Resource Allocation in OFDM Based Wireless Relay Networks ...

4.3 Cooperative Non-Orthogonal Transmission
The optimization can be stated as
max
π,p,w
C , s.t. (4.5), (4.6), (4.4), (4.7)
with π = {π (k,j) }, p = {p k }, w = {w n,j }, and p k ≥ 0, w n,j ≥ 0, for all n =
{1, ..., N}, k = {1, ..., K}, j = {1, ..., K}.
4.3.1 Joint Optimization Algorithm
Finding the power variables p k and the beamforming weights w n,j together seems
difficult.
Fortunately, the structure of the problem permits to first compute the
optimal beamforming coefficients for a given source power p k . Denote ρ j as the
total power allocated to sub-carrier j such that ∑ K
j=1 ρ j ≤ P R . Then (4.7) can be
re-stated as
max
π,p,w C (4.8)
s.t.
K∑ K∑ ∑ N K∑
|w n,j | 2 (p k |h n,k | 2 + σr) 2 ≤ ρ j ,
k=1 j=1
π (k,j)
n=1
K∑
ρ j ≤ P R , (4.6), (4.4).
j=1
Thus, for any sub-carrier pair (k, j) with known p k
problem can be stated as
max
w
s.t.
j=1
the optimum beamforming
log 2 (1 + SNR (k,j) ) (4.9)
N∑
|w n,j | 2 (p k |h n,k | 2 + σr) 2 ≤ ρ j . (4.10)
n=1
Since the logarithm is a monotonically increasing function, maximization of the
objective in (4.9) is equivalent to the maximization of (4.2). Defining
√
γn,k γ n,j
ψ = , ξ = 1 + γ n,k + γ n,j
. (4.11)
1 + γ n,j 1 + γ n,k
with
γ n,k = |h n,k| 2 p k
σ 2 r
, andγ n,j = |g n,j| 2 ρ j
,
σd
2
66