Digital Signal Processing Reference
In-Depth Information
8
−
4
2
)
|
−
4
P
(
d
→
d
)
≤
1
|
Φ(
j
,
1
(2.75)
j
=
rotation matrix
R
such that
j
=
1
|
Φ(
0. The best known rotations for QAM
to maximize the minimum product distance are provided in [
4
]. Similarly, we can
prove that the diversity for User 2 is also 4. Therefore, our scheme can achieve
full diversity for each user. Similarly, it can be shown that the system provides full
diversity when we use Eqs. (
2.61
-
2.64
) to simplify the decoding complexity for
QAM.
j
,
1
)
| =
2.5 Extension to Two Users with More than Two Transmit
Antennas
In this section, we show that the scheme used for 2 users eachwith 2 transmit antennas
can also be extended to 2 users each with more than 2 transmit antennas. Assume we
have 2 users each with
N
=
2
n
transmit antennas. At the first
N
time slots, Users 1
and 2 send codewords
⎛
⎝
⎞
⎠
,
⎛
⎝
⎞
⎠
−
c
1
c
1
···
c
1
−
s
1
s
1
···
s
1
c
2
−
c
2
···
c
2
s
2
−
s
2
···
s
2
C
=
S
=
(2.76)
.
.
.
.
.
.
.
.
.
.
.
.
c
N
c
N
··· −
c
N
s
N
s
N
··· −
s
N
respectively. The received signals at time slot
i
,
i
=
1
,...,
N
, is denoted by
y
1
y
2
y
i
=
(2.77)
Within these
N
time slots, the channel matrices for Users 1 and 2 are
h
11
h
12
···
g
11
g
12
···
h
1
N
g
1
N
H
=
,
G
=
(2.78)
h
21
h
22
···
h
2
N
g
21
g
22
···
g
2
N
respectively. At time slot
i
,
i
=
1
,...,
N
, the precoders for Users 1 and 2 are
⎛
⎝
⎞
⎠
,
⎛
⎝
⎞
⎠
a
i
11
a
i
12
a
i
1
N
b
i
11
b
i
12
b
i
1
N
···
···
a
i
21
a
i
22
a
i
2
N
b
i
21
b
i
22
b
i
2
N
···
···
A
i
B
i
=
=
(2.79)
.
.
.
.
.
.
.
.
.
.
.
.
a
i
N
1
a
i
N
2
···
a
i
NN
b
i
N
1
b
i
N
2
···
b
i
NN
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