Resource Allocation in OFDM Based Wireless Relay Networks ...

4.3 Cooperative Non-Orthogonal Transmission
the problem in (4.9) can be written as
max
w
s.t.
| ∑ N
n=1 w n,jψ| 2
∑ N
n=1 |w (4.12)
n,j| 2 ξ
N∑
|w n,j | 2 (p k |h n,k | 2 + σr) 2 ≤ ρ j . (4.13)
n=1
The problem becomes similar to the optimization in [58] and with some
mathematical calculations, the the optimal value of w n,j cab be computed as
ŵ n,j =
(
σ 2 r|g n,j | 2 + η n,k
ρ j
√
ρj h ∗ n,k g∗ n,j
) ⎛ ⎝
√ ∑ N
n=1
⎞
|h n,k | 2 |g n,j | 2 η n,k
σ
d
2 (
) 2
⎠
σr|g 2 n,j | 2 + η n,k
ρ j
, (4.14)
where η n,k σd 2(p k|h n,k | 2 + σr) 2 and (.) ∗ denotes the complex conjugate.
Substituting w n,j = ŵ n,j in (4.2) yields
N∑ p k |h n,k | 2 ρ j |g n,j | 2
SNR k,j =
p k |h n,k | 2 σd 2 + ρ . (4.15)
j|g n,j | 2 σr 2 + σrσ 2 d
2
n=1
Thus, the problem (4.7) becomes
max
π,p,ρ
1
2
K∑
k=1 j=1
s.t. (4.6), (4.4),
K∑ ( )
π (k,j) log 2 1 + SNRk,j
K∑
ρ j ≤ P R ,
j=1
(4.16)
where ρ = {ρ j }, ∀j.
The Lagrangian of (4.16) under power constraints can be written as
L (p, ρ, π, µ, λ)
K∑ K∑ π (k,j)
=
2
k=1 j=1
(
) (
log 2 1 + SNR (k,j) + µ P S −
K∑
k=1
) (
p k + λ P R −
K∑
ρ j
),
where µ and λ are the Lagrange multipliers associated with the corresponding power
constraints. The dual function of (4.16) can be expressed as [13]
j=1
D (µ, λ) = max
π,p,ρ
L (p, ρ, π, µ, λ) , s.t. (4.4). (4.17)
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