Geology Reference
In-Depth Information
The Levinson algorithm is recursive, each solution building on the previous one.
Thus, we begin with the simplest case of all, that for
m
=
0. With the first subscript
denoting the value of
m
, the system is
f
0,0
r
0
=
g
0
with solution
f
0,0
=
g
0
/
r
0
.Next,
for
m
=
1 the system takes the form
⎝
⎠
=
r
0
r
1
f
1,0
,
f
1,1
=
f
1,0
,
f
1,1
R
1
.
(g
0
,g
1
)
(2.111)
r
∗
1
r
0
Rather than proceeding to the solution of this system directly, we first solve the
auxiliary system
a
1,0
,
a
1,1
R
1
=
(α
1
,0).
(2.112)
The auxiliary system is then of the same form as the prediction error equations
(2.80). To solve the auxiliary system, we consider the system
⎝
⎠
=
r
0
r
1
a
0,0
,0
(
α
0
,β
0
)
.
(2.113)
r
∗
1
r
0
Thus, α
0
=
a
0,0
r
0
and β
0
=
a
0,0
r
1
. From the Hermitian property of the coe
cient
matrix, reversing the order of equations and variables in this system, and taking
complex conjugates, leads to the system
0,
a
∗
0,0
⎝
⎠
=
β
∗
0
,α
∗
0
.
r
0
r
1
(2.114)
r
∗
1
r
0
If we multiply this system by
k
0
and add it to (2.113), we get
a
0,0
,
k
0
a
∗
0,0
R
1
α
0
k
0
α
∗
0
.
k
0
β
∗
0
,β
0
=
+
+
(2.115)
If this is to be identical to the auxiliary system (2.112),
k
0
β
∗
0
,
a
1,0
=
a
0,0
α
1
=
α
0
+
k
0
a
∗
0,0
k
0
=−
β
0
/α
∗
0
.
a
1,1
=
(2.116)
This completes the solution of the auxiliary system (2.112) within the scale factor
a
0,0
.Weset
a
0,0
1 to complete the solution.
Recursive solutions of the successive auxiliary systems can be appended. For
=
m
=
2, the extension of (2.113) is
⎝
⎠
r
0
r
1
r
2
a
1,0
,
a
1,1
,0
r
∗
1
(α
1
,0,β
1
)
.
r
0
r
1
=
(2.117)
r
∗
2
r
∗
1
r
0
Search WWH ::
Custom Search