Given the electric field of a complete amplit | vedclass

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(B) The given expression for the amplitude modulated wave is $\vec E = \hat i E_c (1 + \frac{E_m}{E_c} \cos \omega_m t) \cos \omega_c t$.Expanding thi

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$\omega_c, \omega_c + \omega_m$ and $\omega_c - \omega_m$

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(B) The given expression for the amplitude modulated wave is $\vec E = \hat i E_c (1 + \frac{E_m}{E_c} \cos \omega_m t) \cos \omega_c t$.Expanding this expression,we get $\vec E = \hat i [E_c \cos \omega_c t + E_m \cos \omega_m t \cos \omega_c t]$.Using the trigonometric identity $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$,we can write:$\vec E = \hat i [E_c \cos \omega_c t + \frac{E_m}{2} (\cos(\omega_c + \omega_m)t + \cos(\omega_c - \omega_m)t)]$.This expression shows that the modulated wave consists of three distinct frequency components:$1$. The carrier frequency: $\omega_c$$2$. The upper sideband frequency: $\omega_c + \omega_m$$3$. The lower sideband frequency: $\omega_c - \omega_m$Therefore,the frequencies present are $\omega_c, \omega_c + \omega_m$,and $\omega_c - \omega_m$.

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

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