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Eric Kvaalen (talk | contribs) J.M. Whittaker was teh son of E.T. |
Eric Kvaalen (talk | contribs) →Sampling of non-baseband signals: Clarified some things. |
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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 [[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.)
:<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> we can give an interpolation that reproduces the signal. (There maay be several combinations of <math>N</math> and sampling frequency that work. This formulation includes the normal baseband condition as the case <math>N=0.</math>)
The corresponding interpolation
▲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>
This function is 1 at <math>t=0</math> and zero at any other multiple of <math>T</math> (as well as at other times if <math>N>0</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.
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