Question

A 10 MHz sinusoidal carrier wave of amplitude 10 mV is modulated by a 5 kHz sinusoidal audio signal wave of amplitude 6mV. Find the frequency components of the resultant modulated wave and their amplitudes.

Answer 1

Hint: To find the required frequency components and their amplitudes we have to write all the given data and then use them to calculate the upper level/sideband and the lower level/sideband frequency separately. We will also find the modulating factor (\[\mu \]) by using the formula: \[\mu= \dfrac{{{A_m}}}{{{A_c}}}\]and after that, we will plug the value of modulating factor (\[\mu \]) in the formula: \[A = \dfrac{{\mu {A_c}}}{2}\] to get the value of the amplitude of the resultant modulated wave.

Complete step by step solution:
According to the question, the data are given:
Frequency of carrier wave \[\left( {{{\text{f}}_c}} \right){\text{=10 MHz}}\]
Frequency of the signal \[\left( {{{\text{f}}_s}} \right){\text{=5 kHz}}{\text{=0}}{\text{.005 MHz}}\]
The amplitude of the audio signal \[\left( {{\text{Es}}} \right){\text{=6 mV}}\]
The amplitude of the carrier signal \[\left( {{\text{Ec}}} \right){\text{=10 mV}}\]
Step I
Calculate the frequencies of the modulated carrier wave:
Since the original carrier wave of frequency \[\left( {{{\text{f}}_c}} \right){\text{=10 MHz}}\]
So the upper-level band frequency will be \[\left( {{{\text{f}}_c} + {{\text{f}}_s}} \right){\text{=(10+0}}{\text{.005) MHz}}\]
\[ \Rightarrow \left( {{{\text{f}}_c}{\text{ + }}{{\text{f}}_s}} \right){\text{=10}}{\text{.005 MHz}}\]
Similarly,
The lower- level band frequency will be \[\left( {{{\text{f}}_c}{\text{ - }}{{\text{f}}_s}} \right){\text{=(10-0}}{\text{.005) MHz}}\]
\[ \Rightarrow \left( {{{\text{f}}_c}{\text{ - }}{{\text{f}}_s}} \right){\text{=9}}{\text{.995 MHz}}\]
Step II
Evaluating the value for the modulation factor \[\mu \]
Since the modulation factor \[\mu= \dfrac{{{A_m}}}{{{A_c}}}\]
\[ \Rightarrow \mu= \dfrac{6}{{10}}\]
\[ = 0.6\]
So the modulation factor for the carrier wave is \[0.6\]
Step III:
Calculate the amplitude of the resultant modulated wave:
The required amplitude \[\left( A \right) = \dfrac{{\mu {A_c}}}{2}\]
\[ \Rightarrow \left( A \right) = {\text{3 mV}}\]

$\therefore$ Hence, the required amplitude for the resultant modulated wave is \[{\text{3 mV}}\].

Note:
To answer this type of question, the key is to know the concept of modulation and demodulation of a signal and how the characteristics of the output signal change. One should also remember the important results, short and quick formula to save his or her time during writing the paper. Students must practice a lot on such kind of formula-based questions to master all the conceptual aspects behind the problems.