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Shannon's version of the sampling theorem states that if a function contains frequencies all strictly less than $B$ hertz, then it is completely determined by giving its ordinates at a series of points spaced $\frac{1}{2B}$ seconds apart.

Now, let us suppose we are talking about just one frequency (say a laser with frequency $B-\varepsilon$ hertz, where $\epsilon$ is an arbitrarily small positive number) with an amplitude that is varied over time for relaying a signal. Let's assume that samples of this wave are taken at the following points of time, measured in seconds:

$ \{0, \frac{1}{2B}, \frac{2}{2B}, \frac{3}{2B}, \ldots \} $.

Suppose the laser amplitude is changed to half its value at $\frac{1.4}{2B}$ seconds and is then doubled back to its original amplitude at $\frac{1.6}{2B}$, wouldn't this information go undetected? It seems to me that the theorem assumes that amplitude does not change inside one wavelet.

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    $\begingroup$ Note that, in the case of samping a sine wave, the samples need to be spaced more than $1/2B$ seconds. Otherwise, you might end up sampling the signal at its zero crossings and the samples are useless. $\endgroup$
    – MBaz
    May 3 at 12:42
  • $\begingroup$ @MBaz does the $\epsilon$ subtracted from the frequency of the wave not suffice? $\endgroup$ May 3 at 13:44
  • $\begingroup$ I was referring to your claim in the first paragraph, before you introduce $\varepsilon$. In addition, you should require $\varepsilon >0$; saying it is small is not sufficient. $\endgroup$
    – MBaz
    May 3 at 14:18
  • $\begingroup$ @MBaz: I think you mean “closer than 1/2B” — as written, you get certain undersampling $\endgroup$
    – RLH
    May 4 at 5:03
  • $\begingroup$ @MBaz you are correct, I have made changes in the question accordingly, thanks for suggesting the edits. $\endgroup$ May 4 at 8:02
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Once you start changing the amplitude you are increasing the bandwidth of the signal. That's called "amplitude modulation" and the highest frequency is now the sum of the original frequency and the highest frequency in the modulation signal.

The sampling theorem still holds. You still only need twice the bandwidth but the bandwidth has increased due to the modulation

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A few additional things to note: the original version of the Shannon theorem was later corrected by replacing "greater than B hertz" with "greater than or equal to B hertz", and, the theorem only applies to signals of infinite duration (which a laser's signal isn't), albeit various algorithms have been developed to control levels of accuracy with certain types of thresholds like filter parameters, to provide an approximation of a broadband signal that cannot be perfectly reproduced. Real-world signals are broadband signals. So the result of reconstruction will have artifacts the characteristics and magnitudes of which can tend to engender complex relationships between the samples used to describe the original signal and what factually happens in the real world when such a signal is reproduced.

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  • $\begingroup$ There was no later correction, in Shannon's original paper the only requirement was $L^2$ with support $[-B,B]$ for the spectrum and $L^1\cap L^2$ for the infinite signals. What you mean are probably pop-science interpretations in conditions where Lebesgue-integrable functions are not well-understood. And the ever-lasting failure to understand that the spectrum of pure sine waves is not an integrable function, thus falls outside the scope of this theorem. $\endgroup$ May 4 at 12:23

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