Sampling theorem (Nyquist-Shannon) in Signal Processing - Mini Project: Build & Apply

Sampling Theorem (Nyquist-Shannon) Demonstration

📖 Scenario: Imagine you are working with sound signals. You want to convert a continuous sound wave into digital data so a computer can store and play it back. To do this correctly, you need to understand how often to take samples from the sound wave. This is where the Sampling Theorem, also called the Nyquist-Shannon theorem, helps.This project will guide you through creating a simple example to see how sampling frequency affects the ability to capture a signal without losing information.

🎯 Goal: You will build a small program that creates a simple sine wave signal, sets a sampling frequency, samples the signal, and then shows the sampled values. This will help you understand the importance of sampling frequency in capturing signals accurately.

📋 What You'll Learn

Create a continuous sine wave signal with a specific frequency

Set a sampling frequency variable

Sample the sine wave at the given sampling frequency

Print the sampled values to observe the effect of sampling

💡 Why This Matters

🌍 Real World

Sampling continuous signals like sound or images is essential for digital devices to process and store real-world data.

💼 Career

Understanding sampling is key for roles in audio engineering, telecommunications, and data science involving signal processing.

1

Create a continuous sine wave signal

Create a list called time_points with values from 0 to 1 (exclusive) in steps of 0.01. Then create a list called signal that contains the sine of 2 * pi * 5 * each time point. Use import math at the top.

Signal Processing

import math
# Create time_points list from 0 to 1 with step 0.01
# Create signal list as sine of 2 * pi * 5 * time_points
# Your code here

Hint

Use a list comprehension to create time_points and another to create signal. The frequency of the sine wave is 5 Hz.

2

Set the sampling frequency

Create a variable called sampling_frequency and set it to 8.

Signal Processing

import math

time_points = [i * 0.01 for i in range(100)]
signal = [math.sin(2 * math.pi * 5 * t) for t in time_points]

# Set sampling_frequency to 8
# Your code here

Hint

Sampling frequency should be less than twice the signal frequency to see aliasing effects.

3

Sample the sine wave at the sampling frequency

Create a list called sampled_signal by sampling the signal at intervals of 1 divided by sampling_frequency. Use a list comprehension with range and step size calculated from the sampling frequency.

Signal Processing

import math

time_points = [i * 0.01 for i in range(100)]
signal = [math.sin(2 * math.pi * 5 * t) for t in time_points]

sampling_frequency = 8

# Sample the signal at intervals of 1/sampling_frequency
# Your code here

Hint

Calculate the step size for sampling by dividing 1 second by sampling frequency and converting to index steps for the 0.01 time step.

4

Print the sampled signal values

Print the sampled_signal list to see the sampled values.

Signal Processing

import math

time_points = [i * 0.01 for i in range(100)]
signal = [math.sin(2 * math.pi * 5 * t) for t in time_points]

sampling_frequency = 8

sample_interval = int(1 / (sampling_frequency * 0.01))
sampled_signal = [signal[i] for i in range(0, len(signal), sample_interval)]

# Print the sampled_signal list
# Your code here

Hint

The printed list shows sampled sine values at the chosen sampling frequency.

Practice

1. What does the Nyquist-Shannon sampling theorem state about the sampling rate?

easy

A. Sample at the same rate as the highest frequency

B. Sample at half the highest frequency of the signal

C. Sample at any random rate

D. Sample at least twice the highest frequency of the signal

Solution

1. Step 1: Understand the sampling theorem

   The Nyquist-Shannon theorem says to sample at least twice the highest frequency to capture all information.

2. Step 2: Identify correct sampling rate

   Sampling below twice the highest frequency causes aliasing and loss of information.

3. Final Answer:

   Sample at least twice the highest frequency of the signal -> Option D

4. Quick Check:

   Sampling rate ≥ 2 x highest frequency [OK]

Hint: Remember: sample rate must be double highest frequency [OK]

Common Mistakes:

• Thinking sampling rate can be less than highest frequency
• Confusing sampling rate with signal frequency
• Ignoring aliasing effects

2. Which of the following is the correct Python syntax to calculate the Nyquist frequency for a signal with sampling rate fs?

easy

A. nyquist = fs / 2

B. nyquist = fs * 2

C. nyquist = 2 / fs

D. nyquist = fs + 2

Solution

1. Step 1: Recall Nyquist frequency formula

   The Nyquist frequency is half the sampling rate, so nyquist = fs / 2.

