hose Fourier transform is localized around zero, we obtain the new function <!--\tilde{\hspace{0pt}G}(\omega)=-->

`\[\tilde{\hspace{0pt}g}(t)\equiv g(t) e^{i\omega_o t} \Longleftrightarrow G(\omega-\omega_o). \]`

The bandpass filter `$\tilde{\hspace{0pt}g}(t)$` will localize in the vicinity of frequency `$\omega_o$`.

Thus, if `$\omega_o$` is chosen to be positive (and sufficiently large), we will isolate a band of positive frequencies, and if  `$\omega_o$` is chosen to be negative, we will isolate a band