Editing Bandlimiting (section)

==Sampling bandlimited signals==

A [[Bandlimiting#Bandlimited signals|bandlimited signal]] can be perfectly recreated from its samples if the [[sampling rate]]—how often the signal is measured—is more than twice the signal’s [[Bandwidth (signal processing)|bandwidth]] (the range of frequencies it contains). This minimum rate is called the [[Nyquist rate]], a key idea in the [[Nyquist–Shannon sampling theorem]], which ensures no information is lost during sampling.

In reality, most signals aren’t perfectly bandlimited, and signals we care about—like audio or radio waves—often have unwanted energy outside the desired frequency range. To handle this, [[digital signal processing]] tools that sample or change sample rates use bandlimiting filters to reduce [[aliasing]] (a distortion where high frequencies disguise themselves as lower ones). These filters must be designed carefully, as they alter the signal’s [[frequency domain]] magnitude and phase (its strength and timing across frequencies) and its [[time domain]] properties (how it changes over time).

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

The signal whose Fourier transform is shown in the figure is also bandlimited. Suppose <math>x(t) </math> is a signal whose Fourier transform is <math>X(f), </math> the magnitude of which is shown in the figure. The highest frequency component in <math>x(t) </math> is <math>B. </math> As a result, the Nyquist rate is

:<math> R_N = 2B \, </math>

or twice the highest frequency component in the signal, as shown in the figure.  According to the sampling theorem, it is possible to reconstruct <math>x(t)\ </math> completely and exactly using the samples

:<math>x(nT) = x \left( { n \over f_s  } \right)  </math> for all integers <math>n \, </math> and <math>T \ \stackrel{\mathrm{def}}{=}\  { 1 \over f_s } </math>

as long as

:<math>f_s > R_N  \, </math>

The reconstruction of a signal from its samples can be accomplished using the [[Whittaker–Shannon interpolation formula]].