Game Development Reference
In-Depth Information
From Eq.
4.8
, it is obvious that
N
a
k
R
(
k
,
l
)
=
R
(
0
,
l
),
l
=
1
,
2
,...,
K
.
(4.10)
k
=
1
We can also rewrite it in a matrix way as
⊡
⊣
⊤
⊦
⊡
⊣
⊤
⊦
=
⊡
⊣
⊤
⊦
,
R
(
1
,
1
)
R
(
2
,
1
)
···
R
(
N
,
1
)
a
1
a
2
···
a
N
R
(
0
,
1
)
R
(
1
,
2
)
R
(
2
,
2
)
···
R
(
N
,
2
)
R
(
0
,
2
)
(4.11)
···
···
···
···
···
R
(
1
,
N
)
R
(
2
,
N
)
···
R
(
N
,
N
)
R
(
0
,
N
)
Or simply as
[
R
]
a
=
r
.
(4.12)
Equation
4.12
is also known as Yule-Walker equation, from which we can get
[
R
]
−
1
r
a
=
.
(4.13)
In such a case, the variance is minimized as
E
(
S
=
k
=
0
N
2
˃
p
=
S
−
S
p
)
R
(
0
,
0
)
−
a
k
R
(
k
,
0
)
(4.14)
r
T
[
R
]
−
1
r
r
T
a
=
R
(
0
,
0
)
−
=
R
(
0
,
0
)
−
For a stable source,
[
R
]
is the autocorrelation matrix of the prediction vector
T
.And
r
is the cross correlation of
S
0
and
S
p
. Since
S
p
is
a Toeplitz matrix, its inverse matrix can be calculated in a fast way by applying the
Levinson-Durbin algorithm (Rabiner and Schafer
1978
).
=[
S
1
,
S
2
,
···
,
S
N
]
[
R
]
4.1.3 Gain of the Prediction Coding
To analyze the efficiency of the predictor, we should compare the fidelities of DPCM
and PCM at the same bit-rate. Therefore the gain of the prediction coding
G
DPCM
is
defined as
D
PCM
(
R
)
G
DPCM
=
)
,
(4.15)
D
DPCM
(
R
where
D
PCM
and
D
DPCM
represent the fidelities of PCM and DPCM in a form of
summed squared error (SSE) respectively. In statistics, Eq.
4.15
can be rewritten as
