Modulation concept (AM, FM, PM) in Signal Processing - Mini Project: Build & Apply

Understanding Modulation Concepts: AM, FM, and PM

📖 Scenario: Imagine you are working with radio signals. You want to understand how different modulation methods change a simple message signal so it can be sent over the airwaves. This project will help you see how Amplitude Modulation (AM), Frequency Modulation (FM), and Phase Modulation (PM) work by creating and plotting these signals.

🎯 Goal: You will create a simple message signal and then generate its AM, FM, and PM modulated versions. Finally, you will plot all signals to see how modulation changes the message.

📋 What You'll Learn

Create a time array for the signal

Create a message signal as a sine wave

Define modulation parameters like carrier frequency and modulation index

Generate AM, FM, and PM signals using the message signal

Plot the original and modulated signals for comparison

💡 Why This Matters

🌍 Real World

Radio and communication systems use modulation to send information over long distances by changing a carrier wave.

💼 Career

Understanding modulation is important for roles in telecommunications, signal processing, and electronics engineering.

1

Create the time array and message signal

Create a numpy array called t that goes from 0 to 1 second with 1000 points. Then create a message signal called message as a sine wave with frequency 5 Hz using t.

Signal Processing

# Create time array t and message signal
# Your code here

Hint

Use np.linspace to create t. Use np.sin and multiply by 2 * np.pi * 5 * t to create the message.

2

Define carrier frequency and modulation index

Create a variable called carrier_freq and set it to 50. Create another variable called mod_index and set it to 0.5.

Signal Processing

import numpy as np

t = np.linspace(0, 1, 1000)
message = np.sin(2 * np.pi * 5 * t)

# Define carrier frequency and modulation index
# Your code here

Hint

Just assign the numbers 50 and 0.5 to the variables carrier_freq and mod_index.

3

Generate AM, FM, and PM signals

Create three variables: am_signal, fm_signal, and pm_signal. Use the formulas:
AM: (1 + mod_index * message) * np.cos(2 * np.pi * carrier_freq * t)
FM: np.cos(2 * np.pi * carrier_freq * t + mod_index * np.sin(2 * np.pi * 5 * t))
PM: np.cos(2 * np.pi * carrier_freq * t + mod_index * message)

Signal Processing

import numpy as np

t = np.linspace(0, 1, 1000)
message = np.sin(2 * np.pi * 5 * t)

carrier_freq = 50
mod_index = 0.5

# Create AM, FM, and PM signals
# Your code here

Hint

Use the given formulas exactly to create the three signals.

4

Plot the original and modulated signals

Import matplotlib.pyplot as plt. Plot t vs message, am_signal, fm_signal, and pm_signal on the same figure with labels 'Message', 'AM', 'FM', and 'PM'. Add a legend and show the plot.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

t = np.linspace(0, 1, 1000)
message = np.sin(2 * np.pi * 5 * t)

carrier_freq = 50
mod_index = 0.5

am_signal = (1 + mod_index * message) * np.cos(2 * np.pi * carrier_freq * t)
fm_signal = np.cos(2 * np.pi * carrier_freq * t + mod_index * np.sin(2 * np.pi * 5 * t))
pm_signal = np.cos(2 * np.pi * carrier_freq * t + mod_index * message)

# Plot all signals
# Your code here

Hint

Use plt.plot for each signal with the correct label, then call plt.legend() and plt.show().

Practice

1. Which type of modulation changes the amplitude of the carrier wave to transmit information?

easy

A. Frequency Modulation (FM)

B. Amplitude Modulation (AM)

C. Phase Modulation (PM)

D. Pulse Modulation

Solution

1. Step 1: Understand modulation types

   Amplitude Modulation (AM) changes the amplitude of the carrier wave to carry information.

2. Step 2: Compare with other modulation types

   Frequency Modulation changes frequency, Phase Modulation changes phase, so they are not correct here.

3. Final Answer:

   Amplitude Modulation (AM) -> Option B

4. Quick Check:

   Amplitude change = AM [OK]

Hint: Amplitude change means AM modulation [OK]

Common Mistakes:

• Confusing amplitude change with frequency or phase
• Choosing FM or PM instead of AM
• Thinking pulse modulation is amplitude modulation

2. Which of the following is the correct way to represent Frequency Modulation (FM) mathematically for a carrier wave c(t) = A_c cos(2πf_c t) with message signal m(t)?

easy

A. s(t) = A_c cos(2πf_c t + k_f m(t))

B. s(t) = A_c (1 + k_a m(t)) cos(2πf_c t)

C. s(t) = A_c cos(2π(f_c + k_f m(t)) t)

D. s(t) = A_c cos(2πf_c t) + m(t)

Solution

1. Step 1: Recall FM formula

   Frequency Modulation changes the frequency of the carrier. The instantaneous frequency is f_c + k_f m(t).

