Resource Allocation in OFDM Based Wireless Relay Networks ...

2.4 Generalization to Multiple Relay Scenario
where Q n is the total power of the n-th RS.
With a little bit abuse of notations, we re-define τ = {τ (m,n),(k,j) }, p = {p m,n,k },
q = {q n,j }. Then the optimization can be formulated as
max
τ ,π,p,q
M∑
N∑
K∑
m=1 n=1 k=1 j=1
s.t. (2.31), (2.40), (2.41).
K∑
π k,j τ (m,n),(k,j) r (m,n),(k,j) (2.42)
2.4.1 Joint Optimization
We solve problem (2.42) using the similar techniques in Section 2.3.
function is
¯D (ν, λ n ) = max
π,τ ,p,q
M∑
N∑
K∑
m=1 n=1 k=1 j=1
s.t. (2.31), (2.40), (2.41),
K∑
π k,j τ (m,n),(k,j) e (m,n),(k,j) +
M∑
ν m P m +
where ν = [ν 1 , . . . , ν M ] T and λ = [λ 1 , . . . , λ N ] T are the dual variables, and
m=1
The dual
N∑
λ n Q n
n=1
(2.43)
e (m,n),(k,j) = r (m,n),(k,j) − ν m p m,n,k − λ n q n,j . (2.44)
For a given π, τ and under the high SNR approximation, (3.9) can be
decomposed into following MNK 2 sub-problems
max
p m,n,k ≥0,q n,j ≥0 log 2
(
1 + p )
m,n,ka m,n,k q n,j b n,j
− (ν m p m,n,k + λ n q n,j ) , ∀m, n, k, j.
p m,n,k a m,n,k + q n,j b n,j
(2.45)
Similar to the previous section, the KKT conditions yields the optimal p m,n,k and
q n,j as
p ∗ m,n,k =
q ∗ n,j =
ν m
λn
(1 +
λ n
ν m
(1 +
1
√ )
λn a m,n,k
ν m b n,j
1
√ )
νm b n,j
λ na m,n,k
⎛
⎛
⎜
⎝ 1 λ n
−
⎜
⎝ 1 −
ν m
( √am,n,k
√ ) 2
⎞
+ νm
λn
b n,j
⎟
⎠
a m,n,k b n,j
( √bn,j
+
√
λ n
a m,n,k b n,j
) 2
⎞
ν m
a m,n,k
⎟
⎠
+
+
. (2.46)
. (2.47)
33