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Given the electric field of a complete amplitude modulated wave as \(\vec{E}=\hat{i}E_c\left(1+\frac{E_m}{E_c}\cos\omega_m t\right)\cos\omega_c t\) Where the subscript c stands for the carrier wave and m for the modulating signal. The frequencies present in the modulated wave are

1. ω_c and √(ω_c^2 + ω_m^2)
2. ω_c, ω_c + ω_m and ω_c − ω_m
3. ω_c and ω_m
4. ω_c and √(ω_cω_m)

Correct answer: ω_c, ω_c + ω_m and ω_c − ω_m

Solution

Expanding the AM wave gives a carrier term plus two sideband terms: \cos\omega_c t\cos\omega_m t = \tfrac12[\cos(\omega_c+\omega_m)t+\cos(\omega_c-\omega_m)t]. So the spectrum contains the carrier frequency and the two side frequencies \omega_c+\omega_m and \omega_c-\omega_m.

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