Nyquist–Shannon sampling theorem: Difference between revisions

Revision as of 02:04, 23 June 2025

Latest revision as of 02:26, 31 August 2025

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Using the bandpass condition, where <math>X(f) = 0</math> for all <math>|f|</math> outside the open band of frequencies

:<math>\left(\frac{N}2 f_\mathrm{s}, \frac{N+1}2 f_\mathrm{s}\right),</math>

for some nonnegative integer <math>N</math> and some sampling frequency <math>f_\mathrm{s}</math>, it is possible to find an interpolation that reproduces the signal. Note that there may be several combinations of <math>N</math> and <math>f_\mathrm{s}</math> that work, including the normal baseband condition as the case <math>N=0.</math> The corresponding interpolation filter to be convolutedconvolved with the sample is the impulse response of an ideal "brick-wall" [[bandpass filter]] (as opposed to the ideal [[brick-wall filter|brick-wall]] [[lowpass filter]] used above) with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses:

<math display="block">(N+1)\,\operatorname{sinc} \left(\frac{(N+1)t}T\right) - N\,\operatorname{sinc}\left( \frac{Nt}T \right).</math>