A signal Vm sin 100πt + 2Vm sin 200πt must be sampled and stored in a data acquisition system. To extract the signal effectively, the original sampling frequency should be:
X 1. 

Explanation:

Sampling Frequency Analysis

Definition: Sampling frequency, also known as the sampling rate, is the number of samples per second taken from a continuous signal to convert it into a discrete signal. According to the Nyquist theorem, the sampling frequency must be at least twice the highest frequency present in the signal to accurately reconstruct the original signal without aliasing.

Given Signal:

The signal provided is: Vm × sin(100πt) + 2Vm × sin(200πt)

Here, the frequencies of the components are:

• First Component: sin(100πt), where f = 50 Hz (since ω = 2πf and ω = 100π).
• Second Component: sin(200πt), where f = 100 Hz (since ω = 2πf and ω = 200π).

Highest Frequency in the Signal:

The highest frequency present in the signal is 100 Hz (from the second component).

Nyquist Criterion:

To sample and reconstruct the signal effectively without aliasing, the sampling frequency must be at least twice the highest frequency in the signal. Mathematically:

Sampling Frequency (fs) ≥ 2 × Highest Frequency (fmax)

Substituting the values:

fs ≥ 2 × 100 Hz

fs ≥ 200 Hz

Correct Option:

The correct sampling frequency is 200 Hz, as per the Nyquist criterion. This ensures that the signal can be sampled and reconstructed accurately without any loss of information or occurrence of aliasing.

Additional Information

To analyze the given options:

• Option 1 (50 Hz): This sampling frequency is insufficient as it is less than the highest frequency in the signal (100 Hz). Sampling at 50 Hz would lead to aliasing, making it impossible to reconstruct the original signal accurately.
• Option 2 (200 Hz): This is the correct choice as it satisfies the Nyquist criterion by being at least twice the highest frequency in the signal (100 Hz). Sampling at this frequency ensures accurate signal extraction and reconstruction.
• Option 3 (100 Hz): Although 100 Hz matches the highest frequency in the signal, it does not satisfy the Nyquist criterion, which requires the sampling frequency to be at least twice the highest frequency. Sampling at 100 Hz would result in aliasing and loss of information.
• Option 4 (150 Hz): This sampling frequency is greater than the highest frequency (100 Hz) but still does not meet the Nyquist criterion. Sampling at 150 Hz could lead to partial aliasing and inaccuracies in signal reconstruction.

Conclusion:

To avoid aliasing and ensure accurate signal extraction, the sampling frequency must adhere to the Nyquist criterion. For the given signal Vm × sin(100πt) + 2Vm × sin(200πt), the correct sampling frequency is 200 Hz, as it is twice the highest frequency (100 Hz). Sampling at this rate guarantees effective signal processing without loss of information or distortion.