Geology Reference
In-Depth Information
for
m
=
0,1,...,
N
. Adding the two equations gives
E
⎩
−
⎭
=
N
2
d
∗
j
f
j
−
m
+
f
j
−
m
g
∗
k
f
∗
j
−
k
2
0,
(2.66)
k
=
0
or, on dividing by two and taking complex conjugates,
k
=
0
g
k
E
f
j
−
k
f
∗
j
−
m
N
E
d
j
f
∗
j
−
m
,
m
=
=
0,1,...,
N
.
(2.67)
E
f
j
−
k
f
∗
j
−
m
and the cross-
Recognising that the autocorrelation is φ
ff
(
m
−
k
)
=
E
d
j
f
∗
j
−
m
, the conditional equations for the optimumWiener
correlation isφ
df
(
m
)
=
filter become
N
0
g
k
φ
ff
(
m
−
k
)
=
φ
df
(
m
),
m
=
0,1,...,
N
.
(2.68)
k
=
In matrix form, they are
⎝
⎠
⎝
⎠
⎝
⎠
φ
ff
(0) φ
ff
(
−
1)
···
φ
ff
(
−
N
)
g
0
g
1
.
.
.
g
N
φ
df
(0)
φ
df
(1)
.
.
.
φ
df
(
N
)
φ
ff
(1) φ
ff
(0)
···
φ
ff
(
−
N
+
1)
=
.
(2.69)
.
.
.
.
.
.
.
.
.
.
.
.
φ
ff
(
N
) φ
ff
(
N
−
1)
···
φ
ff
(0)
cient matrix is then seen to be equidiagonal and Hermitian (the com-
plex conjugate of its transpose is equal to the matrix itself). The latter follows from
the properties φ
∗
ff
(
k
)
The coe
k
) and φ
∗
ff
(
=
φ
ff
(
k
). Matrices that are equidi-
agonal and Hermitian are said to be
Toeplitz
matrices.
It is to be noted that if
f
j
and
d
j
are energy signals, the optimum Wiener fil-
ter equations retain the same form using the relaxed definitions of autocorrelation
(2.22) and crosscorrelation (2.26).
=
φ
ff
(
−
−
k
)
2.2.1 Prediction and prediction error filters
Optimum Wiener filters can be applied to the problem of prediction. For example,
we might want to predict the value of a time sequence one time unit after the
previous
N
values. That is, we want to predict
f
j
from
f
j
−
N
,...,
f
j
−
2
,
f
j
−
1
.The
prediction is given by the output
N
h
j
=
1
g
k
f
j
−
k
(2.70)
k
=
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