Databases Reference
In-Depth Information
noise, which is the first term in Equation (
19
). The design process for the uniform quantizer
is a balancing of these two effects. An important parameter that describes this trade-off is the
loading factor
f
l
, defined as the ratio of the maximum value the input can take in the granular
region to the standard deviation. A common value of the loading factor is 4. This is also
referred to as 4
σ
loading
.
Recall that when quantizing an input with a uniform distribution, the SNR and bit rate are
related by Equation (
17
), which says that for each bit increase in the rate there is an increase
of 6.02dB in the SNR. In Table
9.3
, along with the step sizes, we have also listed the SNR
obtained when a million input values with the appropriate
pdf
are quantized using the indicated
quantizer.
From this table, we can see that, although the SNR for the uniform distribution follows
the rule of a 6.02dB increase in the signal-to-noise ratio for each additional bit, this is not
true for the other distributions. Remember that we made some assumptions when we obtained
the 6
02
n
rule that are only valid for the uniform distribution. Notice that the more peaked a
distribution is (that is, the further away from uniform it is), the more it seems to vary from the
6.02dB rule.
We also said that the selection of
.
is a balance between the overload and granular errors.
The Laplacian distribution has more of its probability mass away from the origin in its tails than
the Gaussian distribution. This means that for the same step size and number of levels, there is
a higher probability of being in the overload region if the input has a Laplacian distribution than
if the input has a Gaussian distribution. The uniform distribution is the extreme case, where
the overload probability is zero. If we increase the step size for the same number of levels,
the size of the overload region (and hence the overload probability) is reduced at the expense
of granular noise. Therefore, for a given number of levels, if we are picking the step size to
balance the effects of the granular and overload noise, distributions that have heavier tails will
tend to have larger step sizes. This effect can be seen in Table
9.3
. For example, for eight
levels, the step size for the uniform quantizer is 0.433. The step size for the Gaussian quantizer
is larger (0.586), while the step size for the Laplacian quantizer is larger still (0.7309).
Mismatch Effects
We have seen that for a result to hold, the assumptions we used to obtain the result have to hold.
When we obtain the optimum step size for a particular uniform quantizer using Equation (
19
),
we make some assumptions about the statistics of the source. We assume a certain distribution
and certain parameters of the distribution. What happens when our assumptions do not hold?
Let's try to answer this question empirically.
We will look at two types of mismatches. The first is when the assumed distribution
type matches the actual distribution type, but the variance of the input is different from the
assumed variance. The second mismatch is when the actual distribution type is different from
the distribution type assumed when obtaining the value of the step size. Throughout our
discussion, we will assume that the mean of the input distribution is zero.
In Figure
9.11
, we have plotted the signal-to-noise ratio as a function of the ratio of the
actual to assumed variance of a 4-bit Gaussian uniform quantizer, with a Gaussian input. (To
see the effect under different conditions, see Problem 5 at the end of this chapter.) Remember
