Phase Modulation Study Guide & Revision Exercises

1. Introduction to Phase Modulation

Phase Modulation (PM) is an angle modulation technique where the phase of the carrier signal is varied in proportion to the amplitude of the modulating signal.

s(t) = A_c cos(2πf_ct + k_pm(t))

Where:

A_c = Carrier amplitude
f_c = Carrier frequency
k_p = Phase sensitivity (rad/V)
m(t) = Modulating signal

Figure 1 shows a sinusoidal message signal that is to be transmitted from an end to another, a carrier signal which is to be phase modulated. And the last image in the above figure represents the phase modulated signal.

Figure 1. Phase Modulation waveforms

From Figure 1, it is clear from the above figure that when the amplitude of the sinusoidal signal starts to increase and reaches the maximum value, then the phase lead of the carrier signal gets increased. Due to this a compression in the carrier signal is noticed. This resultantly increases the frequency of the signal. However, when the amplitude of the modulating signal starts falling and attains a minimum value, then the phase lag of the carrier wave occurs. Thereby resultantly causing stretching of the signal. Due to this, the frequency of the signal gets increased. So, in this way, we can say that with the change in phase of the signal during phase modulation, the frequency of the signal also shows some variation.

For a sinusoidal signal, the modulated signal is somewhat similar in case of both frequency and phase modulation. While in the case of the square wave signal, the case is not exactly the same as shown in Figure 2. For the case of the square wave, when the signal moves from positive to negative amplitude then negative phase reversal exists. While, when the message signal shows movement from negative to positive amplitude then positive phase reversal takes place. Therefore, it is not true that the frequency and phase modulated signal are similar for any type of waveform.

Figure 2. Square Wave FM and squarewave PM waveforms

PM vs. Frequency Modulation (FM)

PM: Phase changes with m(t)

FM: Frequency changes with m(t)

Note: PM and FM are related through the derivative/integral of m(t)

2. Mathematical Foundations

2.1 Phase Deviation

The peak phase deviation (Δφ):

Δφ = k_p · max|m(t)|

For sinusoidal m(t) = A_mcos(2πf_mt):

Modulation index (β) = Δφ = k_pA_m

2.2 Spectrum and Bandwidth

PM generates infinite sidebands described by Bessel functions:

s(t) = A_c Σ J_n(β) cos[2π(f_c + nf_m)t]

Carson's Rule for bandwidth estimation:

BW ≈ 2(β + 1)f_m

3. Generation & Demodulation

3.1 PM Generation Methods

Method	Description	Implementation
Direct	Vary phase directly using voltage-controlled phase shifter	Varactor diode, reactance modulator
Indirect	Integrate m(t) → FM modulator (Armstrong method)	Op-amp integrator + VCO

3.2 PM Demodulation Techniques

Phase-Locked Loop (PLL) - Most common method
Differentiator + Envelope Detector - Convert PM to AM
Product Detector - Multiplier + LPF

4. Key Properties

Property	PM Characteristics
Noise Immunity	Better than AM, worse than FM
Bandwidth	Increases with modulation index (β)
Power Efficiency	Constant envelope → good for RF amplifiers
Implementation Complexity	Higher than AM, similar to FM

5. Applications

Satellite Communications (GPS, deep-space telemetry)
Digital Modulation foundation (BPSK, QPSK)
Audio Synthesis (Yamaha DX7 FM synthesis)
Radar Systems (Phase-coded pulses)

6. Python Simulation

import numpy as np
import matplotlib.pyplot as plt

# Parameters
fc = 1000  # Carrier frequency (Hz)
fm = 100   # Modulating frequency (Hz)
beta = 2   # Modulation index
Ac = 1     # Carrier amplitude
t = np.linspace(0, 0.1, 1000)

# Modulating and PM signals
m_t = np.cos(2 * np.pi * fm * t)
pm_signal = Ac * np.cos(2 * np.pi * fc * t + beta * m_t)

# Plotting
plt.figure(figsize=(10,4))
plt.plot(t, pm_signal, label='PM Signal')
plt.plot(t, m_t, 'r--', label='Modulating Signal')
plt.xlabel('Time (s)'); plt.ylabel('Amplitude')
plt.legend(); plt.grid()
plt.show()

7. PM vs. FM

Advantages of PM

Constant envelope → efficient for RF power amps
Better noise immunity than AM
Natural fit for digital PSK

Disadvantages

Harder to demodulate than FM
Requires coherent detection
Larger bandwidth than AM

8. Practice Problems

Calculate β for k_p = 1.5 rad/V and m(t) = 2cos(2π500t)
Estimate bandwidth for β = 4 and f_m = 1 kHz
Explain why PM is less common than FM in analog communication

9. Phase Modulation Quiz

Section 1: Fundamental Concepts

What is the primary parameter varied in Phase Modulation (PM)?
a) Amplitude
b) Frequency
c) Phase angle
d) Wavelength
True or False: In PM, the phase deviation is proportional to the amplitude of the modulating signal.
The instantaneous phase of a PM wave is given by:
a) $_{}_{}$
b) $_{}_{}$
c) $_{}$
d) None of the above
Phase Modulation is a subset of:
a) Amplitude Modulation
b) Angle Modulation
c) Pulse Modulation
d) Digital Modulation
The unit of phase deviation constant
a) Hz/V
b) radians/V
c) V/rad
d) dimensionless

Section 2: Mathematical Analysis

A PM signal is represented as $_{}_{}_{}$ . What is the phase deviation?
a) radians
b) $_{}$
c) $_{}$
d) $_{}$
For a modulating signal $_{}$ , the PM wave’s phase angle is:
a) $_{}_{}_{}$
b) $_{}_{}$
c) $_{}$
d) $_{}$
The bandwidth of a PM wave depends on:
a) Modulating signal frequency only
b) Phase deviation index () only
c) Both and modulating frequency
d) Carrier amplitude
True or False: Carson’s rule is used to estimate the bandwidth of a PM signal.
If for a PM signal, the number of significant sidebands is approximately:
a) 2
b) 4
c) 6
d) 8

Section 3: Comparisons and Applications

PM is less susceptible to noise than AM because:
a) It encodes information in phase, not amplitude
b) It uses higher frequencies
c) It has a narrower bandwidth
d) All of the above
True or False: FM and PM are identical when the modulating signal is a sine wave.
Which of the following is a practical application of PM?
a) Broadcast radio (AM)
b) Wi-Fi (OFDM)
c) Analog TV
d) Satellite communications
The phase deviation index () for PM is analogous to what in FM?
a) Frequency deviation
b) Modulation index
c) Bandwidth
d) Carrier swing
PM is preferred over FM in scenarios where:
a) Constant amplitude is critical
b) Higher bandwidth is available
c) Noise immunity is unimportant
d) Low-frequency signals are used

Section 4: Advanced Concepts (5 Questions)

Narrowband PM has a value of:
a)
b)
c)
d)
Demodulation of PM can be done using a:
a) Envelope detector
b) Phase-locked loop (PLL)
c) Differentiator followed by an envelope detector
d) Low-pass filter
True or False: In PM, the frequency deviation is proportional to the derivative of the modulating signal.
The spectrum of a PM wave with will have:
a) Only the carrier
b) Carrier + one sideband
c) Carrier + multiple sidebands
d) No sidebands
Phase Modulation is used in digital modulation as:
a) ASK
b) PSK
c) FSK
d) QAM

Answer Key

c
True
a
b
b
a
a
c
True
c
a
False (only for specific cases)
d
b
a
a
b
True
b
b