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To capture a frequency f you must sample faster than 2f. Half the sample rate is the Nyquist frequency, the ceiling on what you can record faithfully.
A frequency above Nyquist folds back below it as |f − fs·round(f/fs)|: a tone that was never in the original.
Film at 24 frames a second cannot keep up with a fast wheel, so it appears to slow, stop or spin backwards. Same maths, different sampler.
An analogue low-pass filter removes everything above Nyquist before the converter, so nothing is left to fold down.
An anti-aliasing filter, an analogue low-pass before the sampler. CD audio samples at 44.1 kHz to cover hearing up to about 20 kHz with margin for the filter. Real converters run 2.5× to 10× the signal bandwidth: twice the highest frequency is a floor, not a design target, and assumes a perfect brick-wall filter and infinite record length.
Two samples per cycle: in theory the limit; in practice phase luck means you can catch the peaks or the zero-crossings, so it is unreliable. Stay comfortably below.
Up to a point: it raises Nyquist so more of your signal fits under it. Beyond covering your highest real frequency plus filter margin, extra rate just makes bigger files without capturing anything new.
The true frequency has mirror images at fs − f, fs + f and beyond: sampling does not destroy that energy, it copies it to every one of those positions. Only the copy that lands below Nyquist survives as something you can reconstruct; the rest are shown faded.
The apparent frequency is |f − fₛ·round(f/fₛ)| — the signal reflects ("folds") off multiples
of the sampling rate and off the Nyquist frequency, landing somewhere between 0 and fₛ/2.
With an anti-aliasing filter: an analogue low-pass filter before the sampler that removes anything above the Nyquist frequency, so nothing is left to fold down. CD audio samples at 44.1 kHz to cover hearing up to ~20 kHz with margin for the filter.
You're right at the edge — two samples per cycle. In theory it's the limit; in practice phase luck means you can catch the peaks or the zero-crossings, so it's unreliable. Stay comfortably below fₛ/2.
Up to a point — it raises the Nyquist frequency so more of your signal fits under it. But beyond covering your highest real frequency (plus filter margin), extra rate just makes bigger files without capturing anything new.
The true frequency has mirror images at fs − f, fs + f, and beyond — sampling doesn’t destroy that energy, it copies it to every one of those positions. Only the copy that lands below the Nyquist frequency survives as something you can actually hear or reconstruct; the rest are shown faded because they never make it into the sampled signal.
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Sampling too slowly does not lose the signal, it moves it. Anything above half the sample rate folds back down and reappears somewhere below it, as a perfectly convincing tone that was never there.
| Real signal | Appears at | |
|---|---|---|
| 1 kHz | 1 kHz | unchanged |
| 3 kHz | 3 kHz | unchanged |
| 3.9 kHz | 3.9 kHz | unchanged |
| 4 kHz | 4 kHz | unchanged |
| 5 kHz | 3 kHz | folded down |
| 6 kHz | 2 kHz | folded down |
| 7 kHz | 1 kHz | folded down |
| 9 kHz | 1 kHz | folded down |
| 15 kHz | 1 kHz | folded down |
Sampled at 8 kHz, 7 kHz and 9 kHz both come out as 1 kHz — and so does 15 kHz. Once sampled there is nothing left in the data to say which of them was real, or whether a genuine 1 kHz tone was there all along. That is why the anti-alias filter goes before the converter: afterwards the information is gone rather than merely corrupted, and no amount of processing recovers it.
Twice the highest frequency is a floor, not a design target. It assumes a perfect brick-wall filter and infinite record length. Real converters run 2.5× to 10× the signal bandwidth to leave a filter something to work with.
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