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which implies that
M
l
k
P
k
=
(26)
0
k
=
0
The final step is the selection of
γ
, which involves a significant amount of creativity. The
value we pick for
γ
determines how fast the quantizer will respond to changing statistics. A
large value of
γ
will result in faster adaptation, while a smaller value of
γ
will result in greater
stability.
Example9.5.4:
Suppose we have to obtain the multiplier functions for a 2-bit quantizer with input probabilities
P
0
=
.
,
P
1
=
.
2. First, note that the multiplier value for the inner level has to be less than
1. Therefore,
l
0
is less than 0. If we pick
l
0
=−
0
8
0
4, this would satisfy Equation (
26
),
while making
M
0
less than 1 and
M
1
greater than 1. Finally, we need to pick a value for
1 and
l
1
=
γ
.
in a rather extreme example.
The input is a square wave that switches between 0 and 1 every 30 samples. The input is
quantized using a 2-bit Jayant quantizer. We have used
l
0
In Figure
9.17
, we see the effect of using different values of
γ
2. Notice what
happens when the input switches from 0 to 1. At first the input falls in the outer level of the
quantizer, and the step size increases. This process continues until
=−
1 and
l
1
=
is just greater than 1. If
γ
has been increasing quite slowly and should have a value close to 1 right
before its value increases to greater than 1. Therefore, the output at this point is close to 1.5.
When
is close to 1,
is close to 1, the
output suddenly drops to about 0.5. The step size now decreases until it is just below 1, and
the process repeats, causing the “ringing” seen in Figure
9.17
.As
becomes greater than 1, the input falls in the inner level, and if
γ
increases, the quantizer
adapts more rapidly, and the magnitude of the ringing effect decreases. The reason for the
decrease is that right before the value of
γ
increases above 1, its value is much smaller than
1, and subsequently the output value is much smaller than 1.5. When
increases beyond
1, it may increase by a significant amount, so the inner level may be much greater than 0.5.
These two effects together compress the ringing phenomenon. Looking at this phenomenon,
we can see that it may have been better to have two adaptive strategies, one for when the input
is changing rapidly, as in the case of the transitions between 0 and 1, and one for when the
input is constant, or nearly so. We will explore this approach further when we describe the
quantizer used in CCITT standard G.726.
When selecting multipliers for a Jayant quantizer, the best quantizers expand more rapidly
than they contract. This makes sense when we consider that when the input falls into the
outer levels of the quantizer, it is incurring overload error, which is essentially unbounded.
This situation needs to be mitigated with dispatch. On the other hand, when the input falls
in the inner levels, the noise incurred is granular noise, which is bounded and, therefore, may
be more tolerable. Finally, the discussion of the Jayant quantizer was motivated by the need
for robustness in the face of changing input statistics. Let us repeat the earlier experiment
with changing input variance and distributions and see the performance of the Jayant quantizer
