# Hilbert+Fourier Transform

• Jan 26th 2007, 01:28 PM
Rezast
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 $\hat{\varphi}\left({t}\right)$is the Hilbert transform of $\varphi\left({t}\right)$,show that when $-f_{c}{<}{f}{<}f_{c}$
,the frequency spectrum (i.e. the Fourier transform) of the following signal is zero:

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