Frequency Modulation (FM)

Frequency Modulation (FM):

The carrier signal's frequency varies linearly in response to the voltage variation in the message signal.

Here, modulation index = Δf / fm

Where Δf is the peak frequency deviation and fm = frequency of the message signal.

Here frequency deviation means how much the the frequency of the carrier signal deviates from its carrier's original frequency after frequency modulation.

When performing frequency modulation (FM) with a carrier frequency of 100 Hz and a message frequency of 10 Hz, the resulting peak frequencies are as follows: 80 Hz (100 - 2*10 Hz), 90 Hz (100 - 10 Hz), 100 Hz, 110 Hz (100 + 10 Hz), 120 Hz (100 + 2*10 Hz).

Figure: Frequency Spectrums of FM Signal

MATLAB Code for Frequency Modulation and Demodulation

% The code is written by SalimWireless.Com 
% Frequency Modulation (FM) and Demodulation
clc;
clear all;
close all;
function Frequency_Modulation_Paraphrased
    % Parameters for Frequency Modulation

% Carrier frequency
    carrier_frequency = 100;

% Message signal frequency
    message_frequency = 10;

% Amplitude of the message signal
    message_amplitude = 1;

% Frequency sensitivity (modulation index)
    sensitivity = 50;

% Sampling frequency
    sampling_frequency = 10 * max(carrier_frequency, message_frequency);

% Time vector from 0 to 1 second
    time_vector = 0:1/sampling_frequency:1;

% Generate the message signal
    message_signal = message_amplitude * cos(2*pi*message_frequency*time_vector);

% FM modulation
    modulated_signal = my_fmmod(message_signal, carrier_frequency, sampling_frequency, sensitivity);

% Plot original message signal and the modulated signal
    plot_signals(time_vector, message_signal, modulated_signal);

% Coherent Frequency Demodulation (FM)
    demodulated_signal = my_coherent_fmdemod(modulated_signal, carrier_frequency, sampling_frequency, sensitivity, time_vector, message_frequency);

% Plot the demodulated signal
    plot_demodulated_signal(time_vector, demodulated_signal);
end

function modulated_signal = my_fmmod(message_signal, carrier_frequency, sampling_frequency, sensitivity)
    % Function to perform FM modulation

% Generate the time vector
    time_vector = 0:1/sampling_frequency:(length(message_signal)-1)/sampling_frequency;

% Calculate phase deviation based on the message signal
    phase_deviation = sensitivity * cumsum(message_signal) * 2*pi/sampling_frequency;

% Modulate the signal using FM modulation
    modulated_signal = cos(2*pi*carrier_frequency*time_vector + phase_deviation);

% Scale the modulated signal to match the amplitude of the message signal
    modulated_signal = scale_amplitude(message_signal, modulated_signal);
end

function scaled_signal = scale_amplitude(original_signal, signal_to_scale)
    % Function to scale the amplitude of a signal

% Calculate the amplitude ratio between the original and scaled signals
    amplitude_ratio = rms(original_signal) / rms(signal_to_scale);

% Scale the signal_to_scale to match the amplitude of the original_signal
    scaled_signal = amplitude_ratio * signal_to_scale;
end

function demodulated_signal = my_coherent_fmdemod(modulated_signal, carrier_frequency, sampling_frequency, sensitivity, time_vector, message_frequency)
    % Function to perform coherent frequency demodulation (FM)

% Generate the local oscillator signal
    local_oscillator = cos(2*pi*carrier_frequency*time_vector);

% Demodulate the modulated signal using the local oscillator
    demodulated_signal = modulated_signal .* local_oscillator;

% Apply a low-pass filter (Butterworth filter) to remove high-frequency components
    demodulated_signal = apply_low_pass_filter(demodulated_signal, sampling_frequency, message_frequency);
end

function filtered_signal = apply_low_pass_filter(signal, sampling_frequency, cutoff_frequency)
    % Function to apply a low-pass filter (Butterworth filter)

% Set filter parameters
    filter_order = 4; % Order of the Butterworth filter
    cutoff_frequency = cutoff_frequency * 1.5; % Adjust the cutoff frequency as needed

