Resource Allocation in OFDM Based Wireless Relay Networks ...

2.3 Resource Allocation Schemes
and similarly the solution of (2.23) yields the optimal value of q j as
q ∗ j =
λ
ν m
(1 +
where (x) + max (0, x).
1
√ )
νbj
λa m,k
⎛
⎜
⎝ 1 −
ν m
( √bj
+
√
a m,k b j
) 2
λ
ν m
a m,k
⎞
⎟
⎠
+
, (2.25)
The above discussion shows that there are only two possible case, i.e., either both
p m,k and q j are positive or zero. In other words, if power allocated to a particular
sub-carrier over either hop is zero, no power is allocated to its corresponding
sub-carrier over the other hop.
Substituting (2.24) and (2.25) into (2.12), we obtain
D (ν, λ) = max
π,τ
s.t.
M∑ K∑ K∑
M∑
π k,j τ m,(k,j) F m,(k,j) (ν, λ) + ν m P m + λQ (2.26)
m=1 k=1 j=1
K∑
π k,j = 1, ∀j,
k=1
j=1
m=1
m=1
K∑
M∑
π k,j = 1, ∀k, τ m,(k,j) = 1, ∀(k, j),
where the function F m,(k,j) (ν, λ) is obtained by substituting p ∗ m,k and q∗ j into e m,(k,j) .
Next, we look into the optimum sub-carrier allocation under a given sub-carrier
pairing (k, j). The dual function (2.26) can be written as
D (ν, λ) = max
τ
s.t.
M∑
M∑
τ m,(k,j) F m,(k,j) (ν, λ) + ν m P m + λQ (2.27)
m=1
M∑
τ m,(k,j) = 1,
m=1
∀(k, j).
m=1
The optimal solution is simply choosing a user that has the maximum value of
F m,(k,j) (ν, λ). Let the user index be m ∗ , i.e.,
m ∗ = arg max
m F m,(k,j)(ν, λ), ∀(k, j). (2.28)
Then
⎧
⎨ 1, for m = m ∗
τm,(k,j) ∗ =
⎩ 0, otherwise.
(2.29)
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