FM Bandwidth

The deviation $d$ (amplitude of the modulator) acts as a control of FM bandwidth and is usually set as the product of $f_m$ and the modulation index $I$:

$$d = I f_m$$

so that instantaneous frequency is

$$f_i = f_c + If_m\sin(2\pi f_m t).$$

Modulation index $I$ determines harmonic content:

• in general, the highest sideband that has significant amplitude is approximately
  $$k_{\mbox{max}} = I+1.$$

If $I(t)$ is time varying, the timbre of a tone can change over time (as do musical sounds!).

Though calculating the actual amplitude of sidebands is beyond the scope here, it is interesting to known they can be described by Bessel functions.