Digital Signal Processing Reference
In-Depth Information
forming process will not be dealt with in detail because quantization in accordance
with Model 1 is not really recommended for analog-to-digital conversions of
time-variant signals. This approach is much better suited for quantization of
short time intervals, phase angles and other related physical quantities. For these
applications, the quantization thresholds are in the form of short pulses generated
at proper time instants and the involved techniques are those used for randomized
sampling. They are considered in Chapter 6.
The randomized quantization of time intervals can therefore be analysed using
virtually the same relationships as those derived in Chapter 6 for randomized
sampling. Of course, this is also true for amplitude quantization performed in
accordance with Model 1. For instance, it can written that the expected value of
the quantized signal is
E
[
x
k
]
=
qE
[
n
k
]
.
(4.5)
The expected number of threshold levels within the interval [
x
k
] can be
determined by applying the function derived in Chapter 3. In this case,
−
X
0
,
x
k
q
.
=
[
P
(
X
0
+
−
=
lim
x
0
⇒∞
E
[
n
k
]
lim
x
0
⇒∞
x
k
)
P
(
X
0
)]
(4.6)
Substituting Equation (4.6) into expression (4.5) gives
E
[
x
k
]
=
x
k
.
(4.7)
Therefore the expected value of a quantized signal value is equal to the corre-
sponding signal value. In this sense the quantization is a linear operation. Such
quantization is unbiased, which holds even for very crude quantization.
To ensure the correct performance of quantizers built according to the require-
ments of Model 1, only one parameter of the random threshold level sets, namely
the mean value
q
, should be kept constant at a given level. Other statistical pa-
rameters may slowly drift so long as they remain within some relatively broad
limits. Instantaneous input-output characteristic is given in Figure 4.6.
Model 2
This model is more versatile and practical. In fact, this is a version of the generic
Model 1, characterized by
σ/
q
⇒
0, where
σ
is the standard deviation of the
intervals
q
k
(
i
−
1)
. Under these conditions, the intervals between thresholds
are constant and equal to the quantization step
q
{
q
ki
−
Randomized quantization per-
formed in accordance with this model is also illustrated by Figure 4.3. This time
diagram is given for quantization of a unipolar signal. The equidistant threshold
level sets change their positions randomly at time instants
t
k
−
1
.
,
t
k
,
t
k
+
l
,....
