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

A signal x(t) has a Fourier transform X ($$\omega $$). If x(t) is a real and odd function of t, then X($$\omega $$) is

The amplitude spectrum of a Gaussian pulse is

The Fourier transform of a function x(t) is X(f). The Fourier transform of $${{dx(t)} \over {dt}}$$ will be