% Design the Butterworth filter
    [b, a] = butter(filter_order, cutoff_frequency/(sampling_frequency/2), 'low');

% Apply the filter to the signal
    filtered_signal = filtfilt(b, a, signal);
end

function plot_signals(time_vector, message_signal, modulated_signal)
    % Function to plot the original message signal and the modulated signal

% Plot the original message signal
    subplot(2,1,1);
    plot(time_vector, message_signal);
    title('Message Signal');
    xlabel('Time (s)');
    ylabel('Amplitude');

% Plot the modulated signal
    subplot(2,1,2);
    plot(time_vector, modulated_signal);
    title('FM Modulated Signal');
    xlabel('Time (s)');
    ylabel('Amplitude');
end

function plot_demodulated_signal(time_vector, demodulated_signal)
    % Function to plot the demodulated signal

% Plot the demodulated signal
    figure;
    plot(time_vector, demodulated_signal);
    title('Demodulated Signal (Coherent FM)');
    xlabel('Time (s)');
    ylabel('Amplitude');
end

Output

Fig 2: Frequency Modulation and Demodulation

Q & A and Summary

1. What is the fundamental difference between Frequency Modulation (FM) and Amplitude Modulation (AM), and why is FM more resistant to noise?

In AM, the amplitude of the carrier wave varies according to the message signal, while in FM, the frequency of the carrier varies, and the amplitude remains constant. FM is more resistant to noise because most noise affects amplitude, and FM encodes information in frequency, not amplitude.

2. Explain how the integral \( \int m(\tau) d\tau \) in the FM equation contributes to frequency modulation.

The integral \( \int m(\tau) d\tau \) determines the phase deviation of the carrier. Since frequency is the derivative of phase, integrating the modulating signal controls how the instantaneous frequency of the carrier changes, thus achieving frequency modulation.

3. Why does an FM signal theoretically have infinite bandwidth, and how is this managed in practical systems?

FM generates infinite sidebands due to Bessel function expansion. Practically, only a limited number have significant power. Carson’s Rule estimates usable bandwidth:
\( \text{BW} \approx 2(\Delta f + f_m) \)
This ensures efficient spectrum allocation in real systems.

4. What role does the modulation index (β) play in shaping the spectral characteristics of an FM signal?

The modulation index \( \beta = \frac{\Delta f}{f_m} \) controls how many sidebands the FM signal has.
- Low β (β << 1): Narrowband FM (similar to AM).
- High β (β > 1): Wideband FM with richer spectrum and better fidelity.

5. How do Bessel functions \( J_n(\beta) \) influence the amplitude of FM sidebands?

Each FM sideband's amplitude is determined by \( J_n(\beta) \), a Bessel function of the first kind. The signal's energy spreads across these sidebands, with the function values defining their relative strength and distribution.

6. Why is Narrowband FM considered similar to AM in certain aspects?

When \( \beta \ll 1 \), FM behaves like AM with carrier and first-order sidebands dominating. The FM signal approximates to a form close to amplitude modulation but retains some of FM’s noise immunity.

7. How does a Phase-Locked Loop (PLL) demodulate an FM signal?

A PLL tracks the FM signal’s frequency:

Phase Detector: Compares FM signal and VCO.
Loop Filter: Smooths the error voltage.
VCO: Adjusts frequency based on control voltage.

The control voltage becomes the recovered message signal.

8. How does FM improve signal fidelity and reduce distortion in broadcasting applications like FM radio?

Wideband FM (β > 1) captures more message details, allowing high-fidelity audio. It also resists amplitude noise, making it ideal for applications like FM radio where sound quality is critical.

9. What challenges arise in designing FM systems given the non-linearity of the modulation process?

Challenges include:

Infinite bandwidth and complex spectrum.
Non-linear nature restricts linear superposition.
PLL circuits require precision for accurate demodulation.

10. Why is Carson’s Rule only an approximation, and under what conditions does it fail?

Carson’s Rule assumes significant sidebands fall within 2(Δf + f_m). It may fail when:

β is very small or large (extreme cases).
Message has wide or non-sinusoidal content.
Exact bandwidth control is needed (e.g., military or satellite systems).

Search This Blog