Databases Reference
In-Depth Information
The value of
N
specifies the order of the predictor. Using the fine quantization assumption,
we can now write the predictor design problem as follows: Find the
2
{
a
i
}
so as to minimize
σ
d
:
⎡
2
⎤
⎦
x
n
−
N
2
d
⎣
σ
=
E
a
i
x
n
−
i
(22)
i
=
1
wherewe assume that the source sequence is a realization of a real-valuedwide-sense stationary
process. Take the derivative of
2
d
with respect to each of the
a
i
and set this equal to zero. We
get
N
equations and
N
unknowns:
σ
2
E
x
n
−
x
n
−
1
N
2
d
∂σ
a
1
=−
a
i
x
n
−
i
=
0
(23)
∂
i
=
1
2
E
x
n
−
x
n
−
2
N
2
d
∂σ
a
2
=−
a
i
x
n
−
i
=
0
(24)
∂
i
=
1
.
.
2
E
x
n
−
x
n
−
N
N
2
d
∂σ
a
N
=−
a
i
x
n
−
i
=
0
(25)
∂
i
=
1
Taking the expectations, we can rewrite these equations as
N
a
i
R
xx
(
i
−
1
)
=
R
xx
(
1
)
(26)
i
=
1
N
a
i
R
xx
(
i
−
2
)
=
R
xx
(
2
)
(27)
i
=
1
.
.
N
a
i
R
xx
(
i
−
N
)
=
R
xx
(
N
)
(28)
i
=
1
where
R
xx
(
k
)
is the autocorrelation function of
x
n
:
R
xx
(
k
)
=
E
[
x
n
x
n
+
k
]
(29)
We can write these equations in matrix form as
Ra
=
p
(30)
where
⎡
⎣
⎤
⎦
R
xx
(
0
)
R
xx
(
1
)
R
xx
(
2
)
···
R
xx
(
N
−
1
)
R
xx
(
1
)
R
xx
(
0
)
R
xx
(
1
)
···
R
xx
(
N
−
2
)
R
xx
(
2
)
R
xx
(
1
)
R
xx
(
0
)
···
R
xx
(
N
−
3
)
R
=
(31)
.
.
.
R
xx
(
N
−
1
)
R
xx
(
N
−
2
)
R
xx
(
N
−
3
)
···
R
xx
(
0
)
