Sampling, quantisation and aliasing: how not to record a false measurement

The theorem, and what it does not say

The sampling theorem says that a signal containing no energy above a frequency can be reconstructed exactly from samples taken at more than twice that frequency. Every word of that matters, and two of them are almost always overlooked: "containing no energy above" is a statement about the analog signal, not about the part of it you are interested in; and "reconstructed" means with an ideal interpolation, not by joining the dots on a plot.

So the theorem is a floor, and treating it as a target produces two disappointments. A hundred-hertz sine sampled at two hundred and fifty hertz satisfies Nyquist and can still show a peak amplitude several percent low, because no sample happened to land on the peak. And a signal with energy above your band of interest — which is every real signal — folds that energy down into your band unless something stopped it before the converter.

Aliasing, demonstrated once and for all

That figure is the whole subject. A seven-period signal sampled at eight intervals produces exactly the same nine values as a one-period signal, so no algorithm can tell them apart: the information distinguishing them was destroyed at the moment of sampling, not lost somewhere downstream.

Three consequences follow, and they are the reason this article exists. Aliasing is an analog phenomenon that happens before the converter, so no digital filtering afterwards addresses it. An alias is indistinguishable from a genuine signal, so it does not look like a fault — it looks like a discovery. And the folded frequency is not random: energy at a frequency above half the sample rate appears at the difference, which is why a 2 kHz vibration sampled at 2.4 kHz shows up as a very convincing 400 Hz oscillation.

The anti-alias filter, and why oversampling makes it cheap

The only protection is an analog low-pass filter before the converter, and its job is to attenuate everything above half the sample rate to below the converter noise floor. How hard that is depends entirely on how much room you gave it, and the arithmetic is worth doing once.

A filter rolls off at about six decibels per octave per pole. Sample at ten times your highest frequency of interest and there are 2.3 octaves between the top of your band and half the sample rate — a four-pole filter delivers some fifty-five decibels of attenuation there, which is plenty. Sample at two and a half times instead, and there are only 0.3 octaves available: the same four-pole filter gives eight decibels, and no realisable analog filter will save you.

That is the real reason for oversampling, and it is an economic argument rather than a purist one: the sample rate you can afford in bits per second buys you the filter you do not have to build.

The filter also has a cost that is easy to forget. It delays the signal, and the delay varies with frequency unless the filter is specifically linear-phase. Two channels with different filter settings therefore arrive with different delays — and if you intend to compare them, that delay difference is a phase error you will measure and attribute to the aircraft.

Choosing the rate: the factor of five to ten

Putting the two arguments together gives the working rule. Five to ten times the highest frequency of interest: enough for a readable time history, enough amplitude accuracy without interpolation, and enough room for a modest analog filter followed by digital processing.

Below three times, you are relying on ideal reconstruction and a very steep filter. Above twenty times, you are paying bits per second for resolution you could have obtained by oversampling and decimating — which is the next section, and the more efficient route.

Quantisation: the bits you have, and the bits you use

A converter of N bits over a range R has a step of R divided by two to the N. Sixteen bits over ±10 V is a step of about 305 microvolts, and the quantisation noise that comes with it sets a signal-to-noise ratio of roughly six decibels per bit plus two — about ninety-eight decibels at full scale.

The phrase "at full scale" is where the bits go. A signal that only ever uses five percent of the input range is a signal recorded with 4.3 bits fewer than the datasheet promises: a sixteen-bit converter delivering an eleven-and-a-half-bit measurement. This is the same argument as the conditioning gain, seen from the other end, and it is the most common resolution loss in real installations.

The effective number of bits — ENOB — is the honest figure: what remains after the converter own noise and distortion and, more importantly, after the amplifier and the range choice ahead of it. Nobody should quote a converter resolution for a measurement; the measurement has an ENOB, and it is always lower.

Oversampling and decimation: buying bits with bit rate

Averaging M consecutive samples of a signal with uncorrelated noise improves the signal-to-noise ratio by ten times the logarithm of M — which means every factor of four in sample rate buys one bit of effective resolution. Sample sixteen times faster than you need, average, and you have gained two bits for free in every sense except bit rate on the way in.

That is how a system built on a sixteen-bit converter delivers eighteen-bit performance, and it is also why oversampling followed by a digital decimation filter is usually better than a steep analog filter: the digital filter can be linear-phase, identical on every channel, and changed after the flight if you got the corner frequency wrong.

The condition is that the noise be uncorrelated. Averaging does nothing against a periodic interference at a frequency related to your sample rate, and it does nothing at all against an alias — which was already indistinguishable from signal before you averaged it.

Inter-channel skew, and what it does to a correlation

Two channels are only comparable if they were sampled at the same instant. Front ends that multiplex one converter across several inputs do not do that: they sample in sequence. Sixteen channels sharing a converter at an aggregate hundred kilosamples per second means ten microseconds between neighbours, and a hundred and fifty microseconds between the first channel and the sixteenth.

On a one-kilohertz signal, a hundred and fifty microseconds is fifty-four degrees of phase. Any analysis that compares phase between those two channels — mode shapes, transfer functions, load paths, anything with a coherence plot in it — is measuring the acquisition system and calling it the structure.

The fixes are known and should be specified deliberately: a simultaneous-sampling front end with one converter per channel, or a documented skew that is corrected during processing. What is not acceptable is not knowing which of the two you have.

The mistakes, collected

• Treating Nyquist as a target. It is a floor, and a signal reconstructed at the floor is not a curve anybody can read.
• A filter that does not match the rate that channel was actually assigned — usually because the rate was changed and the filter was not.
• Different filters on two channels you will compare, which manufactures a phase difference out of nothing.
• An input range far larger than the signal, which throws away bits before the first sample.
• Averaging to reduce noise that is not random, or to remove an alias, which is already impossible.
• Assuming a multiplexed front end samples simultaneously.
• And the one that hides everything else: no spectral check of a raw channel before the campaign. Ten minutes with a spectrum on a bench finds every alias you were about to record.

That last item is the whole article in practical form. Look at the spectrum of each sensor family once, at a rate well above what you intend to use, and see what is actually there. Whatever you find above your intended band is what your filter has to remove — and if you skip that ten minutes, the filter is a guess.