Modern Physics 7 Question 10
10. A signal $A \cos \omega t$ is transmitted using $v _0 \sin \omega _0 t$ as carrier wave. The correct amplitude modulated (AM) signal is
(Main 2019, 9 April I)
(a) $\left(v _0 \sin \omega _0 t+A \cos \omega t\right.$
(b) $\left(v _0+A\right) \cos \omega t \sin \omega _0 t$
(c) $v _0 \sin \left[\omega _0(1+0.01 A \sin \omega t) t\right]$
(d) $v _0 \sin \omega _0 t+\frac{A}{2} \sin \left(\omega _0-\omega\right) t+\frac{A}{2} \sin \left(\omega _0+\omega\right) t$
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Solution:
- Given, modulating signal,
$$ A _m=A \cos \omega t $$
Carrier wave, $A _c=v _0 \sin \omega _0 t$
In amplitude modulation, modulated wave is given by
$$ Y _m=\left[A _0+A _m\right] \sin \omega _0 t $$
where, $A _0$ is amplitude of the carrier wave (given as $v _0$ )
$$ \begin{aligned} \therefore Y _m & =\left[v _0+A \cos \omega t\right] \sin \omega _0 t \\ & =v _0 \sin \omega _0 t+A \sin \omega _0 t \cos \omega t \\ & =v _0 \sin \omega _0 t+\frac{A}{2}\left[\sin \left(\omega _0+\omega\right) t+\sin \left(\omega _0-\omega\right) t\right] \\ & =v _0 \sin \omega _0 t+\frac{A}{2} \sin \left(\omega _0-\omega\right) t+\frac{A}{2} \sin \left(\omega _0+\omega\right) t \end{aligned} $$
