Digital Signal Processing Reference
In-Depth Information
v
g
)
†
v
h
|
|
(
2
v
h
,
v
g
>
=
cos
θ
hg
=
<
v
h
|
.
(4.37)
v
g
|·|
|
i
i
Note that the maximum value of cos
θ
hg
is 1 and the corresponding
θ
hg
=
0, which
means
v
i
h
and
v
i
g
are orthogonal to each other.
Now we introduce our feedback scheme with the assumption that User 1 has
already got a codebook
Υ
1
in time slot 2. Also User
Υ
1
in time slot 1 and a codebook
Υ
2
in time slots 1 and 2, respectively. All these
codebooks should possess the property given by (
4.16
) and (
4.20
). In time slot 1,
the receiver selects an index
2 has already got codebooks
Υ
2
and
1
within the range from 0 to
L
1
−
1 and sends it back
v
h
|
to User 1. The selection criterion is that with such an index
1
,
|
is maximized,
v
h
|=|
HA
1
as given by (
4.22
) and
A
1
v
h
|
where
is
equivalent to maximizing the received SINR for User 1. Therefore, full diversity is
also achieved, as shown later. At the same time slot, the receiver also picks an index
2
and sends it back to User 2. The selection criterion is that with such an index
|
(
1
)
|
=
Υ
1
[
1
]
. Maximizing
|
1
hg
2
,
θ
−
B
1
g
21
−
g
22
hg
is given by (
4.36
) in which
v
g
=
1
)
∗
as
is minimized, where
θ
(
1
g
11
g
12
given by (
4.24
),
B
1
. We will show that by doing so, we can also maximize
coding gain within our system framework.
Similarly, in time slot 2, the receiver finds an index
=
Υ
2
[
2
]
2
and sends it back to User 2.
2
,
v
g
|
The selection criterion is that with such an index
|
is maximized. The receiver
1
and sends it back to User 1. The selection criterion is that with
also finds an index
1
,
2
such an index
θ
hg
is minimized.
4.3.2 Diversity Analysis
In what follows, we show that by the above proposed scheme, the diversity for each
user is full as long as our codebooks satisfy some conditions. The diversity is defined
as
log
P
e
log
d
=−
lim
ρ
→∞
(4.38)
ρ
ρ
where
denotes the SNR and
P
e
represents the probability of error. We first consider
Eq. (
4.29
) to analyze the diversity for User 1. We know
R
1
1
and we
c
1
=
c
2
c
2
.By(
4.29
), the pairwise error probability
(PEP) can be given by the Gaussian tail function as [
4
]
1
c
2
c
1
define the error matrix
ε
=
−
c
2
⎛
⎞
ρ
||
HR
1
ε
||
2
F
|
H
⎝
⎠
P
(
d
→
d
)
=
Q
4
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