A modulated signal is given by
\[s(t)=e^{-a t} \cos \left[\left(\omega_{c}+\Delta \omega\right) t\right] u(t),\]
where $a, \omega_{c}$ and $\Delta \omega$ are positive constants, and $\omega_{c} \gg \Delta \omega$.
The complex envelope of $s(t)$ is given by
- $\exp \left(-\right. at)$ $\exp \left[j\left(\omega_{c}+\Delta \omega\right) t\right] u(t)$
- $\exp (-a t) \exp (j \Delta \omega t) u(t)$
- $\exp (j \Delta \omega t) \cdot u(t)$
- $\left.\exp \left[j \omega_{c}+\Delta \omega\right) t\right]$