the sampling theorem

It feels like magic that a continuous sound — a smooth, unbroken wave in the air — can be stored as a finite list of numbers on a CD and played back with no audible loss. How can a handful of dots possibly recapture a curve that has infinitely many points? The sampling theorem is the precise statement that says exactly when this magic is allowed, and it is the theoretical bedrock of all digital audio, video, and communications.

The theorem, due to Nyquist and Shannon, states: if a signal contains no frequencies above some limit B (it is band-limited to B), then it is COMPLETELY and EXACTLY determined by its samples taken at any rate strictly greater than 2B. Not approximately — exactly. The continuous signal can be reconstructed perfectly from the discrete samples by a specific interpolation: replace each sample with a sinc function (sin(pi*t)/(pi*t), scaled) centred at that sample and add them all up; the sincs interlock so their wiggles cancel everywhere except where each sample sits. The threshold rate 2B is the Nyquist rate, and the slogan is simply 'sample faster than twice the highest frequency'.

The two hypotheses are where honesty lives. First, band-limitedness is an idealization: no real signal is perfectly band-limited (a finite recording cannot be), so in practice you apply an anti-aliasing low-pass filter to FORCE the signal below B before sampling, accepting that you discard the genuinely-too-fast content. Second, sampling exactly AT 2B is not enough — you need strictly above, with margin — and reconstruction with the ideal sinc requires infinitely many samples, so real systems use practical approximate filters. Within those caveats the theorem is exact and astonishing: a properly band-limited signal loses nothing when digitized.