In Wikipedia, there is Shannon's proof on Nyquist-Shannon sampling theorem. ( http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem#Shannon.27s_original_proof )
The original proof presented by Shannon is elegant and quite brief, but it offers less intuitive insight into the subtleties of aliasing, both unintentional and intentional. Quoting Shannon's original paper, which uses ''f'' for the function, ''F'' for the spectrum, and ''W'' for the bandwidth limit:Let $\scriptstyle F(\omega)$ be the spectrum of $\scriptstyle f(t).$ Then $f(t)= {1 \over 2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t}\;{\rm d}\omega \ = {1 \over 2\pi} \int_{-2\pi W}^{2\pi W} F(\omega) e^{i\omega t}\;{\rm d}\omega \ $ since $\scriptstyle F(\omega)$ is assumed to be zero outside the band ''W''. If we let $t = {n \over {2W}}\,$ where ''n'' is any positive or negative integer, we obtain $f \left({n \over {2W}} \right) = {1 \over 2\pi} \int_{-2\pi W}^{2\pi W} F(\omega) e^{i\omega {n \over {2W}}}\;{\rm d}\omega.$
What I am not getting is where $\infty$ is replaced with $2\pi W$. Why is it substituted like this? Also, why is $t$ substitued with $n \over 2W$?
Thanks.