Question

(a) Find the 10 point DFT of the sequence r(n) -1,1,1,1,1,0,0,0,0,0], for n- 0,..,9. X(k)-4,1-j*2.414,0,1-j 0.414,0,1+j 0.414,0,1+j 2.414 help (formulas) Hint: You can use the formula used in example 7.1.2 in page 457 Compute the DFT of the following sequences in terms of X(k). Do not copy the entire formula of X(k) from part (a). (b) ri(n) - [0,0,0,1,1,1, 1, 1,0,0], for n-0,...,9. Xi(k)4,1+j 2.413,0,1+j0.414,0,1- 0.414,1-j2.413 You may use X(k) to denote the DFT of x(n). help (formulas) (c) 2'2 (n) = [1, 0, 0, 0, 0, 0, 1, 1, 1, 11, for n = 0, . . . ,9. X2(k)= 4,-2.414-j,0,0414+),0.j-2.414 You may use X(k) to denote the DFT of x(n). help (formulas).

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Answer 1

Given X(k) = sigma^n - 1_n = 0 x(n) e^- ju 2 x n /N k x(n) = [1 1 1 1 1 0 0 0 0 0] k = 0, 1, 2, 3 N - 1 k = 9 N = 10 X(k) = sigma^9_n = 0 x(n) e^- 9 2 x n/10 k = x(0) + x(1) e^- j pi/5 k + x(2) e^- j 2 pi/5 k + x(3) e^- j 3 pi/5 k + x(4) e^- j 4 pi/5 k + x(5) e^- j 5 pi/5 k + -- x(5) & above value are zero because x(n) is zero. X(k) = 1 + e^- j pi/5 k + e^- j 2 pi/5 k + e^- j 3 pi/5 k + e^- j 4 pi/5 k + 0 + 0 X(0) = 5 X(1) = 1 + e^- j pi/5 + e^- j 2 pi/5 + e^-
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