Environmental Engineering Reference
In-Depth Information
When
A
=
1and
a
=
1, in particular, we have
⎧
⎨
1
, |
t
| <
1
,
+
∞
2
π
sin
ω
cos
ω
t
1
2
, |
ω
=
d
t
|
=
1
,
ω
⎩
0
0
, |
t
| >
1
.
This is called the
Dirichlet discontinuous factor
.
The commonly used inverse image function and image functions can be found
from the Table of Fourier transformations (Appendix C ).
Properties
In discussing the properties of Fourier transformations, all functions are assumed to
satisfy the conditions for the Fourier transformation.
1.
Linearity
If
F
f
1
(
ω
)
,
f
2
(
ω
)
,
[
f
1
(
t
)] =
F
[
f
2
(
t
)] =
for any two constants
α
and
β
,
f
1
f
2
F
[
α
f
1
(
t
)+
β
f
2
(
t
)] =
α
(
ω
)+
β
(
ω
)
or
F
−
1
α
f
2
(
ω
)
=
α
f
1
(
ω
)+
β
f
1
(
t
)+
β
f
2
(
t
)
.
2.
Shifting Property
F
−
1
f
(
ω
∓
ω
0
)
=
e
±
iω
t
0
F
e
±
iω
0
t
F
[
f
(
t
±
t
0
)] =
[
f
(
t
)]
,
f
(
t
)
.
The above two properties follows directly from the properties of integration.
3.
Differential Property
F
f
(
[
t
)] =
i
ω
F
[
f
(
t
)]
.
and
+
∞
−
∞
Proof
.Since
f
(
t
)
is continuous in
(
−
∞
,
+
∞
)
|
f
(
t
)
|
d
t
is convergent,
lim
|
t
|→
+
∞
f
(
t
)=
0. Thus
+
∞
F
f
(
)
=
f
(
e
−
iω
t
d
t
t
t
)
−
∞
e
−
iω
t
+
∞
+
∞
−
∞
+
e
−
iω
t
d
t
=
f
(
t
)
i
ω
f
(
t
)
=
i
ω
F
[
f
(
t
)]
.
−
∞
Similarly, if
f
(
t
)
has
n
-th continuous derivatives in
(
−
∞
,
+
∞
)
and
f
(
j
)
(
)=
(
=
,
,
, ··· ,
−
)
lim
t
0,
j
0
1
2
n
1
,
|
t
|→
+
∞
F
f
(
n
)
(
n
F
t
)
=(
i
ω
)
[
f
(
t
)]
.
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