Geology Reference
In-Depth Information
Thus,
n
n
1
e
i
2π
−
j
k
Δ
t
g
=
n
g
j
T
+
2
n
1
j
=−
k
=−
n
j
=−
n
g
j
n
n
1
(
−
j
)
T
t
k
e
i
2π
=
2
n
+
1
k
=−
n
j
=−
n
g
j
(
2
n
n
1
1
)
δ
j
=
g
,
=
+
(2.146)
2
n
+
1
with the use of the orthogonality relation (2.139). The representation is therefore
exact, and the DFT pair
n
G
k
e
i
2π
f
k
t
j
g
j
=Δ
f
(2.147)
k
=−
n
and
j
=−
n
g
j
e
−
i
2π
f
k
t
j
n
G
k
=Δ
t
,
(2.148)
with
f
k
=
k
/
T
, provide a one-to-one mapping between the time and frequency
domains.
The DFT pair is easily generalised to an arbitrary number of points. For a record
of length
T
t
, sampled at
N
equispaced points, the frequency domain repres-
entation is given at
N
points at intervals of
=
N
Δ
Δ
f
=
1/
T
along the positive frequency
axis, and
N
−
1
k
e
i
2
N
kj
g
j
=Δ
f
0
G
,
j
=
0,1,...,
N
−
1,
(2.149)
k
=
N
−
1
g
j
e
−
i
2
N
kj
G
=Δ
t
,
k
=
0,1,...,
N
−
1.
(2.150)
k
j
=
0
Both of these expressions are periodic in the indices
j
and
k
, with period
N
,asare
(2.147) and (2.148), with period 2
n
1. This suggests that we wrap both the time
and frequency records into cylinders, as illustrated in Figure 2.4.
For an odd number of points
N
+
=
+
1, the generalised DFT (2.150) may be
related to the DFT expressed by (2.148). The time sequences (2.147) and (2.149)
are related by
2
n
g
j
=
g
j
+
n
,
j
=−
n
,
−
n
+
1,...,
n
.
(2.151)
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