# Does the Shannon theorem not apply when the amplitude of a wave is changed faster than half the time period of the wave?

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.

• 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.
– MBaz
May 3 at 12:42
• @MBaz does the $\epsilon$ subtracted from the frequency of the wave not suffice? May 3 at 13:44
• 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.
– MBaz
May 3 at 14:18
• @MBaz: I think you mean “closer than 1/2B” — as written, you get certain undersampling
– RLH
May 4 at 5:03
• @MBaz you are correct, I have made changes in the question accordingly, thanks for suggesting the edits. May 4 at 8:02

• 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. May 4 at 12:23