To solve this problem, we need to consider the concept of sampling frequency and sampling theorem (Nyquist-Shannon theorem). The Nyquist-Shannon sampling theorem tells us that in order to perfectly reconstruct a continuous-time signal from its samples, the sampling frequency must be at least twice the highest frequency component in the signal.

Given Signal:

The continuous-time signal is: $x(t) = \sin(4 \pi t)$ This is a sinusoidal signal, and the argument $4 \pi t$ suggests the signal has a frequency of $f = 2$ Hz. (Since $4 \pi t = 2 \pi f t$, we find $f = \frac{4 \pi}{2 \pi} = 2 \, \text{Hz}$).

Nyquist Criterion:

To reconstruct $x(t)$ from its samples without aliasing, the sampling frequency $f_s$ must be at least twice the signal frequency. So: $f_s \geq 2 \times 2 \, \text{Hz} = 4 \, \text{Hz}$

This is the minimum sampling rate required to recover the signal without distortion (aliasing).

Possible Sampling Frequencies:

• Sampling frequency of 4 Hz: This is the minimum required to satisfy the Nyquist criterion. It will sample the signal at exactly twice its frequency.
• Sampling frequencies greater than 4 Hz: Any sampling frequency greater than 4 Hz will also work and allow for the reconstruction of the signal. Some possible sampling frequencies are:
  • 5 Hz
  • 6 Hz
  • 8 Hz
  • 10 Hz
  • 12 Hz
  • 20 Hz, etc.

Sampling Frequencies That Will Recover $x(t)$:

To recover the signal $x(t) = \sin(4\pi t)$ perfectly, any sampling rate $f_s \geq 4 \, \text{Hz}$ will work. Specifically, you can use any of the following:

• 4 Hz
• 5 Hz
• 6 Hz
• 8 Hz
• 10 Hz

The key condition is that the sampling frequency should not be lower than 4 Hz, as lower frequencies would cause aliasing, which means we could no longer recover the original continuous-time signal accurately.

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Would you like more details on how aliasing affects signal recovery, or examples of signal reconstruction techniques? Here are some related questions for further exploration:

1. How does aliasing affect the reconstruction of a continuous-time signal?
2. How do you apply the Nyquist-Shannon sampling theorem in practice for different types of signals?
3. What happens if the sampling rate is less than twice the signal frequency?
4. How would we reconstruct the signal from its samples using a reconstruction filter?
5. How can we calculate the period of a discrete-time signal after sampling?

Tip: When designing systems to sample continuous-time signals, always ensure that the sampling rate is at least twice the maximum frequency component of the signal to avoid aliasing.