Nyquistsamplingtheorema

Nyquist sampling theorem, also known as the sampling theorem, is a foundational principle in digital signal processing. It states that a continuous-time signal that is bandlimited to B hertz can be completely reconstructed from its samples if the sampling frequency fs is greater than or equal to 2B. The minimum such rate is called the Nyquist rate, and half the sampling rate is the Nyquist frequency.

In practical terms, when sampling at or above the Nyquist rate, the spectrum of the sampled signal does not overlap with itself, and an ideal low-pass filter with cutoff B can recover the original signal exactly from the discrete samples. If the sampling rate is below the Nyquist rate, spectral copies overlap in what is known as aliasing, causing distortion that cannot be removed by simple filtering.

Historically, the concept is attributed to Harry Nyquist in the 1920s, with the theorem later formalized and broadened by Claude Shannon in 1949. The term Nyquist frequency is commonly used to denote fs/2, the highest frequency that can be faithfully represented at a given sampling rate.

In practice, real-world signals are not perfectly bandlimited, and systems incorporate anti-aliasing filters before sampling to limit bandwidth. Reconstruction from samples uses practical interpolation methods rather than an ideal sinc function. The theorem underpins digital audio, communications, imaging, and measurement systems, guiding the design of analog-to-digital converters and data reconstruction processes.