Question

Audio sine waves of $3\,KHz$ frequency is used to amplitude modulate a carrier signal of $1.5\,MHz$. Which of the following statements are true?(Multiple answer correct)
(A) The sideband frequencies are $1506\,KHz$ and $1494\,KHz$.
(B) The bandwidth required for amplitude modulation is $6\,KHz$.
(C) The bandwidth required for amplitude modulation is $3\,KHz$.
(D) The sideband frequencies are $1503\,KHz$ and $1497\,KHz$.

Answer 1

In order to solve the above problem, we have to know the formula of side band frequencies of audio sine waves and the carrier signal of an AM wave.

Formula Used:
The side band frequencies of an AM wave are;
$\left( {{f_c} + {f_m}} \right)$
$\left( {{f_c} - {f_m}} \right)$
Where, ${f_m}$ denotes the frequency of audio sine wave, ${f_c}$ denotes the frequency of carrier signal.

Complete step by step answer:
The data given in the problem is;
Audio sine wave frequency, ${f_m} = 3\,KHz$,
Carrier signal frequency, ${f_c} = 1.5\,MHz$.
In the side band frequency formula;
$\left( {{f_c} + {f_m}} \right)$ is known as the upper side band frequency (USB),
$\left( {{f_c} - {f_m}} \right)$ is known as the lower side band frequency (LSB),
By changing $MHz$ into $KHz$,
We get ${f_c} = 1500\,KHz$.

Upper side band frequency (USB);
$\left( {{f_c} + {f_m}} \right)$
substitute the values of ${f_c}$ and ${f_m}$ ;
$\left( {{f_c} + {f_m}} \right) = 1500\,KHz + 3\,KHz$ .
$\left( {{f_c} + {f_m}} \right) = 1503\,KHz$.

Lower side band frequency (LSB);
$\left( {{f_c} - {f_m}} \right)$
substitute the values of ${f_c}$ and ${f_m}$ ;
$\left( {{f_c} - {f_m}} \right) = 1500\,KHz - 3\,KHz$,
$\left( {{f_c} - {f_m}} \right) = 1497\,KHz$.
The upper side band frequency is $1503\,KHz$, and the lower side band frequency is $1497\,KHz$.
Also, the bandwidth of the audio sine wave is;
$2{f_m}$
Substitute the value of audio sine wave frequency ${f_m}$;
$2{f_m} = 2 \times 3\,KHz$
$2{f_m} = 6\,KHz$
The bandwidth of the audio sine wave $6\,KHz$.

Therefore, the sideband frequencies are $1503\,KHz$ and $1497\,KHz$ and the bandwidth required for amplitude modulation is $6\,KHz$.

Hence, the option (B) The bandwidth required for amplitude modulation is $6\,KHz$ and option (D) The sideband frequencies are $1503\,KHz$ and $1497\,KHz$ is the correct answer.

Note: The general formula for side band frequency is $\left( {{f_c} \pm {f_m}} \right)$, which is then further divided into upper side band frequency (USB) and lower sideband frequency (LSB). In case of lower side band frequency (LSB) the modulated signal is subtracted from carrier signal and In case of upper side band frequency (USB) the modulated signal is added from carrier signal.