Digital Signal Processing Reference
In-Depth Information
design precoders
A
1
,
A
2
,
B
1
,
B
2
to achieve interference-free transmission for each
codeword at both receivers as shown in Fig.
5.3
when we have at least six transmit
antennas.
5.3 Decoding
In the last section, we have shown how to achieve interference-free transmission.
Once interference-free transmission as shown in Fig.
5.3
is achieved, it is easy to
realize low-complexity decoding. In this section, we will show how to decode and
analyze the decoding complexity. Based on Eq. (
5.15
), after combining the channel
equations in the first four time slots and making some simple transformation, we
have
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
c
11
c
12
c
13
c
14
c
21
c
22
c
23
c
24
s
11
s
12
s
13
s
14
s
21
s
22
s
23
s
24
⎝
⎠
+
H
2
⎝
⎠
+
G
1
⎝
⎠
+
G
2
⎝
⎠
+
y
1
=
H
1
n
1
(5.36)
where
⎛
⎞
⎛
⎞
H
1
k
(
H
1
k
(
H
1
k
(
H
1
k
(
y
1
(
1
)
2
)
3
)
4
)
⎝
⎠
,
⎝
⎠
H
1
k
(
))
∗
−
(
H
1
k
(
))
∗
(
H
1
k
(
))
∗
−
(
H
1
k
(
))
∗
y
1
)
∗
y
1
(
(
2
1
4
3
H
k
=
y
1
=
H
1
k
(
H
1
k
(
H
1
k
(
H
1
k
(
3
)
4
)
1
)
2
)
H
1
k
(
H
1
k
(
H
1
k
(
H
1
k
(
y
1
)
∗
))
∗
−
(
))
∗
(
))
∗
−
(
))
∗
(
4
3
2
1
(5.37)
⎛
⎝
⎞
⎠
,
⎛
⎝
⎞
⎠
G
1
k
(
G
1
k
(
G
1
k
(
G
1
k
(
n
1
(
1
)
2
)
3
)
4
)
G
1
k
(
))
∗
−
(
G
1
k
(
))
∗
(
G
1
k
(
))
∗
−
(
G
1
k
(
))
∗
n
1
)
∗
n
1
(
(
2
1
4
3
G
k
=
n
1
=
G
1
k
(
G
1
k
(
G
1
k
(
G
1
k
(
3
)
4
)
1
)
2
)
G
1
k
(
))
∗
−
(
G
1
k
(
))
∗
(
G
1
k
(
))
∗
−
(
G
1
k
(
))
∗
n
1
)
∗
(
4
3
2
1
(5.38)
2. By the property of our precoders in Eqs. (
5.18
), (
5.20
-
5.22
),
H
k
and
G
k
can also be written as
where
k
=
1
,
⎛
⎞
H
1
k
(
11
H
1
k
(
12
H
1
k
(
13
H
1
k
(
1
)
α
1
)
α
1
)α
1
)
⎝
⎠
2
11
H
1
k
(
))
∗
−
(
H
1
k
(
))
∗
13
H
1
k
(
2
))
∗
−
(α
12
H
1
k
(
2
))
∗
(α
1
1
(α
1
1
H
k
=
(5.39)
12
H
1
k
(
3
3
13
H
1
k
(
H
1
k
(
3
11
H
1
k
(
α
1
)α
1
)
1
)
α
1
)
4
13
H
1
k
(
))
∗
−
(α
4
12
H
1
k
(
))
∗
(α
11
H
1
k
(
4
))
∗
−
(
H
1
k
(
))
∗
(α
1
1
1
1
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