FM numerical: 102.4 MHz carrier, 75 kHz deviation

Given: $f_c = 102.4$ MHz, $f_m = 5$ kHz, $\Delta f = 75$ kHz.

(a) Carrier swing

The carrier frequency swings between $f_c - \Delta f = 102.325$ MHz and $f_c + \Delta f = 102.475$ MHz:

$$\text{Carrier swing} = 2\Delta f = 2 \times 75 = 150\ \text{kHz}$$

(b) Bandwidth

By Carson's rule:

$$B_T = 2(\Delta f + f_m) = 2(75 + 5) = 160\ \text{kHz}$$

(Equivalently, Carson's rule keeps sidebands up to $n = \beta + 1 = 16$, so $B = 2 \times 16 \times 5 = 160$ kHz. A Bessel-table (1%) estimate would be somewhat wider, but Carson's rule is the standard answer.)

(c) Modulation index

$$\beta = \frac{\Delta f}{f_m} = \frac{75}{5} = 15$$

Answer: carrier swing = 150 kHz; bandwidth = 160 kHz; $\beta = 15$.