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 figure reads from left to right. The blue card is input vibration frequency. The middle card applies this page's rule. The green card is observed alias. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

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.

Walk the arrows. A component above the Nyquist limit folds into the trusted band and can imitate a real slow vibration.

Derive the baseline in four named moves

1. 1

   Name the input. The chapter baseline is 38 Hz.

2. 2

   Name the relationship. alias = |input frequency - 50 x round(input frequency / 50)|

3. 3

   Substitute with units. |38 - 50 x round(38 / 50)| = 12 Hz

4. 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.

Input vibration frequency

38 Hz

Chapter baseline

Observed alias

Observe A component above the Nyquist limit folds into the trusted band and can imitate a real slow vibration. Reset input vibration frequency to 38 and compare observed alias.

Explain A component above the Nyquist limit folds into the trusted band and can imitate a real slow vibration.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only input vibration frequency moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only input vibration frequency. It is not a field prediction.

Try

Choose fs = 50 Hz to locate the 25 Hz Nyquist limit, then test a 40 Hz input and select Check audit.

Observe

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.

Explain

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:

f_Nyquist = f_s / 2 = 50 / 2 = 25 Hz

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:

f_alias = |f − f_s·round(f / f_s)|

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:

SNR improvement = √4 = 2:1; extra bits = log₂(2) = 1 bit

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.

Ready: work the ledger before checking it.