Illustrate the discrete-time fourier transform of the

Question

Question 1:

A certain analog signal x_a (t) has the following real Fourier transform:

                1, |Ω|≤8Π
X_a(jΩ) =
                0, otherwise

(a) Illustrate X_a(jΩ)

(b) Carefully illustrate the discrete-time Fourier transform of the discrete-time signal x(n) define by x(n) = x_a(nT) for the case T = 1/16sec and T = 1/4 sec.

(c) Is x_a(t)band limited? If so, find the Nyquist rate. If not, tell why not.

Question 2:

Let x[n], 0 ≤ n ≤ N - 1 be an even-length sequence with an N-point DFT X[k], 0 ≤ k ≤ N - 1. Determine the N-point DFTs of the following length-N sequences in terms of X[k]:

(a) u[n] = x[n] - x[n - N/2]

(b) y[n] = x[n] + x[n - N/2]

(c) y[n] = (-1)ⁿx[n]