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or code-vectors, we may refer to it as a
K
-level quantizer. Finally, whenever we measure the
performance of the quantizer by computing the distortion, it will always be on a per-sample
basis.
10.3 Advantages of Vector Quantization over
Scalar Quantization
For a given rate (in bits per sample), use of vector quantization results in a lower distortion than
when scalar quantization is used at the same rate, for several reasons. In this section we will
explore these reasons with examples (for a more theoretical explanation, see [
38
,
112
,
118
]).
Note that whenever we compare performance it is always on a per-sample basis rather than on
a per-vector basis.
If the source output is correlated, vectors of source output values will tend to fall in
clusters. By selecting the quantizer output points to lie in these clusters, we have a more
accurate representation of the source output. Consider the following example.
Example10.3.1:
In Example 8.5.1, we introduced a source that generates the height and weight of individuals.
Suppose the height of these individuals varies uniformly between 40 and 80 inches, and the
weight varies uniformly between 40 and 240 pounds. Suppose we are allowed a total of 6
bits to represent each pair of values. We can use 3 bits to quantize the height and 3 bits to
quantize the weight. Thus, the weight range between 40 and 240 pounds will be divided into
eight intervals of equal width of 25, with reconstruction values
{
,
,...,
}
. Similarly,
the height range between 40 and 80 inches can be divided into eight intervals of width 5,
with reconstruction levels
52
77
227
. When we look at the representation of height and
weight separately, this approach seems reasonable. But let's look at this quantization scheme
in two dimensions. We will plot the height values along the
x
-axis and the weight values along
the
y
-axis. Note that we are not changing anything in the quantization process. The height
values are still being quantized to the same eight different values, as are the weight values.
The two-dimensional representation of these two quantizers is shown in Figure
10.2
.
From the figure we can see that we effectively have a quantizer output for a person who
is 80 inches (6 feet 8 inches) tall and weighs 40 pounds, as well as a quantizer output for an
individual who is 42 inches tall but weighs more than 200 pounds. Obviously, these outputs
will never be used, as is the case for many of the other outputs. A more sensible approach
would be to use a quantizer like the one shown in Figure
10.3
, where we take account of the
fact that the height and weight are correlated. This quantizer has exactly the same number of
output points as the quantizer in Figure
10.2
; however, the output points are clustered in the
area occupied by the input. Using this quantizer, we can no longer quantize the height and
weight separately. We have to consider them as the coordinates of a point in two dimensions
in order to find the closest quantizer output point. However, this method provides a much finer
quantization of the input.
{
42
,
47
,...,
77
}
