Question

$\text{Let}x\left(t\right)\text{be a single tone FM signal and}y\left(t\right)=x\left(\frac{t}{2}\right).\text{Consider the following statements in relation to}x\left(t\right)andy\left(t\right):$
S1: Both the signals have same bandwidth according to Carson's rule.
S2: Both the signals have same modulation index.
S3: Both the signals have same maximum frequency deviation. $Selectthecorrectstatement\left(s\right)usingthecodesgivenbelow.$

• A. S2 only
• B. S2 and S3 only
• C. S1 and S3 only
• D. S1 only

Answer 1

The correct option is A S2 only

$\text{Let,}x\left(t\right)=A_c\cos [2\pi f_{c}t+{\beta }_x\sin (2\pi f_{m}t\left)\right]$
$\text{So,}y\left(t\right)=x\left(\frac{t}{2}\right)=A_c\cos [\pi f_{c}t+{\beta }_x\sin \left(\frac{2\pi f_{m}t}{2}\right)]$
$\text{For}x\left(t\right),\Delta f_x={\beta }_x{f}_m$
$(BW{)}_x=(1+{\beta }_x)2f_m$
$\text{For}y\left(t\right),{\beta }_y={\beta }_x$
$\Delta f_y={\beta }_y\left(\frac{f_m}{2}\right)={\beta }_x\left(\frac{f_m}{2}\right)=\frac{\Delta f_x}{2}$

$(\text{BW}{)}_y=(1+{\beta }_y)2\left(\frac{f_m}{2}\right)=(1+{\beta }_x)f_m=\frac{(\text{BW}{)}_x}{2}$
$\text{So,}(\text{BW}{)}_y=\frac{\left(\right(\text{BW}{)}_x)}{2}\Rightarrow \text{S1 is incorrect}$
${\beta }_y={\beta }_x\Rightarrow \text{S2 is correct}$
$\Delta f_y=\frac{\Delta f_x}{2}\text{S3 is incorrect}$