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Concept
Bandlimiting
Résumé
Bandlimiting refers to a process which reduces the energy of a signal to an acceptably low level outside of a desired frequency range.
Bandlimiting is an essential part of many applications in signal processing and communications. Examples include controlling interference between radio frequency communications signals, and managing aliasing distortion associated with sampling for digital signal processing.
A bandlimited signal is, strictly speaking, a signal with zero energy outside of a defined frequency range. In practice, a signal is considered bandlimited if it’s energy outside of a frequency range is low enough to be considered negligible in a given application.
A bandlimited signal may be either random (stochastic) or non-random (deterministic).
In general, infinitely many terms are required in a continuous Fourier series representation of a signal, but if a finite number of Fourier series terms can be calculated from that signal, that signal is considered to be band-limited. In mathematic terminology, a bandlimited signal has a Fourier transform or spectral density with bounded support.
A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the bandwidth of the signal. This minimum sampling rate is called the Nyquist rate associated with the Nyquist–Shannon sampling theorem.
Real world signals are not strictly bandlimited, and signals of interest typically have unwanted energy outside of the band of interest. Because of this, sampling functions and digital signal processing functions which change sample rates usually require bandlimiting filters to control the amount of aliasing distortion. Bandlimiting filters should be designed carefully to manage other distortions because they alter the signal of interest in both its frequency domain magnitude and phase, and its time domain properties.
An example of a simple deterministic bandlimited signal is a sinusoid of the form If this signal is sampled at a rate so that we have the samples for all integers , we can recover completely from these samples.
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