Bandlimited

Bandlimiting is the limiting of a signal's Fourier transform or power spectral density to zero above a certain finite frequency. A band-limited signal is one whose Fourier transform or power spectral density has bounded support.

The signal may be either random () or non-random (deterministic).

In general, infinitely many terms are required in a continuous Fourier series representation, but if a finite number of Fourier series terms can be calculated from that signal, that signal is considered to be band-limited.

A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency in the bandlimited signal. This minimum sampling frequency is called the Nyquist rate. This result, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem.

An example of a simple deterministic bandlimited signal is a sinusoid of the form ${\displaystyle x(t)=\sin(2\pi ft+\theta )\ }$. If this signal is sampled at a rate ${\displaystyle f_{s}={\frac {1}{T}}>2f}$ so that we have the samples ${\displaystyle x(nT)\ }$, for all integers ${\displaystyle n}$, we can recover ${\displaystyle x(t)\ }$ completely from these samples. Similarly, sums of sinusoids with different frequencies and phases are also bandlimited to the highest of their frequencies.

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