Joint Subcarrier Pairing and Power Allocation for OFDM Transmission with Decode-and-Forward Relaying

• The paper addresses joint power allocation and subcarrier pairing for DF relaying with the source–destination link and destination combining, a combination the authors say was not previously studied.Source: §I · p. 3
• The framework considers both total and individual source and relay power constraints.Source: §I · p. 3
• The joint problem is formulated as mixed integer programming and solved in the dual domain after continuous relaxation, with algorithms proposed to obtain feasible allocations.The approach uses system properties and continuous relaxation, then applies a subgradient method to the dual problem.Source: §I · p. 3
• When relay transmission on a subcarrier would not improve weighted sum rate, the model allows the source to send additional messages on idle subcarriers in the second slot.This extension jointly optimizes pairing and power allocation with weighted rates.Source: §I · p. 3
• For the extended setting, simulations show that the duality gap is nearly zero.Source: §I · p. 3
• The proposed algorithms nearly achieve the optimal weighted sum rate and outperform SCP in various channel conditions.The paper identifies SCP as the method proposed in.Source: §I · p. 3

• The system has one source, one half-duplex relay, and one destination; source and relay transmit in successive time slots.Source: §II · p. 4
• Each first-slot source subcarrier is paired with one second-slot relay subcarrier to carry a message, and the paired subcarriers need not be the same.The paper calls such a pair SP (k,m).Source: §II · p. 4
• The destination maximum-ratio combines source and relay signals belonging to the same message, while messages on different pairs are independent.Source: §II · p. 4
• The source is assumed to know the normalized channel gains, which remain constant across a two-slot period, and allocates resources using weighted sum rate.The weights reflect different priorities or quality-of-service requirements.Source: §II · pp. 4–5
• The baseline model leaves a second-slot subcarrier idle when the relay does not use it; the later model permits additional source messages there.Source: §II · p. 5

III. WEIGHTED SUM RATE MAXIMIZATION UNDER TOTAL POWER CONSTRAINT

• The section formulates joint subcarrier pairing and power allocation to maximize weighted sum rate under a total power constraint.Source: §III · p. 5

A. Primal Problem Formulation

• For a selected pair, the rate formulation accounts for relay and direct-link modes, using the equality of relay- and destination-received information to unify the relay-mode expressions.Source: A. Primal Problem Formulation · p. 6
• The resulting optimization is a mixed integer programming problem, so the paper relaxes the binary pairing indicator into a continuous time-sharing factor.Source: A. Primal Problem Formulation · pp. 6–7
• The relaxed objective is concave and forms a convex program, but the relaxed indicator may not be integer-valued; the dual method supplies an upper bound and is used to solve the problem.Source: A. Primal Problem Formulation · p. 7

B. Dual Problem

• The dual procedure computes power for each possible pair, scores each pairing, selects the highest-scoring second-hop subcarrier per first-hop subcarrier, and updates dual variables by subgradient iterations.The dual variables converge to the dual optimum; the resulting pairing and powers satisfy the stated constraints.Source: B. Dual Problem · p. 8
• A higher weighting factor allocates more power to its corresponding subcarrier pair.Source: B. Dual Problem · p. 9

C. Discussion on the Duality Gap

• As total power changes, optimal pairing can change at transition points, producing discrete changes in the weighted-sum-rate curve’s slope and possible nonconcavity.The authors note that this nonconcavity may result in a nonzero duality gap, even when all subcarrier weights are equal.Source: C. Discussion on the Duality Gap · p. 9
• The original, relaxed, and relaxed-dual optimal values satisfy RB ≤ RR ≤ DR, with DR also the dual optimum value for the original problem’s relaxation.Source: C. Discussion on the Duality Gap · p. 9
• When M = 2, about 1% of channel realizations are nonconcave; when M = 4, the probability is about 0.4%.Source: C. Discussion on the Duality Gap · p. 9
• For M ≥6, the maximum weighted sum rate is almost always concave in total power, and RB ≈ RR ≈ DR for most practical OFDM systems.The authors report that the duality gap is virtually zero in this setting.Source: C. Discussion on the Duality Gap · p. 9

