Nyquist–Shannon sampling theorem

A band-limited signal can be reconstructed exactly from uniform samples taken at a rate greater than twice its highest frequency. The theorem establishes the sampling rate needed to avoid loss of information through aliasing.

Band-limited signal: A signal whose Fourier transform is zero outside a bounded frequency range. The theorem's exact reconstruction guarantee applies to signals with this property.
Fourier transform: A mathematical transform that represents a signal in terms of its frequencies. Band limitation is defined by which frequencies appear in this representation.
Analog-to-digital converter: An electronic device that converts a continuous input signal into digital measurements. Its sampling stage must capture the input's bandwidth without aliasing.
Harry Nyquist: A Swedish-American physicist and engineer whose work shaped telegraphy and information transmission. His 1928 analysis established a rate limit for avoiding intersymbol interference in telegraph signaling.
Sampling frequency: The number of samples taken per unit time from a continuous signal. It must exceed twice the signal's highest frequency for the theorem's guarantee.
Dirac comb: A periodic distribution consisting of equally spaced impulses. Multiplying a signal by an impulse train models uniform sampling mathematically.
Anti-aliasing filter: A low-pass filter applied before sampling to suppress frequencies above the usable limit. Real signals often exceed the converter's bandwidth, so filtering protects the theorem's assumptions.
Claude Shannon: An American mathematician and engineer who founded modern information theory. His 1949 paper gave a rigorous reconstruction result for band-limited signals.
Aliasing: The misidentification of distinct frequencies when a signal is sampled too slowly. Sampling below the theorem's bound makes different signals indistinguishable.
Fourier series: A representation of a periodic function as a sum of sinusoidal components. It introduces frequency-based descriptions used to reason about sampling.

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