Digital Signal Processing Reference
In-Depth Information
13.1
Multiplication of Quantized Signals
In order to highlight the problems arising when two or more pseudo-randomly
quantized signals are processed together, multiplication of two quantized signals
x
and
y
are considered, where
x
=
q
(
c
x
+
n
x
)
,
y
=
q
(
c
y
+
n
y
)
and
1
2
1
2
c
x
=
ξ
−
,
c
y
=
ξ
−
.
x
y
13.1.1 Expected Value of Multiplied Quantized Signals
The product
q
2
(
c
x
c
y
z
=
x y
=
+
c
x
n
y
+
c
y
n
x
+
n
x
n
y
)
(13.1)
is obviously a random variable. Its expected value is
q
2
(
E
[
c
x
c
y
]
E
[
z
]
=
E
[
x y
]
=
+
E
[
c
x
n
y
]
+
E
[
c
y
n
x
]
+
E
[
n
x
n
y
])
.
It can easily be shown that the first three expectations are equal to zero. Hence
q
2
E
[
n
x
n
y
]
E
[
z
]
=
E
[
x y
]
=
.
(13.2)
Usually it makes sense to apply pseudo-randomized quantizing only if it can
be coarse. To simplify the following analysis, assume that the quantization con-
sidered is extremely coarse, i.e. that
1
1
for
q
ξ
≤
x
,
for
q
ξ
≤
y
,
x
y
=
=
n
x
n
y
0
for
q
ξ
>
x
,
0
for
q
ξ
>
y
.
x
y
It then follows from Equation (13.2) that
q
2
q
q
x
y
ξ
ξ
d(
q
x
)
d(
q
y
)
E
[
z
]
=
ϕ
(
x
,
y
)d
x
d
y
q
q
0
0
0
0
=
E
[
xy
]
=
a
11
.
(13.3)
To minimize the bias error, the multiplication of
x
and
y
can therefore be
reduced to multiplying
n
x
and
n
y
, without taking into account other terms of
Equation (13.1). However, this does not mean that those other terms are redundant.
In principle, their addition reduces the statistical error. On the other hand, these
additional terms also complicate calculations.