2. Step 2: Match formula to options

   s(t) = A_c cos(2π(f_c + k_f m(t)) t) shows s(t) = A_c cos(2π(f_c + k_f m(t)) t), which correctly represents FM.

3. Final Answer:

   s(t) = A_c cos(2π(f_c + k_f m(t)) t) -> Option C

4. Quick Check:

   Frequency inside cosine = FM [OK]

Hint: FM changes frequency inside cosine argument [OK]

Common Mistakes:

• Choosing amplitude modulation formula instead
• Ignoring the time variable in frequency term
• Adding message signal outside cosine

3. Given the Python code below simulating a simple AM signal, what will be the output array values?

import numpy as np

A_c = 1
k_a = 0.5
f_c = 5
fs = 50
t = np.arange(0, 1, 1/fs)
m_t = np.sin(2 * np.pi * 1 * t)
s_t = A_c * (1 + k_a * m_t) * np.cos(2 * np.pi * f_c * t)
print(np.round(s_t[:5], 3))

medium

A. [1.0, 0.309, -0.809, -1.191, -0.691]

B. [1.0, 0.809, -0.309, -1.191, -0.691]

C. [1.0, 0.309, 0.809, 1.191, 0.691]

D. [0.0, 0.309, -0.809, -1.191, -0.691]

Solution

1. Step 1: Calculate message and carrier values

   Message m_t is a 1 Hz sine wave sampled at 50 Hz. Carrier frequency is 5 Hz. Calculate first 5 samples of s_t using AM formula.

2. Step 2: Compute first 5 samples

   Using formula: s_t = 1 * (1 + 0.5 * m_t) * cos(2π * 5 * t), the first 5 rounded values are approximately [1.0, 0.309, -0.809, -1.191, -0.691].

3. Final Answer:

   [1.0, 0.309, -0.809, -1.191, -0.691] -> Option A

4. Quick Check:

   AM signal first samples = [1.0, 0.309, -0.809, -1.191, -0.691] [OK]

Hint: AM output = carrier * (1 + k_a * message) [OK]

Common Mistakes:

• Mixing up frequency and amplitude terms
• Not rounding values correctly
• Using wrong sampling frequency

4. The following code attempts to generate a Phase Modulated (PM) signal but has an error. What is the error?

import numpy as np

A_c = 1
k_p = 0.5
f_c = 10
fs = 100

t = np.arange(0, 1, 1/fs)
m_t = np.sin(2 * np.pi * 2 * t)
s_t = A_c * np.cos(2 * np.pi * f_c * t + k_p * np.sin(m_t))
print(np.round(s_t[:5], 3))

medium

A. The time vector t is incorrectly defined

B. The carrier frequency f_c is too high for sampling frequency fs

C. The amplitude A_c should be multiplied inside the cosine argument

D. The modulation term should be k_p * m_t, not k_p * np.sin(m_t)

Solution

1. Step 1: Analyze modulation term in PM formula

   Phase modulation uses k_p * m_t inside the cosine argument, not k_p * np.sin(m_t).

2. Step 2: Identify the error in code

   The code incorrectly applies np.sin(m_t) instead of directly using m_t as the phase modulation signal.

3. Final Answer:

   The modulation term should be k_p * m_t, not k_p * np.sin(m_t) -> Option D

4. Quick Check:

   PM uses message signal directly in phase [OK]

Hint: PM uses message signal directly inside cosine phase [OK]

Common Mistakes:

• Applying sine twice on message signal
• Confusing amplitude with phase modulation
• Incorrect time vector or sampling frequency

5. You want to transmit a message signal m(t) that has sudden jumps (discontinuities). Which modulation technique among AM, FM, and PM is best to minimize distortion caused by these jumps, and why?

hard

A. FM, because frequency changes are less affected by amplitude noise

B. PM, because phase changes handle discontinuities smoothly

C. AM, because amplitude changes are less sensitive to sudden jumps

D. None, all modulation types distort sudden jumps equally

Solution

1. Step 1: Understand effect of sudden jumps on modulation

   Sudden jumps cause rapid changes in signal. AM changes amplitude, which is sensitive to noise and distortion.

2. Step 2: Compare FM and PM for handling jumps

   FM changes frequency, which is more robust to amplitude noise and sudden jumps than AM. PM also changes phase but can be more sensitive to rapid phase shifts.

3. Final Answer:

   FM, because frequency changes are less affected by amplitude noise -> Option A

4. Quick Check:

   FM resists distortion from sudden jumps [OK]

Hint: FM resists noise and sudden jumps better than AM or PM [OK]

Common Mistakes:

• Choosing AM thinking amplitude is always best
• Assuming PM handles jumps better than FM
• Ignoring noise sensitivity differences