Digital Signal Processing Reference
In-Depth Information
of the function
1
1
−
0
.
3
z
X
(
z
)
=
−
1
,
ROC:
z
>
0
.
3
is required, we determine that the matching entry in Table 13.1 is given by the
transform pair
1
1
− α
z
−
1
,
←→
α
k
u
[
k
]
ROC :
z
>α.
Substituting
α =
0
.
3, the inverse z-transform of
X
(
z
) is given by
x
[
k
]
=
0
.
3
k
u
[
k
]. The scope of the table look-up method is limited to the list of
z-transforms available in Table 13.1.
13.3.2 Inversion formula method
In this method, the inverse z-transform is calculated directly by solving the
complex contour integral specified in the synthesis equation, Eq. (13.5). This
approach involves contour integration, which is beyond the scope of the text.
13.3.3 Partial fraction method
In LTID signals and systems analysis, the z-transform of a function
x
[
k
] gen-
erally takes the following rational form:
X
(
z
)
=
N
(
z
)
D
(
z
)
=
b
m
z
m
+
b
m
−
1
z
m
−
1
++
b
1
z
+
b
0
z
n
+
a
n
−
1
z
n
−
1
++
a
1
z
+
a
0
(13.9a)
or alternatively
X
(
z
)
=
N
′
−
1
++
b
1
z
−
m
+
1
+
b
0
z
−
m
(
z
)
D
′
(
z
)
=
z
m
−
n
b
m
+
b
m
−
1
z
.
(13.9b)
1
+
a
n
−
1
z
−
1
++
a
1
z
−
n
+
1
+
a
0
z
−
n
Note that the numerator
N
(
z
) and denominator
D
(
z
) in Eq. (13.9a) are polyno-
mials of the complex function
z
. In this case, the inverse z-transform of
X
(
z
) can
be calculated using the partial fraction expansion method. The method consists
of the following steps.
Step 1
Calculate the roots of the characteristic equation of the rational function
Eq. (13.9a). The characteristic equation is obtained by equating the denominator
D
(
z
) in Eq. (13.9a) to zero, i.e.
D
(
z
)
=
z
n
+
a
n
−
1
z
n
−
1
++
a
1
z
+
a
0
=
0
.
(13.10)
For an
n
th-order characteristic equation, there will be
n
first-order roots.
Depending on the value of the coefficients
{
b
l
}
,0
≤
l
≤
n
−
1, roots
{
p
r
}
,
1
≤
r
≤
n
, of the characteristic equation may be real-valued and/or complex-
valued. By expressing
D
(
z
) in the factorized form, the z-transform
X
(
z
)is
represented as follows:
X
(
z
)
z
N
(
z
)
≡
p
n
)
.
(13.11)
z
(
z
−
p
1
)(
z
−
p
2
)
(
z
−
p
n
−
1
)(
z
−
Search WWH ::
Custom Search