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Exam 3 Solutions 13

Exam 3 Solutions 13 - 1(20 points The following values from...

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1. (20 points) The following values from the 8-point DFT of a length-8 real sequence x [ n ] are known: X [0] = 3 , X [2] = 0 . 5 - 4 . 5 j, X [4] = 5 , X [5] = 3 . 5 + 3 . 5 j, X [7] = - 2 . 5 - 7 j. a) (5 points) Find the missing values X [1], X [3], X [6]. b) (5 points) Evaluate x [0]. c) (10 points) Find the 4-point DFT of the length-4 sequence w [ n ] given by: w [ n ] = x [ n ] + x [ n + 4] n = 0 , 1 , 2 , 3 . Hint: Derive a general formula that relates W [ k ] to X [ k ] so you don’t have to calculate x [ n ] and w [ n ]. (Sam) a). Because x [ n ] is real, X [1] = X * [7] = - 2 . 5 + 7 j (1) X [3] = X * [5] = 3 . 5 - 3 . 5 j (2) X [6] = X * [2] = 0 . 5 + 4 . 5 j (3) b). x [0] = 1 8 7 X k =0 X [ k ] (4) = 1 8 (3 + 2( - 2 . 5) + 2(0 . 5) + 2(3 . 5) + 5) (5) = 11 8 (6) c). W [ k ] = 3 X n =0 ( x [ n ] + x [ n + 4]) e - 2 π 4 kn (7) = 7 X n =0 x [ n ] e - 2 π 4 kn (8) = X [2 k ] (9) Thus W [0] = X [0] = 3 (10) W [1] = X [2] = 0 . 5 - 4 . 5 j (11) W [2] = X [4] = 5 (12) W [3] = X [6] = 0 . 5 + 4 . 5 j (13) 1
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5. (20 points) When the input to an LTI is: x [ n ] = 1 2 n u [ n ] + 2 n u [ - n - 1] the output is: y [ n ] = 6 1 2 n u [ n ] - 6 3 4 n u [ n ] . a) (10 points) Find the transfer function of H ( z ) and indicate the region of convergence. b) (5 points) Is the system causal? Is it stable?
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