Phase Modulation (PM)

Phase Modulation (PM):

In phase modulation, the phase of the carrier signal varies linearly in accordance with the message signal voltage.

Where, s(t) is the carrier signal and m(t) is the message signal.

Here in the above equation Kp is the phase sensitivity; the phase of the carrier signal signal is varied in accordance to the amplitude of the message signal where the amplitude of the carrier remain same in the modulation process.

In the phase modulation process,

the modulation index = ΔΦ / fm.T

where, ΔΦ is the peak frequency deviation representing the maximum change in carrier frequency due to modulation

fm and T are the frequency and the period of the message signal respectively

MATLAB Code for Phase Modulation and Demodulation

% The code is written by SalimWireless.Com 
% Phase Modulation (PM) and Demodulation
clc;
clear all;
close all;
function PhaseModulationExample
    % Parameters
    fs = 1000;          % Sampling frequency (Hz)
    fc = 100;           % Carrier frequency (Hz)
    fm = 5;             % Message signal frequency (Hz)
    phasedev = pi/2;    % Phase deviation

% Time vector
    t = 0:1/fs:1;

% Generate message signal
    message_signal = cos(2*pi*fm*t);

% Phase modulation
    modulated_signal = pmmod(message_signal, fc, fs, phasedev);

% Phase demodulation with alternative low-pass filter
    demodulated_signal = pmdemod(modulated_signal, fc, fs, phasedev);

% Plot the original message signal, modulated signal, and demodulated signal
    figure;

subplot(3,1,1);
    plot(t, message_signal);
    title('Original Message Signal');
    xlabel('Time (s)');
    ylabel('Amplitude');

subplot(3,1,2);
    plot(t, modulated_signal);
    title('Phase Modulated Signal');
    xlabel('Time (s)');
    ylabel('Amplitude');

subplot(3,1,3);
    plot(t, demodulated_signal);
    title('Demodulated Signal with Alternative Low-pass Filter');
    xlabel('Time (s)');
    ylabel('Amplitude');
end

% Phase modulation function
function y = pmmod(x, Fc, Fs, phasedev)
    t = 0:1/Fs:(length(x)-1)/Fs;
    y = cos(2*pi*Fc*t + phasedev*x);
end

% Phase demodulation function with alternative low-pass filter
function z = pmdemod(y, Fc, Fs, phasedev)
    t = 0:1/Fs:(length(y)-1)/Fs;
    y_analytic = hilbert(y);
    instantaneous_phase = unwrap(angle(y_analytic));
    demodulated_signal = gradient(instantaneous_phase) / (2*pi*Fc/Fs);
    z = phasedev * demodulated_signal;

% Apply an alternative low-pass filter
    cutoff_frequency = 5;  % Adjust the cutoff frequency as needed
    filter_order = 8;      % Adjust the filter order as needed

% Design a low-pass FIR filter using fir1
    b = fir1(filter_order, cutoff_frequency/(Fs/2), 'low');

% Apply the filter using filtfilt for zero-phase filtering
    z = filtfilt(b, 1, z);
end

Output

Q & A and Summary

1. What distinguishes Phase Modulation (PM) from Frequency Modulation (FM)?

In PM, the phase of the carrier is varied in proportion to the amplitude of the modulating signal \( m(t) \), whereas in FM, the frequency of the carrier is varied according to the amplitude of \( m(t) \). Though both generate sidebands, their modulation index and spectral properties differ.

2. Derive the mathematical expression of a PM signal and explain its parameters.

A PM signal is given by:
\( S(t) = A_c \cos\left(2\pi f_c t + K_p m(t)\right) \)
Where:

\( A_c \): Amplitude of the carrier signal
\( f_c \): Carrier frequency
\( K_p \): Phase sensitivity (radians/volt)
\( m(t) \): Message signal

3. What is the modulation index in PM and how is it interpreted?

The modulation index \( \Delta\phi \) is given by:
\( \Delta\phi = K_p A_m \)
Where \( A_m \) is the peak amplitude of \( m(t) \). It indicates the maximum phase deviation in radians. PM can be categorized into:

Narrowband PM: \( \Delta\phi \ll 1 \)
Wideband PM: \( \Delta\phi \geq 1 \)

4. How is the frequency spectrum of a PM signal described?

When modulated by a sinusoidal signal, the PM spectrum is:
\( S(j\omega) = \sum_{n=-\infty}^{\infty} J_n(\Delta\phi) \cdot \delta(\omega - \omega_c - n\omega_m) \)
Each term represents a sideband located at \( \omega_c \pm n\omega_m \), with amplitudes determined by Bessel functions \( J_n(\Delta\phi) \).

5. What role does a Phase-Locked Loop (PLL) play in PM demodulation?

A PLL is used to demodulate PM by tracking the phase of the incoming signal. The key blocks:

Phase Detector (PD): Produces error voltage based on phase difference.
Loop Filter: Smooths the error signal.
Voltage-Controlled Oscillator (VCO): Adjusts its phase to match the incoming signal.

The output of the loop filter reconstructs the original message signal.

6. Why are Bessel functions relevant in PM spectrum analysis?

In PM, Bessel functions \( J_n(\Delta\phi) \) determine the amplitude of the carrier and its sidebands in the frequency domain. Since phase deviation affects the shape of the spectrum, these functions help in predicting how energy is distributed across sidebands.

7. What are the practical differences between Narrowband PM and Wideband PM?

Narrowband PM (NBPM) behaves similarly to AM and is simpler to implement but has poor noise immunity. Wideband PM (WBPM) offers better noise performance and spectral efficiency but requires more bandwidth and complexity.

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