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0 and 10.0. A simple quantization
scheme would be to represent each output of the source with the integer value closest to it. (If
the source output is equally close to two integers, we will randomly pick one of them.) For
example, if the source output is 2.47, we would represent it as 2, and if the source output is
3.1415926, we would represent it as 3.
This approach reduces the size of the alphabet required to represent the source output; the
infinite number of values between
Consider a source that generates numbers between
−
10
.
−
10
.
0 and 10.0 are represented with a set that contains only
21 values
. At the same time we have also forever lost the original
value of the source output. If we are told that the reconstruction value is 3, we cannot tell
whether the source output was 2.95, 3.16, 3.057932, or any other of an infinite set of values.
In other words, we have lost some information. This loss of information is the reason for the
use of the word “lossy” in many lossy compression schemes.
The set of inputs and outputs of a quantizer can be scalars or vectors. If they are scalars,
we call the quantizers
scalar quantizers
. If they are vectors, we call the quantizers
vector
quantizers
. We will study scalar quantizers in this chapter and vector quantizers in Chapter 10.
(
{−
10
,...,
0
,...,
10
}
)
9.3 The Quantization Problem
Quantization is a very simple process. However, the design of the quantizer has a significant
impact on the amount of compression obtained and loss incurred in a lossy compression scheme.
Therefore, we will devote a lot of attention to issues related to the design of quantizers.
In practice, the quantizer consists of two mappings: an encoder mapping and a decoder
mapping. The encoder divides the range of values that the source generates into a number of
intervals. Each interval is represented by a distinct codeword. The encoder represents all the
source outputs that fall into a particular interval by the codeword representing that interval.
As there could be many—possibly infinitely many—distinct sample values that can fall in any
given interval, the encoder mapping is irreversible. Knowing the code only tells us the interval
to which the sample value belongs. It does not tell us which of the many values in the interval
is the actual sample value. When the sample value comes from an analog source, the encoder
is called an analog-to-digital (A/D) converter.
The encodermapping for a quantizer with eight reconstruction values is shown in Figure
9.1
.
For this encoder, all samples with values between
−
1 and 0would be assigned the code 011. All
Codes
000
001
010
011
100
101
110
111
−3.0
−2.0
−1.0
0
1.0
2.0
3.0
Input
F I GU R E 9 . 1
Mapping for a 3-bit encoder.
