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Sum-Rate Maximization of Uplink Rate Splitting Multiple Access (RSMA) Communication

Zhaohui Yang, Mingzhe Chen, Walid Saad, Wei Xu, Mohammad Shikh-Bahaei

arXiv:1906.04092v1cs.IT

TL;DR

The paper addresses uplink RSMA sum-rate maximization when users split messages and the BS must jointly choose transmit powers and decoding order under proportional rate constraints. It transforms the discrete-order problem into rate-splitting optimization, then recovers powers and searches decoding orders, while also proposing user pairing to reduce complexity. Simulations show RSMA achieves higher sum-rate than NOMA, FDMA, and TDMA.

  • Problem

    Uplink RSMA requires jointly optimizing power allocation and message decoding order to maximize sum-rate under proportional rate constraints.

  • Method

    The method replaces power and decoding variables with rate variables, derives optimal rate splitting, recovers transmit powers for each order, and exhaustively searches decoding orders.

  • Results

    RSMA achieves higher sum-rate than NOMA, FDMA, and TDMA, while user pairing provides a lower-complexity alternative.

  • Takeaways & Limitations

    The paper provides an uplink RSMA optimization framework and a user-pairing scheme for reducing the complexity of sum-rate maximization.

Abstract

from arXiv · show

In this paper, the problem of maximizing the wireless users' sum-rate for uplink rate splitting multiple access (RSMA) communications is studied. In the considered model, each user transmits a superposition of two messages to a base station (BS) with separate transmit power and the BS uses a successive decoding technique to decode the received messages. To maximize each user's transmission rate, the users must adjust their transmit power and the BS must determine the decoding order of the messages transmitted from the users to the BS. This problem is formulated as a sum-rate maximization problem with proportional rate constraints by adjusting the users' transmit power and the BS's decoding order. However, since the decoding order variable in the optimization problem is discrete, the original maximization problem with transmit power and decoding order variables can be transformed into a problem with only the rate splitting variable. Then, the optimal rate splitting of each user is determined. Given the optimal rate splitting of each user and a decoding order, the optimal transmit power of each user is calculated. Next, the optimal decoding order is determined by an exhaustive search method. To further reduce the complexity of the optimization algorithm used for sum-rate maximization in RSMA, a user pairing based algorithm is introduced, which enables two users to use RSMA in each pair and also enables the users in different pairs to be allocated with orthogonal frequency. For comparisons, the optimal sum-rate maximizing solutions with proportional rate constraints are obtained in closed form for non-orthogonal multiple access (NOMA), frequency division multiple access (FDMA), and time division multiple access (TDMA). Simulation results show that RSMA can achieve up to 10.0\%, 22.2\%, and 83.7\% gains in terms of sum-rate compared to NOMA, FDMA, and TDMA.

I. INTRODUCTION

Uplink RSMA remains less studied than downlink RSMA, despite its potential rate-region benefits and the importance of jointly optimizing power and decoding order. This paper develops an optimization framework, a lower-complexity pairing scheme, and closed-form baselines for comparison.

  • Existing RSMA research has largely focused on downlink systems rather than uplink data transmission.
  • Uplink RSMA requires joint optimization of power management and message decoding order because decoding order affects users’ transmission rates.
  • The paper formulates uplink RSMA power allocation and decoding as sum-rate maximization under proportional rate constraints.
  • The proposed solution removes the discrete decoding-order variable through rate-splitting optimization, derives transmit powers for a given order, and exhaustively searches decoding orders.
  • A user-pairing RSMA scheme reduces computational complexity while approaching the sum-rate performance of RSMA without user pairing.
  • The paper also derives closed-form sum-rate solutions for NOMA, FDMA, and TDMA and compares them with RSMA.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models a single-cell uplink in which each user superposes two messages and the BS decodes them by SIC. The optimization maximizes sum-rate under proportional fairness, but discrete decoding orders and nonconvex constraints make the problem difficult.

  • System model: The considered system is a single-cell uplink with one BS serving K users through RSMA.
  • System model: Each user transmits a superposition of two messages with separate message powers, and the BS decodes all messages using SIC.
  • System model: Users have individual maximum transmission-power limits, while the received signal combines channel gains, transmitted messages, and additive white Gaussian noise.
  • Decoding order: The decoding order is represented by a permutation over the 2K transmitted messages, with each message assigned an order position.
  • Optimization problem: The objective is to maximize users’ sum-rate subject to proportional rate constraints defined by predetermined nonnegative fairness values.
  • Motivation: Although RSMA can theoretically reach the optimal rate region, prior work had not provided a practical algorithm for jointly computing uplink decoding order and power allocation.
  • Optimization problem: The formulation is a nonconvex mixed-integer problem because it combines a nonlinear equality constraint with the discrete decoding-order variable.
  • Contribution: The paper develops an algorithm intended to obtain the globally optimal solution to the formulated problem.

