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which is independent of the input
pdf
. This means that if we use a compressor characteristic
whose derivative satisfies Equation (
46
), the signal-to-noise ratio will remain constant regard-
less of the input variance. This is an impressive result. However, we do need some caveats.
Notice that we are not saying that the mean squared quantization error is independent of
the quantizer input. It is not, as is clear from Equation (
48
). Remember also that this result is
valid as long as the underlying assumptions are valid. When the input variance is very small,
our assumption about the
pdf
being constant over the quantization interval is no longer valid,
and when the variance of the input is very large, our assumption about the input being bounded
by
x
max
may no longer hold.
With fair warning, let us look at the resulting compressor characteristic. We can obtain the
compressor characteristic by integrating Equation (
46
):
log
|
x
|
x
max
(
)
=
x
max
+
β
(52)
c
x
where
is a constant. The only problem with this compressor characteristic is that it becomes
very large for small
x
. Therefore, in practice we approximate this characteristic with a function
that is linear around the origin and logarithmic away from it.
Two companding characteristics that are widely used today are
β
μ
-law companding and
A
-law companding. The
μ
-law compressor function is given by
ln
1
x
max
|
x
|
+
μ
c
(
x
)
=
x
max
sgn
(
x
)
(53)
(
+
μ)
ln
1
The expander function is given by
x
max
μ
|
x
|
x
max
c
−
1
(
x
)
=
[
(
1
+
μ)
−
1
]
sgn
(
x
)
(54)
This companding characteristic with
255 is used in the telephone systems in North
America and Japan. The rest of the world uses the
A
-law characteristic, which is given by
μ
=
⎧
⎨
|
|
A
x
|
x
|
1
A
(
)
x
max
ln
A
sgn
x
0
1
+
c
(
x
)
=
(55)
x
max
1
+
ln
A
|
x
|
⎩
1
A
|
x
max
|
x
x
max
sgn
(
x
)
1
1
+
ln
A
and
⎧
⎨
|
x
A
(
|
x
|
1
1
+
ln
A
)
0
x
max
1
+
ln
A
c
−
1
(
x
)
=
exp
|
x
|
1
(56)
⎩
x
max
A
1
ln
A
|
x
|
x
max
(
1
+
ln
A
)
−
x
max
1
1
+
9.7 Entropy-Coded Quantization
In Section
9.3
we mentioned three tasks: selection of boundary values, selection of reconstruc-
tion levels, and selection of codewords. Up to this point we have talked about accomplishment
