Environmental Engineering Reference
In-Depth Information
When f
(
t
)
is an odd or even function, in particular, we have
+
+
f
(
t
)=
b
( ω )
sin
ω
t d
ω ,
f
(
t
)=
a
( ω )
cos
ω
t d
ω ,
0
0
and
+
+
2
π
2
π
b
( ω )=
f
( τ )
sin
ωτ
d
τ ,
a
( ω )=
f
( τ )
cos
ωτ
d
τ ,
0
0
respectively.
B.1.2 Fourier Transformation
Definition
If the function f
(
t
)
satisfies certain conditions, we may express it by using a Fourier
integral. Let
+
f
e t d t
( ω )=
f
(
t
)
.
(B.4)
By Eq. (B.3) we have
+
1
2
f
e t d
f
(
t
)=
( ω )
ω .
(B.5)
π
Thereforewe may express f
f
. Equations (B.4)
and (B.5) are called the Fourier transformation and the inverse Fourier transforma-
tion , respectively. The f
( ω )(
f
(
t
))
by the integral of f
(
t
)(
( ω ))
( ω )
is called the image function of f
(
t
)
.The f
(
t
)
is called
the inverse image function of f
( ω )
. For convenience in applications, Eqs. (B.4)
and (B.5) are written as
f
F 1 f
( ω ) .
( ω )=
F
[
f
(
t
)] ,
f
(
t
)=
Example 1. Find the Fourier transformation and the integral expression of the ex-
ponentially decaying function
0
,
t
<
0
,
f
(
t
)=
e β t
,
0
t
, β >
0
.
Solution. By Eq. (B.4), we obtain the image function
+
1
β +
ω = β
i
ω
f
e ( β + ) t d t
( ω )=
F
[
f
(
t
)] =
=
2 ,
i
β
2
+ ω
0
 
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