Nyquist–Shannon sampling theorem

When you digitize a sound or a sensor reading, you don't store the smooth analog wave — you grab snapshots at regular intervals. The sampling theorem answers the make-or-break question: how often must you snapshot to capture the signal perfectly? Its beautiful answer: you must sample at MORE than twice the highest frequency present in the signal. Sample fast enough and the original continuous wave can be reconstructed exactly, with zero loss, from the dots.

That magic threshold, twice the top frequency, is the Nyquist rate. This is why audio CDs sample at 44.1 kHz: human hearing tops out near 20 kHz, and 44.1 kHz comfortably clears the 40 kHz minimum. Sample too slowly and disaster strikes — high frequencies masquerade as low ones (aliasing), like wagon wheels appearing to spin backwards in old films. To prevent it, every digitizer first passes the signal through an anti-aliasing low-pass filter that chops off anything above half the sample rate before the snapshots are taken.