Shannon-Nyquist sampling theorem

The gist: if you have an analog signal, how many snapshots of it do you have to take to recreate it in a digital domain?

Shannon-Nyquist sampling theorem says that to reconstruct a signal, you must sample it at a rate greater than twice its highest frequency component: $f_s > 2 f_{max}$. Half the sample rate, $f_s / 2$, is called the Nyquist frequency.¹²³

Exactly twice is not enough. Sample a steady sine wave at exactly 2× its frequency and you get two samples per cycle. If they land on the zero crossings, every sample is 0 and the sine vanishes. If they land on the peaks, you can't tell the sine's true amplitude or phase.

In practice you leave some margin. CD audio samples at 44.1 kHz to capture up to roughly 20 kHz, the limit of human hearing. The gap gives the anti-aliasing filter room to roll off.