Spring 2006 Math 308505
7 The Laplace Transform
7.8 Impulses and The Delta Function
Wed, 08/Mar
c 2006, Art Belmonte
Summary
Construction of the delta function
Let
p
≥
0. Recall from §7
.
6 the translated Heaviside function
u
p
(
t
)
=
u
(
t

p
)
=
0
,
t
<
p
;
1
t
≥
p
.
For
p
≥
0, define
δ
p
(
t
)
=
1
(
u
p
(
t
)

u
p
+
(
t
)
)
or
1
,
for
p
≤
t
<
p
+
;
0
for
t
<
p
or
t
≥
p
+
.
The
Dirac delta function
centered at
p
is defined as the limit
δ
p
(
t
)
=
lim
→
0
δ
p
(
t
).
We denote
δ
0
by
δ
. NOTE THAT
δ
p
(
t
)
=
δ(
t

p
)
. The delta
“function” is an example of a
generalized function
or
distribution
.
Colloquially speaking,
δ(
t
)
=
0
,
t
6=
0
,
∞
,
t
=
0
.
Physically, it models a force that concentrates a great energy over
a short duration, such as a hammer hitting a nail or a bat hitting a
baseball. The delta function is known by its properties, which we
now discuss.
Properties of the delta function
In the following
p
is a nonnegative constant,
φ
a function that is
continuous near
p
, and
f
a piecewise continuous function.
Moreover, we define
δ
*
f
=
lim
→
0
(
δ
0
*
f
)
. Then
∞
∞
δ
p
(
t
)φ(
t
)
dt
=
∞
∞
δ(
t

p
)φ(
t
)
dt
=
φ(
p
)
L
δ
p
(
t
)
=
L
{
δ(
t

p
)
}
=
e

ps
L
{
δ
0
(
t
)
} =
L
{
δ(
t
)
}
=
1
d
dt
u
(
t

a
)
=
δ(
t

a
)
f
*
δ
=
δ
*
f
=
f
(The first item is called the
sifting property
. The last item says that
δ
is the
identity
for the convolution product.)
Impulse response functions [revisited]
For constants
a
,
b
,
c
, the
(unit) impulse response function
is the
solution
h
(
t
)
to the system represented by the symbolic initial
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 Spring '08
 comech
 Math, Laplace, Impulse response, Dirac delta function

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