Sampling Physics and FFT Payload

Sampling Physics and FFT Payload

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Sampling Physics and FFT Payload

The chapter derives a sampling rate from physics: a 1800 RPM motor spins at 30 Hz, its eight rolling elements produce a 240 Hz bearing-fault tone, and after harmonics and Nyquist the deployable rate is 4 kHz. Raw vibration then costs 8 KB/s, which an FFT summary cuts about 67x. This audit checks the sampling physics and FFT payload that justify edge compression for this bearing example.

Companion to the chapter Edge Sampling and Compression — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is fft summary size. The middle card applies the page rule. The green card is payload reduction. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

FFT summary size changes payload reduction An input card leads through the rule reduction = 8,000 raw bytes/s / summary bytes/s to the payload reduction result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A smaller summary saves more radio traffic only if it still preserves the fault peaks.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 120 bytes/s.

  2. 2

    Name the relationship. reduction = 8,000 raw bytes/s / summary bytes/s

  3. 3

    Substitute with units. 8,000 / 120 = 66.67 times

  4. 4

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

Predict, then change fft summary size

Try Predict the direction of reduction = 8,000 raw bytes/s / summary bytes/s. Test another fft summary size, then compare payload reduction.

120 bytes/s
Chapter baseline
Payload reduction

Observe A smaller summary saves more radio traffic only if it still preserves the fault peaks. Reset fft summary size to 120 and compare payload reduction.

Explain A smaller summary saves more radio traffic only if it still preserves the fault peaks.

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 fft summary size moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only fft summary size. It is not a field prediction.

TryThe chapter derives a sampling rate from physics: a 1800 RPM motor spins at 30 Hz , its eight rolling elements produce a 240 Hz bearing-fault tone, and after harmonics and Nyquist the deployable rate is 4 kHz . Use Check derivation.
ObserveThe displayed ledger resolves 1800 RPM, 30 Hz, 240 Hz, 4 kHz, 8 KB/s at full precision. This audit checks the sampling physics and FFT payload that justify edge compression for this bearing example. Check derivation shows this.
ExplainThe 4 kHz rate follows the 240 Hz bearing tone through harmonics and Nyquist margin, while the 67x FFT payload cut works because it retains diagnostic frequency peaks instead of every waveform sample. Check derivation confirms it.

Ada: Use the physics first, then the bytes. The rotating shaft sets the frequencies you must preserve; the payload budget decides why the edge should send compact frequency evidence instead of every raw sample.

The worked bearing example above uses only the chapter’s stated values:

  • Motor speed: 1800 RPM / 60 = 30 revolutions per second, so the shaft frequency is 30 Hz.
  • Eight rolling elements passing once per revolution gives 8 x 30 = 240 Hz for the first bearing-fault component.
  • The 2nd and 3rd harmonics are 2 x 240 = 480 Hz and 3 x 240 = 720 Hz; the highest frequency of interest is therefore 720 Hz.
  • Nyquist requires at least twice that frequency: 2 x 720 = 1440 Hz.
  • The chapter’s practical safety factor is 2.5 x Nyquist, so 2.5 x 1440 = 3600 Hz, rounded to a deployable 4 kHz sampling rate.
  • At 4,000 samples/s and 2 bytes/sample, raw vibration data is 4,000 x 2 = 8,000 bytes/s, reported here as 8 KB/s using the chapter’s decimal KB convention.
  • A 1-second FFT window therefore starts from 8,000 bytes; sending the top 10 frequency peaks as a 120 bytes/s summary gives 8,000 / 120 = 66.7, which rounds to the stated 67x reduction.
  • The aliasing warning also checks out: sampling the 240 Hz fault component at 100 Hz gives |240 - round(240 / 100) x 100| = |240 - 2 x 100| = 40 Hz, a false low-frequency pattern below the 50 Hz Nyquist limit of a 100 Hz sampler.

The audit conclusion is narrow: the numbers justify FFT-style edge compression for this vibration example because it preserves the frequency evidence the bearing diagnosis needs while reducing the radio payload by about 67x. A slow environmental trend would need a different calculation, not this one copied blindly.

Technical boundaries
The FFT payload model deliberately does not simulate spectral leakage, window overlap, sensor noise, non-stationary faults, multi-axis coupling, or the CPU energy of the transform; it preserves the chapter's stated frequency peaks and byte counts.

Work the audit first, then check the displayed derivation.

Every number above is taken from the chapter’s own material and re-derived step by step.