Resource Allocation in OFDM Based Wireless Relay Networks ...

2.3 Resource Allocation Schemes
(
)
log 2 1 + a m,kp m,k b j q j
1+a m,k p m,k +b j q j
(3.10) becomes
with concave function log 2
(
1 + a m,kp m,k b j q j
a m,k p m,k +b j q j
). Thus,
(
max log 2 1 + a )
m,kp m,k b j q j
− ν m p m,k − λq j (2.15)
p m,k ,q j a m,k p m,k + b j q j
s.t. p m,k ≥ 0, q j ≥ 0,
for all m, k, j. In particular, [38] demonstrated that the resource allocation obtained
from (2.15) yields very close results to the actual throughput even in the low SNR
scenario.
Since the above problem is now in the convex form, we refer to KKT
conditions and first define the Lagrangian J associated with (2.15) such that
(
J = log 2 1 + a )
m,kp m,k b j q j
+ (α m,k − ν m )p m,k + (β j − λ)q j . (2.16)
a m,k p m,k + b j q j
The KKT conditions can be written as follows
p m,k ≥ 0
α m,k ≥ 0
α m,k p m,k = 0
( (
∂
log
∂p 2 1 + a )
)
m,kp m,k b j q j
+ (α m,k − ν m )p m,k + (β j − λ)q j = 0 (2.17)
m,k a m,k p m,k + b j q j
q j ≥ 0
β j ≥ 0
β j q j = 0
( (
∂
log
∂q 2 1 + a )
)
m,kp m,k b j q j
+ (α m,k − ν m )p m,k + (β j − λ)q j = 0.
j a m,k p m,k + b j q j
From the fourth condition, we get
α m,k =ν m −
From α m,k ≥ 0, we obtain
ν m ≥
a m,k q 2 j b 2 j
(p m,k a m,k + q j b j ) (p m,k a m,k + q j b j + p m,k a m,k q j b j ) . (2.18)
a m,k q 2 j b 2 j
(q j b j ) 2 + p m,k (p m,k a 2 m,k + a m,kq j b j (p m,k a m,k + q j b j + 2)) . (2.19)
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