sm8_15

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Problem 8.15 For the discrete-time process of Problem 8.14, use the discrete Fourier transform to 
approximate the corresponding spectrum. That is, 
 
∑−
=
=
1
0
)()(
N
n
kn
YY WnRkS 
 
If the sampling in the time domain is at n/Ts where n = 0, 1, 2, …, N-1. What frequency does k correspond 
to? 
 
Solution 
 ( ) 2101221200 andLet σααβσααβ =+= . Then 
 
( ) ( ) ( ) ( )( )[ ]
( )
⎟⎠
⎞⎜⎝
⎛+=
⎟⎟⎠
⎞
⎜⎜⎝
⎛ ++=
++=
++−+=
+−
+−
−
=
∑
N
k
ee
WWW
WnnnkS
N
kj
N
kj
kk
kn
N
n
Y
πββ
ββ
ββ
δδβδβ
ππ
2cos2
11
10
22
10
1
0
0
1
0
10
 
 
The term corresponds to frequency ( )kSY N
kf s where 
S
S T
f 1= . 
 
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