D. Algorithm Design

• When multiple source subcarriers select one relay subcarrier, the pairing violates a constraint; the amendment moves redundant assignments to unused columns while retaining the largest-X assignment.It seeks moves with minimal change in α_m and selects destinations with large X_k,m values.Source: D. Algorithm Design · pp. 9–10
• During the final 10% of iterations, the algorithm generates feasible pairings, applies regular water-filling, and outputs the best pairing and power allocation.Source: D. Algorithm Design · p. 10
• The algorithm in Table I obtains a weighted sum rate quite close to the optimal rate.Source: D. Algorithm Design · p. 10

E. Complexity Comparison

• The algorithm’s overall complexity is O(JM^2), compared with O(M·M!) for exhaustive search.J is the number of iterations; the paper characterizes the proposed complexity as more feasible and tractable.Source: E. Complexity Comparison · pp. 10–11

IV. WEIGHTED SUM RATE MAXIMIZATION UNDER INDIVIDUAL POWER CONSTRAINTS

• Relay use depends on channel conditions and indirectly on the source and relay power constraints, so subcarrier pairs cannot be classified into direct-link or relay modes in advance.The section formulates the optimization under separate source and relay power constraints.Source: §IV · p. 11
• The unified weighted-rate and equivalent-channel expressions can be applied to the dual problem, which can then be solved similarly to the total-power-constrained case.The formulation represents the active direct-link and relay-mode subcarrier pairs while satisfying their additional pairing constraints.Source: §IV · p. 11

A. Unified Rate Formulation

• For fixed source and relay multipliers, each subcarrier pair is classified as direct-link, relay, or intermediate mode; intermediate mode equalizes information received at the relay and destination.An intermediate-mode pair contributes the same Lagrangian value whether assigned to direct-link or relay mode, but its powers must still be allocated accordingly.Source: A. Unified Rate Formulation · pp. 12, 14

B. Dual Problem

• When an optimal solution contains an intermediate-mode pair, assigning it to relay mode can make the subgradient updates oscillate around the optimal multipliers.The assignment increases relay power and decreases source power, which can push the multipliers in opposite directions in subsequent iterations.Source: B. Dual Problem · p. 16
• The feasible-solution algorithm uses an amendment procedure, and the resulting subcarrier pairing is reported to be near optimal.With pairing fixed, the zero-crossing method can then obtain the optimal relay-to-source multiplier ratio and corresponding mode and power allocation.Source: B. Dual Problem · p. 17
• The duality gap approaches zero when the number of subcarriers is reasonably large.The paper reports this result from simulation.Source: B. Dual Problem · p. 17

V. WEIGHTED SUM RATE MAXIMIZATION WITH EXTRA DIRECT-LINK TRANSMISSION

• In the preceding system, direct-link-mode pairs leave the second time slot unused because only the relay transmits in that slot.The modification allows source transmission on those idle subcarriers.Source: §V · p. 17
• The source may send extra messages in the second time slot on subcarriers left idle by direct-link-mode pairs.The modified system is considered under both total and individual power constraints.Source: §V · p. 17

A. Total Power Constraint

• The relay-use condition depends on channel gains and power allocation, so the paper introduces a relay-use indicator and jointly optimizes it with the relaxed weighted-sum-rate problem.The indicator is one when the relay is used for a subcarrier pair and zero otherwise.Source: A. Total Power Constraint · p. 17
• The feasible-solution algorithm directly uses the relay-use indicators found during iteration with the amended pairing, making the procedure suboptimal because those indicators depend on power allocation.The algorithm avoids the complexity of jointly optimizing the indicators and power allocation for fixed pairing.Source: A. Total Power Constraint · p. 19
• The duality gap is virtually zero when the number of subcarriers is reasonably large.The paper reports this result for the total-power-constrained problem.Source: A. Total Power Constraint · p. 19

