Resource Allocation in OFDM Based Wireless Relay Networks ...

3.4 Joint Resource Allocation Scheme
We solve (3.10) for all m, k, j, thus there are total MK 2 sub-problems. The power
allocation problem in (3.10) is non-convex and finding the closed form solution is not
trivial. Nevertheless, the optimal solution (ˆp A m,k , ˆpR j , ˆp B m,k ) can be obtained through
searching over p A m,k , pR j , and p B m,k , assuming that each takes discrete values [50], [51].
This approach requires O(Z 3 ) computational complexity where Z is the number of
power levels that can be taken by each of p A m,k , pR j , and p B m,k . Therefore the total
complexity of solving power allocation for all m, (k, j) is O(MK 2 Z 3 ).
Substituting optimal power values ˆp A m,k , ˆpR j , and ˆp B m,k
{
∑ M K∑ K∑
D(ν, λ, η) = max
π (k,j) τ m,(k,j) F m,(k,j) +
π,τ
m=1 k=1 j=1
M∑
∣ ∣∣∣ ∑ K
+ η m P Bm π (k,j) = 1, ∀j,
m=1
k=1
into (3.9), we obtain
M∑
ν m P Am + λP R
m=1
K∑
π (k,j) = 1, ∀k,
j=1
M∑
}
τ m,(k,j) = 1, ∀(k, j) , (3.11)
m=1
where F m,(k,j) is obtained by substituting ˆp A m,k , ˆpR j , and ˆp B m,k
1
C ( )
SNR A 2 m,j +
1
C ( SNR B 2 m,j)
− νm p A m,k − λpR j − η m p B m,k .
becomes
into the objective
To find the optimum sub-carrier allocation under a given tone matching, (3.11)
{
∑ M K∑ K∑
M∑
M∑
max
τ m,(k,j) F m,(k,j) + ν m P Am + λP R + η m P Bm
τ
m=1 k=1 j=1
m=1
m=1
M∑
}
∣ τ m,(k,j) = 1, ∀(k, j) . (3.12)
m=1
The optimal solution of (3.12) is obtained by choosing an MU pair that maximizes
F m,(k,j) , i.e.,
⎧
⎨ 1, for m = arg max m F m,(k,j) , ∀(k, j),
ˆτ m,(k,j) =
⎩ 0, otherwise.
(3.13)
For a given π (k,j) , each maximization operation in (3.13) has the complexity of O(M)
and the total complexity of solving sub-carrier allocation problem thus is O(MK 2 ).
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