Game Development Reference
In-Depth Information
4.1.2 Optimal Predictors
Although DPCM itself does not constrain the predictor to be linear or non-linear,
linear predictor attracts most attention from both academy and industry due to its
simplicity, efficiency and rigorous theoretical foundation in practice. In the following
discussion, we focus on the design of the optimal linear predictor.
For a linear predictor, the prediction signal can be formulated as,
N
a
i
S
i
,
S
p
=
(4.3)
i
=
1
where
N
is called the 'order' of the predictor.
a
i
represents a prediction coefficient.
It plays a key role in the design of the optimal linear predictor to determine the
prediction coefficients. In order tomaximize the coding efficiency, we must minimize
the variance of the error, denoted as
˃
p
. For simplification, we assume
S
i
≈
S
i
,
(4.4)
Then,
N
S
p
=
a
i
S
i
.
(4.5)
i
=
1
When we consider
S
,
S
p
,
S
as stochastic variables, the variance can be calculated
as,
E
2
N
2
2
˃
p
=
E
{|
S
−
S
p
|
}=
|
S
−
a
i
S
i
|
.
(4.6)
i
=
1
2
To obtain the minimal
˃
p
,let
2
p
∂
a
i
=
∂˃
0
,
i
∈
1
,
2
,...,
N
.
(4.7)
So that
E
N
{
S
−
a
i
S
i
}
S
k
=
0
,
k
∈{
1
,
2
,...,
N
}
.
(4.8)
i
=
1
Denote
R
(
k
,
l
)
to be the autocorrelation of
S
k
and
S
l
as
R
(
k
,
l
)
=
E
{
S
k
,
S
l
}
.
(4.9)
