What Is Frequency Modulation (FM)? Formula, Frequency Deviation, Modulation Index with Solved Examples


Definition of Frequency modulation (FM)

Frequency modulation (FM) is a type of angle modulation in which the frequency of the carrier wave (mostly sine wave) is varied in accordance to the frequency of the modulating or message signal while keeping amplitude and phase constant.

Derivation of frequency modulation (FM) formula

In general, angle modulated signal $S(t)$ is represented as $S(t) = A_{c}cos[\theta_{i}(t)]$ and the instantaneous frequency (i.e. the frequency at each instance) is defined as $f_{i}(t) = f_{c} + k_{f}m(t)$.

Also, $m(t) = A_{m}\cos\omega(t)$ is the message or original signal.

From this, here are few things to understand;

The instantaneous frequency has the combination of the frequency of the carrier wave itself, the message signal $m(t)$ and frequency sensitivity kf. But as the angle modulated wave is a function of theta, we need to express frequency in terms of theta.

Recall that angular frequency is equal to the derivative of theta $\omega = \frac{\text{d}\theta}{\text{d}t}$, so to get theta we need to integrate bothside. Also recall that $\omega(t) = 2\pi f$ and substitute.

=2\pi[f_{c}(t) + \frac{k_{f}A_{m}\sin \omega_{m}(t)}{\omega_{m}(t)}]