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required to represent a single quantizer output. We would like to get the lowest distortion for
a given rate or the lowest rate for a given distortion.
Let us pose the design problem in precise terms. Suppose we have an input modeled by a
random variable
X
with
pdf f
X
(
)
. If we wished to quantize this source using a quantizer with
M
intervals, we would have to specify
M
x
+
1 endpoints for the intervals and a representative
value for each of the
M
intervals. The endpoints of the intervals are known as
decision
boundaries
, while the representative values are called
reconstruction levels
. We will often
model discrete sources with continuous distributions. For example, the difference between
neighboring pixels is often modeled using a Laplacian distribution even though the differences
can only take on a limited number of discrete values. Discrete processes are modeled with
continuous distributions because these models can simplify the design process considerably,
and the resulting designs perform well in spite of the incorrect assumption. Several of the
continuous distributions used to model source outputs are unbounded—that is, the range of
values is infinite. In these cases, the first and last endpoints are generally chosen to be
±∞
.
M
i
=
M
i
=
Let us denote the decision boundaries by
{
b
i
}
0
, the reconstruction levels by
{
y
i
}
1
, and
the quantization operation by
Q
(
·
)
. Then
Q
(
x
)
=
y
i
if
b
i
−
1
<
x
b
i
(1)
The mean squared quantization error is then given by
∞
q
2
f
X
(
σ
=
(
x
−
Q
(
x
))
x
)
dx
(2)
−∞
b
i
b
i
−
1
(
M
2
f
X
(
=
x
−
y
i
)
x
)
dx
(3)
i
=
1
, besides being referred
to as the quantization error, is also called the
quantizer distortion
or
quantization noise
.But
the word “noise” is somewhat of a misnomer. Generally, when we talk about noise we mean
a process external to the source process. Because of the manner in which the quantization
error is generated, it is dependent on the source process and, therefore, cannot be regarded as
external to the source process. One reason for the use of the word “noise” in this context is
that from time to time we will find it useful to model the quantization process as an additive
noise process, as shown in Figure
9.4
.
If we use fixed-length codewords to represent the quantizer output, then the size of the
output alphabet immediately specifies the rate. If the number of quantizer outputs is
M
, then
the rate is given by
The difference between the quantizer input
x
and output
y
=
Q
(
x
)
R
=
log
2
M
(4)
For example, if
M
=
8, then
R
=
3. In this case, we can pose the quantizer design problem
as follows:
Given an input
pd f
f
x
(
x
)
and the number of levels
M
in the quantizer, find the
decision boundaries
{
b
i
}
and the reconstruction levels
{
y
i
}
so as to minimize the
mean squared quantization error given by Equation (
3
).
