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
$C_m(t)=A_1\sin \left({\omega }_{1}t\right)+A_2\sin \left({\omega }_{2}t\right)-A_3\sin \left({\omega }_{3}t\right)\dots (i)$
where, ${\omega }_1>{\omega }_2$ and ${\omega }_3>{\omega }_1$
Standard equation of modulating signal is given as
$C_m(t)=A_c\sin \left({\omega }_{c}t\right)+\frac{\mu A_c}{2}\cos \left({\omega }_c-{\omega }_m\right)t-\frac{\mu A_c}{2}$
$\cos \left({\omega }_c+{\omega }_m\right)t\dots (ii)$
Clearly, ${\omega }_c>{\omega }_c-{\omega }_m$ and ${\omega }_c+{\omega }_m>{\omega }_c$
Eq. (i) can also be written as,
$C_m(t)=A_1\sin \left({\omega }_{1}t\right)+A_2\cos \left(\frac{\pi }{2}+{\omega }_{2}t\right)-A_3\cos \left(\frac{\pi }{2}+{\omega }_{3}t\right)\dots (iii)$
Comparing Eq. (ii) and Eq. (iii), we get
$A_1=A_c$
${\omega }_c={\omega }_1$
${\omega }_c-{\omega }_m={\omega }_2\dots (iv)$
and ${\omega }_c+{\omega }_m={\omega }_3\dots .(v)$
Also, $\frac{\mu A_c}{2}=A_2$
$\Rightarrow \mu =\frac{2A_2}{A_1}\dots (vi)$
Substracting Eq. (iv) from Eq. (v), we have
${\omega }_3-{\omega }_2=\left({\omega }_c+{\omega }_m\right)-\left({\omega }_c-{\omega }_m\right)$
$\Rightarrow 2{\omega }_m={\omega }_3-{\omega }_2$
$\Rightarrow {\omega }_m=\frac{{\omega }_3-{\omega }_2}{2}\dots (vii)$