Digital Signal Processing Reference
In-Depth Information
Note that
δ
1
+
δ
2
=
ϕ
1
+
ϕ
2
+
ϕ
3
, then by (
2.96
), it is easy to show that
2
·
min
{
δ
1
,δ
2
}
3
≤
ϕ
i
≤
2
·
max
{
δ
1
,δ
2
}
(2.100)
which results in
|
H
(
→
)
P
d
d
exp
(
|
λ
+|
λ
(
|
λ
+|
λ
2
i
2
i
2
2
i
2
i
2
−
ρ(Φ(
1
,
1
)
11
|
21
|
)
+
Φ(
2
,
1
)
12
|
22
|
))
≤
8
exp
exp
−
ρϕ
i
8
−
ρ
·
min
{
δ
1
,δ
2
}
12
≤
≤
(2.101)
and therefore
E
exp
Pr
E
P
d
|
H
≤
−
ρ
·
δ
2
12
P
(
d
→
d
)
=
→
d
{
δ
1
>δ
2
}
E
exp
Pr
−
ρ
·
δ
1
12
+
{
δ
1
<δ
2
}
(2.102)
Let
V
1
,
V
2
,
V
3
denote the unitary matrices derived from the singular value decompo-
sition in (
2.30
) respectively for the three cases. Conditioned on
V
1
,
V
2
,
V
3
, it can be
shown that
λ
11
,
λ
21
,
λ
21
,
λ
12
,
λ
22
,
λ
22
are i.i.d complex Gaussian random variables
with mean 0 and variance 1. The same claim holds for
λ
11
,
λ
21
,
λ
11
,
λ
12
,
λ
22
,
λ
12
as
well. Then similar to (
2.72
), we have
E
exp
,
V
3
E
exp
V
3
−
ρ
·
δ
i
12
−
ρ
·
δ
i
12
V
1
V
2
=
E
V
1
,
,
,
V
2
1
≤
(2.103)
j
=
1
[
2
3
1
+
(ρ
|
φ
j
|
/
12
)
]
Substituting (
2.103
)in(
2.102
), at high SNRs, we get
12
−
6
2
1
|
φ
j
|
−
6
P
(
d
→
d
)
≤
(2.104)
j
=
As a result, the diversity
d
≥
6. Similarly we can prove that
d
≤
6. Therefore,
d
=
6
and we can achieve full diversity for User 1.
Now we prove that we can also achieve full diversity for User 2. Similar to (
2.50
),
when there are 3 receive antennas, the channel equations can be written as
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