Frequency Modulation Basics (English)

Frequency Modulation is the prominent method of radio broadcasting, invented by Edwin Armstrong in 1933. FM Broadcasting uses Wide-band FM to produce high-fidelity sound over broadcast radio, capable of better sound quality than the leading broadcasting technology at that time: Amplitude Modulation Radio. Wide-band FM is still used to this day as the most popular broadcasting radio scheme, FM radio receivers are available in our mobile phones. Narrow-band FM is also widely used for commercial two-way radio or for amateur radio purposes. In this article, we will talk about the basics of Frequency Modulation scheme.

Modulation in telecommunications is a process of varying a carrier waveform (or carrier signal) to carry an information (or known as baseband signal). Frequency Modulation is one of the three analog modulation schemes, three of them being:

Amplitude Modulation
Frequency Modulation
Phase Modulation

Although the Frequency Modulation and Phase Modulation is usually referred as “Angle Modulation”, leaving only two kinds of analog modulation: angle and amplitude. This is because phase and frequency is tightly coupled, and the relationship of phase and angular frequency can be defined by the equation

$\displaystyle \omega(t) = \frac{d\phi(t)}{dt}$

A carrier frequency is usually defined as a sinusoidal wave, thus can be represented in the following equation

$\displaystyle c(t) = A_c \cos(\omega_ct)$

where $c(t)$ is the instantaneous value of the carrier cosine wave, $A_c$ is the amplitude of the carrier, and $\omega_c$ is the carrier angular frequency

Frequency Modulation is a type of Angle Modulation, in which the name implies, changes the angle of the carrier waveform to contain the information of the baseband signal $m(t)$ . The simplest way to generate an FM signal is through a Voltage Controlled Oscillator. The baseband signal $m(t)$ is used to control the output frequency of the VCO (the FM wave).

direct_method — Figure 1. Direct Wideband Frequency Modulation generation (Source: TutorialsPoint)

The baseband signal controls the frequency of the FM wave, when the baseband signal is zero, the frequency of the output signal is the same as the carrier frequency $f_c$ when the baseband signal is nonzero, the instantaneous frequency of the output signal is expressed by

$\displaystyle f_i(t) = f_c + K_{VCO} m(t)$

where $K_{VCO}$ is the modulation sensitivity of the VCO expressed in units of Hz/V, and $K_{VCO}m(t)$ represents the instantaneous frequency deviation ( $\Delta f$ ).

Since the phase and angular frequency relationship is defined previously, we can derive the instantaneous phase as follows:

$\displaystyle \theta_i(t) = \int_{0}^{t} \omega_i(\tau) d\tau = \omega_ct + 2\pi K_{VCO} \int_{0}^{t} m(\tau) d\tau$

With the instantaneous phase defined, we can write the FM output signal as follows:

$\displaystyle x_{FM}(t) = A_c \cos\bigg(\omega_ct + 2\pi K_{VCO}\int_{0}^{t} m(\tau) d\tau\bigg)$

Mathematically, to analyze the FM signal, a single tone message is usually used, by approximating the baseband signal as a sinusoidal wave shown below:

$\displaystyle m(t) = A_m \cos(\omega_mt)$

where $A_m$ is the amplitude of the baseband signal and $\omega_m$ is the angular frequency of the baseband signal. Substituting the sinusoidal baseband signal to the FM output signals, we find:

$\displaystyle x_{FM}(t) = A_c \cos\bigg(\omega_ct + 2\pi K_{VCO}\int_{0}^{t} A_m \cos(\omega_m\tau) d\tau\bigg)$

$\displaystyle x_{FM}(t) = A_c \cos\bigg(\omega_ct + \frac{K_{VCO}A_m}{f_m}sin(\omega_mt)\bigg)$

The peak frequency deviation ( $\Delta f$ ) is obtained from the modulation sensitivity of the VCO and the amplitude of the baseband signal. The ratio of the peak frequency deviation ( $\Delta f$ ) and the modulating frequency ( $f_m$ ) is called the modulation index, $\beta$ . Thus, we can see the usual FM equation as follows: