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<math>=X(\omega - \omega_c)\!</math> | <math>=X(\omega - \omega_c)\!</math> | ||
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| + | Demodulate the complex exponential | ||
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| + | <math>y(t) \longrightarrow\otimes\longrightarrow x(t) </math> | ||
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| + | <math>\uparrow </math> | ||
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| + | <math>e^{-j\omega_ct}</math> | ||
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| + | <math>y(t) = e^{j\omega_c t}x(t)</math> | ||
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| + | <math>e^{-j\omega_c t}y(t) = e^{-j\omega_c t}e^{j\omega_c t}x(t)</math> | ||
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| + | <math>=x(t)\!</math> | ||
Revision as of 06:34, 17 November 2008
Amplitude Modulation
Def:Amplitude modulation (AM) is a method of impressing data onto an alternating-current (AC) carrier waveform.The highest frequency of the modulating data is normally less than 10 percent of the carrier frequency.
$ x(t) \longrightarrow\otimes\longrightarrow y(t)=x(t)c(t) $
$ \uparrow $
c(t)
x(t) : "information bearing signal"
c(t) : "carrier"
There are two important types of carriers which are "complex exponential" and "sinusoidal"
Complex expenetial
$ c(t) = e ^{j(\omega_c t + \theta_c)} $
$ \omega_c = $ Frequency of carrier
$ \theta_c = $ Phase of carrier
complex exponential modulation
$ y(t) = e^{j\omega_c t}x(t) $
$ y(\omega) = F(e^{j\omega_c t}x(t)) $
$ =\frac{1}{2 \pi}F(e^{j\omega_c t})X(\omega) $
$ =\frac{1}{2\pi} 2\pi \delta (\omega - \omega_c) * X(\omega) $
$ =X(\omega - \omega_c)\! $
Demodulate the complex exponential
$ y(t) \longrightarrow\otimes\longrightarrow x(t) $
$ \uparrow $
$ e^{-j\omega_ct} $
$ y(t) = e^{j\omega_c t}x(t) $
$ e^{-j\omega_c t}y(t) = e^{-j\omega_c t}e^{j\omega_c t}x(t) $
$ =x(t)\! $
