Digital Signal Processing Reference
In-Depth Information
D
13
D
23
D
21
c
21
D
11
c
21
Receiver 1
c
11
Receiver 2
c
11
c
22
c
12
c
22
c
12
c
13
c
23
c
23
c
13
c
24
c
14
c
24
c
14
s
24
s
14
s
13
s
23
s
22
s
21
s
24
s
14
s
12
s
23
D
22
s
11
s
13
s
22
s
21
D
14
s
12
D
24
D
12
s
11
Fig. 5.2
Signal vector illustration at two receivers
y
t
2
=
H
t
21
C
1
(
H
t
22
C
2
(
G
t
21
S
1
(
G
t
22
S
2
(
n
t
2
t
)
+
t
)
+
t
)
+
t
)
+
(5.17)
By Eq. (
5.15
), since the receiver has four receive antennas, each symbol is actually
transmitted along a 4-dimensional vector in a 4-dimensional space. Because each
user sends eight symbols at the same time, at the receiver, there are 16 signal vectors
in the four-dimensional space.
Since we want to send
C
1
,
C
2
,
S
1
,
S
2
without any interference from each other,
we let each one of
C
1
,
S
2
occupy only one dimension. In other words, for
any codeword, we should transmit each of the corresponding four symbols in the
same direction. In this way, there are only four transmit directions. Once we can
align the four transmit directions of
C
1
,
C
2
,
S
1
,
S
2
properly, we can separate them
completely. This is our first step to achieve interference-free transmission.
This idea is illustrated in Fig.
5.2
, where
D
ij
is the
j
th direction at Receiver
i
.By
Eq. (
5.15
),
c
11
,
C
2
,
S
1
,
c
14
are transmitted along
H
t
11
(
H
t
11
(
H
t
11
(
H
t
11
(
c
12
,
c
13
,
1
),
2
),
3
),
4
)
,
respectively. In order to make
H
t
11
(
H
t
11
(
H
t
11
(
H
t
11
(
1
),
2
),
3
),
4
)
along the same direc-
tion, by Eq. (
5.14
), we need
1
α
1
α
1
α
A
t
1
(
A
t
1
(
A
t
1
(
A
t
1
(
1
)
=
2
)
=
3
)
=
4
)
(5.18)
t
11
t
12
t
13
t
t
t
13
A
t
1
||
2
F
1
where
α
11
,α
12
,α
are constants that we will determine later. From
||
=
2
,
we know
1
A
t
1
(
2
||
)
||
F
=
1
(5.19)
t
t
t
2
2
2
2
(
1
+
(α
11
)
+
(α
12
)
+
(α
13
)
)
So when we design precoder
A
t
1
,Eqs.(
5.18
) and (
5.19
) should be satisfied. Similarly,
precoders
A
t
2
,
B
t
1
,
B
t
2
should also satisfy the following conditions:
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