Contents
How it works
$ x(t)c(t)=y(t) $
Where $ x(t) $ is the "information signal" and $ c(t) $ is the "carrier"
Two Major Carriers
Complex Exponential
$ c(t) = e^{j(\omega_ct+\theta_c)} $
Sinusoidal
$ c(t) = cos(\omega_ct+\theta_c) $
Where $ \omega_c $ is the frequency and $ \theta_c $ is the phase
Complex Exponential Modulation
$ y(t) = e^{j\omega_ct}x(t) $
$ Y(\omega)=F(e^{j\omega_ct}x(t)) $
$ Y(\omega)=\frac{1}{2\pi}F(e^{j\omega_ct})X(\omega) $
$ Y(\omega)=\frac{1}{2\pi}(2\pi \delta(\omega-\omega_c)X(\omega) $
$ Y(\omega)=X(\omega-\omega_c) $
What happens with this modulation is that the original signal $ x(t) $ and shifted in the frequency domain by $ \omega_c $
