Digital Signal Processing Reference
In-Depth Information
performance than that based on Z
2
in terms of central and side distortions. A
2
is
equivalent or similar to the hexagonal latt
ic
e [
12
]. The hexagonal lattice can be
spanned by the vectors (1, 0) and .
1=2;
p
3=2/, and the generator matrix is
2
3
10
p
3
2
4
5
:
G
D
(4.1)
1
2
Every pair of coefficients in each subband is formed as a two-dimensional
vector according to the grouping way in step 2. A lattice vector quantizer with
a quantization “volume-size” (like the step-size in scalar quantization) is applied
to such two-dimensional vectors, thus producing a quantized symbol ,
A
2
.
It is known that the VQ encoding complexity increases with dimensionality and
codebook size. Here we use the lowest dimension vector, that is, two-dimensional
vector. Moreover, LVQ encoding can be implemented by a fast quantizing algorithm
[
12
] which does not require performing the computation-intensive nearest neighbor
search based on squared distance calculation. In the fast encoding algorithm [
12
],
only two matrix multiplications are required for vector mapping between a two-
dimensional vector and a three-dimensional vector, and a modification may be
needed for the mapped three-dimensional vector to make the sum of its three-
dimensional values zero. In this way, the complexity of LVQ on A
2
is considered
very low. In addition, another fast quantizing algorithm in [
12
] may be a better
choice to accelerate LVQ encoding further. Considering that the hexagonal lattice
is the union of two rectangular lattices, the encoding can be simply achieved by
finding the nearest point in each rectangular sub-lattice and selecting the nearer of
these two points.
Step 4: Labeling Function with Alternative Transmission
Information about a quantized point is mapped to two representations and
then sent across two channels, subjected to bit rate constraints imposed by each
individual channel. This is done by a labeling function [
7
] followed by arithmetic
encoding. The labeling function maps
ƒ to a pair .
0
1
;
0
2
/
2
ƒ
0
ƒ
0
,whereƒ
0
is a sub-lattice of ƒ with the index N , N
D j
ƒ=ƒ
0
j
. The index N determines the
coarse degree of the sub-lattice which can control the amount of redundancy in the
MD coder [
4
]. In Sect.
4.1.3
, optimization for the index N and the LVQ quantization
“volume-size” (in step 3) will be presented in detail.
Figure
4.3
is an example of an A
2
sub-lattice with index N
D
13. In the case of
N
D
13, we can obtain a labeling function as in Table
4.1
, where each fine lattice
point is mapped to a unique label .
0
1
;
0
2
/, with
0
1
and
0
2
being two sub-lattice
points as close to as possible. Note that the proposed mapping scheme shown
in the table is slightly different from the index assignment developed by Servetto,
Vaishampayan, and Sloane [
4
] (known as SVS technique). In our proposed scheme,
0
1
is always closer to , thus
0
1
is denoted as the near sub-lattice point and
0
2
the
