BandLimited

Bandlimited refers to a property of a signal whose frequency spectrum is nonzero only within a finite interval. In continuous-time signals, a signal x(t) is bandlimited with bandwidth B if its Fourier transform X(f) = 0 for all |f| > B (equivalently X(ω) = 0 for |ω| > Ω, with Ω = 2πB). In discrete-time signals, a sequence is bandlimited if its spectrum is confined to a limited range of normalized frequencies.

A key implication is that a nonzero signal cannot be both time-limited and bandlimited, by the relevant version of the Paley–Wiener theorem. In practice, bandlimited models are idealizations used to analyze and design systems.

Sampling and reconstruction are central to bandlimited signals. If a continuous-time bandlimited signal with bandwidth B is sampled at a rate fs > 2B (the Nyquist rate), the original signal can be perfectly reconstructed from its samples, typically via sinc interpolation. Sampling below this rate causes aliasing, where higher-frequency components masquerade as lower frequencies.

Practically, real signals are not strictly bandlimited, and real filters are not ideal. Bandlimiting is often achieved with low-pass filters that approximate a desired cutoff. An ideal low-pass filter has a rectangular frequency response, yielding a sinc impulse response in time, which is noncausal and infinitely long. Real filters approximate this behavior with finite impulse responses.

Common terminology includes baseband-limited signals, whose spectrum is nonzero near zero, and bandpass-limited signals, whose energy lies in a higher-frequency band. Bandlimited assumptions underpin much of digital signal processing, communications, and data acquisition.