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Cross-correlation function problem

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anderyza

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Hello everyone,

I have a following problem:

given cross-correlation function \[{C_{12}}{\left(\tau\right)}\] associated with a pair of time functions \[{f}_{1}(t)\] 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 DC and two waveforms with frequencies of σ and 3σ, 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?
 

Can you try solve by Matlab program or Mathematical software ? , may must to adapt littlely that eq before calculation.
 

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