Similarly when xn is odd symmetric as cos is an even

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Unformatted text preview: 1. Then: ! !! !! = ! !! When x[N ­n]=x[n], sin !!"# ! − !!"# ! 2!"# ! ! cos +! ! =  −sin( !!"# ! ! !! ! ! sin ! !! ), so x[N ­n]sin 2!"# ! !!"# ! − !!"# ! + x[n]sin( !!"# ! )=0. Therefore all the imaginary part of X[k] will be cancelled. Then X[k] would only have cos elements left, so it is real valued. Similarly when x[n] is odd symmetric, as cos is an even symmetric function, all the cos terms will be cancelled to become zero. And only sin terms would be left, therefore X[k] would be imaginary valued. iii): As ! ! = (−1)! ! ! , then the DFT of the sequence is as below: !−1 ! !! = (− 1) ! ! != 0 So a =1, b=0. !2!"# !− ! !−1 = ! != 0 !"# !! !2!"# !− ! !−1 = !!! != 0 − ! !2!"(!− ) 2 ! ! = ! [! − ] 2 Q5. Answer: i): As the frequency spacing is equal to Fs/N, then: N = 20k/2 = 10000 samples. For 1 Hz, N = 20000 samples. ii): From the lecture notes, we know that: if we want to have no spec...
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