The basic principle behind FM is that the amplitude of an analog baseband signal can be represented by a slightly different frequency of the carrier. We represent this relationship in the graph below.
Figure 1. Frequency Modulation
As this graph illustrates, various amplitudes of the baseband signal (shown in white) relate to specific frequencies of the carrier signal (shown in red). Mathematically, we represent this by describing the equations which characterize FM.
First, we represent our message, or baseband, signal by the simple designation m(t). Second, we represent a sinusoidal carrier by the equation:
xc(t) = Ac cos (2πfct).
The actual mathematical process to modulate a baseband signal, m(t), onto the carrier requires a two-step process. First, the message signal must be integrated with respect to time to get an equation for phase with respect to time, θ(t). This integration enables the modulation process because phase modulation is fairly straightforward with typical I/Q modulator circuitry. A block diagram description of an FM transmitter follows.
Figure 2. FM Transmitter Block Diagram
As the block diagram above illustrates, the integration of a message signal results in an equation for phase with respect to time. This equation is defined by the following equation:
where kf is the frequency sensitivity. Again, the resulting modulation that must occur is phase modulation, which involves changing the phase of the carrier over time. This process is fairly straightforward and requires a quadrature modulator, shown below.
Figure 3. Quadrature Modulator
As a result of phase modulation, the resulting FM signal, s(t), now represents the frequency modulated signal. This equation is shown below.
where m(τ) = M cos (2πfmτ). More simply, we can also represent this equation as: