DTFT, DFT, ZT, and FFT
465
and
g(n) → H[f(n)]
Then
g n
H
f k n k
K
( )
( ) (
)
ϭ
Ϫ
ϭϪϱ
ϩϱ
∑





 , from R.5.16
g n
K
( )
( ) [ (
)]
ϭ
Ϫ
ϭϪϱ
ϩϱ
f k H n k
∑






and
g n
( )
( ) (
)
ϭ
Ϫ
ϭϪϱ
ϩϱ
f k h n k
K
∑






R.5.18 Note that the output of a linear system is given by
g n
f k h n k
n
( )
( ) (
)
ϭ
Ϫ
ϭϪϱ
ϩϱ
∑
or in short
g(n) = h(n) ⊗ f(n)
Recall that ⊗ denotes convolution.
R.5.19 For example, let f(n) = 2
n and h(n) = (1/3)
n u(n). Then g(n) = h(n) ⊗ f(n) or
g n
f k h n k
h k f n k
k
k
( )
( ) (
)
( ) (
)
ϭ
Ϫ ϭ
Ϫ
ϭϪϱ
ϩϱ
ϭϪϱ
ϩϱ
∑
∑
g n
u k
k
k
n k
( )
( )
(
)
ϭ
ϭϪϱ
ϩϱ
Ϫ
1
3
2

 

 
∑
g n
k
k
n
k
n
k
k
k
n
k
( ) ϭ
ϭ
ϭ
ϭ
ϩϱ
Ϫ
ϭ
ϩϱ
Ϫ
1
3
2 2
2
1
3
2
2
1
3
0
0

 

 

 

 

 

 
∑
∑
k k
k
ϭ
ϩϱ
0
1
2
∑

 

 
g n
n
k
k
n
n
n
( )
( / )
ϭ
ϭ
Ϫ
ϭ
ϩϱ
ϭ
ϩϱ
ϭ
ϩϱ
2
1
6
2
1
1 1 6
0
0
0

 

 

 

 
∑
∑
∑
CRC_47760_CH005.indd 465
CRC_47760_CH005.indd 465
7/23/2008 1:47:17 PM
7/23/2008 1:47:17 PM