Digital Signal Processing Reference
In-Depth Information
where
h
1
(
and
h
2
(
are elements of the equivalent channel matrices in Equa-
tion (
3.6
). Equation (
3.7
) can be rewritten as
i
,
j
)
i
,
j
)
⎛
⎝
⎞
⎠
=
H
1
⎛
⎝
⎞
⎠
∗
a
2
(
a
1
(
1
,
1
)
1
,
1
)
a
2
(
a
1
(
2
,
1
)
2
,
1
)
H
2
(3.8)
a
2
(
a
1
(
3
,
1
)
3
,
1
)
a
2
(
a
1
(
4
,
1
)
4
,
1
)
where
⎛
⎝
⎞
⎠
−
h
1
(
2
,
1
)
−
h
1
(
2
,
2
)
−
h
1
(
2
,
3
)
−
h
1
(
2
,
4
)
h
1
(
1
,
1
)
h
1
(
1
,
2
)
h
1
(
1
,
3
)
h
1
(
1
,
4
)
H
1
=
(3.9)
−
h
1
(
4
,
1
)
−
h
1
(
4
,
2
)
−
h
1
(
4
,
3
)
−
h
1
(
4
,
4
)
h
1
(
3
,
1
)
h
1
(
3
,
2
)
h
1
(
3
,
3
)
h
1
(
3
,
4
)
Now let
H
1
=
H
−
1
2
V
H
Q
=
U
(3.10)
where we have made the singular value decomposition. It has been proved in [
2
] that
⎛
⎝
⎞
⎠
=
⎛
⎝
⎞
⎠
=
a
1
(
a
2
(
1
,
1
)
1
,
1
)
a
1
(
a
2
(
1
(
2
,
1
)
2
,
1
)
)
∗
,
v
(
i
u
(
i
),
η
=
)
,
i
=
1
,
2
,
3
,
4
a
1
(
a
2
(
3
,
1
)
3
,
1
)
i
,
i
a
1
(
a
2
(
4
,
1
)
4
,
1
)
(3.11)
⎛
⎞
a
1
(
1
,
1
)
⎝
⎠
a
1
(
2
,
1
)
will satisfy Equation (
3.8
). There are four different choices for
and
a
1
(
,
)
3
1
a
1
(
4
,
1
)
⎛
⎞
a
2
(
1
,
1
)
⎝
⎠
a
2
(
2
,
1
)
depending on which
i
we pick. Different choices of
i
result in different
a
2
(
3
,
1
)
a
2
(
4
,
1
)
performances. For given channel matrices
H
1
and
H
2
,attimeslot1,welet
v
=
)
∗
,
v
(
i
, such that the norm of
H
1
v
is the largest, i.e.,
i
∈{
1
,
2
,
3
,
4
}
v
=
)
∗
||
2
F
arg
max
4
||
H
1
v
(
i
(3.12)
)
∗
,
v
(
i
i
=
1
,
2
,
3
,
ThenforUser1,attimeslot1,welet
⎛
⎝
⎞
⎠
=
⎛
⎝
⎞
⎠
=
⎛
⎝
⎞
⎠
,
a
1
(
a
1
(
,
i
)
a
1
(
,
)
,
)
1
1
1
1
1
v
1
+
j
=
1
k
j
,
a
1
(
a
1
(
i
)
a
1
(
2
,
1
)
2
,
2
,
1
)
i
=
k
i
−
1
·
2
,
3
,
4 (3.13)
a
1
(
a
1
(
i
)
a
1
(
4
,
i
)
a
1
(
)
a
1
(
4
,
1
)
3
,
1
3
,
)
a
1
(
4
,
1
)
3
,
1
For User 2, at time slot 1, we let
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