Databases Reference
In-Depth Information
2
q
J
=
H
(
Q
)
+
λσ
(60)
then find the compressor characteristic to minimize it.
For the distortion
2
q
, we will use the Bennett integral shown in Equation (
45
). The
quantizer entropy is given by Equation (
57
). For high rates, we can assume (as we did before)
that the
pdf f
X
(
σ
x
)
is constant over each quantization interval
i
, and we can replace Equation
(
58
)by
P
i
=
f
X
(
y
i
)
i
(61)
Substituting this into Equation (
57
), we get
f
X
(
H
(
Q
)
=−
y
i
)
i
log
[
f
X
(
y
i
)
i
]
(62)
f
X
(
f
X
(
=−
y
i
)
log
[
f
X
(
y
i
)
]
i
−
y
i
)
log
[
i
]
i
(63)
f
X
(
f
X
(
log
2
x
max
/
M
=−
y
i
)
log
[
f
X
(
y
i
)
]
i
−
y
i
)
i
(64)
c
(
y
i
)
where we have used Equation (
43
)for
i
. For small
i
we can write this as
log
2
x
max
/
M
H
(
Q
)
=−
f
X
(
x
)
log
f
X
(
x
)
dx
−
f
X
(
x
)
dx
(65)
c
(
x
)
log
2
x
max
log
c
(
=−
f
X
(
x
)
log
f
X
(
x
)
dx
−
M
+
f
X
(
x
)
x
)
dx
(66)
c
(
where the first term is the differential entropy of the source
h
(
X
)
. Let's define
g
=
x
)
.
Then substituting the value of
H
(
Q
)
into Equation (
60
) and differentiating with respect to
g
,
we get
f
X
(
x
max
g
−
1
3
M
2
g
−
3
x
)
[
−
2
λ
]
dx
=
0
(67)
This equation is satisfied if the integrand is zero, which gives us
2
3
x
max
M
=
g
=
K
(
constant
)
(68)
Therefore,
c
(
)
=
(69)
x
K
and
c
(
x
)
=
Kx
+
α
(70)
x
,
which is the compressor characteristic for a uniform quantizer. Thus, at high rates the optimum
quantizer is a uniform quantizer.
Substituting this expression for the optimum compressor function in the Bennett integral,
we get an expression for the distortion for the optimum quantizer:
If we now use the boundary conditions
c
(
0
)
=
0 and
c
(
x
max
)
=
x
max
, we get
c
(
x
)
=
x
max
3
M
2
2
q
σ
=
(71)
