Geology Reference
In-Depth Information
The autocorrelation of the output then reduces to
∞
H
(
f
)
H
∗
(
f
)
S
gg
(
f
)
e
i
2π
f
τ
df
φ
oo
(τ)
=
−∞
∞
−∞
|
2
S
gg
(
f
)
e
i
2π
f
τ
df
.
=
H
(
f
)
|
(2.314)
The autocorrelation of the output at zero lag is
T
/2
T
/2
−
T
/2
|
o
(
t
)
1
T
1
T
o
(
t
)
o
∗
(
t
)
dt
=
2
dt
.
φ
oo
(0)
=
lim
T
→∞
lim
T
→∞
|
(2.315)
−
T
/2
It represents the power of the output. From expression (2.314), it is also
∞
−∞
|
2
S
gg
(
f
)
df
.
=
|
φ
oo
(0)
H
(
f
)
(2.316)
2
is the
power transfer function
of the linear system;
S
gg
(
f
)repres-
ents the
power spectrum
or
power spectral density
of the signal g(
t
). It gives the
distribution of power over the frequency axis, and has the units of power per unit
frequency.
The power spectrum and autocorrelation are Fourier transform pairs,
Function
|
H
(
f
)
|
∞
φ
gg
(τ)
e
−
i
2π
f
τ
d
τ,
S
gg
(
f
)
=
(2.317)
−∞
∞
S
gg
(
f
)
e
i
2π
f
τ
df
.
φ
gg
(τ)
=
(2.318)
−∞
This is sometimes referred to as the
Wiener-Khintchine theorem
.
By definition, the autocorrelation of g(
t
)is
T
/2
−
T
/2
g(
t
)g
∗
(
t
φ
gg
(τ)
=
lim
T
−
τ)
dt
.
(2.319)
→∞
If we define
⎩
g(
t
),
|
t
|≤
T
/2,
h
(
t
)
=
(2.320)
0,
|
t
|
>
T
/2
then we can write
∞
1
T
h
(
t
)
h
∗
(
t
φ
gg
(τ)
=
lim
T
→∞
−
τ)
dt
.
(2.321)
−∞
The Fourier integral representation of
h
(
t
)is
∞
H
(
f
)
e
i
2π
ft
df
,
h
(
t
)
=
(2.322)
−∞
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