Hilbert+Fourier Transform

Hi,

This is my first post here..

Well this was one the problems in my final exam that I need some help with.

I didn’t exactly know where to post this so I hope this is the right place.

Assuming that $\displaystyle \hat{\varphi}\left({t}\right)$is the Hilbert transform of $\displaystyle \varphi\left({t}\right)$,show that when $\displaystyle -f_{c}{<}{f}{<}f_{c}$

,the frequency spectrum (i.e. the Fourier transform) of the following signal is zero:

$\displaystyle \displaystyle{v\left({t}\right)=e^{-\hat{\varphi}\left({t}\right)}\cos{\left(2\pi f_{c}{t}+\varphi\left({t}\right)\right)}}$