Digital Signal Processing Reference
In-Depth Information
The quantizer shown in Figure 5.1(c) differs distinctly from the other schemes
illustrated. There are two components in the output. The quantized signal values
provided are defined as a function depending both on
n
k
and
Evi-
dently the auxiliary pseudo-random process in this case is used in quite a different
way. It is added to the input signal (this is equivalent to shifting the set of ref-
erence threshold levels up or down) and then the same process is used to form
the output signal values. As shown below, such a pseudo-randomized quantizer
has outstanding properties that cannot be obtained either from deterministic or
randomized quantizers. It has the advantages of randomized quantizers without
their basic disadvantages.
The question, of course, is exactly how the
ξ
k
(or
q
0
k
)
.
k
values should be used in
the definition of the pseudo-randomized quantization output. To find the an-
swer to this question, an attempt will be made to discover the conditions under
which the pseudo-ramdomized quantization operation might be considered to be
optimal.
ξ
5.2
Optimal Quantizing
Quantizing will be considered optimal if this operation provides the best condi-
tions for estimating the mean value
m
x
of a signal
x
(
t
)
Without loss
of generality, an analysis of optimal quantization can be carried out under the
assumption that
x
(
t
)
∈
[0
,
X
]
.
=
x
=
constant and that only a single threshold level is used
for quantization.
5.2.1 Single-threshold Quantizing
Consider pseudo-randomized quantizing with the information provided by
{
ξ
}
k
taken into account. The quantized signal then can be defined as
f
1
(
ξ
k
)
for
ξ
≤
x
k
,
k
x
k
=
(5.1)
f
2
(
ξ
k
)
for
ξ
>
x
k
,
k
and the estimate of
m
x
as
N
1
N
m
x
=
x
k
.
k
=
1
Such quantizing should be performed in a way ensuring that the estimate
m
x
is
unbiased, i.e. that the following equality holds:
E
[
m
x
]
=
E
[
x
k
]
=
x
.
(5.2)
