The Nyquist, Alias, and Oversampling Numbers
The Nyquist, Alias, and Oversampling Numbers
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
The Nyquist, Alias, and Oversampling Numbers
Three checks, ~4 minutes
Sampling math is a physics boundary: once the ADC folds a fast component into the low band, later software sees a perfectly believable but false waveform.
In the chapter’s own worked example, a machine-vibration sensor logging at 50 SPS has a Nyquist limit of just 25 Hz, so a genuine 30 Hz vibration component folds down into a phantom 20 Hz tone, while a 55 Hz component appears as a false 5 Hz wander. Both look like real signal, and the chapter insists the fix must sit before the ADC as an analog anti-alias filter, not as digital cleanup afterward. This audit asks the question that insistence invites: once a fast component has folded into the low band, can any digital filtering or averaging tell it apart from a true low-frequency signal?
Companion to the chapter Sampling and Anti-Alias Limits — every number here comes from that chapter.
See the relationship before changing it
The chapter figure below shows: Sampling a 50 Hz machine-vibration signal three ways: undersampling at 80 Hz aliases it to a false 30 Hz, while sampling at 500 Hz with an anti-alias filter reconstructs it faithfully.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only input vibration frequency, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 38 Hz.
- 2
Name the relationship. alias = |input frequency - 50 x round(input frequency / 50)|
- 3
Substitute with units. |38 - 50 x round(38 / 50)| = 12 Hz
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change input vibration frequency
Try Predict the direction of alias = |input frequency - 50 x round(input frequency / 50)|. Test another input vibration frequency, then compare observed alias.
Observe This exercise uses the descending fold from Nyquist (sample rate / 2) to the sample rate. Predict a decrease in the observed alias as the input increases; across wider ranges aliasing repeats and is non-monotonic. Reset to the chapter baseline and compare.
Explain Increasing input vibration frequency above 25 Hz brings motor-drive ripple closer to the logger's 50 samples/s clock, so folding produces progressively slower false pump motion.
Source: content/sensors/sensor-data-processing-sampling-aliasing.qmd — Phoebe's Field Notes: Why Four Samples Buys 2x, Not 4x. Use the chapter equation and fixed ledger above; only the named input varies.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Choose fs = 50 Hz to locate the 25 Hz Nyquist limit, then test a 40 Hz input and select Check audit.
The alias readout folds 40 Hz to 10 Hz; a 90 Hz tone sampled at 100 SPS also appears at the same false 10 Hz.
After folding to 10 Hz, sampling has lost the original spectral identity, so a digital filter cannot distinguish the alias from a genuine component there.
1. Nyquist sets the first limit
The chapter’s 50 SPS logger can only represent components below half the sample rate:
That means the 12 Hz pump feature is inside the measurable band, but the 38 Hz motor-drive ripple is outside it and must either be filtered before the ADC or it will fold down.
2. The alias calculation explains the false 12 Hz feature
Use the same formula introduced above:
| Check | Calculation | Result |
| 38 Hz ripple at 50 SPS | round(38 / 50) = 1; |38 − 50·1| | 12 Hz alias |
| Same ripple at 80 SPS | fNyquist = 80 / 2 = 40 Hz; round(38 / 80) = 0; |38 − 80·0| | 38 Hz, no fold |
| Safer 200 SPS capture with 40 Hz analog cutoff | fNyquist = 200 / 2 = 100 Hz; guard band = 100 − 40 | 60 Hz roll-off room |
3. Oversampling helps noise, not aliases
The chapter’s four-sample averaging statement follows from independent random noise shrinking by the square root of the number of samples:
That arithmetic is useful only after the analog path is safe. A coherent 90 Hz tone sampled at 100 SPS still aliases to |90 − 100·1| = 10 Hz, so averaging would preserve the false low-frequency component instead of removing it.
Every number above is taken from the chapter’s own sampling examples and re-derived step by step.
Technical boundaries: The single-tone alias formula omits anti-alias filter roll-off, aperture jitter, spectral leakage, finite record length, broadband noise, nonuniform sampling, and quantisation.