Digital Signal Processing Reference
In-Depth Information
performed by using relatively few threshold levels. An essential characteristic is
the probability density function of errors defining the distribution of the quanti-
zation errors of the quantization models.
Model 1
The probability density function for an input signal
x
(
t
)
∈
[0
,
X
] is denoted by
ϕ
Then the
probability that there will be
n
threshold levels inside the interval [0,
x
]is
(
x
). Assume that at the quantization instant the signal is equal to
x
.
Pr[
n
x
=
n
]
=
P
n
(
x
)
−
P
n
+
1
(
x
)
,
(4.16)
where
P
(
n
) is the probability distribution function of the
n
th threshold level. For
any value of the signal,
x
can be written. Hence the probability density
function of the quantization error is defined as
=
n q
+
ε
∞
Ψ
1
(
ε
=
+
ε
−
+
ε
ϕ
+
ε
.
)
[
P
n
(
n q
)
P
n
+
1
(
n q
)]
(
n q
)
(4.17)
n
=
0
Note that the indices at
) and other parameters following indicate the number
of the corresponding quantization model considered. The expected value of the
quantization error may be given as
Ψ
1
(
ε
∞
−∞
ε
∞
E
1
[
ε
]
=
[
P
n
(
n q
+
ε
)
−
P
n
+
1
(
n q
+
ε
)]
ϕ
(
n q
+
ε
) d
ε
n
=
0
∞
−∞
ε
∞
=
(
x
−
n q
) [
P
n
(
x
)
−
P
n
+
1
(
x
)]
ϕ
(
x
) d
x
.
(4.18)
=
n
0
To simplify the equation it is worthwhile to introduce and use the follow-
ing function:
∞
n
[
P
n
(
x
)
−
P
n
+
1
(
x
)
]
=
H
(
x
)
.
(4.19)
n
=
0
When
−
X
⇒∞
,
H
(
x
)
=
x
/
q
. In addition,
∞
[
P
n
(
x
)
−
P
n
+
1
(
x
)]
=
1
.
(4.20)
n
=
0
If these considerations are taken into account,
x
∞
q
x
q
E
1
[
ε
]
=
−
ϕ
(
x
) d
x
=
0
.
(4.21)
0
