Geology Reference
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arising on the right side of (2.135), with
t
j
=
j
Δ
t
. We recognise it as the sum of a
finite geometric progression with first term
(
k
−
)
T
e
−
i
2π
n
Δ
t
a
=
(2.136)
and common ratio
(
k
−
)
T
Δ
t
e
i
2π
r
=
.
(2.137)
Thus, for
T
=
(
2
n
+
1
)
Δ
t
,
a
1
r
2
n
+
1
n
−
(
k
−
)
T
e
i
2π
t
j
=
1
−
r
j
=−
n
t
1
t
(
k
−
)
T
(
k
−
)
T
e
−
i
2π
n
Δ
−
e
i
2π
(2
n
+
1)
Δ
=
(
k
−
)
T
Δ
t
e
i
2π
1
−
e
−
i
2π
(
k
−
T
n
Δ
t
1
e
i
2π(
k
−
)
−
=
.
(2.138)
(
k
−
)
T
Δ
t
e
i
2π
1
−
If
k
, the sum is zero. If
k
=
,byl'Hopital's rule, the sum is 2
n
+
1. The sinusoids
are orthogonal under addition over the sample points, giving
n
e
i
2π
(
k
−
)
t
j
k
=
(2
n
+
1)δ
.
(2.139)
T
j
=−
n
From (2.135), we then find that
n
n
1
T
2
n
+
1
h
j
e
−
i
2π
T
j
Δ
t
k
=
H
k
(2
n
+
1)δ
=
H
.
(2.140)
T
j
=−
n
k
=−
n
Thus, the DFT of the sequence
h
j
is
n
h
j
e
−
i
2π
T
t
j
H
=Δ
t
,
(2.141)
j
=−
n
for
=−
1,
n
.
As illustrated in Figure 2.3, the frequency domain sequence,
H
j
, representing
the DFT, may be regarded as resulting from a function of frequency being sampled
n
,
−
n
+
1,...,
n
−
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