Digital Signal Processing Reference
In-Depth Information
where
H
11
(
2
1
1
1
2
λ
=||
1
)
||
F
|
e
1
+
α
11
e
2
+
α
12
e
3
+
α
13
e
4
|
H
11
(
2
2
2
2
2
+||
1
)
||
F
|
α
11
e
1
−
e
2
+
α
12
e
3
−
α
13
e
4
|
H
11
(
2
3
3
3
2
+||
1
)
||
F
|
α
12
e
1
+
α
13
e
2
+
e
3
+
α
11
e
4
|
H
11
(
2
4
4
4
2
+||
1
)
||
F
|
α
13
e
1
−
α
12
e
2
+
α
11
e
3
−
e
4
|
(5.52)
Since
2
F
F
≥
||
H
1
||
1
H
11
(
2
||
1
)
||
·
(5.53)
1
1
1
4
2
2
2
1
+|
α
11
|
+|
α
12
|
+|
α
13
|
Inequality (
5.51
) can be written as
exp
(5.54)
exp
2
F
1
11
e
2
1
12
e
3
1
13
e
4
2
−
ρ
4
−
ρ
||
H
1
||
|
e
1
+
α
+
α
+
α
|
P
(
c
→
c
|
H
1
)
≤
1
11
1
12
1
13
16
(
1
+|
α
|
2
+|
α
|
2
+|
α
|
2
)
Therefore, we have
P
(
c
→
c
)
=
E
[
P
(
c
→
c
|
H
1
)
]
E
exp
2
1
1
1
2
−
ρ
||
H
1
||
F
|
e
1
+
α
11
e
2
+
α
12
e
3
+
α
13
e
4
|
=
1
1
1
16
(
1
+|
α
11
|
2
+|
α
12
|
2
+|
α
13
|
2
)
1
=
(5.55)
16
j
+
ρτ
1
[
1
16
]
=
where
1
1
1
2
τ
=
|
e
1
+
α
11
e
2
+
α
12
e
3
+
α
13
e
4
|
(5.56)
11
|
2
12
|
2
13
|
2
1
+|
α
+|
α
+|
α
At high SNR region, (
5.55
) can be written as
ρτ
16
−
16
P
(
c
→
c
)
≤
(5.57)
So the diversity is 16, full diversity, as long as
τ
=
0. Also the coding gain is affected
1
1
1
13
by
τ
and we can choose
α
11
,
α
12
,
α
properly to maximize
τ
. The best choice for
1
1
1
parameters
13
depends on the adopted constellation. Such an optimization
is a straightforward optimization that has been discussed in many existing literature
[
3
]. Similarly, we can prove that the diversity for other codewords is also full.
α
11
,
α
12
,
α
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