Frequency Modulation

Frequency modulation is commonly for public broadcasting purposes due to several advantages. An FM transmitter can always operate at peak power and any disruptions to or fading of the signal can be corrected at the receiver. However, frequency modulation will in most circumstances use greater bandwidth than amplitude modulation.

Narrowband Frequency Modulation

Begin with a signal: $ x(t) = Acos(w_{m}t) \ $

The instantaneous frequency is: $ w_{i}(t) = w_{c} + k_{f}Acos(w_{m}t)\ $

which oscillates between $ w_{c} + k_{f}A\ $ and $ w_{c} - k_{f}A\ $

with $ \Delta w = k_{f}A\ $.

The resulting equation is: $ w_{i}(t) = w_{c}+\Delta wcos(w_{m}t)\ $

and

$ y(t) = cos[w_{c}t + \int x(t)dt] = cos(w_{c}t + \frac{\Delta w}{w_{m}}sin(w_{m}t)+ \theta_{0})\ $

where $ \theta_{0}\ $ is a constant as a result of integration. $ \frac{\Delta w}{w_{m}} $ is the modulation index $ m\ $ for frequency modulation so $ y(t) = cos[w_{c}t + msin(w_{m}t)\ $