sampling and the Nyquist criterion

Every digital recording is a deception that works: a microphone produces a smooth, continuous voltage, but a computer can only store a finite list of numbers. Sampling is the act of measuring that continuous signal at regularly spaced instants and keeping just those readings. The surprising fact is that, under the right condition, those discrete dots contain everything — the original smooth curve can be reconstructed from them with no loss at all. The Nyquist criterion is exactly that condition.

The Nyquist-Shannon theorem says: if a signal contains no frequencies above some highest frequency B (it is band-limited), then sampling it at a rate greater than 2B samples per second captures it perfectly, and the original can be rebuilt by interpolating with sinc functions. The threshold rate 2B is the Nyquist rate; half the sampling rate is the Nyquist frequency, the highest frequency a given rate can represent. The Fourier picture explains why: sampling in time multiplies the signal by a train of deltas, which in frequency convolves the spectrum with a train of deltas, producing infinitely many shifted copies of the spectrum. If the copies do not overlap (the Nyquist condition), a single low-pass filter cuts out one clean copy and recovers the signal exactly.

Sample too slowly and the spectral copies overlap; high frequencies fold down and masquerade as low ones, an irreversible corruption called aliasing — the reason a fast-spinning wheel appears to turn backwards in a film, and the reason recordings carry an anti-aliasing filter before the digitizer to throw away frequencies above the Nyquist limit. This theorem is the foundation of all digital audio, imaging, and instrumentation. The honest fine print: perfect reconstruction assumes an exactly band-limited signal and ideal (sinc) interpolation, neither of which is fully realizable, so real systems sample comfortably above 2B and accept tiny, controlled errors.