Digital Signal Processing Reference
In-Depth Information
The noise is now to be calculated as mean-square quantization error,
e
2
where
e
is the difference between the actual and approximated value of voltage/current.
The peak to peak amplitude of the sampled signal
x
s
(
is divided into M equal
levels each of width S. At the centre of each level of width the quantization levels
are located as
x
1
,
x
2
,
t
)
happens
to be the closest to the level
x
k
, the quantizer output will be
x
k
and obviously, the
quantization error is
e
...
x
M
as shown the Fig.
2.14
. Since, in the figure
x
s
(
t
)
=
x
s
(
t
)
−
x
k
.
Fig. 2.14
Interpretation of
quantization error
x
S
x
k
x
3
x
2
x
1
Let,
p
(
x
)
dx
be the probability that
x
s
(
t
)
lies in the voltage/current range
x
+
dx
/
2
to
x
−
dx
/
2. Then the mean square quantization error [
3
]is
x
1
+
S
/
2
x
2
+
S
/
2
2
dx
2
dx
e
2
=
p
(
x
)
(
x
−
x
1
)
+
p
(
x
)
(
x
−
x
2
)
+
...
(2.11)
x
1
−
S
/
2
x
2
−
S
/
2
Now, the PDF (probability density function)
p
(
x
) of the message signal will not
certainly be constant throughout each division. Assuming large
M
, i.e., with suffi-
ciently small
S
,
p
(
x
) can be taken as constant throughout each division. Then, the 1st
term of the right hand side of the Eq. (
2.11
),
p
(
x
)
p
(
1
)
=
=
constant. The 2nd term is
p
(
2
)
, and so on. Hence, the constant terms may be taken out of the integration
sign. If we now substitute
y
=
p
(
x
)
≡
−
x
x
k
, the expression in Eq. (
2.11
) becomes
S
/
2
p
(
1
)
+
p
(
2
)
+
...
y
2
dy
e
2
=
−
S
/
2
p
(
1
)
+
S
3
12
(2.12)
p
(
2
)
+
...
=
p
(
1
)
S
S
2
12
p
(
2
)
S
=
+
+
...
Now, according to definition,
p
(
1
)
S
be the probability that the signal lies within the
1st quantization range,
p
(
2
)
S
be the probability that the signal lies within the 2nd
quantization range, and so on. Hence, the sum of the terms in the bracket in the
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