Question

[20 total pts) Consider the sequences x1n] = { 1, 2,-2, 1}, and x2 [n] = { 1, 2,-2, 1, 0, 0, 0, 0). The sequence x2In] is known as a zero-padded version of x,[n]. When answering the questions below, please use Table 1, provided on Page 3. a. [3 pts] Compute X1(eo), the Discrete-Time Fourier Transform (DTFT) of x1 [n], and evaluate it for the following values of normalized frequency: a-63 笎哮,쯤뀨 write the values in the table b. 13 pts] Compute X [k], the Discrete Fourier Transform (DFT) of x1 [n]. Write the values in the table. c. [3 pts] Determine the normalized frequencies that correspond to the values of k from Part b. d. 13 pts] Compute X2[k], the Discrete Fourier Transform (DFT) of x2[n]. Write the values in the table. e. (3 pts] Determine the normalized frequencies that correspond to the values of k from Part d. f. 13 pts] Compare the results obtained forX(e), X],and X2k]. What similarities do you see? . 12 pts] Comment on what you would expect to happen if we continued adding zeros to x2 In]. 4'2'4 Specifically, what would happen to X2 k] (compared to X, Celo) as the zero-padded length of x2 In] approaches infinity?

Answer 1

つ2ew ICe 3 IT/ 千) = 在[S 2 C) (-1 xi(e

fft(x) 千 4- 千 intervals T /