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<math>y(t) = e^{j(\omega_c t + \theta_c)}x(t)\!</math> | <math>y(t) = e^{j(\omega_c t + \theta_c)}x(t)\!</math> | ||
| − | <math> = | + | <math> = F(e^{j(\omega_c t + \theta_c)} * x(t)\!</math> |
| − | <math> =\ | + | <math> =\frac{1}{2\pi} F(e^{j(\omega_c t + \theta_c)}) * X(\omega)\!</math> |
| − | <math> =\ | + | <math> =\frac{1}{2\pi} 2\pi \delta(\omega-\omega_c) * X(\omega)\!</math> |
| + | |||
| + | <math> =X(\omega-\omega_c)\!</math> | ||
Revision as of 16:22, 16 November 2008
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)\! $
