DTFT, DFT, ZT, and FFT
471
f. Convolution in time
f 1 (n) ⊗ f 2 (n) ↔ F 1 (e
jW ) . F 2 (e
jW )
g. Convolution in frequency
f n f n
F e
F e
jW
jW
1
2
1
2
1
2
( ) ( )
{ (
)
(
)}
F e F e
d
j
j
↔
⋅
⊗
ϭ
Ϫ
Ϫ
1
W
2
( ) (
)
1
2
(
)
n n
n
ϩ
∫
h. Parseval’s theorem is given by
"
"
"
"
f n
F e
dW
n
jW
( )
( )
2
2
1
2
ϭϪϱ
ϩϱ
Ϫ
ϩ
ϭ
∑
∫
i. The initial time value is given by
f
Fe d W
jW
( )
( )
0
1
2
ϭ
Ϫ
ϩ
∫
j. The initial frequency value is given by
F e
f n
j
n
( )
( )
0
ϭ
ϭϪϱ
ϩϱ
∑
k. Subtraction of shifted sequences
f(n) − f(n − 1) ↔ [1 − e
jW ] . F(e
jW )
R.5.39 A partial list of the most frequently used transform pairs are summarized in
Table 5.1.
TABLE 5.1
DTFT Pairs
Time
Frequency
f(n)
F(e
jW )
δ(n)
1
δ(n − n 0 )
e
−jn 0 W
e
−jw 0 n
2πδ(W − w 0 )
1
2πδ(W)
cos(w 0 n)
π[δ(W + w 0 ) + δ(W − w 0 )]
sin(w 0 n)
jπ[δ(W + w 0 ) − δ(W − w 0 )]
u(n)
[1/1 − e
−jw ] + πδ(W)
a
n
u(n)
1/(1 − ae
−jW ), (|a|< 1)
(a + 1)a
n
u(n)
1/(1 − e
−jw )
2 , (|a|< 1)
(n + k − 1)!
___________
n!(k − 1)!
a
n u(n)
1
__________
(1 − ae
−jW )
k
, (|a| < 1)
CRC_47760_CH005.indd 471
CRC_47760_CH005.indd 471
7/23/2008 1:47:19 PM
7/23/2008 1:47:19 PM