B. Individual Power Constraints

• For individual constraints, the paper combines earlier results with modifications to relay-mode power allocation and the Lagrangian comparison.The procedure updates µS, µR, and α using the power allocation; if an intermediate-mode subcarrier pair occurs, it must satisfy (32), and a zero-crossing method obtains the optimal µR/µS.Source: B. Individual Power Constraints · pp. 20–21
• The intermediate mode has the relay receiving more information than the destination, with (31) strict; the relay mode satisfies (31) with equality.The direct-link rate with extra second-slot transmission is at least its earlier rate formulation at the same power, since zero second-slot power recovers the earlier case.Source: B. Individual Power Constraints · p. 20
• Because variable pairing and rate weights make an exact equal-power-cost comparison complicated, the paper compares direct-link and relay contributions to the Lagrangian instead.With extra second-slot source-destination transmission, the direct-link mode may be selected even when the earlier condition based on a_SR and µS holds.Source: B. Individual Power Constraints · p. 20
• If the direct-link allocation gives zero power to the second-slot source-destination transmission, mode selection reverts to Section IV, and the intermediate mode can be assigned to the relay mode.The unified rate formulation from Section IV-A can then be applied.Source: B. Individual Power Constraints · p. 20
• Mode selection can generally be reduced to direct-link and relay modes; an intermediate mode may remain when the conditions in (32) apply.The intermediate mode can disappear when direct-link selection covers its region, except when the optimal direct-link allocation gives zero power to the second-slot transmission.Source: B. Individual Power Constraints · pp. 20–21

VI. SIMULATION RESULTS

• Iterations before the amendment algorithm is triggered range from a few hundred for fewer than 10 subcarriers to slightly more than 10,000 for 64 subcarriers.Source: §VI · p. 22
• The proposed algorithms nearly achieve the dual optimum across cases; even with 4 subcarriers, the averaged duality gap is hardly noticeable and is zero with high probability.The results support a virtually zero duality gap when the number of subcarriers is reasonably large.Source: §VI · p. 22
• Across cases, fixed pairing loses substantially, rates rise with subcarrier count, total-power constraints outperform individual constraints, and extra direct-link transmission always improves performance.The rate increase is attributed to frequency diversity and greater pairing flexibility; the total-power advantage reflects more flexible power allocation.Source: §VI · p. 22
• The SCP is nearly optimal in Figs. 3 and 9 but notably worse in Fig. 6, where ignoring the SD link in pairing makes its mode selection resemble random pairing.In Fig. 6, the SCP still outperforms fixed pairing because it helps subcarrier pairs in relay mode.Source: §VI · p. 23
• Under individual power constraints or weighted-sum-rate evaluation, gaps between the SCP and proposed algorithms become larger; the original SCP is unsuitable for weighted sum rate, and Fig. 7’s weighted-SCP gap stems from random pairing.Source: §VI · p. 23
• When extra direct-link transmission is allowed, the SCP performs worse than the proposed algorithm because it does not account for the benefits of that transmission.The passage identifies additional diversity as one such benefit.Source: §VI · p. 23

• Under total power constraints, SCP is similar to or slightly worse than fixed pairing because it relays more SPs, foregoing extra direct-link messages; this effect is absent when the SD link is weak.Source: §VI · p. 24
• Under individual power constraints, SCP always outperforms fixed pairing, but its advantage is smaller in Fig. 11 than Fig. 9 because more SPs use direct-link transmission, where better SR-RD pairing adds no rate.Source: §VI · p. 24

VII. CONCLUSION

• The paper jointly optimizes subcarrier pairing and power allocation for weighted sum rate, accounting for the source-destination link and destination combining.The authors state that this joint problem had not previously been solved.Source: §VII · p. 24
• Both total and individual source and relay power constraints are considered.Source: §VII · p. 24
• The paper also investigates sending additional source messages over subcarriers left idle by the relay in the second time slot.Source: §VII · p. 24
• The optimization uses continuous relaxation and a dual method; a subgradient method finds Lagrange multipliers and helps obtain primal feasible solutions.Source: §VII · p. 24
• The proposed algorithms achieve nearly optimal weighted sum rates and outperform the method in under various channel conditions.The algorithms provide tractable-complexity feasible subcarrier-pairing schemes and corresponding power allocations.Source: §VII · p. 24