Environmental Engineering Reference
In-Depth Information
Similarly, we can show that
+
∞
+
∞
1
2
f
1
(
ω
)
f
2
(
ω
)
f
1
(
t
)
f
2
(
t
)
d
t
=
d
ω
.
π
−
∞
−
∞
This establishes the relation between the integral of the product of two inverse im-
age functions and the integral of image functions, and plays an important role in
calculating various forms of energies. For example, in electricity,
R
+
∞
−
∞
+
∞
1
R
i
2
v
2
w
=
(
t
)
d
t
=
(
t
)
d
t
−
∞
is the total energy passing through an electric resistance
R
.Here
i
are the
electric current passing through
R
and the electric voltage acting on
R
, respectively.
The integral of
+
∞
−
∞
(
t
)
and
v
(
t
)
f
2
(
t
)
d
t
is often called the
energy integral
.
f
Corollary 1.
If
(
ω
)=
F
[
f
(
t
)]
, the energy integral is
+
∞
+
∞
+
∞
(
ω
)
1
2
1
2
2
d
f
2
f
(
t
)
d
t
=
ω
=
S
(
ω
)
d
ω
,
π
π
−
∞
−
∞
−
∞
(
ω
)=
(
ω
)
2
is called the
density of the energy spectrum
. It can be shown
f
where
S
that the
S
(
ω
)
is a real-valued function of
ω
and is an even function such that
S
. Thus we may obtain the total energy by integrating the density
of the energy spectrum with respect to the frequency
(
−
ω
)=
S
(
ω
)
ω
.
+
∞
Corollary 2.
Define
R
(
t
)=
f
(
τ
)
f
(
τ
+
t
)
d
τ
. Thus
−
∞
+
∞
+
∞
1
2
e
iω
t
d
e
−
iω
t
d
t
R
(
t
)=
S
(
ω
)
ω
,
S
(
ω
)=
R
(
t
)
,
π
−
∞
−
∞
so that the
R
(
t
)
and the
S
(
ω
)
form a Fourier transformation couple. Here
R
(
t
)
is
called the
self-correlation function
of
f
(
t
)
.
Proof.
Let
f
(
ω
)=
F
[
f
(
t
)]
. By the shifting and multiplying properties, we have
+
∞
+
∞
+
∞
(
ω
)
1
2
1
2
2
e
iω
t
d
f
e
iω
t
d
R
(
t
)=
f
(
τ
)
f
(
τ
+
t
)
d
τ
=
ω
=
S
(
ω
)
ω
.
π
π
−
∞
−
∞
−
∞
Thus, by the Fourier integral,
+
∞
e
−
iω
t
d
t
S
(
ω
)=
R
(
t
)
.
−
∞
Convolution Theorem
. The integral
+
∞
−
∞
Consider two known functions
f
1
(
t
)
and
f
2
(
t
)
f
1
(
τ
)
f
2
(
t
−
τ
)
d
τ
is called the
convolution
of functions
f
1
(
t
)
and
f
2
(
t
)
and is denoted by
f
1
(
t
)
∗
f
2
(
t
)
,
i.e.
+
∞
(
)
∗
(
)=
(
τ
)
(
−
τ
)
τ
.
f
1
t
f
2
t
f
1
f
2
t
d
−
∞
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