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and determine the coefficients such that the mean square
error between the estimated signal
a
i
and the real signal
is
ˆ
s
s
minimal. In other words, the aim here is to minimize
(
)
{
}
2
2
ˆ
[3.11]
eE
=−=
ss
E
ε
In these relations,
ˆ
s
is the estimation error,
is the
ε =
−
s
E
ˆ
E
overall mean in the statistical sense of the word and
is the
linear estimation of
on the basis of the data
. The
x
i
s
projection theorem states that
is minimal if the error is
e
()
orthogonal to
. It arises directly from the fact that
,
x
i
e
=
ea
i
and that
is minimal if
[PAP 84]. Consequently
∂
e
∂
a
i
=
0
e
⎧
⎫
⎡
n
⎤
∑
xx
Es
−
a
*
=
0
[3.12]
⎨
⎬
⎢
⎥
ii
i
⎣
⎦
⎩
⎭
i
=
1
which is nothing but the expression of the orthogonality
between the error and the data . The application of
equation [3.12] for to gives us the Yule-Walker
linear system of equations, which is written in matrix form
as
x
i
ε
i
=
1
i
=
n
[3.13]
RA
=
R
0
The correlation matrix
is
R
(
)
[3.14]
*
R
≡=
RE
x x
ij
i
j
The vectors
and
are written as
R
0
A
R
⎛
⎞
01
a
⎜
⎟
⎛⎞
.
1
⎜⎟
⎜
⎟
.
(
)
⎜⎟
⎜
⎟
*
[3.15
]
A
=
,
R
=
⎜
RE
=
sx
⎟
⎜⎟
0
0
i
i
.
⎜⎟
⎜
⎟
.
a
⎜
⎟
⎝⎠
⎜
⎟
n
R
⎝
⎠
0
n
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