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In-Depth Information
Substituting this into Equation (
41
) and solving for
k
, we get
2
x
max
Mc
(
k
=
(43)
y
k
)
Finally, substituting this expression for
k
into Equation (
40
), we get the following relationship
between the quantizer distortion, the
pdf
of the input, and the compressor characteristic:
2
x
max
Mc
(
3
M
1
12
2
q
σ
=
f
X
(
y
i
)
y
i
)
i
=
1
M
x
max
3
M
2
f
X
(
y
i
)
2
x
max
Mc
(
=
y
i
)
·
c
2
(
y
i
)
i
=
1
M
x
max
3
M
2
f
X
(
y
i
)
=
y
i
)
i
(44)
c
2
(
i
=
1
which for small
i
can be written as
3
M
2
x
max
x
max
f
X
(
x
)
2
q
σ
=
2
dx
(45)
c
(
(
x
))
−
x
max
This is a famous result, known as the Bennett integral after its discoverer, W.R. Bennett
[
130
], and it has been widely used to analyze quantizers. We can see from this integral that
the quantizer distortion is dependent on the
pdf
of the source sequence. However, it also tells
us how to get rid of this dependence. Assume
x
max
α
|
c
(
x
)
=
(46)
x
|
where
α
is a constant. From the Bennett integral we get
x
max
x
max
3
M
2
2
x
max
α
2
q
x
2
f
X
(
σ
=
x
)
dx
(47)
−
x
max
2
3
M
2
σ
α
2
x
=
(48)
where
x
max
2
x
x
2
f
X
(
σ
=
x
)
dx
(49)
−
x
max
q
Substituting the expression for
σ
into the expression for SNR, we get
x
10 log
10
σ
SNR
=
(50)
σ
q
3
M
2
=
10 log
10
(
)
−
20 log
10
α
(51)
