Sampling theorem (Nyquist-Shannon) in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Sampling Theorem (Nyquist-Shannon)?

It states that a continuous signal can be perfectly reconstructed from its samples if the sampling rate is at least twice the highest frequency present in the signal.

beginner

What is the Nyquist rate?

The Nyquist rate is twice the highest frequency of a signal. Sampling at this rate or higher prevents loss of information.

intermediate

What happens if you sample below the Nyquist rate?

Sampling below the Nyquist rate causes aliasing, where different signals become indistinguishable and distort the reconstructed signal.

beginner

Why is the Sampling Theorem important in real life?

It helps in converting real-world signals like sound and images into digital form without losing information, enabling technologies like music streaming and digital cameras.

intermediate

How can you avoid aliasing when sampling a signal?

Use a low-pass filter to remove frequencies above half the sampling rate before sampling, ensuring the signal meets the Nyquist criterion.

What is the minimum sampling rate to perfectly reconstruct a signal with a highest frequency of 5 kHz?

A 20 kHz

B 5 kHz

C 2.5 kHz

D 10 kHz

The minimum sampling rate is twice the highest frequency, so 2 × 5 kHz = 10 kHz.

What is aliasing in signal sampling?

A Distortion caused by sampling below Nyquist rate

B Perfect reconstruction of signal

C Loss of signal amplitude

D Increasing sampling rate

Aliasing happens when sampling below the Nyquist rate, causing different signals to overlap and distort.

Which filter is used to prevent aliasing before sampling?

A High-pass filter

B Band-pass filter

C Low-pass filter

D Notch filter

A low-pass filter removes frequencies above half the sampling rate to prevent aliasing.

If a signal contains frequencies up to 8 kHz, what is the Nyquist rate?

A 16 kHz

B 8 kHz

C 32 kHz

D 4 kHz

The Nyquist rate is twice the highest frequency, so 2 × 8 kHz = 16 kHz.

Why is sampling important in digital audio?

A To increase signal frequency

B To convert sound into digital data for storage and processing

C To reduce sound quality

D To remove noise from sound

Sampling converts continuous sound waves into digital data so computers can store and process audio.

Explain the Nyquist-Shannon Sampling Theorem and why the sampling rate must be at least twice the highest frequency.

Think about how sampling too slowly causes signal overlap.

Describe how aliasing occurs and how it can be prevented in practical signal processing.

Consider what happens when different frequencies become indistinguishable.

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