To capture a wave, you have to sample it more than twice per cycle. That threshold — fₛ > 2f — is the Nyquist criterion, and half the sampling rate (fₛ/2) is the Nyquist frequency, the highest frequency you can honestly record. Sample any faster wave than that and it doesn't just get lost: it masquerades as a lower frequency that was never there. That impostor is an alias.
Sample a 900 Hz tone at 1000 Hz. The Nyquist frequency is only 500 Hz, so 900 Hz is over the limit. It folds down to |900 − 1000| = 100 Hz — you'd record and play back a 100 Hz hum that the original never contained.
It's the same effect as a car wheel appearing to spin backwards on film: the frame rate (sampling) is too slow for the wheel (signal), so your eye reconstructs a slower — even reversed — rotation.
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.
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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