Archi-Civilarchi-civil.com
Structural Health Monitoring · SHM-01  |  Confidence: A (closed-loop verified)
Figures are produced by engine/dsp.js and pinned in tests/dsp.test.js.

Home / Articles / Structural Health Monitoring SHM-01

Sampling, Nyquist and Aliasing: The Error You Cannot Fix Afterwards

The first link in the measurement chain, and the only decision in it that cannot be undone. A badly chosen sampling rate embeds a falsehood in the data that can never be separated out again — and no amount of software will help.

Confidence: A Engine: engine/dsp.js Tests: tests/dsp.test.js — 26/26 Reading: ~11 min Last verified: 2026-09-10 Türkçe: bu yazının Türkçesi

1. The measurement chain

In structural health monitoring, data passes through these links before it becomes a mode:

structure → accelerometer → anti-aliasing filter → sampler (ADC) → record → windowing → FFT → peak picking → modal parameters The two links in bold are in hardware and cannot be changed after the measurement is over. The rest of the chain is software and can be repeated as often as you like.

That distinction is the whole subject of this article. If you chose the wrong window, you recompute the spectrum. If you chose the wrong sampling rate, you have no option but to repeat the measurement.

2. What sampling does

A continuous acceleration signal is read at equal intervals and converted to numbers. The number of readings per second is the sampling rate fs. What happens between readings is unknown to us — and the data does not know it is unknown either. The digital record carries no trace of the gap.

Intuitively: to recognise a cycle of vibration you need at least two points per cycle. One peak, one trough. With fewer than that, there is no mathematical way to tell a fast oscillation from a slow one.

3. The Nyquist limit and folding

That intuition is made precise by the Nyquist–Shannon sampling theorem. In a signal sampled at fs, the highest frequency that can be represented correctly is the Nyquist limit:

fNyquist = fs / 2 Every component above this limit does not vanish — it folds into the spectrum and appears as a lower-frequency component.

The apparent frequency follows from the folding relation:

fapparent = | f − fs · round( f / fs ) | round(·) is rounding to the nearest integer. The relation says the spectrum folds like a mirror about fs/2.

4. Worked case: folded components

The table below is computed for fs = 50 Hz; the figures are produced by the engine when the page loads. The Nyquist limit is —.

True frequencyAppears in spectrum atStatus
computing…
 Every component above 25 Hz has been folded down. The 45 Hz component appears at 5 Hz — that is, in a position where it could land directly on top of a genuine low-frequency structural mode.

The row to watch is the 45 Hz one. A high-frequency noise source — a machine, a pump, a mains harmonic — appears in the spectrum as though it were a low-frequency structural mode. The engineer looking at the spectrum has no information with which to tell them apart.

5. Why it cannot be fixed afterwards

The irreversible link in the chain Aliasing is not a loss of information but a mixing of it. The folded component does not appear as a separate peak standing on its own; it is added on top of whatever content already exists at that frequency. Two contributions have become a single number. There is no mathematical way to recover the parts from the sum — no trace remains of the proportions in which they were added.

This is why the remedy is not numerical but analogue: before sampling, while the signal is still continuous, you suppress content above Nyquist with a low-pass filter. This is the anti-aliasing filter, and it sits in hardware between the accelerometer and the ADC.

In practice the filter is not ideal; suppression after the cut-off is gradual. The sampling rate is therefore usually chosen appreciably higher than twice the highest frequency of interest — to leave room for the filter's transition band.

6. Collision: the worst case

The most insidious form of aliasing is when several components fold to the same apparent frequency.

Collisioncomputing…

In that case you see a single peak in the spectrum, and that peak is the sum of three separate physical events with nothing to do with one another. Interpreting its amplitude is meaningless; even attributing the existence of the peak to a mode is wrong.

7. Record length is a separate decision

The sampling rate determines how high you can look. The record length determines how finely you can discriminate. These are two independent decisions and they are often confused.

fs → upper limit: fNyquist = fs/2   ·   T → resolution: Δf = 1 / T Sampling faster does not improve resolution. The only way to separate two modes closer than 1/T is a longer record.

A worked figure: separating two modes 0.02 Hz apart requires a record of at least 50 seconds; raising the sampling rate from 100 to 1000 Hz changes nothing about that. This distinction is treated in detail in SHM-02.

8. Setting up a measurement

  1. Identify the highest mode you care about. In structural health monitoring the first three to five modes are usually enough; for building periods that is a region of a few Hz.
  2. Choose a sampling rate at least twice that, and in practice appreciably above. Leave room for the filter's transition band.
  3. Confirm that an anti-aliasing filter exists. This is not a software setting; it is looked up in the instrument's datasheet. Without one, no sampling rate will save you.
  4. Set the record length from the resolution you need, not from the sampling rate.
  5. Note the high-frequency sources in the environment. Machines, generators, traffic. Work out in advance where they will land when folded — which is exactly what the table in this article is for.

9. Test yourself

  1. You are sampling at fs = 200 Hz. At what frequency does a 130 Hz component appear in the spectrum?
  2. In that situation, does raising the sampling rate to 400 Hz solve the problem? What about doubling the record length?
  3. You see an unexpected peak at 5 Hz in a spectrum. How would you test whether it is folded noise? (Hint: change the sampling rate and measure again — a genuine mode stays put, a folded component moves.)
  4. Why must the anti-aliasing filter be analogue rather than digital?
  5. What is the minimum record length needed to separate two modes 0.01 Hz apart?

References

  1. Nyquist–Shannon sampling theorem. The derivation is universal and belongs to no national code.
  2. The folding relation fapparent = |f − fs·round(f/fs)| follows directly from the periodicity of the spectrum.

All figures in this article are produced by engine/dsp.js. Verification is not by comparison with third-party output but by closed loop: we generate a known signal ourselves, pass it through the chain and measure whether the engine recovers the known values.

Open the analysis tool → SHM-02: FFT and windowing → Türkçe okuyun →
archi-civil.com — Sampling, Nyquist and Aliasing · SHM-01 · Confidence A · Printed:
Aliasing cannot be corrected by any numerical operation after the recording. The remedy is in hardware.