(New page: ==Modulation== x(t) is the information bearing signal and c(t) is the carrier signal. The modulated signal y(t) is given by <math>x(t)\!</math> ----------> x --------> <mat...) |
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| − | This gives copies of our signal centered at <math>-\omega and \omega</math> and one half the magnitude of the original signal. | + | This gives copies of our signal centered at <math>-\omega</math> and <math>\omega</math> and one half the magnitude of the original signal. |
If the signal is not band limited, the copies will always overlap. | If the signal is not band limited, the copies will always overlap. | ||
Revision as of 08:37, 17 November 2008
Modulation
x(t) is the information bearing signal and c(t) is the carrier signal. The modulated signal y(t) is given by
$ x(t)\! $ ----------> x --------> $ y(t)\! $ ^ | | $ c(t) = cos(\omega_c t+\theta_c)\! $
so $ y(t)=x(t)c(t) $. Then by taking the Fourier Transform of both sides of this equations yields $ y(\omega) $. Also, for convenience, we choose $ \theta_c=0 $.
$ F(y(t))=F(x(t)c(t))=F(x(t)cos(\omega_c t)) $
$ y(\omega)=\frac{1}{2\pi}F(x(t))*F(cos(\omega_c t)) $
$ y(\omega)=\frac{1}{2\pi}\chi(\omega)*(\pi(\delta(\omega+\omega_c)+\delta(\omega-\omega_c))) $
$ y(\omega)=\frac{1}{2}(\chi(\omega+\omega_c)+\chi(\omega-\omega_c)) $
This gives copies of our signal centered at $ -\omega $ and $ \omega $ and one half the magnitude of the original signal.
If the signal is not band limited, the copies will always overlap.
How to recover
The first step is to modulate y(t) with the same carrier signal.
$ y(t)\! $ ----------> x --------> $ x(t)cos^2(\omega_c t)\! $ ^ | | $ c(t) = cos(\omega_c t)\! $
So now our signal looks like this:
To recover x(t) from $ x(t)cos^2(\omega_c t) $, we run the signal through a low pass filter with a gain of 2 and cut off frequency $ \omega_c $.
