Game Development Reference
In-Depth Information
s
D
PCM
(
R
)
˃
G
DPCM
=
)
=
p
,
(4.16)
D
DPCM
(
R
˃
s
p
are variances of errors for PCM and DPCM respectively with the
where
˃
and
˃
bit-rate R.
Based on the theory of quantization (Jayant and Noll
1984
; Berger
1971
), the
relationship between the variance and the bit-rate can be formulated as
2
2
2
−
2
R
2
˃
q
=
ʵ
˃
x
,
(4.17)
2
2
where
˃
x
are the quantization error and the variance of the source respectively.
R
represents the bit-rate.
q
and
˃
2
is a constant variable depending on the characteristics
of the quantize, which is usually called the 'quantizer performance factor'. By intro-
ducing Eq.
4.17
into Eq.
4.16
we can get,
2
PCM
2
s
˃
ʵ
PCM
˃
G
DPCM
=
DPCM
=
p
,
(4.18)
2
2
˃
ʵ
DPCM
˃
DPCM
are the quantizer performance factor for the PCM and DPCM
coding system respectively. If the source is stable and its power spectrum density
function is
S
PCM
and
where
e
j
w
)
2
(
, then the minimum
˃
p
can be calculated as (Jayant and Noll
1984
),
⊛
⊞
ˀ
ln
S
e
j
ˉ
d
ˉ
1
2
ˀ
⊝
⊠
.
p
p
=
˃
=
˃
lim
N
exp
(4.19)
,
min
ₒ∞
−
ˀ
The spectral flatness measure is defined as
p
s
ʳ
=
˃
/˃
.
(4.20)
,
min
It is obvious that the gain of prediction coding is inversely proportional to the
spectral flatness measure. Intuitively, a signal is easier to predict if its spectrum is
sharper. Conversely, a signal is harder to predict if its spectrum is flatter. It can be
proved that, Eq.
4.19
can be rewritten as (Jayant and Noll
1984
)
N
2
p
˃
min
=
lim
N
ʻ
k
,
(4.21)
,
ₒ∞
k
where
ʻ
k
is the
k
th eigenvalue of the
N
-order autocorrelation matrix for the signal.
On the other hand, because traces of similar matrices are identical, it is obvious that,
N
k
1
2
s
˃
=
lim
N
ʻ
k
.
(4.22)
ₒ∞
