Geology Reference
In-Depth Information
the points
f
t
the quotient is dominated by the behaviour of the numerator.
The numerator may be written
sin
π
f
(2
N
+
=
n
/
Δ
Δ
t
=
sin
π(
f
−
n
/
Δ
t
)(2
N
+
1)
1)
1)
Δ
t
+
π
n
(2
N
+
sin
π(
f
t
)
T
cos [π
n
(2
N
=
−
n
/
Δ
+
1)]
cos
π(
f
t
)
T
sin [π
n
(2
N
+
−
n
/
Δ
+
1)]
1)
n
(
2
N
+
1
)
sin
π(
f
t
)
T
=
(
−
−
n
/
Δ
1)
n
sin
π(
f
t
)
T
.
=
(
−
−
n
/
Δ
(2.292)
Near
f
=
n
/
Δ
t
, the denominator becomes
sin
π(
f
n
π
=
sin
π(
f
t
cos(
n
π)
−
n
/
Δ
t
)
Δ
t
+
−
n
/
Δ
t
)
Δ
cos
π(
f
t
sin(
n
π)
+
−
n
/
Δ
t
)
Δ
1)
n
sin
π(
f
t
=
(
−
−
n
/
Δ
t
)
Δ
1
)
n
≈
(
−
π
(
f
−
n
/
Δ
t
)
Δ
t
.
(2.293)
Thus, near
f
=
n
/
Δ
t
, the Fourier transform of the finite Dirac comb sampler is
closely
1
)
n
sin
π
(
f
−
n
/
Δ
t
)
T
(
(
−
1) sinc
π(
f
t
)
T
.
Δ
t
=
(2
N
+
−
n
/
Δ
(2.294)
1)
n
−
π(
f
−
n
/
Δ
t
)
Hence, as illustrated in Figure 2.12, the transform resembles a series of sinc func-
tions of amplitude 2
N
+
1, centred on the frequencies
f
=
n
/
Δ
t
=
2
nf
N
.
To isolate the e
ff
ects of discrete sampling from finite record e
ff
ects, we take
the limit as
N
→∞
,
T
→∞
. From equation (2.279),
T
sinc(π
fT
) contains unit
1) sinc
π(
f
t
)
T
contains area (2
N
area. Hence, (2
N
+
−
n
/
Δ
+
1)/
T
=
1/
Δ
t
.As
N
,
T
t
constant, each of the sinc functions in the Fourier transform of
the finite Dirac comb sampler become distinct and very narrow. In the limit, each
one approaches
→∞
, with
Δ
1
Δ
t
δ(
f
−
n
/
Δ
T
).
(2.295)
The Fourier transform of the infinite Dirac comb,
∞
δ(
t
−
j
Δ
t
),
(2.296)
j
=−∞
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