Modulation of Signals
Modulation is defined as the embedding of an information-bearing signal into a second signal. The process of recovering the information is known as demodulation. Both processes are used in many of the devices commonly used today, including radios, phones, and televisions to name just a few.
A modulated signal y(t) can be seen as the product of two other signals, the information-bearing (aka Modulating) signal x(t) and the carrier signal c(t). In general, c(t) is a complex exponential or sinusoidal function that has its amplitude multiplied by x(t).
$ y(t) = x(t) c(t) \! $
c(t) is usually equal to:
$ e ^{j(\omega_c t + \theta_c)} \! $ or $ cos(\omega_c t + \theta_c)\! $
The value $ \omega_c $ is known as the carrier frequency. In order for the information to be successfully demodulated this value must always be less than the maximum frequency of the modulating signal $ \omega_m $
When using a complex exponential as the carrier signal,
$ Y(j\omega ) = \Chi (j\omega - j\omega_c)\! $
this results in a modulated signal which is simply the input shifted in frequency by an amount equal to $ \omega_c $
In order to recover the original signal x(t) simply multiply by $ e^{-j\omega_c t} \! $
When using a sinusoidal carrier,
$ y(j\omega ) = \frac{1}{2} [\Chi(j\omega - j\omega_c ) + \Chi(j\omega + j\omega-c )]\! $
this results in two copies of the input signal, each with only half of the magnitude of the original and shifted in frequency +- $ \omega_c $
In order to recover this signal one of the signals so that it is centered at zero and use a lowpass filter with a gain of 2 and cutoff frequency greater than $ \omega_m $ but less than $ 2\omega_m - \omega_c $
