Editing Bandlimiting (section)

=== 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]].