I am stuck with basic understanding of the Auto-correlation derivation of a simple signal and I would be pleased if you could help me out with that.
Lets have a signal $x(t)=\cos(2\pi{f_{0}}{t})$.
By simple integral, I am able to find the theoretical result I should get for the Auto-correlation :
$R_{x}=\frac{1}{2}\cos(2\pi{f_{0}}{t})$
However, I cannot calculate it by Fourier Transform even if it should be possible.
Given $F[x*x] = F[x].F[x] $ and given $R_{x} = x(t)*x(-t)$ here is my derivation :
$R_{x} = F^{-1}[F[x(t)].F[x(-t)]]$
As $F[x(t)] = \frac{\delta(f-f_{0})+\delta(f+f_{0})}{2}$ and $F[x(-t)] = \frac{\delta(f-f_{0})+\delta(f+f_{0})}{2}$, whe have :
$R_{x} = \frac{\delta(f-2{f_{0}})+\delta(f+2{f_{0}})+2\delta(f)}{4}$
$R_{x} = \frac{1}{2}(\cos(4\pi{f_{0}t)}+1)$
$R_{x} = \cos^2{(2\pi{f_{0}t})}$
As you can see, my second result is quite strange. I assume I made a mistake somewhere and I would be glad if you could point out where it is !
Best regards