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

The Nyquist-Shannon sampling theorem states that to accurately reconstruct a band-limited signal, the sampling frequency ($f_s$) must be at least twice the maximum frequency component ($f_m$) of the signal. 
This minimum sampling frequency, $2f_m$, is known as the Nyquist rate. So, the condition for valid sampling is: $f_s \ge 2f_m$ Given the maximum frequency of the signal, $f_m = 5 \text{ kHz}$. 
The Nyquist rate for this signal is: $f_{Nyquist} = 2 \times f_m = 2 \times 5 \text{ kHz} = 10 \text{ kHz}$. According to the sampling theorem, the sampling frequency must be greater than or equal to 10 kHz. 
Let's check the given options: 
(A) $12 \text{ kHz}$: $12 \text{ kHz} \ge 10 \text{ kHz}$. This is a valid sampling frequency. 
(B) $5 \text{ kHz}$: $5 \text{ kHz}<10 \text{ kHz}$. 
This is not a valid sampling frequency. 
(C) $15 \text{ kHz}$: $15 \text{ kHz} \ge 10 \text{ kHz}$. This is a valid sampling frequency. 
(D) $20 \text{ kHz}$: $20 \text{ kHz} \ge 10 \text{ kHz}$. This is a valid sampling frequency. 
Therefore, the sampling frequency which is not valid is 5 kHz. Sampling at this frequency would lead to aliasing, where higher frequency components in the signal appear as lower frequencies in the sampled data.