Resource Allocation in OFDM Based Wireless Relay Networks ...

4.3 Cooperative Non-Orthogonal Transmission
For a known p k , the structure of T (ρ j ) motivates us to apply the simple fixed point
iterative algorithm [67].
Firstly, observe the following properties of T (ρ j ). For ρ j > 0,
• Positivity: T (ρ j ) > 0.
• Monotonicity: if ρ j > ρ ′ j, then T (ρ j ) > T (ρ ′ j).
• Scalability: For all α > 1, αT (ρ (k,j) ) > T (αρ (k,j) ).
Thus, the fixed point iterations
√
ρ (i+1) 1
j =
λ T (ρ(i) j ). (4.23)
will converge to a unique value, denoted as ρ j [67]. Now that we have obtained the
optimum value ρ j for some given value of p k . Now for the obtained ρ j , we can easily
find the optimum value of p k , denoted as p k , through gradient descent method [13]
1 . Finally, an alternate optimization over p k and ρ j is considered and at convergence
the solutions are dented by ˆp k and ˆρ j .
Substituting ˆp k and ˆρ j into (4.18) yields
D (ν, λ) = max
π
K∑ K∑
π (k,j) ∆ (k,j) + µP s + λP R (4.24)
k=1 j=1
s.t. (4.4),
where ∆ (k,j) represents the objective value in (4.19) at ˆp k and ˆρ j where the analytical
expression for ∆ (k,j) is missing due to unavailability of closed-form solutions for
ˆp k and ˆρ j . We define a matrix with row indices being sub-carriers over the first
hop and column indices being sub-carriers over the second hop . The values of
each entry is the throughput obtained by pairing the corresponding sub-carriers.
Problem (4.24) is equivalent to maximizing the sum throughput by choosing the best
1 The relay power ρ j could also be obtained through gradient descent search, however fixed point
algorithm in (4.23) significantly reduces the number of iterations require for convergence, as will
be shown later in the simulations.
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