The two methods to compute the bandwidth of an FM station are by using the modulation index and by using the Carson’s rule.
1) The formula for bandwidth calculation using modulation index is given as
where, N is the number of significant sidebands, and fm is the highest modulating frequency
Assuming a modulation index of 2, the number of significant sidebands will be 4. The highest modulating frequency fm is given as 15 kHz. Thus the bandwidth can be calculated as
2) The formula for calculating the bandwidth using Carson’s rule is given as
where fd is the maximum frequency deviation and fm is the maximum modulating frequency.
fd is given as 75 kHz and fm is given as 15 kHz. Thus substituting these values in the formula the bandwidth is calculated as
Discussion on bandwidth and channel spacing:
As mentioned the stations are spaced every 200 kHz. As per the calculation of the bandwidth using Carson’s rule there will be very narrow spacing between two channels. Whereas with the modulation index method the spacing will be larger compared to the Carson’s rule. However it should be noted that if a higher modulation index is taken the spacing between two channels will reduce as the number of sidebands will increase, thus increasing the bandwidth.