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k
1
Ff z
¦
i
,
f
0
(2.16)
0
i
i
0
Gg z
n
¦
,
(2.17)
g
j
j
0
and
n
max(
n
1,
n
, the CARMA equation becomes
k
)
g
a
c
B
G
ª
º
yt k
(
)
ut
()
et
()
Fet k
(
)
¼ .
(2.18)
«
»
A
A
¬
Taking e ( t + k ) from Equation (2.14) as
A
B
k
etk
(
)
ytk z
(
)
utk
(
)
C
C
and using the Diophantine equation, Equation (2.18) becomes
BF
G
yt k
(
)
ut
()
yt
()
Fet k
(
)
(2.19)
C
C
or
ˆ
yt k
(
.
)
yt
(
kt
)
Fet k
(
)
(2.20)
The resulting output variance
JEyt
{[ (
ˆ
kt
)
Fet
(
k
)] }
2
is now minimized for ˆ(
yt kt
)
, to become
0
J
V
2
.
min
e
From Equations (2.19) and (2.20) it follows that
BF
G
ut
()
yt
()
0
C
C
or,
BFu t
()
Gy t
()
0
which finally results in
 
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