Digital Signal Processing Reference
In-Depth Information
3.3.3.
Practical algorithm
Equation [3.5] is not directly usable. In effect, the number of bits
b
k
obtained
is not always an integer number, nor is it guaranteed to be a positive number. One
solution involves using a sub-optimal algorithm
2
which progressively distributes the
bits to where they have the greatest effect [GER 92]:
- Initialization
-
b
0
=
=
b
M−
1
=0
-
σ
Q
0
=
σ
Y
0
···
···
σ
Q
M
−
1
=
σ
Y
M
−
1
- As long as
M−
1
k
=0
b
k
<bM
-
l
=argmax
k
σ
Q
k
-
b
l
=
b
l
+1
-
σ
Q
l
=
σ
Q
l
/
4
3.3.4.
Further information
In the standard approach to optimal bit allocation as shown earlier,
σ
Q
k
is written
as a function of
σ
Y
k
and of the number of bits
b
k
which are dedicated to quantizing the
sub-band signal
Y
k
(
m
). Equation [3.3] restates that this is within the high-resolution
hypothesis. This hypothesis is too constraining in real-world applications.
Assume that for each sub-band signal, we have a certain number of possible
quantizers available and that the selected quantizer is written
Q
k
(
i
k
). We choose, for
example, to quantize each signal
Y
k
(
m
) at different resolutions by a uniform scalar
quantizer. In this very simple case, the number
i
k
is the resolution
b
k
. Assume also that
for each quantizer we can deduce (or measure) the quantization error power
σ
Q
k
(
i
k
)
that it generates. The number of bits that it requires is
b
k
(
i
k
).WewillseeinChapter4
that an entropy coding can be carried out after a uniform quantization. In this case, we
show that the necessary number of bits required to quantize the signal can be reduced,
which explains the notation
b
k
(
i
k
) and the fact that
b
k
(
i
k
) can be a non-integer.
If we create any combination of
M
quantizers:
i
=[
i
0
,...,i
M−
1
]
t
from all the possible combinations, the quantization error power is calculated:
M−
1
σ
Q
(
i
)=
1
M
σ
Q
k
(
i
k
)
[3.9]
k
=0
2. The algorithm is described as
greedy
, a term which is used every time that the optimum
choice is made at each algorithm step in the hope of obtaining a globally optimized result.
