Error for Infinite-Length Predictor:
We now characterize
σ
e
2
|
min
,
N
σ
e
2
|
min
,
N
as N→∞N→∞.
Note that
r
x
(
0
)
r
x
t
r
x
R
N
︸
R
N
+
1
1
-
h
⋆
=
σ
e
2
|
min
,
N
0
r
x
(
0
)
r
x
t
r
x
R
N
︸
R
N
+
1
1
-
h
⋆
=
σ
e
2
|
min
,
N
0
(7)
Using Cramer's rule,
1
=
σ
e
2
|
min
,
N
r
x
t
0
R
N
R
N
+
1
=
σ
e
2
|
min
,
N
R
N
R
N
+
1
⇒
σ
e
2
|
min
,
N
=
R
N
+
1
R
N
.
1
=
σ
e
2
|
min
,
N
r
x
t
0
R
N
R
N
+
1
=
σ
e
2
|
min
,
N
R
N
R
N
+
1
⇒
σ
e
2
|
min
,
N
=
R
N
+
1
R
N
.
(8)
Given matrix equation Ay=bAy=b, where
A=(a1,a2,⋯,aN)∈RN×NA=(a1,a2,⋯,aN)∈RN×N,
y
k
=
a
1
,
⋯
,
a
k
-
1
,
b
,
a
k
+
1
,
⋯
,
a
N
A
y
k
=
a
1
,
⋯
,
a
k
-
1
,
b
,
a
k
+
1
,
⋯
,
a
N
A
(9)
where
|·||·| denotes determinant.
σ
e
2
|
min
=
lim
N
→
∞
R
N
+
1
R
N
=
exp
1
2
π
∫
-
π
π
ln
S
x
(
e
j
ω
)
d
ω
σ
e
2
|
min
=
lim
N
→
∞
R
N
+
1
R
N
=
exp
1
2
π
∫
-
π
π
ln
S
x
(
e
j
ω
)
d
ω
(10)
where Sx(ejω)Sx(ejω) is the
power spectral density
of
the WSS random process x(n)x(n):
S
x
(
e
j
ω
)
:
=
∑
n
=
-
∞
∞
r
x
(
n
)
e
-
j
ω
n
.
S
x
(
e
j
ω
)
:
=
∑
n
=
-
∞
∞
r
x
(
n
)
e
-
j
ω
n
.
(11)
(Note that, because rx(n)rx(n) is conjugate symmetric for stationary x(n)x(n),
Sx(ejω)Sx(ejω) will always be non-negative and real.)
Prediction Error Whiteness:
We can also demonstrate that the MSE-optimal prediction error is
white when N=∞N=∞.
This is a simple fact of the orthogonality principle seen earlier:
0
=
E
{
e
(
n
)
x
(
n
-
k
)
}
,
k
=
1
,
2
,
⋯
.
0
=
E
{
e
(
n
)
x
(
n
-
k
)
}
,
k
=
1
,
2
,
⋯
.
(14)
The prediction error has autocorrelation
E
{
e
(
n
)
e
(
n
-
k
)
}
=
E
e
(
n
)
x
(
n
-
k
)
+
∑
i
=
1
∞
h
i
x
(
n
-
k
-
i
)
=
E
{
e
(
n
)
x
(
n
-
k
)
}
︸
→
0
for
k
>
0
+
∑
i
=
1
∞
h
i
E
{
e
(
n
)
x
(
n
-
k
-
i
)
}
︸
→
0
=
σ
e
2
|
min
δ
(
k
)
.
E
{
e
(
n
)
e
(
n
-
k
)
}
=
E
e
(
n
)
x
(
n
-
k
)
+
∑
i
=
1
∞
h
i
x
(
n
-
k
-
i
)
=
E
{
e
(
n
)
x
(
n
-
k
)
}
︸
→
0
for
k
>
0
+
∑
i
=
1
∞
h
i
E
{
e
(
n
)
x
(
n
-
k
-
i
)
}
︸
→
0
=
σ
e
2
|
min
δ
(
k
)
.
(15)
