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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 :


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

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You should clarify which definition you are using, and the domain of the signals. If you just define $R_{x} = x(t)*x(-t)$, then in your example $R_0=\int_{-\infty}^{\infty} \cos^2(t) dt$ diverges. – leonbloy Dec 15 '12 at 0:07
Obviously, for the derivation using the "integral" derivation, the following definition has been used : $R_{x} = \frac{1}{2\pi}\int_{0}^{2\pi}cos(\omega{t})cos(\omega(t-\tau))dt$ as the integral of a (co)sine is not defined over infinity. However, I do not get how it should affect the "fourier" derivation ? – Al_th Dec 15 '12 at 1:00
Obviously, the period of $cos(ωt)$ is not $2 \pi$. – leonbloy Dec 15 '12 at 1:10
Oh my god yeah I was tired when I answered you. $R_{x}(\tau) = \frac{1}{T}\int_{\frac{-T}{2}}^{\frac{T}{2}}{cos(\omega{t})cos(\omega(t-\tau))dt‌​}$. Anyway, it doesn't change the result of the auto-correlation. However, My question is more about the fourier derivation of the auto-correlation, no matter how much I could be wrong/inexact with the integral derivation. I would be pleased if you could comment on that in priority ! – Al_th Dec 15 '12 at 10:18
It's not about priorities (or being pedant), it's about getting the right definitions to start with (And, BTW, your new definition is still wrong - the limits on the integration cannot depend on the period of the signal! what would be the autocorrelation of a sum of two unrelated sinusoids?). I'm still waiting for a simple link that points to the autocorrelation defintion you are using and the relation with the Fourier transform. – leonbloy Dec 15 '12 at 12:14

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