hwork4 - e jn x 2 [ n ]? (c) What is 3 n =0 | x 1 [ n ] | 2...

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ECE 410 H.W. # 4 HCMUT 2010 1. Calculate by hand the DFT of the following sequences: (a) Length-3 DFT of (2 , 1 , 1) (b) Length-4 DFT of (2 , 1 , 1 , 0) (c) Length-3 DFT of (2 ,e j 2 π/ 3 ,e j 4 π/ 3 ) 2. Compute the length-4 circular convolution of x [ n ] = (2 , 0 , 1 , 0) and h [ n ] = (1 , 1 , 0 , 0) by hand using the “fast convolution” method: (a) Compute X [ k ], the length-4 DFT of x [ n ] (b) Compute H [ k ], the length-4 DFT of h [ n ] (c) Compute Y [ k ] = X [ k ] H [ k ] (d) Compute y [ n ] via the inverse DFT of Y [ k ] (e) Compute y [ n ] directly by circular convolution and con±rm that you get the same answer both ways 3. The length-4 sequences x 1 [ n ] and x 2 [ n ] have the DFTs X 1 = { 1 , 2 j, 1 , - 1 } and X 2 = { 1 , 2 , 1 , 1 } , respectively. (a) What is the DFT of 2 x 1 [ n ] + x 2 [ n ]? (b) What is the DFT of
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Unformatted text preview: e jn x 2 [ n ]? (c) What is 3 n =0 | x 1 [ n ] | 2 ? (d) What is x 1 [0]? (e) What is x 2 [1]? 4. A discrete-time signal x [ n ] is nonzero only for 0 n < N , and has the DTFT X d ( ) = | | 2 + 3 j for- < . Determine the values of the length-400 DFT X [ k ] for the following k : (a) X [0] =? (b) X [50] =? (c) X [100] =? (d) X [193] =? (e) X [200] =? (f) X [300] =? (g) X [400] =? (h) X [500] =? 5. Let X [ k ] denote the DFT of the length-N discrete-time signal x [ n ], 0 n < N . Prove that X [0] = 0 if x [ n ] =-x [ N-1-n ]....
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