Math Bridge: Why can downsampling turn vibration into drift?

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Why can downsampling turn vibration into drift?

Follow one vibration through the sample grid before any smoothing or scaling touches it.

Data Dora, the guideData Dora guides
The one targetExplain why 100 Hz can look like 0 Hz after a 20 Hz downsample.
The chapter case100 Hz vibration, 20 Hz output, 10-bit 3.3 V ADC.
What it buys youA preprocessing order that does not preserve a lie.

A field team faces an unresolved physical question: Why can downsampling turn vibration into drift? They must answer it before changing downsample rate on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is downsample rate. The middle card applies this page's relationship. The green card is nyquist ceiling. 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 added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Downsample rate changes nyquist ceiling An input card leads through the page relationship to the nyquist ceiling result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The fold formula computes the visible result from the chosen grid. A low-pass filter must remove frequencies above the new Nyquist ceiling before decimation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for downsample rate is 20.

  2. 2

    Name the relationship. fs,min=2(100)=200 Hz; 20/200=0.10x n=round(100/20)=5; falias=|100-5(20)|=0 Hz 2¹⁰=1,024; q=3.3/1,024=3.22 mV; qrms=0.930 mV; SNR=62.0 dB

  3. 3

    Substitute the chapter fixture. Set downsample rate to 20. The page ledger gives nyquist ceiling as 10 Hz.

  4. 4

    Read the result. Keep Hz beside the value. Use it only inside the technical boundary on this page.

Predict, then change downsample rate

Try Predict the direction of nyquist ceiling. Move one control, calculate, then check your prediction.

20
Chapter baseline
Nyquist ceiling

Observe The fold formula computes the visible result from the chosen grid. A low-pass filter must remove frequencies above the new Nyquist ceiling before decimation. Reset the control to 20 and compare nyquist ceiling.

Explain Only downsample rate moves here. The other chapter fixtures remain fixed.

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 downsample rate moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only downsample rate. It replays the chapter fixture; it is not a field prediction.

1. Start with the sample grid

A sample is one point in time. To distinguish the fastest cycle, the grid needs at least two points per cycle. A rate change creates a new grid and must obey the rule again.

Data Dora: Downsampling is another sampling event, not a harmless file-size setting.

2. Name every algebra move

1

Double the fastest frequencyThe minimum safe rate is fs,min=2f.

2

Choose the nearest copyUse n=round(f/fs).

3

Measure the foldfalias=|f−nfs|.

4

Count ADC codesLevels=2ᴺ and q=Vref/2ᴺ.

5

Turn step into RMS noiseqrms=q/√12; ideal SNR=6.02N+1.76.

3. Reproduce the chapter case

fs,min=2(100)=200 Hz; 20/200=0.10×
n=round(100/20)=5; falias=|100−5(20)|=0 Hz
2¹⁰=1,024; q=3.3/1,024=3.22 mV; qrms=0.930 mV; SNR=62.0 dB

The vibration lands on the same phase every sample, so its changing part disappears and can be mistaken for a fixed offset.

4. Try the new sample rate

TryMove the downsample rate away from 20 Hz and watch the same 100 Hz input fold to a different apparent frequency.

Downsample rate
Nyquist ceiling
Required rate
Requirement fraction
Nearest copy n
Apparent frequency
Temperature ceiling
ADC levels
ADC step
ADC RMS noise
Ideal ADC SNR

ObserveAt 20 Hz, the 100 Hz input becomes 0 Hz. At 30 Hz it appears at 10 Hz. The ADC numbers stay fixed because timing loss and amplitude rounding are separate limits.

ExplainThe fold formula computes the visible result from the chosen grid. A low-pass filter must remove frequencies above the new Nyquist ceiling before decimation.

Technical boundaries.

This compact engine follows one ideal tone and one ideal ADC.

Tone
Real vibration contains bands, transients, leakage, and sensor resonances
Filter
A real anti-alias filter needs transition width, attenuation, and phase limits
ADC
Ideal quantization omits thermal noise, nonlinearity, clipping, and reference error

Validate the real spectrum, filter response, converter, and rate-change chain.

5. Put operations in a safe order

Measure the input band, low-pass below the new ceiling, decimate, record the rate change, and only then smooth, impute, normalize, or aggregate.

6. Keep the preprocessing record

Store original and output rates, filter version, passband, stopband, converter range and bits, missing-data rule, timestamp policy, owner, and retest trigger.

7. Check yourself

Why is 200 Hz the minimum for a 100 Hz component?
Answer: Nyquist needs at least two samples per cycle, so 2×100=200 Hz.
Why does 100 Hz become DC at 20 Hz?
Answer: The nearest copy is n=5, so |100−5×20|=0 Hz.
Can normalization recover the lost vibration?
Answer: No. It can rescale recorded values, but it cannot identify which frequencies produced an aliased sequence.
Honesty boundary.

The signal and downsample rates come from the chapter; the ADC is a labelled teaching assumption.

100 Hz and 20 Hz
The chapter's vibration and downsample case
1 Hz
The chapter's temperature-stream rate
10-bit, 3.3 V
A catalog-typical low-cost data-logger example, not a chapter-specified converter

Correct, not complete: production preprocessing needs measured spectra and filter evidence.

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