Databases Reference
In-Depth Information
Example9.4.2:
Suppose our input fell within the interval
[−
1
,
1
]
with probability 0.95, and fell in the intervals
[−
, (1, 100] with probability 0.05. Suppose we wanted to design an eight-level uniform
quantizer. If we followed the procedure described in the previous section, the step size would be
25. This means that inputs in the
100
,
1
)
5, and
inputs in the interval [0, 1) would be represented by the value 12.5. The maximum quantization
error that can be incurred is 12.5. However, at least 95% of the time, the
minimum
error that
will be incurred is 11.5. Obviously, this is not a very good design. A much better approach
would be to use a smaller step size, which would result in better representation of the values
in the
[−
1
,
0
)
interval would be represented by the value
−
12
.
interval, even if it meant a larger maximum error. Suppose we pick a step size
of 0.3. In this case, the maximum quantization error goes from 12.5 to 98.95. However, 95%
of the time the quantization error will be less than 0.15. Therefore, the average distortion, or
msqe
, for this quantizer would be substantially less than the
msqe
for the first quantizer.
[−
1
,
1
]
We can see that when the distribution is no longer uniform, it is not a good idea to obtain
the step size by simply dividing the range of the input by the number of levels. This approach
becomes totally impractical when we model our sources with distributions that are unbounded,
such as the Gaussian distribution. Therefore, we include the
pdf
of the source in the design
process.
Our objective is to find the step size that, for a given value of
M
, will minimize the distortion.
The simplest way to do this is to write the distortion as a function of the step size, and then
minimize this function. An expression for the distortion, or
msqe
, for an
M
-level uniform
quantizer as a function of the step size can be found by replacing the
b
i
s and
y
i
s in Equation
(
3
) with functions of
. As we are dealing with a symmetric condition, we need only compute
the distortion for positive values of
x
; the distortion for negative values of
x
will be the same.
From Figure
9.8
, we see that the decision boundaries are integral multiples of
, and the
is simply
2
k
−
1
2
(
−
),
)
representation level for the interval [
k
1
k
. Therefore, the expression
for
msqe
becomes
M
2
−
i
x
2
1
2
i
−
1
2
q
σ
=
2
−
f
X
(
x
)
dx
2
(
i
−
1
)
i
=
1
2
∞
x
2
M
−
1
+
−
f
X
(
x
)
dx
(18)
2
−
1
2
, we simply take a derivative of this equation and set it equal
to zero [
122
] (see Problem 1 at the end of this chapter):
To find the optimal value of
M
2
−
i
1
q
σ
d
2
i
−
1
=−
1
(
2
i
−
1
)
(
x
−
)
f
X
(
x
)
dx
d
2
(
i
−
1
)
i
=
∞
2
−
x
f
X
(
M
−
1
−
(
M
−
1
)
−
x
)
dx
=
0
(19)
1
2
