Nyquist-Shannon sampling theorem

The Nyquist-Shannon sampling theorem is the foundational result of digital signal theory, stating that a band-limited continuous signal can be perfectly reconstructed from discrete samples taken at a rate more than twice the signal's maximum frequency. The theorem establishes the theoretical boundary between the analog and digital worlds: it guarantees that no information is lost in the sampling process, provided the signal's bandwidth is known and finite. In practice, the theorem is an idealization: real signals are not perfectly band-limited, real filters are not ideal, and the reconstruction requires infinite computation. The theorem is therefore not a recipe for perfect conversion but a boundary condition that defines the minimum sampling rate for a given fidelity target. See aliasing for what happens when the theorem's conditions are violated and Reconstruction filter for the practical challenges of implementing the theorem's promise.