Geoscience Reference
In-Depth Information
3.
Multiply the shifted function
f
2
(
t − τ
) by
f
1
(
τ
)
.
4.
The area under the product of
f
1
(
τ
) and
f
2
(
t−τ
) is the value of convolution at
t.
The convolution in the time domain is multiplication in the frequency domain and
vice versa. Convolution operation has commutative, associative and distributive
properties, like most linear systems. We encounter convolution operations in
filtering processes, truncating lengthy data using window functions and sampling a
signal with a comb function (digitization).
b.
The Multiplication Theorem - If f
1
(t)↔F
1
(ω) and f
2
(t)↔F
2
(ω) are Fourier pairs, then
(
)
(
)
=
∗
(
)
()
Where
∗
()
is the complex conjugate of F
1
(ω). The product
∗
()
()
is called the
cross power spectrum. If we put
f
1
(t) = f
2
(t) = f(t)
and
F
1
(ω) = F
2
(ω) = F(ω),
then the previous equation reduces to
[
()
]
=
[
()
]
This is called Parseval's theorem and the real quantity
[()]
represents the power
spectrum (or energy spectrum) of the function f(t). Thus, it is interesting to note that if
the amplitude spectrum F(ω) of a given is known, it is possible to compute its power
spectrum
[()]
and its total energy,
=
[()]
.
c.
The Correlation Theorem - The Fourier transform of the cross correlation function
ϕ
12
(τ) is the cross power spectrum, E
12
(ω) and that of the auto-correlation function,
ϕ
11
(τ) is the power spectrum, E
11
(ω). Thus
ϕ
12
(τ) =
()
(+)=
(−)
()
and
ϕ
11
(τ) =
()(+)
and so the correlation theorem says that
ϕ
12
(τ)↔E
12
(ω) or
ϕ
12
(τ)↔F
1
(ω).
∗
()
ϕ
11
(τ)↔E
11
(ω) or
ϕ
11
(τ)↔
[
()
]
The cross correlation function behaves in accordance with the degree of similarity between
the two correlated functions. It grows larger when the two functions are similar and
diminishes otherwise. This function becomes zero in the case of completely random data.
We see that the time domain cross correlation and auto-correlation functions are reduced to
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