Question
A signal $$A\cos \omega t$$ is transmitted using $${v_0}\sin \omega $$ modulated $$\left( {AM} \right)$$ signal is:
A.
$${v_0}\sin {\omega _0}t + \frac{A}{2}\sin \left( {{\omega _0} - \omega } \right)t + \frac{A}{2}\left( {{\omega _0} + \omega } \right)t$$
B.
$${v_0}\sin \left[ {{\omega _0}\left( {1 + 0.01A\sin \omega t} \right)t} \right]$$
C.
$${v_0}\sin {\omega _0}t + A\cos \omega t$$
D.
$$\left( {{v_0} + A} \right)\cos \omega t\sin {\omega _0}t$$
Answer :
$${v_0}\sin {\omega _0}t + \frac{A}{2}\sin \left( {{\omega _0} - \omega } \right)t + \frac{A}{2}\left( {{\omega _0} + \omega } \right)t$$
Solution :
The equation of amplitude modulated wave
$$m = \left( {{v_0} + A\cos \omega t} \right)\sin \omega t$$
$$\eqalign{
& = {v_0}\sin {\omega _0}t + A\cos \omega t\sin {\omega _0}t \cr
& = {v_0}\sin {\omega _0}t + \frac{A}{2}\left[ {\sin \left( {{\omega _0} - \omega } \right)t + \sin \left( {{\omega _0} + \omega } \right)t} \right] \cr} $$