Editing Nyquist–Shannon sampling theorem (section)

==Sampling of non-baseband signals==
As discussed by Shannon:<ref name="Shannon49"/>

{{blockquote|A similar result is true if the band does not start at zero frequency but at some higher value, and can be proved by a linear translation (corresponding physically to [[single-sideband modulation]]) of the zero-frequency case. In this case the elementary pulse is obtained from <math>\sin(x)/x</math> by single-side-band modulation.}}

That is, a sufficient no-loss condition for sampling [[signal (information theory)|signal]]s that do not have [[baseband]] components exists that involves the ''width'' of the non-zero frequency interval as opposed to its highest frequency component. See ''[[Sampling (signal processing)|sampling]]'' for more details and examples.

For example, in order to sample [[FM broadcasting|FM radio]] signals in the frequency range of 100–102&nbsp;[[megahertz|MHz]], it is not necessary to sample at 204&nbsp;MHz (twice the upper frequency), but rather it is sufficient to sample at 4&nbsp;MHz (twice the width of the frequency interval).

A bandpass condition is that <math>X(f) = 0,</math> for all nonnegative <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>. This formulation includes the normal baseband condition as the case <math>N=0.</math>

The corresponding interpolation function 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>

Other generalizations, for example to signals occupying multiple non-contiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse. That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.