480
Practical MATLAB
® Applications for Engineers
and
r z
1
1
1
Ϫ
Ͻ
or
z
r
Ͼ 1
Then the convergence conditions exist for the values of z that are located outside
the circle of radius r 1 , centered at the origin of the complex plane.
R.5.60 The region given in R.5.59 is the ROC and must be specifi ed as part of ZT, defi ning
in this way a one-to-one-relation between f(n) and F(z).
R.5.61 For example, let us explore the ROC of the noncausal exponential function f 2 (n),
given by
f n
r n
n
n
2
2
1
( )
(
)
ϭ
Ϫ Ϫ
ϭϪϱ
ϩϱ
∑
ANALYTICAL Solution
Z f n
F z
r u n
z
n
n
n
[ ( )]
( )
(
)
2
2
2
1
ϭ
ϭ
Ϫ Ϫ
Ϫ
ϭϪϱ
ϩϱ
∑
F z
r z
r z
n
n
n
n
n
2
2
1
2
1
1
( )
(
)
ϭ
ϭ
Ϫ
ϭϪ
Ϫ
Ϫ
ϭϪϱ
Ϫ
∞
∑
∑
F z
r z
z
r
z
r
n
n
n
n
n
n
2
2
1
1
2
1
2
0
1
( ) ϭ
ϭ
ϭ Ϫϩ
Ϫ
ϭ
ϩϱ
ϭ
ϩϱ
Ϫ
ϩϱ
( )

 

 

 

 
∑
∑
∑
F z
z r
r
r
z
z
z r
2
2
2
2
2
1
1
1
1
( )
(
)
ϭ Ϫ ϩ Ϫ
ϭ Ϫ ϩ Ϫ
ϭ Ϫ
or
r u n
z
z r
n
2
2
1
(
)
Ϫ Ϫ
Ϫ
⇔
with the ROC given by|z|< r 2 .
R.5.62 Let us evaluate the ZT and the ROC of the discrete-step function f 3 (n) = u(n). Recall
that
f n
u n
n
n
3
1
0
0
0
( )
( )
ϭ
ϭ
Ն
Ͻ



CRC_47760_CH005.indd 480
CRC_47760_CH005.indd 480
7/23/2008 1:47:22 PM
7/23/2008 1:47:22 PM