2. Step 2: Match formula to options

   Only nyquist = fs / 2 correctly divides fs by 2.

3. Final Answer:

   nyquist = fs / 2 -> Option A

4. Quick Check:

   Nyquist frequency = sampling rate / 2 [OK]

Hint: Nyquist frequency is half the sampling rate [OK]

Common Mistakes:

• Multiplying instead of dividing by 2
• Confusing Nyquist frequency with sampling rate
• Using addition or reciprocal incorrectly

3. Given a signal with highest frequency 500 Hz, what is the minimum sampling rate to avoid aliasing? Consider the code:

highest_freq = 500
sampling_rate = ?

medium

A. sampling_rate = 1000

B. sampling_rate = 750

C. sampling_rate = 250

D. sampling_rate = 500

Solution

1. Step 1: Apply Nyquist theorem to highest frequency

   Minimum sampling rate = 2 x highest frequency = 2 x 500 = 1000 Hz.

2. Step 2: Compare with options

   Only sampling_rate = 1000 matches 1000 Hz.

3. Final Answer:

   sampling_rate = 1000 -> Option A

4. Quick Check:

   Minimum sampling rate = 2 x 500 = 1000 [OK]

Hint: Double the highest frequency for minimum sampling rate [OK]

Common Mistakes:

• Choosing sampling rate equal to highest frequency
• Selecting a rate less than twice the frequency
• Confusing frequency units

4. Identify the error in this Python code snippet that calculates the Nyquist frequency:

fs = 800
nyquist = fs * 0.5
print("Nyquist frequency is", nyquist)

medium

A. The multiplication should be division by 2

B. The variable fs should be lowercase

C. The code is correct and will print 400.0

D. The print statement syntax is wrong

Solution

1. Step 1: Understand calculation of Nyquist frequency

   Nyquist frequency = sampling rate divided by 2, which equals fs * 0.5.

2. Step 2: Check code correctness

   Code multiplies fs by 0.5 correctly and prints the value 400.0.

3. Final Answer:

   The code is correct and will print 400.0 -> Option C

4. Quick Check:

   fs * 0.5 = 400.0, print works fine [OK]

Hint: Multiplying by 0.5 equals dividing by 2 [OK]

Common Mistakes:

• Thinking multiplication by 0.5 is wrong
• Assuming variable case matters here
• Believing print syntax is incorrect in Python 3

5. You have a signal with components up to 1200 Hz. You sample it at 2000 Hz. What problem will occur and why?

Options:
A) No problem, sampling rate is sufficient
B) Aliasing will occur because sampling rate is less than twice highest frequency
C) Aliasing will occur because sampling rate is more than twice highest frequency
D) Signal will be perfectly reconstructed without aliasing

hard

A. No problem, sampling rate is sufficient

B. Aliasing will occur because sampling rate is less than twice highest frequency

C. Aliasing will occur because sampling rate is more than twice highest frequency

D. Signal will be perfectly reconstructed without aliasing

Solution

1. Step 1: Calculate minimum sampling rate needed

   Minimum sampling rate = 2 x 1200 Hz = 2400 Hz.

2. Step 2: Compare given sampling rate with minimum

   Given sampling rate is 2000 Hz, which is less than 2400 Hz, so aliasing will occur.

3. Step 3: Understand aliasing cause

   Sampling below twice the highest frequency causes signal distortion called aliasing.

4. Final Answer:

   Aliasing will occur because sampling rate is less than twice highest frequency -> Option B

5. Quick Check:

   Sampling rate < 2 x highest frequency causes aliasing [OK]

Hint: Sampling rate must be ≥ 2 x highest frequency to avoid aliasing [OK]

Common Mistakes:

• Assuming 2000 Hz is enough for 1200 Hz signal
• Confusing aliasing cause with oversampling
• Thinking aliasing occurs when sampling rate is too high