Digital Signal Processing Reference
In-Depth Information
the unit impulse function
d(t)
is
q
f(t)d(t - t
0
) dt = f(t
0
),
(7.17)
L
-
q
with continuous at From (2.40), a nonrigorous, but very useful, defini-
tion of the unit impulse function is
f(t)
t = t
0
.
q
d(t - t
0
)dt = 1
with d(t - t
0
) = 0,
t Z t
0
.
(7.18)
L
-
q
From (7.17), for
t
0
G 0
(see Ref. 3), the Laplace transform of the unit impulse
function is given by
q
d(t - t
0
)e
-st
dt = e
-st
`
t = t
0
= e
-t
0
s
.
l
[d(t - t
0
)] =
L
0
Hence, we have the Laplace transform pair
d(t - t
0
)
Î
l
"
e
-t
0
s
.
(7.19)
For the unit impulse function occurring at
t = 0 (t
0
= 0),
d(t)
Î
l
"
1.
Next, we derive some other transform pairs. Recall the pair
1
s + a
.
e
-at
Î
l
"
[eq(7.14)]
We now use this transform to find the transforms of certain sinusoidal functions. By
Euler's relation,
e
jbt
+ e
-jbt
2
cos
bt =
.
Hence,
1
2
[
l
[e
jbt
] +
l
[e
-jbt
]]
l
[cos
bt] =
by the linearity property, (7.10). Then, from (7.14),
s + jb + s - jb
2(s - jb)(s + jb)
=
1
2
1
s - jb
+
1
s + jb
s
B
R
l
[cos
bt] =
=
+ b
2
.
s
2
sin bt = (e
jbt
- e
-jbt
)/2j,
By the same procedure, because
1
2j
[
l
[e
jbt
] -
l
[e
-jbt
]] =
1
2j
1
s - jb
-
1
s + jb
l
[sin bt] =
B
R
s + jb - s + jb
2j(s - jb)(s + jb)
=
b
=
+ b
2
.
s
2
