Abtasttheorem

Abtasttheorem, commonly called the sampling theorem, is a cornerstone of digital signal processing. It asserts that a continuous-time signal that is band-limited to B Hz can be exactly reconstructed from its samples if the sampling frequency f_s exceeds 2B (the Nyquist rate). If the signal contains higher frequencies or is undersampled, aliasing occurs, causing distortion.

In the standard form, let x(t) be band-limited to B, and take samples x[n] = x(nT) with T = 1/f_s. Then the original signal can be recovered by x_hat(t) = sum_{n=-∞}^{∞} x[n] sinc((t - nT)/T), where sinc(u) = sin(πu)/(πu). Equivalently, reconstruction can be viewed as passing the sampled signal through an ideal low-pass filter with cutoff B.

The theorem was proposed by Harry Nyquist in 1928 and later generalized and formalized by Claude Shannon in 1949 in the context of digital communications. It underpins how analog signals are captured and processed digitally and is widely used in audio, video, and telemetry systems.

Practical implementations require non-ideal filters, finite data, and quantization. Real signals are rarely perfectly band-limited, so anti-aliasing prefilters are used before sampling, and reconstruction uses approximate interpolation. Non-idealities such as jitter, noise, and quantization error introduce distortion. Generalizations exist for non-uniform sampling and for representations beyond sinc-based reconstruction using frames and other basis expansions.

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