A. Optimal Sum-Rate Maximization

The proposed solution first replaces power and decoding variables with rate variables, obtains optimal rate splitting in closed form, and then reconstructs powers and decoding order. Fixed-order power feasibility is handled through nonconvex optimization and exhaustive order search.

  • Rate-variable transformation: The original problem is transformed into an equivalent formulation whose variables are users’ rates rather than transmit powers and decoding order.
  • Rate-variable transformation: Replacing the discrete decoding order with rate variables reduces the optimization dimension and removes the discrete variable from the transformed problem.
  • Closed-form rate solution: The optimal sum-rate and user rates are characterized through a closed-form solution of the transformed problem.
  • Power and order recovery: After obtaining optimal rates, the method reconstructs the original transmit powers and decoding order through a feasibility problem.
  • Power and order recovery: For a fixed decoding order, power allocation is obtained first, and the method then exhaustively searches the possible decoding orders.
  • Feasibility test: The fixed-order feasibility problem is tested through an auxiliary optimization whose objective threshold determines feasibility.
  • Nonconvex optimization: Because the auxiliary problem is nonconvex, the method uses difference-of-convex optimization and multiple initial points to obtain a near-global solution.
  • Nonconvex optimization: A concave first-order approximation converts the nonconvex constraints into a convex subproblem solvable by an interior-point method.

B. Complexity Analysis

The algorithm’s complexity is driven mainly by exhaustive decoding-order search, motivating user pairing for larger networks. The paper also identifies a theoretical ordering of the compared access schemes’ sum-rate performance.

  • Complexity drivers: The main computational burden lies in solving the transformed rate problem and the original problem across decoding orders.
  • Complexity drivers: The rate-variable problem has complexity O(2^K − 1), reflecting enumeration of nonempty user subsets.
  • Complexity drivers: For N initial points, solving the fixed-order problem has complexity O(NK^3(2K)!/2^K).
  • Complexity drivers: The total complexity of Algorithm 1 is O(2^K + NK^3(2K)!/2^K).
  • Complexity reduction: The algorithm is considered practical for small K, where the number of users is limited to reduce SIC complexity.
  • Complexity reduction: For larger networks, users can be divided into small groups occupying different frequency bands, with RSMA applied within each group.
  • Special case: For K = 2, the optimal decoding order and power allocation can be obtained in closed form.

C. RSMA with Two Users

For two-user RSMA, the paper characterizes the achievable rate region and derives optimal power allocation under boundary-rate cases. It also introduces a lower-complexity pairing-based scheme that applies RSMA within pairs and orthogonal frequency allocation across pairs.

  • Two-user rate region: The two-user RSMA rate region is expressed through boundary-rate cases covering fixed individual rates and the maximum-sum-rate boundary.The cases include r1 = R1, r2 = R2, and r1 + r2 = Rmax.
  • Low-complexity method: The boundary-point computation has high complexity, motivating a low-complexity method for obtaining all boundary points.
  • Two-user rate region: Only one user needs to transmit a superposition code of two messages in two-user RSMA, while the other transmits one message.Without loss of generality, user 1 transmits only s11, with zero power assigned to s12.
  • Two-user rate region: The optimal decoding order for two-user RSMA is s21, s11, and s22.
  • Low-complexity method: The proposed pairing scheme groups K = 2M users into M pairs, uses RSMA within each pair, and allocates different frequency bands to different pairs.

V. SUM-RATE MAXIMIZATION FOR UPLINK NOMA/FDMA/TDMA

The paper evaluates RSMA against uplink NOMA, FDMA, and TDMA by solving the corresponding sum-rate maximization problems for comparison.

  • For performance evaluation, the paper solves sum-rate maximization problems for uplink NOMA, FDMA, and TDMA.

A. NOMA

The paper formulates uplink NOMA sum-rate maximization under successive interference cancellation and maximum transmit-power constraints. It provides an optimal solution characterization for this problem.

  • In uplink NOMA, the BS decodes users in descending channel-gain order, first decoding high-gain users and then low-gain users after interference subtraction.
  • Each NOMA user's transmit power is constrained by its maximum power limit qk ≤ Pk.
  • The paper formulates the NOMA sum-rate maximization problem and states its optimal solution in Theorem 2.
  • The optimal NOMA solution uses bisection to effectively obtain the relevant τk solution because the governing right-hand side increases monotonically with τk.
  • Comparison formulation: In FDMA, users receive bandwidth fractions, transmit at maximum power because there is no inter-user interference, and optimize the resulting sum-rate problem.
  • Comparison formulation: The FDMA optimization has an optimal solution characterized by Theorem 3.

C. TDMA

The section formulates the TDMA sum-rate problem through time-allocation fractions and states its optimal solution. It then compares the achievable rate regions and sum-rates of RSMA, NOMA, FDMA, and TDMA under proportional rate constraints.

