Sampling Theorem

The sampling theorem — more precisely, the Nyquist-Shannon Sampling Theorem — establishes that a continuous signal bandlimited to frequency W can be perfectly reconstructed from discrete samples taken at a rate of at least 2W samples per second. The theorem is not merely a practical guideline for engineers but a claim about the information-theoretic completeness of discrete representation: no information is lost in the transition from continuous to sampled form, provided the sampling rate exceeds the Nyquist limit.

The theorem was first stated by Harry Nyquist in 1928 in the context of telegraph transmission and later proved rigorously by Claude Shannon in 1948 as part of the foundations of Information Theory. The mathematical content is an application of the Whittaker-Shannon interpolation formula: the Fourier transform of a bandlimited signal is supported on a finite interval, and the sinc function provides an orthogonal basis for reconstructing the original from its samples.

The practical consequence is that the analog world, with its infinite degrees of freedom, can be captured digitally without loss — a claim that underlies all of Digital Communication, digital audio, digital imaging, and scientific measurement. The theorem's failure mode, aliasing, occurs when the sampling rate is insufficient and high-frequency components masquerade as low-frequency ones, producing irreversible distortion.

The sampling theorem is often taught as an engineering convenience. It is better understood as a boundary theorem in the geometry of function spaces: bandlimited functions live in a subspace with countable basis, and sampling is the projection onto that basis. The infinite is reducible to the countable, and the continuous to the discrete, not approximately but exactly.