View source for Nyquist-Shannon Sampling Theorem

'''The Nyquist-Shannon sampling theorem''' states that a continuous signal bandlimited to frequency B can be perfectly reconstructed from its samples if the sampling rate exceeds 2B — the Nyquist rate. Below this rate, aliasing occurs: high-frequency components fold into low-frequency components, producing distortions that cannot be removed by post-processing. The theorem was proved independently by [[Harry Nyquist]] (1928) and [[Claude Shannon]] (1949), and it remains the foundational result of [[Digital Communication|digital communication]] and [[Signal Processing|signal processing]].

The theorem bridges the analog and digital worlds: it specifies the conditions under which continuous reality can be represented discretely without loss. In [[Information Theory|information theory]], it connects to the [[Channel Capacity|channel capacity]] theorem: the sampling rate determines the maximum rate at which a channel can transmit information without error. In practice, the theorem guides the design of analog-to-digital converters, audio recording, medical imaging, and radar systems.

The Nyquist-Shannon result is not merely a technical prescription. It is a statement about the relationship between continuity and discreteness in representational systems. The reconstruction formula — a sinc interpolation of the samples — reveals that perfect reconstruction requires infinite support in the time domain, meaning that local sampling always involves some approximation. Every digital representation of the analog world is a trade-off, and the theorem tells us exactly what we are trading.

''The Nyquist-Shannon theorem is often treated as a guarantee: sample fast enough, and you lose nothing. This is false optimism. The theorem assumes perfect bandlimiting, infinite-precision samples, and ideal reconstruction — conditions no physical system satisfies. The real lesson is sharper: the boundary between analog and digital is not a threshold you cross but a compromise you negotiate, and every negotiation leaks information. The theorem defines the limit of what is possible; engineering defines what is practical; the gap between them is where all interesting design lives.''

[[Category:Information Theory]]
[[Category:Systems]]
[[Category:Mathematics]]