Nyquist–Shannon sampling theorem: Difference between revisions

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For example, in order to sample [[FM broadcasting|FM radio]] signals in the frequency range of 100–102 [[megahertz|MHz]], it is not necessary to sample at 204 MHz (twice the upper frequency), but rather it is sufficient to sample at 4 MHz (twice the width of the frequency interval). (Reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequency-shifted to near baseband, and digital demodulation can proceed on that basis.)

Using the "bandpass condition", where that <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>, weit canis givepossible to find an interpolation that reproduces the signal. (ThereNote that there maaymay be several combinations of <math>N</math> and sampling frequency<math>f_\mathrm{s}</math> that work. This formulation, includesincluding the normal baseband condition as the case <math>N=0.</math> The corresponding interpolation filter to be convolved 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:

The corresponding interpolation filter to be convoluted 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>