How Does Frequency Modulation Work?

After explaining how amplitude modulation works, the next step is Frequency Modulation (FM), which you might be more familiar with, since it is widely used for radio broadcasting.

Carrier and Message

I’ve used the same carrier and message as in the amplitude-modulation article, and I show them again in Figures 1 and 2.

Figure 1. Carrier wave.

In practice, a carrier for FM broadcast radio would have a frequency of around 100 MHz (in most countries, the frequency range is 87.5 to 108 MHz). How to transmit the following message signal by modulating the frequency of the carrier wave?

Figure 2. Message.

Frequency Modulation

Mathematically, the modulation amounts to changing the frequency of the carrier wave proportionally with the amplitude of the message. The carrier wave is defined as

\[c(t) = A\cos(2 \pi f_ct),\]

with \(A\) the amplitude of the wave and \(f_c\) the frequency. If the message is \(m(t)\), with \(|m(t)|\leq1\), then the FM-modulated carrier is given by

\[c_\mathrm{AM}(t)=A\cos\left(2 \pi f_ct+2\pi f_D\!\int_0^t\!\!m(\tau )\,d\tau\right).\]

The parameter \(f_D\) is the frequency deviation, which determines how far the frequency of the carrier is shifted away from the nominal frequency \(f_c\).

The result is shown in Figure 3. It is not possible to directly “see” the message in the modulated carrier, as it was for amplitude modulation, so I’ve drawn the two signals on top of each other. It is clear that the frequency of the carrier varies with the amplitude of the message. Also note that an FM-modulated carrier has a constant envelope.

Figure 3. FM-modulated carrier.

As for AM, the ratio between the frequency of the carrier and the frequencies that are present in the message is typically much larger than shown in this illustration. For broadcast radio, the frequency of the carrier is around 100 MHz, as mentioned before, while the bandwidth of the transmitted message is around 50 kHz (for stereo audio).

And this is the basic principle that underlies your FM radio…