Question

A modulated signal $C_m\left(t\right)$ has the form $C_m\left(t\right)=30sin300\pi t+10(cos200\pi t-cos400\pi t)$. The carrier frequency $f_c^{\prime }$ the modulating frequency (message frequency) $f_{{\omega }^{\prime }}$ and the modulation index $\mu$ are respectively given by

• A. $f_c=200Hz;f_{\omega }=50Hz;\mu =\frac{1}{2}$
• B. $f_c=150Hz;f_{\omega }=50Hz;\mu =\frac{2}{3}$
• C. $f_c=150Hz;f_{\omega }=30Hz;\mu =\frac{1}{3}$
• D. $f_c=200Hz;f_{\omega }=30Hz;\mu =\frac{1}{2}$

Answer 1

Answer: (B)

$f_c=150Hz;f_{\omega }=50Hz;\mu =\frac{2}{3}$

Amplitude modulated wave is represented as:

$y\left(t\right)=A\sin \left(2\pi f_{c}t\right)+A\mu \sin \left(2\pi f_{c}t\right)sin\left(2\pi f_{w}t\right)$

$=A\sin \left(2\pi f_{c}t\right)+\frac{A\mu }{2}\left[\cos \right(2\pi (f_c-f_w)t)-\cos (2\pi (f_c+f_w)t\left)\right]$

Comparing this with the given equation gives:

$2\pi f_c=300\pi ,⟹f_c=150Hz$

$2\pi (f_c-f_w)=200\pi ,⟹f_w=50Hz$

$A=30$

$A\mu /2=10,⟹\mu =2/3$