DTFT, DFT, ZT, and FFT
481
ANALYTICAL Solution
Z f n
F z
u n z
z
z
n
n
n
n
[ ( )]
( )
( )
3
3
0
1
1
1
ϭ
ϭ
ϭ
ϭ Ϫ
Ϫ
ϭϪϱ
ϩϱ
Ϫ
ϭ
ϩϱ
Ϫ
∑
∑
then
f n
u n
z
z
z
z
3
1
( )
( )
ϭ
Ϫ
Ϫ
⇔
for | | 1 or | | 1
1
Ͻ
Ͼ
R.5.63 To gain some insight and experience in the evaluation of the ZT, additional transformation examples of various functions and their ROC are presented as follows.
For example, evaluate ZT and its ROC of the discrete function f 4 (n) given by
f n
u n
n
n
4
1
1
0
0
0
( )
(
)
ϭ Ϫ Ϫ Ϫ ϭ
Ϫ
Ͻ
Ն



ANALYTICAL Solution
F z
u n
z
n
n
4
1
( )
(
)
ϭ
Ϫ Ϫ Ϫ
Ϫ
ϭϪϱ
ϩϱ
∑
F z
z
z
n
n
n
n
4
1
1
1
( )
( )
ϭ
Ϫ
ϭϪ
Ϫ
ϭϪϱ
Ϫ
ϭ
ϩϱ
∑
∑
F z
z
z
z
z
4
1
1
1
1
( ) ϭ Ϫ Ϫ
ϭ Ϫ
Խ Խ
for Ͻ
R.5.64 Let us evaluate the ZT and ROC of the discrete function f 5 (n) given by
f n
u n
u n
n
n
5
2
3
1
( )
( )
(
)
ϭ
Ϫ
Ϫ Ϫ
ANALYTICAL Solution
Z f n
Z u n
Z u n
n
n
[ ( )]
[
( )]
[
(
)]
5
2
3
1
ϭ
Ϫ
Ϫ Ϫ
Z u n
z
z
z
n
[
( )]
2
2
2
ϭ Ϫ
for Ͼ
and
Z u n
z
z
z
n
[
(
)]
| | |
3
1
3
3
Ϫ Ϫ ϭ Ϫ
for
Ͻ |
Then ROC is given by |2| < |z| < |3|.
CRC_47760_CH005.indd 481
CRC_47760_CH005.indd 481
7/23/2008 1:47:23 PM
7/23/2008 1:47:23 PM