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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