Ada Audits the Sampling Numbers

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

ADA · CALCULATION AUDIT

Ada Audits the Sampling Numbers

Digitizing human voice with content up to 3400 Hz needs at least 6800 Hz of sampling, so telephony’s 8000 Hz leaves a 1200 Hz guard band — a 17.6% margin — while an unfiltered 800 Hz tone sampled at 1000 Hz folds to a false 200 Hz artifact. The same sensor can read safe, ambiguous, or plainly wrong depending only on the rate. This audit re-derives each Nyquist margin and fold and asks what evidence a given sample rate is actually capable of preserving.

Companion to the chapter Sampling Rate and Aliasing — every number here comes from that chapter.

Sampling is where mathematics meets the physics of a moving sensor signal: the equations decide whether a real vibration stays visible or folds into a false low-frequency story.

1. The Nyquist test is a one-line inequality. If the highest trusted signal component is f_max, the sample rate must satisfy:

f_sample ≥ 2 × f_max

For the voice example already used in this chapter, f_max = 3400 Hz, so the minimum is 2 × 3400 = 6800 Hz. Telephony at 8000 Hz leaves 8000 − 6800 = 1200 Hz of guard band, and 1200 / 6800 = 0.176 = 17.6% margin.

2. Aliasing is a fold, not a random error. Once a component is above f_sample/2, the sampled data can report the folded frequency instead of the physical one:

f_alias = |f_signal − f_sample|   for the first fold used in these examples

Design question	Arithmetic shown	Audit result
Minimum for a 500 Hz bearing-defect signal	2 × 500 Hz	1000 Hz minimum
Practical 2500 Hz sampling choice	2500 / 500 = 5 and 2500 / 2 = 1250 Hz	5× oversampling, with a 1250 Hz Nyquist frequency
Unfiltered 800 Hz component at 1000 Hz sampling	|800 − 1000|	200 Hz false low-frequency artifact
1 kHz continuous logging volume	1000 samples/s × 86,400 s/day = 86,400,000 samples/day; × 4 bytes	345,600,000 bytes/day, about 345.6 MB/day
1 Hz environmental logging volume	1 sample/s × 86,400 s/day = 86,400 samples/day; × 4 bytes	345,600 bytes/day, about 345.6 KB/day
ESP32 ADC current estimate	2 mA + 0.001 mA per sample/s	100 Hz: 2.1 mA; 1 kHz: 3 mA; 10 kHz: 12 mA

What the audit buys you: the same physical sensor can be safe at 2500 Hz, ambiguous at 1000 Hz without filtering, and plainly wrong at 500 Hz. The equations do not replace field evidence; they tell you what evidence the ADC is capable of preserving.

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