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.

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.

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