Question

A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is

• A. \( 12 \text{ kHz} \)
• B. \( 5 \text{ kHz} \)
• C. \( 15 \text{ kHz} \)
• D. \( 20 \text{ kHz} \)

Hint:

The Nyquist-Shannon sampling theorem is fundamental in digital signal processing. Always remember the critical condition: $f_s \ge 2f_m$, where $f_s$ is the sampling frequency and $f_m$ is the maximum frequency present in the analog signal. Sampling below the Nyquist rate ($f_s<2f_m$) causes aliasing, making it impossible to perfectly reconstruct the original signal.

Answer 1

The Correct Option is B

To solve this problem, we need to apply the Nyquist-Shannon sampling theorem to determine the valid sampling frequencies for a band-limited signal.

1. Understanding the Concepts:

- Band-limited Signal: A signal that has a finite range of frequencies, and its highest frequency component is known.
- Nyquist-Shannon Sampling Theorem: The theorem states that a continuous signal with a maximum frequency \( f_{\text{max}} \) must be sampled at a frequency \( f_s \) at least twice that of the maximum frequency to avoid aliasing. This is called the Nyquist rate, i.e., \( f_s \geq 2 \times f_{\text{max}} \).

2. Given Values:

\( f_{\text{max}} = 5 \text{ kHz} \)

The Nyquist rate (minimum sampling frequency) is:

\( f_s = 2 \times 5 \text{ kHz} = 10 \text{ kHz} \)

3. Calculating the Valid Sampling Frequencies:

The sampling frequency must be at least 10 kHz to avoid aliasing. Therefore, any frequency less than 10 kHz is invalid for sampling the given signal. Let’s analyze the options:

• \( 12 \text{ kHz} \) is valid because it is greater than 10 kHz.
• \( 5 \text{ kHz} \) is invalid because it is less than the Nyquist rate of 10 kHz.
• \( 15 \text{ kHz} \) is valid because it is greater than 10 kHz.
• \( 20 \text{ kHz} \) is valid because it is greater than 10 kHz.

Final Answer:

The sampling frequency which is not valid is \( 5 \text{ kHz} \).