Geology Reference
In-Depth Information
in order for the z-transforms
(
z
) to have matching terms where they
overlap. Returning to the frequency domain, the truncated form of 1/
S
(
f
) becomes
1
S
(
f
)
=
Ψ
(
z
)and
Φ
Γ
∗
(
z
)
P
N
+
1
,
z
2
f
N
Γ
(
z
)
e
−
i
2π
f
Δ
t
=
.
(2.468)
Hence, the
maximum entropy spectral density
is found to be
P
N
+
1
S
(
f
)
=
2
f
N
Γ
exp(
t
)
Γ
∗
exp(
t
)
.
(2.469)
−
i
2π
f
Δ
−
i
2π
f
Δ
In terms of the prediction error filter coe
cients it becomes
P
N
+
1
S
(
f
)
=
2
.
(2.470)
2
f
N
+
j
=
1
γ
j
e
−
i
2π
fj
Δ
t
1
Calculation of the maximum entropy spectral density then involves determination
of the prediction error filter and the error power of the prediction. Burg (1968)
has demonstrated a recursive method of solving the prediction error system of
equations (2.80).
2.6.3 The Burg algorithm
We begin with the finite, equispaced data sequence (
f
1
,
f
2
,...,
f
M
−
1
,
f
M
)asthe
available record. First, consider the prediction error system of equations for
N
=
0.
It is simply
φ
ff
(0)
=
P
1
.
(2.471)
For the estimate of the autocorrelation at zero lag we only have
M
1
M
f
j
f
∗
j
=
φ
ff
(
0
)
=
P
1
.
(2.472)
j
=
1
Then, we write the system for
N
=
1 to be compatible with that for
N
=
0,
⎝
⎠
⎝
⎠
=
⎝
⎠
.
P
1
φ
ff
(
−
1)
1
γ
1,1
P
2
0
(2.473)
φ
ff
(1)
P
1
A second subscript 1 is added to γ
1
to indicate that it is the prediction error coe
-
cient for
N
1.
There is now the possibility to consider backward prediction as well as for-
ward prediction. Backward prediction is equivalent to forward prediction of the
sequence
f
M
−
j
+
1
,
j
=
1,2,...,
M
. Assuming that the given sequence is stationary,
the autocorrelation φ
ff
(
k
) is replaced by φ
ff
(
=
k
) for the new sequence. From the
Hermitian property of the autocorrelation for stationary sequences, (2.20), φ
ff
(
k
)
−
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