Resource Allocation in OFDM Based Wireless Relay Networks ...

Appendix A
Derivations of Closed-Form
Expression for Power Allocation
at the Relays
The Lagrangian ¯J associated with (4.37) is
(
)
¯J = 1
N∑
N + 1 log a n,k p k b n,j q n,j
2 1 +
− λp k −
a n,k p k + b n,j q n,j
n=1
N∑
ν n q n,j + α k p k +
n=1
N∑
β n q n,j ,
n=1
(A.1)
where α k and β n are the Lagrange multipliers associated with the power constraints
in (4.37). Taking the derivative of ¯J w.r.t. q n,j and setting
∂ ¯J
∂q n,j
= 0, we get
β n = ν n −
From slackness condition, we obtain
⎛
q n,j
⎝ν n −
|a n,k | 2 p 2 k b n,j
(a n,k p k +b n,j q n,j ) 2
(N + 1)(1 + ∑ N
n=1
|a n,k | 2 p 2 k b n,j
(a n,k p k +b n,j q n,j ) 2
(N + 1)(1 + ∑ N
n=1
a n,k p k b n,j q n,j
a n,k p k +b n,j q n,j
) .
⎞
a n,k p k b n,j q n,j
a n,k p k +b n,j q n,j
) .
⎠ = 0.
Now, the KKT condition β n ≥ 0 along with (A.3) implies that for q n,j > 0
ν n =
|a n,k | 2 p 2 k b n,j
(a n,k p k +b n,j q n,j ) 2
(N + 1)(1 + ∑ N
n=1
a n,k p k b n,j q n,j
a n,k p k +b n,j q n,j
) .
and q n,j = 0 otherwise. Solving (A.4) for q n,j , we obtain expression in (4.38).
(A.2)
(A.3)
(A.4)
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