Nyquisttétel

The Nyquist–Shannon sampling theorem, commonly referred to as the Nyquist theorem or Nyquist–Shannon interpolation formula, is a fundamental principle in signal processing and information theory. It establishes the conditions under which a continuous-time signal can be perfectly reconstructed from its samples. The theorem was developed by Harry Nyquist in 1928 and later formalized by Claude Shannon in 1949.

The core idea of the theorem is that to accurately reconstruct a continuous-time signal from its discrete samples, the sampling frequency must be at least twice the highest frequency present in the signal. This minimum sampling rate is known as the Nyquist rate, which is defined as twice the bandwidth of the signal. For example, if a signal contains frequencies up to 4 kHz, the Nyquist rate would be 8 kHz, meaning the signal must be sampled at least 8,000 times per second.

The theorem assumes that the original signal is band-limited, meaning it contains no frequencies higher than a certain maximum frequency. If this condition is violated, aliasing occurs, which distorts the reconstructed signal by introducing lower-frequency components that were not present in the original signal. The Nyquist theorem also relies on the use of an ideal low-pass filter for reconstruction, which in practice may not always be perfectly achievable.

In practical applications, the Nyquist theorem provides a theoretical foundation for digital audio, telecommunications, and other fields where signals are converted from analog to digital form. However, real-world implementations often require slight modifications, such as oversampling, to account for non-ideal filters and noise. The theorem remains a cornerstone of digital signal processing, influencing how signals are sampled, stored, and transmitted efficiently.