  • TDMA formulation: TDMA assigns each user a fraction of time to use the full base-station bandwidth.The allocation vector is a = [a1, a2, · · · , aK]T.
  • TDMA formulation: The uplink TDMA sum-rate maximization problem is formulated using the users’ time-allocation fractions.The section states that the optimal solution is given in a theorem.
  • Sum-rate comparison: The optimal sum-rates satisfy τ_RSMA ≥ τ_NOMA and τ_RSMA ≥ τ_FDMA ≥ τ_TDMA.This ordering follows by constructing feasible solutions in RSMA from NOMA or FDMA solutions, and in FDMA from TDMA solutions.
  • Rate-region comparison: For two users, RSMA’s rate region is characterized by individual-rate bounds and a sum-rate bound, while NOMA, FDMA, and TDMA satisfy a different feasibility relation.The comparison considers the special case K = 2.
  • Rate-region comparison: RSMA has the largest rate region, whereas TDMA has the smallest among the illustrated multiple-access schemes.The observation is made for the example shown in Fig. 2.
  • Sum-rate comparison: Under proportional rate constraints, the optimal rate point is found by intersecting each scheme’s rate region with the corresponding line through the origin.The resulting comparison shows RSMA with the best sum-rate performance and TDMA with the worst.

VI. NUMERICAL RESULTS

The numerical results compare RSMA with NOMA, FDMA, TDMA, and paired RSMA across transmit power, proportional-rate parameters, bandwidth, and network size. RSMA generally achieves the highest sum-rate, while RSMA-UP-SW offers a lower-complexity alternative with similar performance.

  • Two-user comparisons: RSMA achieves the best sum-rate among the evaluated multiple-access schemes in the two-user transmit-power experiment.It improves sum-rate by up to 4.1%, 10.2%, and 28.8% over NOMA, FDMA, and TDMA, respectively.
  • Two-user comparisons: RSMA remains best as the proportional-rate parameter D2 varies, while TDMA has the worst sum-rate.NOMA is below FDMA for D2 < 0.2 but above FDMA for D2 > 0.2.
  • Bandwidth effects: RSMA achieves higher sum-rate than NOMA, FDMA, and TDMA across the tested BS bandwidths, especially at large bandwidth.Sum-rate rises rapidly at small bandwidths and more slowly at larger bandwidths because higher bandwidth increases noise power.
  • User pairing: For ten users, strong-weak pairing gives RSMA-UP the highest sum-rate among the evaluated pairing methods.The results indicate that pairing users with distinctive channel gains benefits sum-rate maximization.
  • User pairing: RSMA, RSMA-UP-SW, and NOMA outperform FDMA particularly in high-sum-rate regions, and RSMA-UP-SW performs similarly to RSMA.RSMA-UP-SW has much lower complexity than full RSMA.
  • Network-size effects: 10.0%, 22.2%, and 83.7% are RSMA's maximum sum-rate gains over NOMA, FDMA, and TDMA, respectively.RSMA-UP-SW achieves gains of up to 4.1%, 11.6%, and 66.8% over the same baselines.
  • Network-size effects: RSMA's advantage over conventional schemes becomes more pronounced as the number of users increases.The paper attributes this to RSMA's power splitting, whereas FDMA and TDMA allocate less bandwidth or time per user at larger user counts.

VII. CONCLUSION

The paper formulates uplink RSMA sum-rate maximization with proportional rate constraints and optimizes rate splitting, transmit power, and decoding order. It also proposes a low-complexity user-pairing scheme, and simulations show higher sum-rate than NOMA, FDMA, and TDMA.

  • The paper investigates decoding-order and power optimization for uplink RSMA under proportional rate constraints.
  • The optimization is transformed into an equivalent problem with only rate-splitting variables, yielding a closed-form optimal solution.
  • Given optimal rate requirements and a decoding order, the method obtains transmit powers and searches decoding orders exhaustively.
  • A low-complexity RSMA scheme with user pairing is proposed to reduce the computational complexity of sum-rate maximization.
  • Simulation results show that RSMA achieves higher sum-rate than NOMA, FDMA, and TDMA.

APPENDIX A

The appendix establishes optimality through feasibility and contradiction arguments, then derives optimal power, bandwidth, and time-sharing solutions. It also gives rate regions for NOMA, FDMA, and TDMA.

  • Optimality proof: A contradiction argument shows that any alternative feasible solution with a better objective would violate the optimality of (τ*, r*).The constructed solution satisfies the relevant constraints while improving the objective, establishing the lemma's claim.
  • Power allocation cases: When user 1 reaches its maximum rate, p11 = P1 and p22 = 0, with p21 calculated from (B.2).
  • Power allocation cases: When user 2 reaches its maximum rate, p21 = 0 and p22 = P2, with p11 calculated from (B.1).
  • Power allocation cases: When r1 + r2 = Rmax, both users transmit maximum power, with p11 = P1 and p21 + p22 = P2.The individual powers are then obtained by first calculating p22 and setting p21 = P2 − p22.
  • Continuous-resource optimization: The optimal τ* is selected from the power constraints because the relevant expression increases monotonically with τ, and optimal transmission power follows from (38).The appendix identifies τk through (39) and gives τ* in (37).
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