DTFT, DFT, ZT, and FFT 465
and
g(n) → H[f(n)]
Then
gn H f k n k
K
()()( )
∑
, from R.5.16
gn
K
()() [( )]
∑ fkH n k
and
gn()() ( )
fk hn k
K
∑
R.5.18 Note that the output of a linear system is given by
gn f khn k
n
()()( )
∑
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)nu(n). Then g(n) = h(n) ⊗ f(n) or
gn f khn k hk f n k
kk
()()( ) () ( )
∑∑
gn uk
k
k
() ()()nk
(^1)
3
2
∑
gn
k
k
nk n
k
k
kn
k
()
(^1)
3
22 2
1
3
22
1
003
∑∑kk
k
0
1
∑ 2
gn n
k
k
n
n
n
()
(/)
2
1
6
2
1
000 116
∑∑∑