The autocorrelation of sin(t) is defined as $$\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)d\tau$$
I've tried using the Wiener-Khinchin theorem which says that
$$Corr(g,g)\Longleftrightarrow|G(f)|^2$$
I've tried reverting the FT squared of the sine wave, and don't see any solution.
I can't find a derivation online for this. So I came here wondering if anyone of you could offer a hint; or perhaps go through a derivation.
One of my problems is calculating the inverse fourier transform of:
$$\frac{1}{4}(\delta(f+\frac{1}{2\pi})-\delta(f-\frac{1}{2\pi}))^2$$
In words: What is the inverse transform of the modulus squared of the FT of the sine function?
Thanks!
Best Answer
The autocorrelation is a function of $\tau$, not $t$ The function is $Aut(\sin(t),\tau)=\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)dt$
where $t$ is a dummy variable.
Expand the sum in the argument of the sine function: $$\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)dt=\int_{-\infty}^\infty (\sin t \cos \tau+ \sin\tau \cos t)\sin(t)dt\\=\int_{-\infty}^{\infty} (\sin^2t\cos \tau + \sin t \cos t \sin \tau) \;dt$$
Over an interval of $2\pi $ in $t$, the second term averages out to $0$ and the first term becomes $\frac 12\cos \tau$, which is what you seek