My Math Forum Cross-correlation function problem

 July 2nd, 2013, 11:37 AM #1 Newbie   Joined: Jul 2013 Posts: 1 Thanks: 0 Cross-correlation function problem Hello everyone, I have a following problem: given cross-correlation function ${C_{12}}{\left(\tau\right)}$ associated with a pair of time functions and ${f}_{1}(t)$ and ${f}_{2}(t)$: ${C_{12}}{\left(\tau\right)}={3 cos}^{2 }{\sigma}{t sin}{\sigma} {t}$ if ${f}_{1}(t)=\frac{ 1}{2 }+\frac{1}{4}cos {\sigma}t+ \frac{1}{2}sin 2{\sigma}t- \frac{3}{2}sin 3{\sigma}t+ 4 cos 4{\sigma}t$ find ${f}_{2}(t)$. The idea I thought of is to apply Fourier transform to cross-correlation function and to the first time function to evaluate spectrum of the second function and then obtain the function itself by applying inverse Fourier transform to the spectrum. But I didn't manage to do it technically. I believe it has a simple solution like comparison of two time series, say by applying trigonometric relations to cross-correlation function we have: ${C_{12}}{\left(\tau\right)}=\frac{3}{2}+\frac{3}{4 } sin {\sigma}t+\frac{3}{4} sin 3{\sigma}t$. Then it can be assumed that the second function should include some constant and two sinusoids with frequencies of ? and 3?, but again I don't know how to solve it technically. Unfortunately I have no answer to the problem, so please could anyone help me to solve it?

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