Frequency Modulation examples/problems

1) A music signal with frequency components from 50 Hz to 21000 Hz is Frequency modulated.  If the maximum allowed frequency deviation is 50 kHz. (i)       What is the modulation index?
ii)What is the signal bandwidth using Carson’s rule?

Solution: i) Modulation index = frequency deviation/modulating frequency = 50 kHz/21 kHz = 2.38

               

(ii) Bandwidth is 2 (modulating frequency + frequency deviation) = 2 (21 kHz + 50 kHz) = 142 kHz

2)A 200 MHz carrier is frequency modulated by a 10 V peak-to-peak signal of 10 kHz. The instantaneous carrier frequency varies between 199.90 and 200.10 MHz.

 Calculate (i) The modulator sensitivity.

                        (ii) The modulation index.

                        (iii) The signal bandwidth using Carson’s rule.

(i)

2 * Frequency deviation = 200.10 – 199.9 MHz = 200 kHz.

Frequency deviation = 100kHz

This is caused by a 10 V signal.

Therefore sensitivity is 100/10 kHz/V = 10 kHz/V

(ii)

The modulation index is 200 kHz / 10 kHz = 20.

(iii)

BW      =          2 (frequency of modulating signal + frequency deviation)                

            =          2 ( f_m + k * f_d)                                                                                     

            =          2 ( 1 + b ) * f_m                                                                                     

Using any of the above

            BW = 420kHz

3) A speech signal with frequency components from 300 Hz to 4 kHz is Frequency modulated.  If the maximum allowed frequency deviation is 30 kHz (i)                 What is the modulation index? (ii)               What is the signal bandwidth using Carson’s rule?

(i)

Modulation index = frequency deviation/modulating frequency = 30 kHz/4 kHz = 7.5

(ii)

Bandwidth is 2 (modulating frequency + frequency deviation) = 2 (30 kHz + 4 kHz) = 68 kHz