Digital Signal Processing Reference
In-Depth Information
The threshold levels at all the comparator reference inputs follow random voltage
changes, thus providing the correct operational mode for pseudo-randomized
quantizing.
Note that the schemes in Figures 5.3(b) and 5.4 have two rather than one output.
This is essential. Output signals of such quantizers can often be processed in two
differing ways. They might be passed partly through one processing channel and
partly through another. Separate algorithms are used for processing the informa-
tion carried by the sequence
{
n
k
q
}
and the information given by the sequence
{
ξ
. It is possible to speed up processing and to simplify the hardware in this
way, lessening the negative impact of the main disadvantage of this kind of quan-
tizing: the increased quantity of bits needed for encoding the pseudo-randomly
quantized signals.
}
k
5.3
Input-Output Relationships
Both instantaneous and expected input-output relationships for pseudo-
randomized quantizing are linear. The point is that the input-expected output
relationship is defined in a remarkable way, specifically
E
[
x
]
=
x
. Instantaneous
values of
x
for any value of
x
all are within a certain area. This means that for all
values of
x
, the respective
x
values are distanced from the corresponding values
of
E
[
x
] by an interval not exceeding
x
0
, the quan-
tized value
x
0
can with equal probability assume any value within the interval
[
x
0
±
±
0.5
q
. For instance, when
x
=
0
.
5
q
].
5.4 Quantization Errors
Assume that the quantized signal
1
2
q
x
=
nq
+
q
0
−
.
(5.16)
Then the conditional probability density function
d
q
0
ϕ
(
x
0
+
ε
)
Ψ
(
ε/
x
=
x
0
)
=
x
0
]
,
(5.17)
q
Pr[
x
=
where Pr[
x
=
x
0
] is the probability that the quantized signal
x
is equal to
x
0
.
It
is obvious that
nq
+
q
0
d
q
0
q
q
Pr[
x
=
x
0
]
=
q
0
ϕ
(
x
)d
x
.
(
n
−
1)
q
+
To obtain the unconditional probability density function of
ε
, function (5.17)
should be averaged over all possible digital values of
n
=
0
,
1
,
2
,...
and over all
