Question: Q. 10. A sinusoidal carrier wave of amplitude $A_{c}$ and angular frequency $\omega_{c}$ is modulated in accordance with a sinusoidal information signal of amplitude $A_{m}$ and angular frequency $\omega_{m}$. Show that the amplitude modulated signal contains three frequencies centered around $\omega_{c}$. Draw the frequency spectrum of the resulting modulated signal.
[CBSE -SQP-2018]
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Solution:
Ans. The modulated signal :
$$ \begin{aligned} & C_{m}(t)=\left(A_{c}+A_{m} \sin \omega_{m} t\right) \sin \omega_{c} t \ & =A_{c}\left(1+\frac{A_{m}}{A_{c}} \sin \omega_{m} t\right) \sin \omega_{c} t \ & C_{m}(t)=A_{c} \sin \omega_{c} t+\mu A_{c} \sin \omega_{m} t \sin \omega_{c} t \ & C_{m}(t)=A_{c} \sin \omega_{c} t+\frac{\mu A_{c}}{2} \cos \left(\omega_{c}-\omega_{m}\right) t-\frac{\mu A_{c}}{2} \cos \left(\omega_{c}-\omega_{m}\right) t \end{aligned} $$
Frequency Spectrum :
[CBSE Marking Scheme, 2018]
