3.3 Designs for Relay-Noise Dominant Conditions
4.2.3 Iterative Greedy Optimization
To handle the optimization for (4.27) we consider iterative greedy algorithm which grad-ually improve the cost function in (4.27). The idea is to select one element in θ, say θ(k), as varying variable and optimize it in each iteration, then pick another θ(j) j 6= k in next iteration. To clearly separate and express the varying and invariant parts, we define
u , (Σ−1A−1)(k), (4.29)
uo , X
∀˜k6=k
(Σ−1A−1)(˜k)θ(˜k), (4.30)
θ∆ ∈ CM, θ∆(i) , exp(j∠u(i) − j∠uo(i)), (4.31)
where ∠u(i) denotes the phase of u(i). With these definitions we rewrite the square of
Differentiate (4.32) with respect to ∠θ(k) yields
∂kΣ−1A−1θk2
To find optimized ∠θ(k) such that (4.32) is minimized, we derive the root of (4.33). The solution would be
Note due to the nature of trigonometric functions we would find two solutions (roots) in the range 0 ≤ ∠θ(k) ≤ 2π, and the one corresponding to smaller kΣ−1A−1θk is the desired θ(k) optimizer. As described earlier, in each iteration we pick one θ(k) as varying variable and repeat (4.32)-(4.34) closed-forms to optimize kΣ−1A−1θk.
4.2.4 Algorithms Summary and Complexity Comparison
Similar to what we presented in Sec. 4.1.4, now we summarize the procedures and compare the complexity of two algorithms for destination-noise dominating model. First, the algorithm based on relay selection in Sec. 4.2.1 could be described as follows:
1. Set ML combinations to represent all possible selection of choosing M relays out of L.
2. For the ith combination (1 ≤ i ≤ ML), we define Fi and Gi as the submatrix of F and G respectively and is corresponding to the selected M relays for the ith selection. Also we denote ri as the corresponding M relay gains.
3. For the ith combination, we compute the individual relay gains ri following (4.16) then calculate the capacity with (4.15).
4. Repeat step 3 for all the combinations then selected the one with largest capacity and set relay gains accordingly.
The computationally demanding step 3 requires twice matrix determinant compu-tation for M × M matrices, and would be repeated for ML
combinations. Thus the computation burden is of order O(LMM3).
For the algorithm discussed in Sec. 4.2.2 and 4.2.3, the procedures could be summarized as follows:
1. Define HR, HN, S, VN, VR, A and Σ as denoted in Sec. 4.2.2.
2. Define θ by randomly choosing all the terms from [0, 2π]
3. Set iteration index i = 1
4. Set index k = mod (i, M ) + 1, where mod (i, M ) means the value of i modulo M .
5. With index k, define u, uo and θ∆ based on (4.29), (4.30) and (4.31).
6. Calculate θ(k) based on (4.34).
7. Increase i by one and go back to step 4, or stop if certain criteria are satisfied.
8. Using the final result of θ, we set r based on (4.27).
The iterative operations between step 4 and 7 do not require complicate computa-tion, and typically would converge within 10 iterations. Thus we may ignore this part for complexity assessment. In step 8 there are multiple matrix multiplication and inversion.
Note A is unitary, and S, Σ are diagonal matrices. Thus the corresponding matrix com-putation is simple. The maximal matrix size for the remaining matrix multiplication in step 8 is L, so the overall complexity of step 8 is of order O(L3). In step 1 we need SVD computation for a M × (L − M (M −1)2 ) matrix , which require [42, p. 234]
{L − M (M − 1)
2 }2(2M − 1) + 6{L − M (M − 1)
2 }3 (4.35)
flops and is of order O(L3). Thus we could conclude that the order of overall computation is of O(L3).
4.3 Numerical Results
To evaluate the performance of algorithms described in Sec. 4.1 and Sec. 4.2, we simulate random channels and perform capacity optimization by the two selection-based methods and proposed efficient designs. Selection-based method for relay-noise dominating and destination-noise dominating are marked as ’relay dom selec’ and ’desti dom selec’, re-spectively. The approach maximizing ratio of norms in Sec. 4.1.3 is marked as ’relay dom MRN’, while the algorithm based on lower-triangular matrix zero forcing in Sec. 4.2.2 is marked as ’desti dom ZF’. For all the simulations we set M = 3. The resulting capacity measures are used to generate cumulative distribution function (CDF) curves so that we could compare the capacity distribution of various system configurations and algorithms.
Each curve shows the distribution of 103 channel realizations. For benchmarking purpose we also simulate a simple relay gains design which set all gains with identical value and scale relay the transmission power according to (2.5).
Since handling relay network with larger size is one of our motivations to develop efficient designs, two relay sizes are simulated so that we could examine if any worth noting difference shown between the sizes, where solid and dash lines are for L = 9 and L = 18, respectively.
In Fig. 4.3 we simulate the relay system with relay-noise dominating condition by setting PR = 100, σR = 0.1 and σD = 0.01. Not surprisingly the algorithms devel-oped for relay-noise dominating, as described in Sec. 4.1, performs better than those for destination-noise dominating. For L = 18 the algorithm maximizing ratio of norms shows slight performance degradation compared to selection-based method, but would save con-siderable computation. Hence the proposed algorithm would be beneficial for large relay systems given noise modeling fit presumed condition.
Next we consider relaying under destination-noise dominating condition and set PR = 1, σR = 0.01 and σD = 0.1. Again algorithms with mismatched model perform worse than those with correct noise model. Note with larger relay size , the approach based on lower-triangle matrix zero forcing shows better results than selection-based method, which suggests that for large relay system the proposed algorithm not only works efficiently but also demonstrates powerful performance by collaborating the whole relay network, while selection-based method could only utilize a small portion of relays thus results in inferior capacity.
Finally in Fig. 4.5, a particular noise condition, set by PR = 10, σR = 0.1 and σD = 0.01, is chosen to examine if the algorithms presented in Sec. 4.1 and Sec. 4.2 still work properly when noise model is neither relay-noise or destination-noise dominant. Note that noise dominant condition is affected not only by ratio of σR and σD, but also PR and L. Thus for generic noise condition (none noise dominates) it would not be easy to fully investigate and conclude the superiority between algorithms. But for this particular noise condition we could observe that in general the proposed algorithms results in better
18 20 22 24 26 28 30 32 10−2
10−1 100
capacity (bps/Hz)
CDF
PR=100, σR=0.1, σD=0.01
relay dom selec relay dom MRN desti dom selec desti dom ZF equal gains dash: L = 18
solid: L = 9
Fig. 4.3: Simulate under relay-noise dominating condition.
5 10 15 20 25 30
10−2 10−1 100
capacity (bps/Hz)
CDF
PR=1, σR=0.01, σD=0.1
relay dom selec relay dom MRN desti dom selec desti dom ZF equal gains dash: L = 18
solid: L = 9
Fig. 4.4: Simulate under destination-noise dominating condition.
system capacity than equal gains design. It is shown in this simulation that the proposed algorithms performs robustly against model mismatch.
14 16 18 20 22 24 26 28 30 32 34 10−2
10−1 100
capacity (bps/Hz)
CDF
PR=10, σR=0.1, σD=0.1
relay dom selec relay dom MRN desti dom selec desti dom ZF equal gains dash: L = 18
solid: L = 9
Fig. 4.5: Simulation with two noise terms having equal power level