Digital Signal Processing Reference
In-Depth Information
where
2
ζ
=
|
γ
1
+
k
1
γ
2
+
k
2
γ
3
+
k
3
γ
4
|
(3.38)
+
j
=
1
k
j
1
At high SNRs, one can neglect the one in the denominator and get
ρζ
16
−
16
P
(
d
→
d
)
≤
(3.39)
Then, it is easy to show that the diversity is 16 if we simply choose
KR
such that
ζ
=
KR
ε
=
0
(3.40)
1
k
1
k
2
k
3
is a normalized vector. Therefore, by using
1
where
K
=
1
+
j
=
1
k
j
our scheme, User 1 can achieve full diversity. In addition, in order to maximize the
coding gain, we need to choose
KR
such that the minimum possible norm of
KR
is
maximized. For QAM, it is not hard to do so. For example, when QPSK is adopted,
we can simply choose
KR
ε
85
1248
. It is easy to check that the minimum
1
=
possible norm of
KR
is maximized. Similarly, we can also prove that the diversity
for Users 2, 3, 4, is 16 as well. Therefore, our scheme can achieve full diversity for
each user. When we use Equations (
3.28
), (
3.29
) to simplify the decoding complexity,
similar techniques can be used to complete the proof of full diversity. Note that our
precoding design procedure itself does not rely on the channel statistics. So using
our scheme, the pairwise error probability can always be upper bounded by
ε
E
exp
2
F
·
ζ
−
ρ
·||
H
1
||
(
→
)
≤
P
d
d
(3.41)
16
like Equation (
3.37
), where
H
1
is the channel matrix for User 1. This means that the
proposed procedure is universal in that it can achieve the maximum possible diversity
over any fading distribution.
3.4 Extension to
J
Users with
N
Transmit Antennas and One
Receiver with
M
Receive Antennas
In this section, we show that the presented scheme can be extended to a general case
of
J
users each with
N
transmit antennas and one receiver with
M
receive antennas.
For the simplification of presentation, we discuss 3 cases where among parameters
M
,
N
and
J
, two are the same and the third one is larger than the other two. It is easy
to extend the results to a general case. In addition, we just show our schemes when
Search WWH ::
Custom Search