anderyza
Newbie level 1
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?
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?