Digital Signal Processing Reference
In-Depth Information
l
-1
[
#
]
where indicates the inverse Laplace transform. The reason for omitting the
subscript on in the inverse transform is given later. The parameter
c
in the limits
of the integral in (7.2) is defined in Section 7.3. Equation (7.2) is called the
complex
inversion integral
. Equations (7.1) and (7.2) are called the
bilateral Laplace-transform
pair
. The bilateral Laplace transform is discussed more thoroughly in Section 7.8.
We now modify Definition (7.1) to obtain a form of the Laplace transform
that is useful in many applications. First, we express (7.1) as
F(s)
q
0
f(t)e
-st
dt +
L
f(t)e
-st
dt.
l
b
[f(t)] = F
b
(s) =
L
(7.3)
-
q
0
Next, we define
f
(
t
) to be zero for such that the first integral in (7.3) is zero. The
resulting transform, called the
unilateral
, or
single-sided Laplace transform
, is given by
t 6 0,
q
f(t)e
-st
dt,
l
[f(t)] = F(s) =
L
(7.4)
0
l
[
#
]
where denotes the unilateral Laplace transform. This transform is usually
called, simply, the Laplace transform, and we follow this custom. We refer to the
transform of (7.1) as the bilateral Laplace transform. We take the approach of mak-
ing the unilateral transform a special case of the bilateral transform. This approach
is not necessary; we could start with (7.4), with for as a definition.
The equation for the inverse Laplace transform, (7.2), is the same for both the
bilateral and unilateral Laplace transforms, and thus is not subscripted. In ad-
dition, the inverse Laplace transform of the unilateral Laplace transform, (7.4),
gives the function for all time and, in particular, gives the value
[3]. Equations (7.2) and (7.4) form the
Laplace-transform pair
.
The Laplace-transform variable
s
is complex, and we denote its real part as
and its imaginary part as
f(t) = 0
t 6 0,
F(s)
f(t)
f(t) = 0, t 6 0
s
v;
that is,
s = s + jv.
Figure 7.1 shows the complex plane commonly called the
s
-plane.
If is Laplace transformable [if the integral in (7.4) exists], evaluation of
(7.4) yields a function
f(t)
F(s).
Evaluation of the inverse transform with
F(s)
, using the
complex inversion integral, (7.2), then yields
f(t).
We denote this relationship with
f(t)
Î
l
"
F(s).
(7.5)
As we see later, we seldom, if ever, use the complex inversion integral (7.2) to
find the inverse transform, because of the difficulty in evaluating the integral. A
simpler procedure is presented in Section 7.6.
If
f(t)
has a discontinuity at
t = t
a
,
the complex inversion integral gives the av-
erage of the discontinuity; that is,
f(t
-
)
+
f(t
+
)
2
f(t
a
) =
,
(7.6)
f(t
-
)
f(t
+
)
where
is the limiting value of
f(t)
from the left as
t
approaches
t
a
,
and
is
the limiting value from the right. For example, suppose that
f(t)
is the function
