Digital Signal Processing Reference
In-Depth Information
1,
n
F
10
b
u[-n + 10] =
n 7 10
.
0,
A second example is given in Figure 11.14(b).
3.
A function
f
[
n
] is
two sided
if it is neither right sided nor left sided. For ex-
ample, cos(
n
) is two sided.
4.
A function is of
finite duration
if it is both right sided and left sided. For
example,
(u[n] - u[n - 10])
is of finite duration. A second example is given in
Figure 11.14(c).
We find these definitions useful when working with bilateral transforms.
In Chapter 7, a procedure was given for finding bilateral Laplace transforms from
unilateral Laplace-transform tables. An equivalent procedure can be developed for
finding bilateral
z
-transforms from unilateral
z
-transform tables. However, this pro-
cedure is complex and prone to error. Instead, a table of bilateral
z
-transforms is
given as Table 11.5. A procedure is now given for using this table.
TABLE 11.5
Bilateral
z
-Transform
f
[
n
]
F(z)
ROC
1.
d[n]
1
All
z
z
-n
0
2.
d[n - n
0
]
z Z 0, n
0
G 0
z Z
q
,
n
0
6 0
z
z - 1
3.
u
[
n
]
ƒ z ƒ 7 1
z
(z - 1)
2
4.
nu
[
n
]
ƒ z ƒ 7 1
z
z - a
a
n
u[n]
5.
ƒ z ƒ 7 ƒ a ƒ
az
(z - a)
2
na
n
u[n]
6.
ƒ z ƒ 7 ƒ a ƒ
az sin b
a
n
sin(bn)u[n]
7.
ƒ z ƒ 7 ƒ a ƒ
z
2
- 2az cos b + a
2
z(z - a cos b)
a
n
cos(bn)u[n]
8.
ƒ z ƒ 7 ƒ a ƒ
z
2
- 2az cos b + a
2
z
z - 1
9.
-u[-n - 1]
ƒ z ƒ 6 1
z
z - a
-a
n
u[-n - 1]
10.
ƒ z ƒ 6 ƒ a ƒ
az
(z - a)
2
-na
n
u[-n - 1]
11.
ƒ z ƒ 6 ƒ a ƒ
z
z - a
-
z
z - 1/a
a
ƒnƒ
, ƒ a ƒ 6 1
12.
ƒ a ƒ 6 z 6 ƒ 1/a ƒ
