Question

A message signal is super-imposed with a carrier signal. The resulting modulating signal $C_{m}(t)$ is given by $C_{m}(t)=A_{1} \sin \left(\omega_{1} t\right)+A_{2} \sin \left(\omega_{2} t\right)-A_{2} \sin \left(\omega_{3} t\right)$ where $\omega_{2}<\omega_{1}<\omega_{3}$. The modulation index and the angular frequency of the message signal respectively are

• A. $\frac{ A _{2}}{ A _{1}}, \frac{\omega_{3}-\omega_{2}}{2}$
• B. $2 \frac{ A _{2}}{ A _{1}}, \omega_{3}-\omega_{2}$
• C. $\frac{ A _{1}}{2} A _{2}, \frac{\omega_{3}-\omega_{2}}{2}$
• D. $2 \frac{ A _{2}}{ A _{1}}, \frac{\omega_{3}-\omega_{2}}{2}$

Answer 1

Answer: (D)

Modulating signal is given as
$C_{m}(t)=A_{1} \sin \left(\omega_{1} t\right)+A_{2} \sin \left(\omega_{2} t\right)-A_{3} \sin \left(\omega_{3} t\right) \ldots(i)$
where, $\omega_{1}>\omega_{2}$ and $\omega_{3}>\omega_{1}$
Standard equation of modulating signal is given as
$C_{m}(t)=A_{c} \sin \left(\omega_{c} t\right)+\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 \ldots (ii) $
Clearly, $\omega_{c}>\omega_{c}-\omega_{m}$ and $\omega_{c}+\omega_{m}>\omega_{c}$
Eq. (i) can also be written as,
$C_{m}(t)=A_{1} \sin \left(\omega_{1} t\right)+A_{2} \cos \left(\frac{\pi}{2}+\omega_{2} t\right)-A_{3} \cos \left(\frac{\pi}{2}+\omega_{3} t\right) \ldots (iii)$
Comparing Eq. (ii) and Eq. (iii), we get
$A_{1}=A_{c}$
$\omega_{c}=\omega_{1}$
$\omega_{c}-\omega_{m}=\omega_{2} \ldots (iv)$
and $\omega_{c}+\omega_{m}=\omega_{3} \ldots .( v )$
Also, $\frac{\mu A_{c}}{2}=A_{2}$
$ \Rightarrow \mu=\frac{2 A_{2}}{A_{1}} \ldots (vi)$
Substracting Eq. (iv) from Eq. (v), we have
$\omega_{3}-\omega_{2}=\left(\omega_{c}+\omega_{m}\right)-\left(\omega_{c}-\omega_{m}\right)$
$\Rightarrow 2 \omega_{m}=\omega_{3}-\omega_{2}$
$\Rightarrow \omega_{m}=\frac{\omega_{3}-\omega_{2}}{2} \ldots(vii )$