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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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 Fall '08
 STAFF
 Frequency

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