Digital Signal Processing Reference
In-Depth Information
⎛
⎝
⎞
⎠
+
c
1
c
2
s
1
s
2
E
s
(
H
†
H
1
2
H
†
H
1
2
)
−
(
y
=
)
n
(2.55)
H
†
H
H
†
n
1
2
)
−
where
n
=
(
(
)
has uncorrelated elements
∼
CN
(
0
,
1
)
. If we define
H
†
H
†
H
=
1
H
1
+
2
H
2
(2.56)
G
†
G
†
G
=
1
G
1
+
2
G
2
(2.57)
n
1
n
(
1
,
1
)
n
(
3
,
1
)
n
=
,
n
1
=
,
n
2
=
(2.58)
n
2
n
(
2
,
1
)
n
(
4
,
1
)
Then (
2.55
) is equivalent to the following two equations
2
c
1
c
2
E
s
H
1
2
1
H
−
y
1
=
+
n
1
(2.59)
2
s
1
s
2
E
s
G
1
2
1
G
−
y
2
=
+
n
2
(2.60)
So we can realize interference cancellation and pairwise complex symbol decoding
for each user. If we use real symbols, instead of complex symbols, we can achieve
symbol-by-symbol decoding using orthogonal designs instead of quasi-orthogonal
designs. In other words, we can design precoders such that all columns of the equiv-
alent matrix
H
in Eq. (
2.14
) are orthogonal to each other.
When QAM is adopted, we show that we can further reduce the decoding
αα
β
−
β
,
complexity as follows. Note that for 2
×
2 complex matrix
Z
=
|
α
|
, which is a real matrix. So matrices
H
and
G
2
2
2
2
+|
β
|
|
α
|
−|
β
|
Z
†
Z
=
2
2
2
2
|
α
|
−|
β
|
|
α
|
+|
β
|
in (
2.59
), (
2.60
) are all real matrices. Then (
2.59
), (
2.60
) are equivalent to the fol-
lowing four equations
2
c
1
R
c
2
R
E
s
H
1
2
Real
1
H
−
{
y
1
}=
+
Real
{
n
1
}
(2.61)
2
c
1
I
c
2
I
E
s
H
2
Imag
H
−
{
y
1
}=
+
Imag
{
n
1
}
(2.62)
2
s
1
R
s
2
R
E
s
G
2
Real
G
−
{
y
2
}=
+
Real
{
n
2
}
(2.63)
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