Digital Signal Processing Reference
In-Depth Information
We make the change of variable
t
+
b
a
dt
a
.
t = at - b Q t =
;
dt =
Then, from (7.41),
q
f(t)u(t)e
-s(t+b)/a
dt
a
F
t
(s) =
L
-b
q
e
-sb/a
a
L
e
-sb/a
a
s
a
f(t)e
-(s/a)t
dt =
¢
≤
=
F
,
0
and the time-transformation property is shown by the transform pair
e
-
b
a
a
l
·
f(at - b)u(at - b)
F(s/a).
(7.42)
This property is now illustrated with an example.
Illustration of time shifting and time transformation
EXAMPLE 7.11
Consider the function sin 3
t
. From Table 7.2,
3
f(t) = sin 3t
Î
l
"
= F(s).
s
2
+ 9
We wish to find the Laplace transform of
p
6
p
6
B
¢
≤R
¢
≤
f
t
(t) = sin
3
4t -
u
4t -
.
From (7.40),
a = 4
and
b = p/6.
Then, from (7.42),
e
-sp/24
4
e
-sp/24
4
12e
-sp/24
s
2
s
4
3
(s/4)
2
F
t
(s) =
F
¢
≤
=
+ 9
=
+ 144
.
To check this result, consider
p
6
p
6
p
24
p
24
sin
B
3
¢
4t -
≤R
u
¢
4t -
≤
= sin
B
12
¢
t -
≤R
u
¢
t -
≤
,
since
u(at - b) = u(t - b/a).
From Table 7.2 and the real-shifting property, (7.22),
p
6
p
6
p
24
p
24
B
B
¢
≤R
¢
≤R
B
B
¢
≤R
¢
≤R
l
sin
3
4t -
u
4t -
=
l
sin
12
t -
u
t -
12e
-sp/24
s
2
= e
-sp/24
l
[sin
12t] =
+ 144
,
and the transform is verified.
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