
Joint Beamfor ming and Power Allocation 127
tive kth substream power allocation factors for user 1,user2, and the relay,
respectively.
The joint power allocation problem using sum-rate criterion
2
can be for-
mulated as
maximize
˜a
1
,...,˜a
M
,
˜
b
1
,...,
˜
b
M
,˜c
1
,...˜,c
M
1
2
M
k=1
log
2
1+
t
1,k
˜
b
k
˜c
k
t
2,k
˜a
k
+ t
3,k
˜
b
k
+ t
4,k
˜c
k
+ t
5,k
+
1
2
M
k=1
log
2
1+
u
1,k
˜a
k
˜c
k
u
2,k
˜a
k
+ u
3,k
˜
b
k
+ u
4,k
˜c
k
+ u
5,k
,
(9.16)
subject to
M
k=1
˜a
k
+
˜
b
k
+˜c
k
≤ P, ˜a
k
≥ 0,
˜
b
k
≥ 0, ˜c
k
≥ 0,
∀
k
= {1,...,M}, (9.17)
where the constants
t
1,k
= λ
2
k
,t
2,k
= σ
2
1
β
k
λ
k
,t
3,k
= σ
2
1
α
k
λ
k
,
t
4,k
= σ
2
r
β
k
λ
k
,t
5,k
= σ
2
1
σ
2
r
α
k
β
k
,u
1,k
= t
1,k
= λ
2
k
,
u
2,k
= σ
2
2
β
k
λ
k
,u
3,k
= σ
2
2
α
k
λ
k
,u
4,k
= σ
2
r
α
k
λ
k
,
u
5,k
= σ
2
2
σ
2
r
α
k
β
k
,α
k
=
M
j=1
|
ˆ
F
1
(j, k)|
2
,β
k
=
M
j=1
|