Digital Signal Processing Reference
In-Depth Information
⎛
⎝
⎞
⎠
G
1
k
(
11
G
1
k
(
12
G
1
k
(
13
G
1
k
(
1
)
β
1
)
β
1
)β
1
)
11
G
1
k
(
2
))
∗
−
(
G
1
k
(
))
∗
13
G
1
k
(
2
))
∗
−
(β
12
G
1
k
(
2
))
∗
(β
(β
1
1
1
1
G
k
=
(5.40)
12
G
1
k
(
3
13
G
1
k
(
3
G
1
k
(
11
G
1
k
(
3
β
1
)β
1
)
1
)
β
1
)
13
G
1
k
(
4
))
∗
−
(β
12
G
1
k
(
4
))
∗
(β
11
G
1
k
(
4
))
∗
−
(
G
1
k
(
))
∗
(β
1
1
1
1
where
k
=
1
,
2. In addition, by using our precoders, we have
H
t
11
(
†
H
t
12
(
H
t
11
(
†
G
t
11
(
H
t
11
(
†
G
t
12
(
G
t
11
(
†
H
t
12
(
(
))
)
=
(
))
)
=
(
))
)
=
(
))
)
1
1
1
1
1
1
1
1
G
t
11
(
†
G
t
12
(
=
(
1
))
1
)
=
0
(5.41)
where
t
=
1
,
2
,
3
,
4. Equation (
5.41
) means that at time slots 1, 2, 3, 4, codewords
c
11
,...,
c
14
are transmitted along a direction that is orthogonal to the directions of
other codewords. Also codewords
s
11
,...,
s
14
are transmitted along a direction that
is orthogonal to the directions of other codewords. So we can multiply
y
1
by
(
H
11
(
1
))
†
,
H
11
(
|
1
))
|
by
(
H
11
(
1
))
T
by
(
H
11
(
1
))
†
by
(
H
11
(
1
))
T
y
1
)
∗
, multiply
y
1
and multiply
y
1
multiply
(
to
H
11
(
H
11
(
H
11
(
|
1
))
|
|
1
))
|
|
1
))
|
remove the interference. The result will be
⎛
⎞
c
11
c
12
c
13
c
14
⎝
⎠
+
y
1
=
H
1
n
1
(5.42)
where
⎛
⎞
1
1
1
a
α
11
a
α
12
a
α
13
a
⎝
⎠
11
)
∗
2
2
13
)
∗
b
2
12
)
∗
b
(α
−
b
(α
−
(α
H
1
=
(5.43)
3
3
3
α
12
c
α
13
c
c
α
11
c
13
)
∗
d
4
12
)
∗
d
4
4
11
)
∗
d
(α
−
(α
(α
−
d
H
11
(
H
11
(
H
11
(
H
11
(
a
=||
1
)
||
F
,
b
=||
1
)
||
F
,
c
=||
1
)
||
F
,
d
=||
1
)
||
F
(5.44)
⎛
⎞
⎛
⎞
H
11
(
H
11
(
†
†
(
1
))
(
1
))
y
1
n
1
|
H
11
(
|
H
11
(
⎝
))
|
⎠
⎝
))
|
⎠
1
1
H
11
(
H
11
(
T
T
1
)
(
1
))
y
1
)
∗
n
1
)
∗
))
|
(
))
|
(
|
H
11
(
|
H
11
(
1
1
y
1
=
,
n
1
=
(5.45)
H
11
(
H
11
(
†
†
(
1
))
(
1
))
y
1
n
1
|
H
11
(
|
H
11
(
))
|
))
|
1
1
H
11
(
H
11
(
T
T
(
1
))
(
1
))
y
1
)
∗
n
1
)
∗
))
|
(
))
|
(
|
H
11
(
|
H
11
(
1
1
t
If
4, are all real, from (
5.43
), it is easy to see that the
equivalent channel matrix
H
1
is real. So if QAM is used, Eq. (
5.42
) is equivalent to
the following two equations
α
1
j
,
j
=
1
,
2
,
3,
t
=
1
,
2
,
3
,
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