Geology Reference
In-Depth Information
Expression (2.433) for the entropy rate then becomes
log
(det
T
N
)
1/(
N
+
1)
h
=
lim
N
→∞
f
N
1
2
f
N
log
S
(
f
)
df
.
=
log(2
f
N
)
+
(2.446)
−
f
N
Substituting the infinite Fourier series expansion of the spectral density (2.336),
the entropy rate can be expressed as
⎝
log
⎩
φ(
k
)
e
−
i
2π
fk
Δ
t
⎭
df
⎠
f
N
f
N
∞
1
2
f
N
=
+
Δ
+
h
log(2
f
N
)
log(
t
)
df
−
f
N
−
f
N
k
=−∞
⎝
−
log
⎩
φ(
k
)
e
−
i
2π
fk
Δ
t
⎭
df
⎠
f
N
∞
1
2
f
N
=
+
·
+
log(2
f
N
)
log(2
f
N
)
2
f
N
−
f
N
k
=−∞
log
⎩
φ(
k
)
e
−
i
2π
fk
Δ
t
⎭
f
N
∞
1
2
f
N
=
df
,
(2.447)
−
f
N
k
=−∞
where φ(
k
) is the autocorrelation at lag
k
and
Δ
t
=
1/2
f
N
.
2.6.2 The maximum entropy spectrum
The starting point for the Burg maximum entropy method of spectral analysis is
expression (2.447) for the entropy rate. For a finite record of
N
1 sample points,
only autocorrelations for lags in magnitude less than
N
are available. Burg suggests
that the most reasonable choice for the unknown autocorrelations is the one that
adds no information or entropy. Hence, the entropy rate
h
is made stationary with
respect to the unknown autocorrelations. Thus, we set
+
∂
h
∂φ(
k
)
=
0,
|
k
|≥
N
+
1.
(2.448)
The conditions that the unknown autocorrelations add no entropy, on taking deriv-
atives, are found to be
f
N
e
−
i
2π
fk
Δ
t
S
(
f
)
df
=
0,
|
k
|≥
N
+
1.
(2.449)
−
f
N
Thus, the Euler coe
cients of the Fourier series expansion of the real, periodic
function 1/
S
(
f
) truncate, and its Fourier series expansion becomes
N
1
S
(
f
)
=
1
2
f
N
c
k
e
i
2π
fk
Δ
t
.
(2.450)
k
=−
N
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