Graphics Programs Reference
In-Depth Information
j
ω
where
z
=
r
, and for most cases,
r
=
1
. It follows that Eq. (13.102) can
be rewritten as
∞
∑
Xe
j
ω
jn
ω
(
)
=
x
()
e
(13.103)
n
=
∞
In the z-domain, the region over which is finite is called the Region of
Convergence (ROC). Appendix 13C has a list of most common Z-transform
pairs. The Z-transform properties are (the proofs are left as an exercise):
X
()
Linearity:
1.
Zax
1
{
()
bx
2
+
()
}
=
aX
1
()
bX
2
+
()
(13.104)
Right-Shifting Property:
2.
k
Zxn k
{
(
)
}
=
z
X
()
(13.105)
Left-Shifting Property:
3.
k
∑
1
z
k
X
()
x
()
z
kn
Zxn k
{
(
+
)
}
=
(13.106)
n
=
0
Time Scaling:
4.
∞
∑
n
Za
n
x
()
Xa
1
1
{
}
=
(
z
)
=
(
a
z
)
x
()
(13.107)
n
=
0
Periodic Sequences:
5.
z
N
z
N
--------------
Zx
()
Zx
()
{
}
=
{
}
(13.108)
1
where
N
is the period.
Multiplication by
n
:
6.
d
X
()
Znx
()
{
}
=
z
(13.109)
d
z
Division by
na
+
; a is a real number:
7.
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