20112ee113_1_Hw7-Solution

20112ee113_1_Hw7-Solution - EE113 Digital Signal Processing...

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Unformatted text preview: EE113: Digital Signal Processing Spring 2011 Prof. Mihaela van der Schaar Homework #7 Solutions Prepared by Yuanzhang Xiao and Khoa T Phan Problems from Prof. Sayed’s lecture notes: 15.1 i) When x ( n ) = δ ( n ), we have X ( k ) = 1 , k = 0 , 1 , ··· ,N- 1 . ii) Since x ( n ) = u ( n )- u ( n- N ) = 1 0 ≤ n ≤ N- 1 0 otherwise , we have X ( k ) = N k = 0 k = 1 , 2 , ··· ,N- 1 iii) x ( n ) = δ ( n- n o ) + δ ( n- N + n o ) , n o < N X ( k ) = N- 1 X k =0 x p ( n ) e- j 2 πkn N = e- j 2 πkno N + e- j 2 πk ( N- no ) N = e- j 2 πkno N + e j 2 πkno N = 2cos 2 πk N n o , ≤ k ≤ N- 1 iv) x ( n ) =- δ ( n- n o ) + δ ( n- N + n o ) , n o < N From (iii), we see that X ( k ) =- e- j 2 πno N k + e j 2 πno N k = 2 j sin 2 πn o N k v) x ( n ) = cos 2 πk o N n , k o < N = 1 2 e j 2 πko N n + 1 2 e- j 2 πko N n X ( k ) = 1 2 N- 1 X n =0 h e j 2 πko N n e- j 2 πk N n + e- j 2 πko N n e- j 2 πk N n i = N 2 [ δ ( k- k o ) + δ ( k- N + k o ))] 1 vi) x ( n ) = sin 2 πnk o N , k o < N Similar to (v), we can show that X ( k ) =- j 2 N- 1 X n =0 h e- j 2 πn ( k- ko ) N- e- j 2 πn ( k + ko ) N i =- j N 2 [ δ ( k- k o )- δ ( k- N + k...
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20112ee113_1_Hw7-Solution - EE113 Digital Signal Processing...

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