PS6_soln - Note: X()=4 when =0 Cannot Recover the original...

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Note: X( )=4 when =0
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Cannot Recover the original x[n] in this case
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original x[n] is recovered in this case
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original x[n] is recovered in this case The 8 point DFT provides the best spectral representation overall. However the 4 point DFT would suffice and would be faster to compute
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Problem 2 Simplify the format of , for ] [ k A 1 N ..., , 2 , 1 , 0 = k , p is an integer and . ] 1 N , 0 [ p a) = = + + = = 1 0 1 0 ]} ) ( 2 sin[ ] ) ( 2 {sin[ 2 1 ] 2 sin[ ] 2 cos[ ] [ N n N n N n k p N n k p N pn N kn k A π ππ Note that in = + 1 0 ] ) ( 2 sin[ N n N n k p , these N items are evenly distributed on the rad interval of ] ) ( 2 , 0 [ k p + , so the sum goes to 0 for any combination of p and k in this problem. Apply this to = 1 0 ] ) ( 2 sin[ N n N n k p , we have . 0 ] [ = k A b) )). ( ) ( ( 2 ]} ) ( 2 cos[ ] ) ( 2 {cos[ 2 1 ] 2 cos[ ] 2 cos[ ] [ 1 0 1 0 k p k p N N n k p N n k p N pn N kn k A N n N n + + = + + = = = = δδ
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PS6_soln - Note: X()=4 when =0 Cannot Recover the original...

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