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## Q. 1.1.6

The amplitude of a sinusoidal carrier is modulated by a single sinusoid to obtain the amplitude modulated signal S(t) = 5 cos 1600πt + 20cos1800πt + 5cos2000πt. The value of the modulation index is _____.

## Verified Solution

AM

\begin{aligned}S(t) &=A_{c} \cos 2 p f_{c} t\left[1+k_{a} m(t)\right] \\S(t) &=A_{c} \cos 2 p f_{ c } t\left[1+k_{a} A_{m} \cos 2 \pi f_{m} t\right] \\S(t) &=A_{c} \cos 2 p f_{c} t+\frac{A_{c} \mu}{2} \cos 2 \pi\left(f_{c}+f_{m}\right) t \\&+\frac{A_{c} \mu}{2} \cos 2 \pi\left(f_{c}-f_{m}\right) t\end{aligned}

Comparing the above standard equation with given equation, we get

\begin{aligned}f_{c} &=900 Hz \\f_{m} &=100 Hz \\A_{c} &=20 \\\frac{A_{c} \mu}{2} &=5 \\A_{c} \mu &=10\end{aligned}

\begin{aligned}&\mu=\frac{10}{A_{c}}=\frac{10}{20} \\&\mu=\frac{1}{2}=0.5.\end{aligned}

Hence, the correct Answer is (0.5).