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Frequency Modulation Basics (English)

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:

1. Amplitude Modulation
2. Frequency Modulation
3. 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).

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:

$\displaystyle x_{FM}(t) = A_c \cos \big(\omega_ct + \beta\sin(\omega_mt)\big)$

In the next article, I will be talking about the Bessel function and how to use them to analyze a Frequency Modulation wave.

Oleh josefmtd

Electronics Engineer

One reply on “Frequency Modulation Basics (English)”

[…] I mentioned previously on my Frequency Modulation Basics post, I would be taking a practical approach to explain the concept of Frequency Modulation. This […]

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