Digital Signal Processing Reference
In-Depth Information
(ii) Applying the initial-value theorem, Eq. (6.27), to
X
2
(
s
), we obtain
s
(
s
+
5)
s
3
+
5
s
2
+
17
s
+
13
+
x
2
(0
)
=
t
→
0
+
x
2
(
t
)
=
lim
lim
s
→∞
sX
2
(
s
)
=
lim
s
→∞
2
6
s
=
lim
s
→∞
=
0
.
Applying the final-value theorem, Eq. (6.28), to
X
2
(
s
) yields
s
(
s
+
5)
s
3
+
5
s
2
+
17
s
+
13
x
2
(
∞
)
=
lim
t
→∞
x
2
(
t
)
=
lim
s
→
0
sX
2
(
s
)
=
lim
=
0
.
s
→
0
The initial and final values of
x
(
t
) can be verified from the following inverse
Laplace transform of
X
1
(
s
) derived in Example 6.7(ii):
x
1
(
t
)
=
(0
.
4e
−
t
−
0
.
4e
−
2
t
cos(3
t
)
+
0
.
2e
−
2
t
sin(3
t
))
u
(
t
)
.
(iii) Applying the initial-value theorem, Eq. (6.27), to
X
3
(
s
), we obtain
5
s
s
2
+
25
5
2
s
x
3
(0
+
)
=
lim
t
→
0
+
x
3
(
t
)
=
lim
s
→∞
sX
3
(
s
)
=
lim
=
lim
s
→∞
=
0
.
s
→∞
Applying the final-value theorem, Eq. (6.28), to
X
3
(
s
) yields
5
s
s
2
+
25
x
3
(
∞
)
=
lim
t
→∞
x
3
(
t
)
=
lim
s
→
0
sX
3
(
s
)
=
lim
=
0
.
s
→
0
To confirm the initial and final values obtained in (iii), we determine these values
directly from the inverse transform of
X
3
(
s
)
=
5
/
(
s
2
+
25). From Table 6.1, the
inverse Laplace transform of
X
3
(
s
) is given by
x
3
(
t
)
=
sin(5
t
)
u
(
t
). Substituting
t
=
0
+
, the initial value
x
3
(0
+
)
=
0, which verifies the value determined from
the initial-value theorem. Applying the limit
t
→∞
to
x
3
(
t
), the final value
of
x
3
(
t
) cannot be determined due to the oscillatory behavior of the sinusoidal
wave. As a result, the final-value theorem provides an erroneous answer. The
discrepancy between the result obtained from the final-value theorem and the
actual value
x
3
(
∞
) occurs because the point
s
=
0 is not included in the ROC
of
sX
3
(
s
)
R
3
:Re
{
s
>
0. As such, the expression for the Laplace transform
sX
3
(
s
) is not valid for
s
=
0. In such cases, the final-value theorem cannot be
used to determine the value of the function as
t
→∞
. Similarly, the point
s
=∞
must be present within the ROC of
sX
3
(
s
) to apply the initial-value
theorem.
6.5 Solution of diff
erential equations
An important application of the Laplace transform is to solve linear, constant-
coefficient differential equations. In Section 3.1, we used a time-domain
approach to obtain the zero-input, zero-state, and overall solution of differ-
ential equations. In this section, we discuss an alternative approach based on
the Laplace transform. We illustrate the steps involved in the Laplace-transform-
based approach through Examples 6.16 and 6.17.
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