Complex Exponential Modulation
Many communication systems rely on the concept of sinusoidal amplitude modulation, in which a complex exponential or a sinusoidal signal, $ c(t)\! $, has its amplitude modulated by the information-bearing signal, $ x(t)\! $. $ x(t)\! $ is the modulating signal, and $ c(t)\! $ is the carrier signal. The modulated signal, $ y(t)\! $, is the product of these two signals:
An important objective of amplitude modulation is to produce a signal whose frequency range is suitable for transmission over the communication channel that is to be used.
One important for of modulation is when a complex exponential is used as the carrier.
$ \omega_c\! $ is called the carrier frequency, and $ \theta_c\! $ is called the phase of the carrier.
Graphically, this equation looks as follows,
$ x(t)\! $ ----------> x --------> $ y(t)\! $ ^ | | $ c(t) = e^{j(\omega_c t + \theta_c)}\! $
Mathematically, we can solve for $ Y(\omega)\! $ as follows,
$ y(t) = e^{j(\omega_c t + \theta_c)}x(t)\! $
$ = F(e^{j(\omega_c t + \theta_c)} * x(t))\! $
$ =\frac{1}{2\pi} F(e^{j(\omega_c t + \theta_c)}) * X(\omega)\! $
$ =\frac{1}{2\pi} 2\pi \delta(\omega-\omega_c) * X(\omega)\! $
$ =X(\omega-\omega_c)\! $
It is apparent from this equation for $ Y(\omega)\! $ that the signal is simply sent as a shifted copy of the original signal $ X(\omega)\! $. To be exact, the spectrum of the modulated output $ y(t)\! $ is simply that of the input, shifted in frequency by an amount equal to the carrier frequency, $ \omega_c\! $.
How to Demodulate
If we reverse the graph shown above, it should become obvious how to demodulate the signal.
$ y(t)\! $ ----------> x --------> $ x(t)\! $ ^ | | $ e^{-j(\omega_c t + \theta_c)}\! $
To recover the $ x(t)\! $ from $ y(t)\! $, simply multiply by the reciprocal of the original $ c(t)\! $. In the general case, multiply by $ e^{-j(\omega_c t + \theta_c)}\! $. In the frequency domain, this has the effect of shifting the spectrum of the modulated signal back to its original position on the frequency axis.
Mathematically, this looks as follows,
Since, $ y(t)= e^{j(\omega_c t + \theta_c)}x(t)\! $, if we multiply both sides by $ e^{-j(\omega_c t + \theta_c)}\! $, we get,
$ e^{-j(\omega_c t + \theta_c)}y(t)= e^{-j(\omega_c t + \theta_c)} e^{j(\omega_c t + \theta_c)}x(t)\! $
Since $ e^{-j(\omega_c t + \theta_c)} e^{j(\omega_c t + \theta_c)} = 1\! $, we get,
$ e^{-j(\omega_c t + \theta_c)}y(t) = x(t)\! $, which proves this property.
Sources
Signals & Systems, 2nd edition, Oppenheim, Willsky
