Consider the representation of the process of sampling followed by reconstruction shown...

Consider the representation of the process of sampling followed by reconstruction shown in Figure P4.28.

Assume that the input signal is

The frequency response of the reconstruction filter is

(a) Determine the continuous-time Fourier transform X_c(jΩ) and plot it as a function of Ω.

(b) Assume that fs = 1/T = 500 samples/sec and plot the Fourier transform Xs (jΩ) as a function of Ω for −2π/T ≤ Ω ≤ 2π/T. What is the output x_r (t) in this case? (You should be able to give an exact equation for x_r(t).)

(c) Now, assume that fs = 1/T = 250 samples/sec. Repeat part (b) for this condition.

(d) Is it possible to choose the sampling rate so that

x _r (t) = A + 2 cos(100πt − π/4)

where A is a constant? If so, what is the sampling rate f_s = 1/T, and what is the numerical value of A?