DTFT, DFT, ZT, and FFT 481
ANALYTICAL Solution
Zf n F z unz z
z
n
n
n
n
[()] 33 () ()
0
1
1
1
∑∑
then
fn un
z
z
3 zz
1
() ()
⇔ for |^1 |1 or | | 1
R.5.63 To gain some insight and experience in the evaluation of the ZT, additional trans-
formation 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
fn u n
n
(^4) n
1
10
00
() ( )
ANALYTICAL Solution
Fz u n zn
n
4 ()(^1 )
∑
Fz zn z
n
n
n
4
1
1
()( ) 1
∑∑
Fz
z
z
z
4 z
1
11
() 1
for
R.5.64 Let us evaluate the ZT and ROC of the discrete function f 5 (n) given by
fn un u n
nn
5 ()23 1() ( )
ANALYTICAL Solution
Zf n[ ( )] 5 Z un[ 231 nn( )] Z u n[ ( )]
Zun
z
z
[()]2n z
2
2
for
and
Zun
z
z
[(n )] |||31z
3
3
for |
Then ROC is given by |2| < |z| < |3|.