Consider X[k] be the N-point DFT of an N-point sequence x[n]. (20 %) x [n] =...

 

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Consider X[k] be the N-point DFT of an N-point sequence x[n]. (20 %) x₁ [n] = {1, -2, 1, −3} ↑ x₂ [n] = {0, 2, -1,0,0,4} ↑ (a) Determine the linear convolution x₁ [n] * x₂ [n] (5%) (b) Determine the circular convolution x₁ [n] x₂ [n] (Please note that to perform circular convolution on two sequences with different lengths, the shorter sequence needs to be zero-padded to match the length of the longer sequence.) (5%) (c) Determine the smallest value of N so that N-point circular convolution is equal to the linear convolution. (10%) Consider X[k] be the N-point DFT of an N-point sequence x[n]. (20 %) x₁ [n] = {1, -2, 1, −3} ↑ x₂ [n] = {0, 2, -1,0,0,4} ↑ (a) Determine the linear convolution x₁ [n] * x₂ [n] (5%) (b) Determine the circular convolution x₁ [n] x₂ [n] (Please note that to perform circular convolution on two sequences with different lengths, the shorter sequence needs to be zero-padded to match the length of the longer sequence.) (5%) (c) Determine the smallest value of N so that N-point circular convolution is equal to the linear convolution. (10%)