Sampling theorem

The sampling theorem states that a band-limited signal can be reconstructed exactly from uniformly spaced samples when the sampling rate exceeds twice its highest frequency.

Nyquist–Shannon sampling theorem: A theorem specifying exact reconstruction of a band-limited signal from uniform samples taken above twice its highest frequency. This is the standard form of the sampling theorem, with its precise rate condition.
Band-limited signal: A signal whose Fourier transform is zero outside a finite frequency interval. Exact reconstruction requires the signal to have a bounded frequency spectrum.
Digital audio: Audio represented and processed as sequences of discrete numerical samples. Audio systems choose sample rates to preserve the frequencies humans can hear.
Aliasing: The misidentification of distinct signal frequencies as the same frequency after sampling. Sampling below the theorem’s threshold causes frequency components to overlap indistinguishably.
Nyquist rate: The minimum sampling frequency required to represent a band-limited signal without aliasing, equal to twice its highest frequency. It states the theorem’s threshold in frequency units.
Uniform sampling: Sampling in which measurements are taken at constant time or spatial intervals. The standard theorem assumes evenly spaced sample locations.
Image sampling: The conversion of a spatially varying image into discrete pixel measurements. The theorem also applies to spatial frequencies in images.
Nyquist frequency: Half the sampling frequency, marking the highest frequency representable without aliasing in sampled data. It is a limit set by the sampling process, not the signal’s own maximum frequency.
Ideal low-pass filter: A filter that passes frequencies below a cutoff and completely rejects frequencies above it. An ideal reconstruction filter isolates the original frequency band from repeated spectral copies.
Sampling frequency: The number of samples taken per unit time, usually measured in hertz. Its value determines whether the samples can distinguish the signal’s frequencies.

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