Digital Signal Processing Reference
In-Depth Information
, we can choose
b
11
b
21
as described
1
So we know that in order to maximize
|
cos
θ
hg
|
by (
4.56
) if we have perfect feedback. Since we only have quantized feedback, we
should design a codebook in which we can find a vector as close to the one described
by (
4.56
) as possible. So, first, we need to determine the distribution of the optimal
b
11
b
21
in (
4.56
). Note that Eq. (
4.56
) can also be written as
b
11
b
21
g
22
(
h
21
)
∗
−
(
h
11
)
∗
−
g
12
=
η
−
g
21
g
11
g
22
(
h
21
)
∗
+
g
12
(
h
11
)
∗
=
η
g
21
(
h
21
)
∗
−
g
11
(
h
11
)
∗
−
α
1
α
2
=
η
(4.57)
g
11
g
12
g
21
g
22
−
1
(
h
21
)
∗
−
(
h
11
)
∗
−
1
g
21
g
12
|
−
1
. Let us assume that
where
η
=
|
g
11
g
22
−
the singular value decomposition of
α
1
α
2
is
F
α
1
α
2
λ
1
0
U
α
α
V
†
=
λ
1
·
=
α
=
U
α
·
1
U
α
(
1
).
(4.58)
Since
h
11
and
h
21
are independent from
G
, conditioned on
h
11
and
h
21
, elements
of
α
1
α
2
are all Guassian distributed random variables with the same mean and
will be an isotropically distributed
unitary vector [
6
]. Further, we can conclude that
α
1
α
2
and thus
α
1
α
2
variance, so any column of
U
α
and thus
b
11
are all
b
21
isotropically distributed unitary vectors.
Therefore, in order to maximize
1
, the codebook for User 2 should pro-
vide the best approximation to any isotropically distributed unitary vector and the
problem becomes exactly the same as the one we discussed before, i.e., to pack one-
dimensional subspaces of a complex space known as Grassmannian line packing.
Therefore, the resulting codebook for User 2 will be the same as the codebook
|
cos
θ
hg
|
Υ
1
for User 1 at time slot 1.
So far, we have shown that by using our codebook, we can maximize
v
h
|
|
and
1
|
at the same time. From (
4.43
), it is easy to see that the coding gain is
maximized.
Similarly, we can prove that in time slot 2, both User 1 and User 2 should adopt
the above codebook.
cos
θ
hg
|
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