In order to recover the original signal from the Sampled one, what is the condition to be satisfied for sampling frequency ω_s and highest frequency component ω_m? 

Explanation:

Condition for Sampling Theorem

Definition: The sampling theorem, also known as the Nyquist-Shannon sampling theorem, is a fundamental principle in signal processing. It states that a continuous-time signal can be completely represented and reconstructed from its samples if the sampling frequency satisfies a specific condition in relation to the highest frequency component of the signal.

Statement: To accurately recover the original signal from its sampled version, the sampling frequency (ω_s) must be at least twice the highest frequency component (ω_m) of the signal. Mathematically, this condition is expressed as:

ω_s ≥ 2ω_m

This condition ensures that no aliasing occurs during the sampling process, allowing the original signal to be reconstructed without distortion.

Correct Option Analysis:

The correct option is:

Option 2: ω_s ≥ 2ω_m

This option correctly represents the Nyquist criterion. According to the theorem, the sampling frequency must be at least twice the highest frequency component of the signal (ω_m) to prevent aliasing and ensure complete reconstruction of the original signal. If this condition is satisfied, the sampled signal contains all the necessary information to recover the original continuous-time signal.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: ω_m < ω_s ≤ 2ω_m

This option is incorrect because it suggests that the sampling frequency can be less than twice the highest frequency component (ω_m) and still allow accurate signal reconstruction. However, if ω_s is less than 2ω_m, aliasing occurs, making it impossible to recover the original signal without distortion. The condition ω_s ≤ 2ω_m does not guarantee the Nyquist criterion is met.

Option 3: ω_s < ω_m

This option is incorrect because it violates the Nyquist criterion. If the sampling frequency is less than the highest frequency component of the signal, severe aliasing occurs, and the original signal cannot be reconstructed. This condition leads to significant distortion in the sampled signal.

Option 4: ω_s = ω_m

This option is also incorrect. If the sampling frequency is equal to the highest frequency component of the signal, the Nyquist criterion is not satisfied. In such a case, aliasing still occurs, and the original signal cannot be accurately reconstructed. The sampling frequency must be at least twice the highest frequency component to meet the Nyquist criterion.

Conclusion:

The correct condition for accurately recovering the original signal from its sampled version is ω_s ≥ 2ω_m. This ensures that the sampling theorem is satisfied, preventing aliasing and enabling complete reconstruction of the original signal. Other options either violate this condition or fail to guarantee the accurate recovery of the signal, leading to distortion or loss of information.