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F I GU R E 10 . 23
Possible quantization regions.
10.6.3 Lattice Vector Quantizers
Recall that quantization error is composed of two kinds of error, overload error and granular
error. The overload error is determined by the location of the quantization regions furthest
from the origin, or the boundary. We have seen how we can design vector quantizers to reduce
the overload probability and thus the overload error. We called this the boundary gain of
vector quantization. In scalar quantization, the granular error was determined by the size of
the quantization interval. In vector quantization, the granular error is affected by the size and
shape of the quantization interval.
Consider the square and circular quantization regions shown in Figure
10.23
.Weshow
only the quantization region at the origin. These quantization regions need to be distributed in
a regular manner over the space of source outputs. However, for now, let us simply consider
the quantization region at the origin. Let's assume they both have the same area so that we
can compare them. This way it would require the same number of quantization regions to
cover a given area. That is, we will be comparing two quantization regions of the same “size.”
To have an area of one, the square has to have sides of length one. As the area of a circle is
given by
r
2
, the radius of the circle is
1
√
π
. The maximum quantization error possible with
the square quantization region is when the input is at one of the four corners of the square. In
this case, the error is
π
1
√
2
, or about 0.707. For the circular quantization region, the maximum
error occurs when the input falls on the boundary of the circle. In this case, the error is
1
√
π
,or
about 0.56. Thus, the maximum granular error is larger for the square region than the circular
region.
In general, we are more concerned with the average squared error than the maximum error.
If we compute the average squared error for the square region, we obtain
2
d
x
1666
Square
x
=
0
.
For the circle, we obtain
2
d
x
Circle
x
=
0
.
159
Thus, the circular region would introduce less granular error than the square region.
Our choice seems to be clear; we will use the circle as the quantization region. Unfor-
tunately, a basic requirement for the quantizer is that for every possible input vector there
should be a unique output vector. In order to satisfy this requirement and have a quantizer
with sufficient structure that can be used to reduce the storage space, a union of translates
