Question

What is the modulation index of an FM signal having a carrier swing of $100kHz$ when the modulating signal has a frequency of $8kHz$?

Answer 1

Hint: The modulation frequency of an FM can be defined as the ratio of the deviation of frequency of carrier signal to the frequency of the modulating signal. Here, we have given the value of carrier swing and the frequency of modulating signal. The formula used for calculating the modulation index of an FM signal is given below.

Formula used:
The formula of modulation index of an FM signal is given below
${m_f} = \dfrac{{\Delta f}}{{{f_a}}}$
Here, ${m_f}$ is the modulation index of FM signal, $\Delta f$ is the deviation of frequency of carrier signal and ${f_a}$ is the frequency of modulating signal.
Also, the formula of deviation of frequency of carrier signal is given below
$\Delta f = \dfrac{{carrier\,signal}}{2}$

Complete step by step answer:
The following terms are given in the question;
Carrier swing or frequency of carrier signal, $ = 100kHz$
Frequency of modulating signal, ${f_a} = 8kHz$
Here, we will first determine the formula of deviation of frequency as given below
$\Delta f = \dfrac{{carrier\,signal}}{2}$
$ \Rightarrow \,\Delta f = \dfrac{{100kHz}}{2}$
$ \Rightarrow \,\Delta f = \dfrac{{100 \times {{10}^3}}}{2}$
$ \Rightarrow \,\Delta f = 50 \times {10^3}$
Now, the modulation frequency of an FM can be defined as the ratio of the deviation of frequency of carrier signal to the frequency of the modulating signal. The formula used for calculating the modulation index of an FM signal is given below
${m_f} = \dfrac{{\Delta f}}{{{f_a}}}$
Now, substituting the values in the above formula, we get
${m_f} = \dfrac{{50 \times {{10}^3}}}{{8 \times {{10}^3}}}$
$ \Rightarrow \,{m_f} = \dfrac{{50}}{8}$
$ \therefore \,{m_f} = 6.25$

Therefore, the modulation index of an FM signal is $6.25$.

Note: The carrier signal is the high frequency signal that has a certain phase, frequency and amplitude but does not contain any information. Here, it is important to convert the bigger units into smaller units. Here, the frequency of deviation of carrier signal is not given, therefore, we have first calculated it.