Math Bridge: Sampling Before Filtering

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Math BridgeSensorsStruggle-friendly runway

What can a 100 Hz accelerometer really reveal?

One thread from sample timing to aliases, code steps, and the noise a filter must actually remove.

Phoebe, the physics guidePhoebe guides
The one targetSeparate acquisition limits from filtering.
The chapter case100 Hz, 3.9 mg/LSB, 0.3 g spikes.
What it buys youStop a filter from polishing false frequencies.

A field team faces an unresolved physical question: What can a 100 Hz accelerometer really reveal? They must answer it before changing accelerometer sample rate on the real device. Predict the direction first.

See the relationship before changing it

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.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for accelerometer sample rate is 100.

  2. 2

    Name the relationship. fN=fs/2; falias=|fsignal-kfs|; σq=q/√12; ratio=A/σq

  3. 3

    Substitute the chapter fixture. Set accelerometer sample rate to 100. The page ledger gives 70 hz appears at as 30 Hz.

  4. 4

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

Predict, then change accelerometer sample rate

Try Predict the direction of 70 hz appears at. Move one control, calculate, then check your prediction.

100
Chapter baseline
70 Hz appears at

Observe Changing fs moves the alias but not the transducer's code step. Sampling design and digital smoothing solve different problems. Reset the control to 100 and compare 70 hz appears at.

Explain Only accelerometer sample 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 accelerometer sample rate moves. Field effects named in the page's technical boundary stay fixed.

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

1. Sample first, filter second

An accelerometer turns motion into one number every sample interval. A digital median or moving average can only work with those numbers after sampling has already decided which frequencies they represent.

Phoebe: A later filter cannot recover a frequency that the sampler already folded into the wrong place.

2. Name the ceiling

1

Halve the rateNyquist says the largest representable frequency is fs/2.

fs = 100 Hz ⇒ fNyquist = 100/2 = 50 Hz

A real component above 50 Hz needs analogue anti-alias filtering before conversion, or it can masquerade as a lower one.

3. Turn one code into a noise floor

2

Use the code stepThe ADXL345 full-resolution scale is q = 3.9 mg/LSB = 0.0039 g.

3

Convert rounding to RMSUniform rounding within one step has σq = q/√12.

σq = 0.0039/√12 = 0.00113 g RMS

4. Try the sample rate

fN=fs/2; falias=|fsignal−kfs|; σq=q/√12; ratio=A/σq

TryMove the sample rate while the chapter's ADXL345 scale, spike, and drift stay fixed. The 70 Hz tone is a labelled interaction value used only to make folding visible.

Nyquist ceiling
70 Hz appears at
Quantisation RMS
0.3 g spike / floor
0.15 g drift / floor
Spike separation

ObserveAt 100 Hz, the ceiling is 50 Hz and the illustrative 70 Hz tone appears at 30 Hz. The chapter's 0.3 g spike is about 266 times the 0.00113 g quantisation floor.

ExplainChanging fs moves the alias but not the transducer's code step. Sampling design and digital smoothing solve different problems.

Technical boundaries.

The alias calculation assumes perfectly periodic sampling, and the rounding-noise estimate assumes errors spread evenly within each code step and unrelated across samples. Changing the sample rate leaves the accelerometer’s code step fixed; timing irregularities and correlated rounding errors require measurements of the real acquisition chain.

The q/√12 model assumes approximately uniform, uncorrelated quantisation error
Needs separate evidence
measurement
Needs separate evidence

Use field evidence or a deeper model before release.

5. Compare the chapter's disturbances

0.3/0.00113 ≈ 266 (48.5 dB); 0.15/0.00113 ≈ 133

Those electrical spikes and slow mechanical drift are much larger than ADC rounding. The median-plus-moving-average chain is therefore aimed at physical and electrical disturbances, not at the quantisation floor.

6. Put each defence in order

Choose sensor bandwidth and sample rate first. Place an analogue anti-alias filter before the ADC. Then use the chapter's five-sample median for impulses and its 50-sample moving average for a stable 0.5-second summary.

7. Check yourself

What is the Nyquist ceiling at 100 Hz?
Answer: 50 Hz, because the ceiling is half the sample rate.
Why is the ADXL345 rounding floor about 0.00113 g?
Answer: Divide its 0.0039 g code step by √12.
Can the moving average restore an aliased vibration?
Answer: No. Once sampling labels it as a lower frequency, the digital filter has no record of its original frequency.
Honesty boundary.

The alias calculation assumes perfectly periodic sampling, and the rounding-noise estimate assumes errors spread evenly within each code step and unrelated across samples. Changing the sample rate leaves the accelerometer’s code step fixed; timing irregularities and correlated rounding errors require measurements of the real acquisition chain.

100 Hz rate
Accelerometer sampling rate
3.9 mg/LSB step
Acceleration represented by one digital code
0.3 g spikes
Transient acceleration spike magnitude
0.15 g drift
Slow acceleration drift magnitude
five-sample median
Median-filter window length
50-sample average reproduce the chapter
Moving-average window length
70 Hz tone is explicitly an interaction value
Illustrative tone used to demonstrate aliasing

This runway does not claim a complete ADXL345 or bearing-fault error model; Under the Hood keeps the real filter and deployment trade-offs.

Return to the chapter with the maths unpacked →