I want to compute the fourier Transform of the form:
$$ y =[cos(w_ot)+cos(2w_ot)]x(t)$$ where the fourier transform of $x(t)$ is given as: \begin{equation} {X(jw)}= \begin{cases} \frac{1}{w_o}[\frac{w}{w_o}+1] & \text{if $ -1<= w/w_o <= 0$}\\ \frac{1}{w_o}[-\frac{w}{w_o}+1] & \text{if $ -1<= w/w_o <= 0$}\\ 0 & \text{else} \end{cases} \end{equation} My attempt:
I computed the fourier transform of $f(t)= [cos(w_ot)+cos(2w_ot)]$. Since, the product of $f(t)$ and $x(t)$ in time-domain is convulation in frequency domain, I tried to convolate $F(jw)$ and $X(jw)$. I am not getting success as the convulation is now involving dirac-delta function from $F(jw)$. I also tried Inverse fourier transform of $X(jw)$, but this looked weird to me because the Inverse Fourier transform now carries imaginary part also. Thank you!! Any hints would be appreciated.