Geology Reference
In-Depth Information
−
T
/2
≤
t
≤
T
/2. The function of time we have access to is not g(
t
)but
h
(
t
)
=
b
(
t
)g(
t
), where
b
(
t
) is the boxcar function (2.253)
⎩
0,
|
t
|
>
T
/2,
b
(
t
)
=
(2.275)
1,
|
t
|≤
T
/2.
Thus, in attempting to calculate the Fourier transform
G
(
f
) from the finite record,
we find instead the Fourier transform of the product,
h
(
t
)
=
b
(
t
)g(
t
), of two func-
tions of time. By our result (2.274), the Fourier transform, found from the finite
record, is the convolution of the true transform with the sinc function of frequency,
representing the Fourier transform of the boxcar (2.256). It is
∞
T
sinc(π
f
T
)
G
(
f
−
f
)
df
H
(
f
)
=
−∞
∞
=
T
sinc[π(
f
−
λ)
T
]
G
(λ)
d
λ,
(2.276)
−∞
on changing the integration variable from
f
to λ
=
f
−
f
. The Fourier transform
found from the finite record for frequency
f
is the average of the true transform,
taken with the sinc averager centred on
f
. The whole of
G
(λ) contributes to each
value of
H
(
f
). The only case in which the true transform is returned is that in which
G
(λ) is everywhere unity. In that case, the limit of the sinc integral gives
∞
sin
x
x
dx
=
π
2
,
(2.277)
0
or
π
T
∞
0
=
π
2
,
sinc(π
fT
)
df
(2.278)
and
∞
T
sinc(π
fT
)
df
=
1.
(2.279)
−∞
Otherwise, the finite length of the actual record causes an undesirable mixture of
frequency content.
The question then arises as to how to mitigate the frequency mixing e
ects of
finite record length. We can obtain the finite record by multiplying g(
t
)bya
win-
dow
other than the boxcar. Suppose we use the triangular function,
ff
Δ
(
t
), called
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