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Therefore, the rate
R
is a function of the decision boundaries and is given by the expression
l
i
b
i
b
i
−
1
M
R
=
f
X
(
x
)
dx
(6)
i
=
1
From this discussion and Equations (
3
) and (
6
), we see that for a given source input, the
partitions we select and the representation for those partitions will determine the distortion
incurred during the quantization process. The partitions we select and the binary codes for the
partitions will determine the rate for the quantizer. Thus, the problem of finding the optimum
partitions, codes, and representation levels are all linked. In light of this information, we can
restate our problem statement:
Given a distortion constraint
D
∗
2
q
σ
(7)
find the decision boundaries, reconstruction levels, and binary codes that minimize the
rate given by Equation (
6
) while satisfying Equation (
7
).
Or, given a rate constraint
R
∗
R
(8)
find the decision boundaries, reconstruction levels, and binary codes that minimize the
distortion given by Equation (
3
), while satisfying Equation (
8
).
This problem statement of quantizer design, while more general than our initial statement,
is substantially more complex. Fortunately, in practice there are situations in which we can
simplify the problem. We often use fixed-length codewords to encode the quantizer output.
In this case, the rate is simply the number of bits used to encode each output, and we can use
our initial statement of the quantizer design problem. We start our study of quantizer design
by looking at this simpler version of the problem and later use what we have learned in this
process to attack the more complex version.
9.4 Uniform Quantizer
The simplest type of quantizer is the uniform quantizer. All intervals are the same size in the
uniform quantizer, except possibly for the two outer intervals. In other words, the decision
boundaries are spaced evenly. The reconstruction values are also spaced evenly, with the same
spacing as the decision boundaries; in the inner intervals, they are themidpoints of the intervals.
This constant spacing is usually referred to as the step size and is denoted by
. The quantizer
shown in Figure
9.3
is a uniform quantizer with
1. It does not have zero as one of its rep-
resentation levels. Such a quantizer is called a
midrise quantizer
. An alternative uniform quan-
tizer could be the one shown in Figure
9.5
. This is called a
midtread quantizer
. As the midtread
quantizer has zero as one of its output levels, it is especially useful in situations where it is im-
portant that the zero value be represented—for example, control systems inwhich it is important
to represent a zero value accurately, and audio coding schemes in which we need to represent
=
