Image Processing Reference
In-Depth Information
and
z(z
0
:
7)
z(z
0
:
7)
x(
1
)
¼
lim
z!
1
(z
1)X(z)
¼
lim
z!
1
(z
1)
25)
¼
lim
z!
25
¼
0
:
4
(z
1)(z
0
:
z
0
:
1
3.5.2 I
NVERSE Z
-T
RANSFORM
An important application of the z-transform is in the analysis of linear discrete-time
systems. This analysis involves computing the response of the systems to a given
input using z-transform. Once the z-transform of the output signal is determined,
inverse z-transform is used to
find the corresponding time-domain sequence. There
are different techniques to
find the inverse z-transform from a given algebraic
expression. In this section, we consider techniques such as inversion integral,
power series expansion, and partial fraction.
(a)
Inversion Integral
: The inverse z-transform of X
(
z
)
is given by the integral
þ
1
j2
X
(
z
)
z
n
1
x
(
n
) ¼
d
z
(
3
:
71
)
p
C
where C is any closed contour in the ROC of X
(
z
)
excluding the origin, as
shown in Figure 3.5. If X
(
z
)
is a rational function of its argument z, then
using the residue theorem, we have
X
Residues of
X
(
z
)
z
n
1
x
(
n
) ¼
at poles of
X
(
z
) inside
C
(
3
:
72
)
Im(
z
)
Contour
C
a
Re(
z
)
b
FIGURE 3.5
Closed contour C.
Search WWH ::
Custom Search