Question: Q. 9. A (sinusoidal) carrier wave
$C(t)=A_{\mathrm{c}} \sin \omega_{c} t$
is amplitude modulated by a (sinusoidal) message signal
$$ m(t)=A_{m} \sin \omega_{m} t $$
Write the equation of the (amplitude) modulated signal. Use this equation to obtain the values of the frequencies of all the sinusoidal waves present in the modulated signal.
U] [CBSE SQP 2014]
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Solution:
Ans. The equation of the (amplitude) modulated signal is
$$ C_{m}(t)=\left[\left(A_{c}+A_{m} \sin \omega_{m} t\right)\right] \sin \omega_{c} t \quad \mathbf{1} $$
This can be rewritten as
$$ C_{m}(t)=\left[A_{c}\left(1+\mu \sin \omega_{m} t\right)\right] \sin \omega_{c} t $$
where, $\mu=A_{m} / A_{c}=$ modulation index
$$ \begin{aligned} \therefore \quad C_{m}(t) & =A_{c} \sin \omega_{c} t+\frac{\mu A_{c}}{2} 2 \sin \omega_{m} t \cdot \sin \omega_{c} t \quad 1 / 2 \ & =A_{c} \sin \omega_{c} t+\frac{\mu A_{c}}{2}\left[\cos \left(\omega_{c}-\omega_{m}\right) t\right. \end{aligned} $$
$$ \left.-\cos \left(\omega_{c}+\omega_{m}\right) t\right]_{1 / 2}^{1 / 2} $$
There are the three sinusoidal waves present in the amplitude modulated signal.
The frequencies of these three waves are
and
$$ \left.\begin{array}{c} f_{1}=\frac{\omega_{c}}{2 \pi} \ f_{2}=\frac{\omega_{c}-\omega_{m}}{2 \pi} \ f_{3}=\frac{\omega_{c}+\omega_{m}}{2 \pi} \end{array}\right} $$
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