Digital Signal Processing Reference
In-Depth Information
property listed shows that time delay of
k
units of time can be represented by
multiplication by
z
-
k
in the
z
-domain. And finally, multiplication by
n
in the time
domain can be represented by differentiation in the
z
-domain. For further
discussion of these and other properties of the
z
-transform or for a more
comprehensive list of
z
-transform pairs, please refer to one of the reference texts
listed in Appendix A.
Table 5.3
Common
z
-Transform Pairs
Time Domain Function
Frequency Domain Function
A
δ
⋅
(
n
)
A
A
⋅
z
A
⋅
u
(
n
)
( −
z
1
A
⋅
z
A
⋅
n
⋅
u
(
n
)
2
( −
z
1
A
⋅
z
n
⋅
A
⋅
a
u
(
n
)
(
z
−
a
)
]
[
A
⋅
z
⋅
z
−
cos(
Ω
)
A
⋅
cos(
Ω
⋅
n
)
⋅
u
(
n
)
2
z
−
2
⋅
cos(
Ω
)
⋅
z
+
1
A
⋅
z
⋅
sin(
Ω
)
A
⋅
sin(
Ω
⋅
n
)
⋅
u
(
n
)
2
z
−
2
⋅
cos(
Ω
)
⋅
z
+
1
[
]
A
⋅
z
⋅
z
⋅
cos(
φ
)
−
a
⋅
cos(
φ
−
Ω
)
n
A
⋅
a
⋅
cos(
Ω
⋅
n
+
φ
)
⋅
u
(
n
)
2
2
z
−
2
⋅
a
⋅
cos(
Ω
)
⋅
z
+
a
Table 5.4
Common
z
-Transform Properties
Time Domain Function
Frequency Domain Function
h
(
n
)
H
(
z
)
∞
−∞
x
(
k
)
⋅
x
(
n
−
k
)
X
(
z
)
⋅
X
(
z
)
1
2
1
2
k
=
−
k
⋅
x
−
(
k
)
z
X
(
z
)
dF
(
z
)
n
⋅
x
(
n
)
−
z
⋅
dz
