bandbegrenzt

Bandbegrenzt, or band-limited in English, is a property of a signal whose spectrum is confined to a finite frequency range. A signal is bandbegrenzt if its Fourier transform X(f) is zero for all frequencies |f| greater than a given limit B. The parameter B is called the bandwidth, indicating the highest frequency content present in the signal.

In mathematical terms, a band-begrenzter signal satisfies X(f) = 0 for |f| > B. This means the signal has no spectral components beyond ±B Hz. Common practical distinctions use two-sided bandwidth (±B) or one-sided bandwidth (B Hz), depending on the convention for positive frequencies.

A fundamental consequence is the sampling theorem: a band-begrenzt signal with maximum frequency B can be perfectly reconstructed from samples taken at a rate fs > 2B, using an ideal reconstruction filter. In real systems, strict band-limitation is rarely achieved; practical signals are only approximately band-limited due to imperfect filters and tail components, leading to potential aliasing if undersampled.

There is also a time–frequency constraint: a nonzero signal that is strictly time-limited cannot be strictly band-limited, and vice versa, according to the uncertainty principle. In practice, engineers work with approximately band-limited signals and design anti-aliasing filters to manage spectral leakage.

Applications of band-limitation include digital audio and video processing, communications, and filter design, where restricting the bandwidth enables efficient sampling, transmission, and reconstruction.