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.
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.