Digital Signal Processing Reference
In-Depth Information
EXAMPLE 7.10
Illustrations of initial- and final-value properties
Examples of the initial-value property and the final-value property are now given. Consider
first the unit step function, which has both an initial value and a final value of unity. Because
from (7.36) we have
l
[u(t)] = 1/s,
s
1
f(0
+
) =
lim
s:
q
sF(s) =
lim
s:
q
s
=
lim
s:
q
1 = 1
and from (7.39),
lim
t:
q
f(t) =
lim
s:0
sF(s) =
lim
s:0
1 = 1.
Consider next the function sin
bt
, where, from Table 7.2,
b
l
[sin
bt] =
+ b
2
.
s
2
The initial value of sin
bt
is zero, and the final value is undefined. From (7.36), the initial
value is
bs
s
2
+ b
2
=
bs
f(0
+
) =
lim
s:
q
sF(s) =
lim
s:
q
lim
s:
q
s
2
= 0,
which is the correct value. Application of the final-value property (7.39) yields
bs
s
2
+ b
2
= 0,
lim
t: q
f(t) =
lim
s:0
sF(s) =
s:0
lim
which is not correct. Recall that the final-value property is applicable only if has a final
value. This example illustrates that care must be exercised in applying the final-value property.
■
f(t)
Time transformations were introduced in Section 2.1. We now consider the effect of
these transformations on the Laplace transform of a function; the result is a com-
bined property of
real shifting
and
time scaling
.
For a function
f(t),
the general independent-variable transformation is given
by
t = (at - b),
yielding
`
f(at - b) = f(t)
t=at -b
= f
t
(t).
(7.40)
Since we are considering the single-sided Laplace transform, we require that
and
a 7 0
b G 0.
As in real shifting, (7.22), we also require that
f(at - b)
be multiplied
by the shifted unit step function
We wish to express
u(at - b).
F
t
(s)
as a function of
F(s) =
l
[f(t)].
From (7.40),
F
t
(s) =
l
[f(at - b)u(at - b)]
q
f(at - b)u(at - b)e
-st
dt.
=
L
(7.41)
0
