Modulation
The general process of embedding an information bearing signal into a second signal is typically referred as modulation.
Amplitude Modulation
Amplitude modulation (AM) is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent. For example, changes in the signal strength can be used to reflect the sounds to be reproduced by a speaker.
Amplitude modulation with a Complex Exponential Carrier
In this case the carrier signal is a complex exponitial.
i.e. $ c(t) = e ^{j(\omega_c t + \theta_c)} $
let us take $ \theta_c=0 $ to make the computations easier then, the modulated signal is :
$ y(t) = e^{j\omega_c t}x(t) $
taking the fourier transform
$ Y(\omega) = F(e^{j\omega_c t}x(t)) $
- $ =\frac{1}{2 \pi}F(e^{j\omega_c t})X(\omega) $
- $ =\frac{1}{2\pi} 2\pi \delta (\omega - \omega_c) * X(\omega) $
- $ =X(\omega - \omega_c)\! $
