Databases Reference
In-Depth Information
Output
3
2
1
−
3
−
2
−
1
1
2
3
4
−4
Input
−1
−
2
−3
−4
F I GU R E 9 . 25
Nonuniform companded quantizer.
with the same uniform quantizer results in an output of 1.5, with an apparent error of 0.3. The
expander then maps this to the final reconstruction value of 0.75, which is 0.15 away from
the input. Comparing 0.15 with 0.4, we can see that relative to the input we get a substantial
reduction in the quantization error. In fact, we will not get any increase in the quantization
error for all values in the interval
, and for most values, we will get a decrease in the
quantization error (see Problem 6 at the end of this chapter). Of course, this will not be true
for values outside the
[−
1
,
1
]
interval. Suppose we have an input of 2.7. If we quantized this
directly with the uniform quantizer, we would get an output of 2.5 with a corresponding error
of 0.2. Applying the compressor mapping, the value of 2.7 would be mapped to 3.13 resulting
in a quantized value of 3.5. Mapping this back through the expander, we get a reconstructed
value of 3.25, which differs from the input by 0.55.
As we can see, the companded quantizer effectively works like a nonuniform quantizer
with smaller quantization intervals in the interval
[−
1
,
1
]
and larger quantization intervals
outside this interval. What is the effective input-output map of this quantizer? Notice that all
inputs in the interval [0, 0.5] get mapped into the interval [0, 1], for which the quantizer output
is 0.5, which in turn corresponds to the reconstruction value of 0.25. Essentially, all values in
the interval [0, 0.5] are represented by the value 0.25. Similarly, all values in the interval [0.5,
1] are represented by the value 0.75 and so on. The effective quantizer input-output map is
shown in Figure
9.25
.
[−
1
,
1
]
If we bound the source output by some value
x
max
, any nonuniform quantizer can always
be represented as a companding quantizer. Let us see how we can use this fact to come up
with quantizers that are robust to mismatch. First we need to look at some of the properties of
high-rate quantizers, or quantizers with a large number of levels.
Define
k
=
b
k
−
b
k
−
1
(37)
