NyquistShannonsamplingsteoremet

Nyquist-Shannon sampling theorem, commonly referred to as the Nyquist–Shannon theorem, is a foundational result in digital signal processing and communications. It states that a continuous-time signal that is bandlimited to a maximum frequency B Hz can be perfectly reconstructed from its samples if it is uniformly sampled at a rate fs greater than 2B samples per second. The minimum sampling rate that guarantees perfect reconstruction is the Nyquist rate, fs = 2B. Under ideal conditions, reconstruction is achieved by passing the sampled data through an ideal low-pass filter with cutoff frequency B, or equivalently by sinc interpolation in the time domain.

If the sampling rate is at or below the Nyquist rate, spectral replicas overlap in the frequency domain, causing aliasing. Aliasing corrupts the original signal so that its samples do not contain enough information to recover the true waveform. In practice, signals are pre-filtered with an anti-aliasing filter to limit bandwidth before sampling, and reconstruction uses non-ideal filters and interpolation methods that introduce some error.

Origin and impact: the result originated with Harry Nyquist in 1928, who identified a limit on reconstructability for bandlimited signals, and was formalized by Claude Shannon in 1949 within information theory, linking sampling to channel capacity and data representation. Applications span digital audio, imaging, telecommunications, and instrumentation, and the theorem underpins the design of analog-to-digital converters, digital signal processing, and data acquisition systems.