Digital Signal Processing Reference
In-Depth Information
We now let
f(t)
equal zero for
t 6 0;
that is, we can write
f(t) = f(t)u(t).
Thus, we
can express (7.4) as
q
f(t)e
-st
dt.
l
[f(t)] = F(s) =
L
(7.78)
-
q
As an aside, we note that this is the equation of the bilateral Laplace transform.
The definition of the Fourier transform is
q
f(t)e
-jvt
dt.
[eq(5.4)]
[f(t)] = F(v) =
L
-
q
Evaluating (7.78) for
s = jv
yields
q
`
f(t)e
-jvt
dt.
F(s)
jv
= F(jv) =
L
(7.79)
-
q
The integrals in (5.4) and (7.79) are equal, and we see an inconsistent notation in
these two equations:
F(v)
F(jv).
(7.80)
However, this conflicting notation is standard; the reader should be aware of it.
A second point is that, for
f(t)
as defined earlier (7.78), we can write
`
[f(t)u(t)] =
l
[f(t)u(t)]
,
(7.81)
s = jv
provided
that each transform exists.
The unilateral and the bilateral Laplace transforms are introduced in this chapter.
We take the approach of developing the unilateral transform as a special case of the
bilateral transform. The unilateral transform is used in the analysis and design of
linear time-invariant (LTI) continuous-time systems that are causal. This transform
is especially useful in understanding the characteristics and in the design of these
systems.
The unilateral transform is emphasized in this chapter. A table of transforms
and a table of properties are developed for the unilateral transform. System analy-
sis using the unilateral transform is then demonstrated.
The bilateral Laplace transform is useful in the steady-state analysis of LTI
continuous-time systems, and in the analysis and design of noncausal systems. A pro-
cedure is developed for finding bilateral transforms from a unilateral transform
table. Then some properties of the bilateral transform are derived. See Table 7.4.
