Digital Signal Processing Reference
In-Depth Information
found under which
Nz
1
<
Nz
2
if
z
1
<
z
2
. To do this the following inequality
should be satisfied:
Nz
1
Nz
2
=
A
1
z
1
/α
A
2
z
2
/α
A
1
z
1
A
2
z
2
=
/
12
x
z
1
z
1
+
X
2
σ
=
/
12
x
z
2
<
1
,
(12.26)
z
2
+
X
2
σ
where
A
1
and
A
2
correspond to
z
=
z
1
and
z
=
z
2
. It follows from Equation
(12.26) that
X
2
12
z
1
z
2
>
x
.
(12.27)
σ
The inequality obtained can be applied to find the value of
z
that provides the
minimum of criterion
Nz.
This procedure can be explained by an example.
Example 12.2
Let
X
/σ
=
10. Substituting this value into the inequality (12.27) yields
x
8
3
.
z
1
z
2
>
(12.28)
Let
z
1
=
1. Then it follows from expression (12.28) that it is better to use one
threshold level than
z
≥
9. At the same time, it is better to use from two to eight
levels than one. Now let
z
1
=
2. In this case it is better to use two threshold levels
than
z
≥
5. On the other hand, application of two threshold levels is less desirable
than
z
4, it is found
that, under the given conditions, the best solution minimizing the criterion
Nz
is
z
=
3or
z
=
4. By comparing the cases when
z
=
3 and
z
=
=
3. The relationship of
z
versus the ratio
σ
/
X
, minimizing this criterion, is
x
shown in Figure 12.1.
12.2.3 Minimizing Bit Flow
Criterion Nn
Minimizing this criterion allows the optimal number of quantization threshold
levels to be determined, which guarantees that the estimate ˆ
x
will be obtained
with the required accuracy by processing the minimum bits. Since values of
z
and
n
are connected by
z
μ
2
n
1, optimization can be carried out in the same
way as in the previous case. If the minimum of
N
is reached by processing
uncorrelated signal samples, as in the case of optimizing with regard to crite-
rion
Nz
, the minimum of criterion
Nn
can be determined on the basis of the
=
−
