Question

Sampling: At what minimum sampling frequency f s should you sample the following signals to make sure they can be reconstructed exactly according to the sampling theorem?

x(t)= π² t² /sin(40πt)sin(60πt) 1

Answer 1

To sample the signal x(t) = π² t² / sin(40πt)sin(60πt) properly, the signal should be sampled at a minimum of twice the highest frequency component, which is 30 Hz. Therefore, the minimum sampling frequency fs should be at least 60 Hz according to the Sampling Theorem.

Step-by-step explanation:

To determine the minimum sampling frequency fs for the signal x(t) = π² t² / sin(40πt)sin(60πt), we must use the Sampling Theorem, which states that a continuous signal can be properly sampled and perfectly reconstructed if it is sampled at a frequency that is at least twice the maximum frequency component of the signal, known as the Nyquist rate. This maximum frequency is also known as the highest frequency component of the signal.

For the given signal, the two sinusoidal components sin(40πt) and sin(60πt) have angular frequencies of 40π rad/s and 60π rad/s respectively. To convert these angular frequencies to regular frequencies, we divide by 2π. Therefore, we have frequencies of 20 Hz and 30 Hz respectively.

The highest frequency component in the signal is 30 Hz, so to satisfy the sampling theorem, the signal should be sampled at a minimum frequency that is twice this value, which gives us a sampling frequency fs ≥ 60 Hz.