Proof of Fourier inversion formula

I think I knew how to show it before. But suddenly just realize I can’t. Say let’s define $latex F(w) = \frac{1}{2 \pi} \int f(t) e^{-i wt} dt$ and the inverse transform $latex \hat{f}(t)= \int F(w) e^{i w t}dw = f(t)$. So we expect $latex f(t)=\frac{1}{2 \pi} \int \int f(\tau) e^{-i w\tau} d\tau e^{iwt}dw$ $latex…

Bandlimited channel

By Shannon-Nyquist theorem, a bandlimited signal can be completely reconstructed with $latex 2W$ samples per second. Think of the opposite way, this means that each second we have $latex 2W$ independent components that we can vary for each second. The total noise power per second is $latex 2 W (N_0/2) = N_0 W$ and as…