| Line 2: | Line 2: | ||
To perform amplitude modulation, we need a carrier <math>c(t)</math>. Specifically, we need either a complex exponential <math>c(t) = e^{w_{c}t+\theta_c}</math> or a sinusoid <math>c(t) = cos(w_{c}t+\theta_c)</math>. | To perform amplitude modulation, we need a carrier <math>c(t)</math>. Specifically, we need either a complex exponential <math>c(t) = e^{w_{c}t+\theta_c}</math> or a sinusoid <math>c(t) = cos(w_{c}t+\theta_c)</math>. | ||
| + | |||
| + | <math>w_c</math> is referred to as the '''carrier frequency'''. | ||
| + | |||
| + | When <math>c(t)</math> is a complex exponential, <math>C(jw) = 2\pi\delta(w-w_c)</math>. | ||
| + | |||
| + | Therefore, <math>Y(jw) = X(jw-jw_c)</math>. | ||
| + | |||
| + | When <math>c(t)</math> is a sinusoid, <math>C(jw) = \pi[\delta(w-w_c)+\delta(w+w_c)]</math>. | ||
| + | |||
| + | Therefore, <math>Y(jw) = \frac{1}{2}[X(jw-jw_c)+X(jw+jw_c)]</math>. | ||
Latest revision as of 16:39, 17 November 2008
Amplitude Modulation
To perform amplitude modulation, we need a carrier $ c(t) $. Specifically, we need either a complex exponential $ c(t) = e^{w_{c}t+\theta_c} $ or a sinusoid $ c(t) = cos(w_{c}t+\theta_c) $.
$ w_c $ is referred to as the carrier frequency.
When $ c(t) $ is a complex exponential, $ C(jw) = 2\pi\delta(w-w_c) $.
Therefore, $ Y(jw) = X(jw-jw_c) $.
When $ c(t) $ is a sinusoid, $ C(jw) = \pi[\delta(w-w_c)+\delta(w+w_c)] $.
Therefore, $ Y(jw) = \frac{1}{2}[X(jw-jw_c)+X(jw+jw_c)] $.
