Geology Reference
In-Depth Information
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
n
Δ
t
a
=
(2.136)
and common ratio
( k
)
T Δ t
e i
r
=
.
(2.137)
Thus, for T =
( 2 n +
1 )
Δ t ,
a 1
r 2 n + 1
n
( k
)
T
e i
t j
=
1
r
j =− n
t 1
t
( k
)
T
( k
)
T
e i
n
Δ
e i
(2 n
+
1)
Δ
=
( k
)
T Δ t
e i
1
e i ( k T n Δ t 1
e i 2π( k )
=
.
(2.138)
( k
)
T Δ t
e i
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 ( 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 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 T t j
H
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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