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HW6_2005

# Digital Signal Processing: A Practical Approach (2nd Edition)

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Homework Set #6: Discrete Fourier Transform 1. Compute the 16-point DFTs of (a) x 1 [ n ]=[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; (b) x 2 [ n ]=[0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]; (c) x 3 [ n ]=[1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0]; (d) x 4 [ n ]=[1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0] (d) x 5 [ n ]=[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0] 2. An 8-point sequence is given as x [ n ] = [3 3 -2 5 -4 1 -4 2], its 8-point DFT is expressed by X 8 [ k ]. By 16-point DFT program, we can compute X 16 [ k ] = DFT 16 ( x [ n ]), where we pad 8 extra zeros in the end). Please compute the following results, (a) = 7 0 8 ] [ ) 1 ( k k k X ; (b) DFT 16 (DFT 16 ( x [ n ])); (c) X 8 [4]; (d) X 16 [8]; (e) 11 ) 8 / 2 ( 7 0 8 ] [ = = n kn j k e k X π . 3. Followed by Problem 1, in terms of X 16 ( k ) or X 8 ( k ), please find the DFTs of the following sequences. (a)
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