(New page: == How it works == <math>x(t)c(t)=y(t)</math> Where <math>x(t)</math> is the "information signal" and <math>c(t)</math> is the "carrier" == Two Major Carriers == === Complex Exponenti...) |
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Where <math>\omega_c</math> is the frequency and <math>\theta_c</math> is the phase | Where <math>\omega_c</math> is the frequency and <math>\theta_c</math> is the phase | ||
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| + | == Complex Exponential Modulation == | ||
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| + | <math>y(t) = e^{j\omega_ct}x(t)</math> | ||
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| + | <math>Y(\omega)=F(e^{j\omega_ct}x(t))</math> | ||
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| + | <math>Y(\omega)=\frac{1}{2\pi}F(e^{j\omega_ct})X(\omega)</math> | ||
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| + | <math>Y(\omega)=\frac{1}{2\pi}(2\pi \delta(\omega-\omega_c})X(\omega)</math> | ||
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| + | <math>Y(\omega)=X(\omega-\omega_c)</math> | ||
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| + | What happens with this modulation is that the original signal <math>x(t)</math> and shifted in the frequency domain by <math>\omega_c</math> | ||
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| + | === Demodulation ie. How the Heck do I get back my original signal === | ||
Revision as of 15:34, 17 November 2008
Contents
How it works
$ x(t)c(t)=y(t) $
Where $ x(t) $ is the "information signal" and $ c(t) $ is the "carrier"
Two Major Carriers
Complex Exponential
$ c(t) = e^{j(\omega_ct+\theta_c)} $
Sinusoidal
$ c(t) = cos(\omega_ct+\theta_c) $
Where $ \omega_c $ is the frequency and $ \theta_c $ is the phase
Complex Exponential Modulation
$ y(t) = e^{j\omega_ct}x(t) $
$ Y(\omega)=F(e^{j\omega_ct}x(t)) $
$ Y(\omega)=\frac{1}{2\pi}F(e^{j\omega_ct})X(\omega) $
$ Y(\omega)=\frac{1}{2\pi}(2\pi \delta(\omega-\omega_c})X(\omega) $
$ Y(\omega)=X(\omega-\omega_c) $
What happens with this modulation is that the original signal $ x(t) $ and shifted in the frequency domain by $ \omega_c $
