Frequency Modulation (FM)

Theory :

Frequency modulation (FM) is the process where the frequency of a carrier wave, denoted as \( c(t) \), is varied in accordance with the instantaneous amplitude of the modulating signal \( m(t) \). Unlike amplitude modulation, where the amplitude of the carrier wave is varied, in FM, the amplitude remains constant while the frequency changes. The modulated signal can be mathematically expressed as:

\( S(t) = A_c \cos\left[ 2\pi f_c t + 2\pi K_f \int m(\tau) d\tau \right] \)

Where:

• \( A_c \) is the amplitude of the carrier signal.
• \( f_c \) is the carrier frequency.
• \( K_f \) is the frequency sensitivity of the modulator, defining how much the frequency of the carrier varies with the modulating signal.
• \( m(t) \) is the modulating (baseband) signal.
• \( \int m(\tau) d\tau \) represents the integral of the modulating signal, which determines the frequency deviation.

This expression shows that the instantaneous frequency of the carrier wave is proportional to the amplitude of the modulating signal. The carrier frequency \( f_c \) is shifted by an amount proportional to the modulating signal \( m(t) \), and this shift results in a frequency-modulated signal.

Block Diagram

Fig: Frequency Modulation

Modulation Index (β) in Frequency Modulation

In Frequency Modulation, the modulation index, denoted by β, represents the relationship between the frequency deviation and the frequency of the modulating signal. Unlike in AM, it's not a simple ratio of amplitudes.

It is defined as the ratio of the peak frequency deviation to the modulating frequency:
β = Δf / f_m
where:

• Δf = Peak frequency deviation. This is the maximum shift in the carrier's frequency from its resting state (f_c). It is determined by Δf = K_f A_m.
• f_m = The maximum frequency of the message signal m(t).
• K_f = The frequency sensitivity of the modulator (in Hz/Volt).
• A_m = The peak amplitude of the message signal.

The value of β distinguishes between two main classes of FM:

• β << 1 (typically β < 0.3): Narrowband FM (NBFM). Its properties are similar to AM.
• β > 1: Wideband FM (WBFM). This is used in broadcasting for high-fidelity audio, as it offers superior noise immunity.

Frequency Domain Description:

An FM signal produces an infinite number of sidebands spaced at intervals of ω_m from the carrier ω_c, both above and below.

S(jω) = ∑_n=-∞^∞ J_n(β) · δ(jω - ω_c - n ω_m)

Where:

• J_n(β): Bessel function of the first kind of order n
• δ(·): Dirac delta (impulse) at each spectral line
• ω_c: Carrier angular frequency (rad/s)
• ω_m: Message angular frequency (rad/s)

Frequency Demodulation

Fig: Frequency Demodulation

The diagram illustrates a Phase-Locked Loop (PLL) used for demodulating Frequency Modulated (FM) signals. The working of each block is described below:

• Input signal: This is the received FM signal, typically denoted as s(t), which carries the frequency variations corresponding to the original message m(t).
• PD (Phase Detector): Compares the phase of the input signal with that of the feedback signal generated by the VCO. The output is a signal proportional to the phase difference or error.
• F(s) (Loop Filter): Processes the phase error signal from the PD. It smooths the signal to eliminate high-frequency components and to ensure loop stability.
• VCO (Voltage Controlled Oscillator): Generates a signal whose frequency is controlled by the filtered phase error. The VCO attempts to match the instantaneous frequency of the input FM signal.
• Demodulated FM Output (∼ dθ_i(t)/dt): The output of the loop (or after taking the derivative of the instantaneous phase) approximates the original message signal m(t).