker noise. A noise whose density rises without limit towards zero frequency has a square whose fluctuations do too, and there is no $B\tau$ large enough: the variance of the averaged square does not converge as the averaging lengthens. Whether a true-RMS reading of flicker noise has a bias that falls at all with averaging, and at what rate, is unmeasured.

The thermal converter. A heater and a thermocouple average power in a thermal mass and take no root inside the average — the root is taken by the calibration, once. Its reading of noise should therefore have no bias of this kind, and a scatter set by its thermal time constant. Solving it as a thermal network driven by noise would test whether the instrument that calibrates every other converter is also the one that reads noise correctly.