Question

Consider an ideal sampling and reconstruction system described as follows: - A signal x(t) is fed as input to an ideal C-to-D converter with sampling period Tₛ . The output signal is x[n]. - The signal x[n] from the C-to-D converter is fed as input to an ideal D-to-C converter with sampling period Tₛ. The output signal is y(t). For the above system, let x(t)=2cos(2π(50)t+π/2)+cos(2π(150)t). (a) If the output of the ideal D-to-C converter is equal to the input x(t), that is, y(t)=2cos(2π(50)t+π/2)+cos(2π(150)t), what general statement can you make about the sampling frequency fₛ in this case?

Answer 1

The sampling frequency, fₛ, in this case, should be greater than 300 Hz based on the Nyquist-Shannon sampling theorem.

Step-by-step explanation:

In this case, if the output of the ideal D-to-C converter is equal to the input x(t), y(t) = x(t), it means that the reconstructed signal is identical to the original signal. This occurs when the sampling frequency, fₛ, satisfies the Nyquist-Shannon sampling theorem:

fₛ > 2fmax

where fmax is the highest frequency component in the original signal x(t).

In the given signal x(t) = 2cos(2π(50)t+π/2)+cos(2π(150)t), the frequency components are 50 Hz and 150 Hz. Therefore, the general statement about the sampling frequency in this case is fₛ > 2(150) = 300 Hz.