Question

What is the modulation index of an FM signal having a carrier swing of 100 kHz and modulating frequency of 8 kHz?
1. 4.75
2. 5.50
3. 6.25
4. 7.50

Answer 1

Correct Answer - Option 3 : 6.25

Concept:

A general expression of an FM signal is given by:

\({S_{FM}}\left( t \right) = \cos \left( {\omega_ct + {K_f}\;\mathop \smallint \limits_{ - \infty }^t m\left( \tau \right)d\tau } \right)\)

The instantaneous frequency will be:

\({f_{i\left( {FM} \right)}}\left( t \right) = \frac{1}{{2\pi }} \times \frac{d}{{dt}}\left[ {\omega _c t + {K_f}\mathop \smallint \limits_{ - \infty }^t m\left( \tau \right)d\tau } \right]\)

\({f_{i\left( {FM} \right)}}\left( t \right) = \frac{1}{{2\pi }}\left[ {\omega_c + {K_f}m\left( t \right)} \right]\)

f_i(FM) = f_c + k_f m(t)

Let, m(t) = A_mcos2πf_ct or A_msin2πf_ct

Maximum frequency of FM signal f_max = f_c + k_f A_m

Minimum frequency of FM signal f_min = f_c - k_f Am

Maximum frequency deviation of FM signal is given as:

δ = | max[k_f m(t)] |

δ = k_f A_m Hz

fmax = fc + δ 

fmin = fc -  δ 

Frequency Carrier swing of FM signal is given as:

f_max - f_min = 2δ -----(1)

Calculation:

Carrier swing = 100 kHz

f_m = 8 kHz

From  equation (1):

δ = 50 kHz

Now, modulation index is given as:

\(β=\frac{\delta}{f_m}=\frac{50 \ kHz}{8 \ kHz}\)

β = 6.25