Digital Signal Processing Reference
In-Depth Information
⎛
⎞
†
R
1
(
H
HR
1
ε
ρ
ε
)
†
⎝
⎠
=
Q
4
exp
†
R
1
(
H
HR
1
ε
†
−
ρ
ε
)
≤
(4.39)
8
x
2
where we have used the inequality
Q
(
x
)
≤
exp
(
−
2
)
. Now we assume
γ
1
γ
2
. Substituting
R
1
ε
and
H
from Eq. (
4.30
)in(
4.39
), we have
R
1
ε
=
exp
2
−
8
|
H
|
h
11
|
+|
h
21
|
2
2
2
P
(
d
→
d
)
≤
|
γ
1
+
γ
2
|
|
γ
1
−
γ
2
|
⎛
⎝
−
8
⎛
⎞
⎞
2
2
v
g
)
†
v
g
)
†
(
(
⎝
v
h
2
v
h
2
⎠
⎠
=
exp
|
γ
1
+
γ
2
|
+
|
γ
1
−
γ
2
|
v
g
|
v
g
|
|
|
2
⎛
⎝
v
h
⎞
⎠
2
v
g
)
†
(
ρ
|
γ
1
+
γ
2
|
v
g
|
|
≤
−
.
exp
(4.40)
8
Let us define
2
v
g
)
†
v
g
)
†
v
h
|
2
(
=
|
(
v
h
Δ
=
.
(4.41)
v
g
|
v
g
|
2
|
|
Using (
4.36
), we can rewrite
Δ
as
1
2
v
h
|
2
Δ
=|
cos
θ
hg
|
·|
.
(4.42)
Substituting (
4.42
)in(
4.40
), we have
exp
1
2
v
h
|
2
2
ρ(
|
cos
θ
hg
|
·|
|
γ
1
+
γ
2
|
)
|
H
P
(
d
→
d
)
≤
−
.
(4.43)
8
Since we choose our precoder
A
1
from the codebook
v
h
|
2
is maximized,
Υ
1
such that
|
it is easy to see
2
≥
|
H
Υ
1
|
v
h
|
2
HA
1
2
|
=|
(
1
)
|
(4.44)
L
Υ
1
is
a matrix satisfying
Υ
1
(
)
=
Υ
1
[
]
(
)
=
,...,
where
i
i
1
,
i
1
L
, i.e., the
i
th column
of matrix
Υ
1
is the same as the first column of the
i
th matrix in the codebook
Υ
1
.
We assume
Υ
1
has the following Singular Value Decomposition
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