Geoscience Reference
In-Depth Information
Fig. A3
Illustration of the limit with a triangular
impulse, making
Δ
t
gradually approach zero
to obtain a (Dirac) delta function.
2
Δ
t
Δ
t
/ 2
t
0
t
Equations (A4) and (A5) represent singular behavior, and they indicate that
δ
(
t
−
t
0
)is
not continuous and not differentiable at
t
t
0
. Therefore, this definition cannot be taken
literally, but it must be interpreted as suggestive of the limiting process involved. A better
way to define the delta function is in the following integral form
=
+∞
δ
(
t
−
t
0
)
f
(
t
)
dt
=
f
(
t
0
)
(A7)
−∞
in which
f
(
t
) is a continuous and smooth function. While the Dirac delta function is
not a well-behaved function in the usual sense, it is classified as a
generalized function
.
As explained in Greenberg (1971), one never talks about the values of a generalized
function, but only about its action on the function
f
(
x
), as indicated in Equation (A7).
The unit step function
A function closely related to the Dirac delta function is the Heaviside step function which
can defined as follows
0
for
t
<
t
0
H
(
t
−
t
0
)
=
(A8)
1
for
t
>
t
0
and is illustrated in Figure A4. It is often convenient to think of the unit step function
as the integral of the unit impulse function or, vice versa, of the impulse function as the
derivative of the step function.
Unit response and actual response of a system
The transformation of input into output is called the response of a system. In general,
a transformation or an operation
T
is said to be
linear
when the operation on the sum of
two functions is the sum of the operations of each individual function; in mathematical
terms,
T
is linear if the equality
T
[
ax
(
t
)
+
by
(
t
)]
=
a
T
[
x
(
t
)]
+
b
T
[
y
(
t
)]
(A9)
