In angle modulation, information changes the instantaneous phase of a carrier rather than its amplitude.

A general waveform is

$$s(t)=A_c\cos\theta(t).$$

Its instantaneous frequency is proportional to the rate of phase change:

$$f_i(t)=\frac{1}{2\pi}\frac{d\theta}{dt}.$$

In phase modulation (PM), the message directly changes phase:

$$\theta(t)=2\pi f_ct+k_pm(t).$$

In frequency modulation (FM), the message changes instantaneous frequency, so phase contains an integral of the message:

$$\theta(t)=2\pi f_ct+2\pi k_f\int_0^t m(\tau),d\tau.$$

Unlike ordinary AM, ideal FM and PM keep the carrier amplitude constant. Their spectra generally contain more than two sidebands, so modulation strength affects occupied bandwidth.

FM is resistant to amplitude variations that can be removed before demodulation, but this robustness is obtained by using additional bandwidth and a more complex receiver.