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Quantization noise
Quantizer input
Quantizer output
F I GU R E 9 . 4
Additive noise model of a quantizer.
T A B L E 9 . 2
Codeword assignment for an
eight-level quantizer.
y
1
1110
y
2
1100
y
3
100
y
4
00
y
5
01
y
6
101
y
7
1101
y
8
1111
However, if we are allowed to use variable-length codes, such as Huffman codes or arith-
metic codes, along with the size of the alphabet, the selection of the decision boundaries will
also affect the rate of the quantizer. Consider the codeword assignment for the output of an
eight-level quantizer shown in Table
9.2
.
According to this codeword assignment, if the output
y
4
occurs, we use 2 bits to encode
it, while if the output
y
1
occurs, we need 4 bits to encode it. Obviously, the rate will depend
on how often we have to encode
y
4
versus how often we have to encode
y
1
. In other words,
the rate will depend on the probability of occurrence of the outputs. If
l
i
is the length of the
codeword corresponding to the output
y
i
, and
P
(
y
i
)
is the probability of occurrence of
y
i
, then
the rate is given by
M
R
=
l
i
P
(
y
i
)
(5)
i
=
1
However, the probabilities
{
P
(
y
i
)
}
depend on the decision boundaries
{
b
i
}
. For example,
the probability of
y
i
occurring is given by
b
i
P
(
y
i
)
=
f
X
(
x
)
dx
b
i
−
1
