FM Modulation

In angle modulation, the amplitude of the signal is held constant and the phase is being varied with the message. An angle modulated signal is of the form:

$$x_c(t)=A_c\cos(\omega_c t+\phi(t)).$$ (1)

The instantaneous phase of $x_c(t)$ is given by

$$\theta_i(t)=\omega_c t+\phi(t),$$ (2)

and the instantaneous frequency is given by

$$\omega_i(t)=\frac{d\theta_i(t)}{dt}=\omega_c+\frac{d\phi(t)}{dt}.$$ (3)

Using this approach, if the message is proportional to $\phi(t)$, which is the phase deviation, then we have phase modulation. If the message is proportional to $\frac{d\phi(t)}{dt}$, which is the frequency deviation, then we have frequency modulation. In order to have phase modulation,

$$\phi(t)=k_p m(t),$$ (4)

where $k_p$ is known as the deviation constant. For frequency modulation,

$$\frac{d\phi(t)}{dt}=k_f m(t),$$ (5)

where $k_f$ is known as the frequency deviation constant. Consequently, an FM modulated signal is of the form

$$x_c(t)=A_c\cos(\omega_c t + k_f\int_t m(\alpha)d\alpha).$$ (6)

An FM signal is shown in Figure 1

Figure 1: Frequency modulation