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Optimization of Rate Allocation and Power Control for Rate Splitting Multiple Access (RSMA)

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

arXiv:1903.08068v1eess.SP

TL;DR

The paper addresses challenges in splitting common and private messages, resource management, and rate allocation for RSMA, formulating sum-rate optimization under rate and SIC constraints. It derives closed-form private-message power and a finite rate-allocation solution space, then uses one-dimensional search; simulations show higher sum-rate than NOMA and OFDMA, especially in demanding settings.

  • Problem

    RSMA faces challenges involving common/private message splitting, resource management, and rate allocation in SISO systems.

  • Method

    The paper formulates sum-rate optimization under rate and SIC constraints, derives closed-form private-message power and a finite rate-allocation solution space, and applies one-dimensional search.

  • Results

    RSMA achieves higher sum-rate than NOMA and OFDMA, with reported data-rate gains of up to 21.5% over OFDMA.

  • Takeaways & Limitations

    RSMA's sum-rate advantage is especially pronounced at low BS maximum transmit power and high minimum rate demand.

Abstract

from arXiv · show

In this paper, the sum-rate maximization problem is studied for wireless networks that use downlink rate splitting multiple access (RSMA). In the considered model, each base station (BS) divides the messages that must be transmitted to its users into a `private' part and a `common' part. Here, the common message is a message that all users want to receive and the private message is a message that is dedicated to only a specific user. The RSMA mechanism enables a BS to adjust the split of common and private messages so as to control the interference by decoding and treating interference as noise and, thus optimizing the data rate of users. To maximize the users' sum-rate, the network can determine the rate allocation for the common message to meet the rate demand, and adjust the transmit power for the private message to reduce the interference. This problem is formulated as an optimization problem whose goal is to maximize the sum-rate of all users. To solve this nonconvex maximization problem, the optimal power used for transmitting the private message to the users is first obtained in closed form for a given rate allocation and common message power. Based on the optimal private message transmission power, the optimal rate allocation is then derived under a fixed common message transmission power. Subsequently, a one-dimensional search algorithm is proposed to obtain the optimal solution of common message transmission power. Simulation results show that the RSMA can achieve up to 15.6\% and 21.5\% gains in terms of data rate compared to non-orthogonal multiple access (NOMA) and orthogonal frequency-division multiple access (OFDMA), respectively.

I. INTRODUCTION

The paper addresses optimal rate allocation and power control for downlink SISO RSMA, where messages are split into common and private parts to manage interference and maximize network sum-rate. It derives an optimized scheme and reports gains over NOMA and OFDMA.

  • RSMA motivation: RSMA divides each transmitted message into a common part decoded by all users and a private part intended for one user.Common-message decoding accounts for interference, whereas private-message decoding treats other users’ private messages as noise.
  • RSMA motivation: Adjusting the common–private split can control interference and affect both computational complexity and achieved data rate.The paper motivates RSMA as an alternative to NOMA, whose users must decode all interference, increasing signal-processing complexity.
  • Research gap: Existing RSMA studies generally obtain only suboptimal power control and do not jointly optimize rate allocation and power control for downlink SISO systems.They also do not consider the SIC constraint needed to guarantee successful common-message decoding.
  • Proposed approach: The paper proposes an optimized rate allocation and power control scheme for downlink SISO RSMA that maximizes network sum-rate under rate and SIC constraints.It optimizes common-message rate allocation and the transmit powers for common and private messages.
  • Proposed approach: The solution derives closed-form private-message power, characterizes a finite rate-allocation solution space, and uses a one-dimensional search algorithm.The search algorithm is shown to have linear complexity for equal rate demands.
  • Results: 15.6% and 21.5% gains in data rate are achieved over NOMA and OFDMA, respectively.The gains are reported for the optimized RSMA algorithm in simulation results.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system model uses RSMA, in which a base station transmits one common message decoded by all users and private messages decoded individually. Users first decode the common message, then their private messages, with rate constraints and SIC requirements defining feasible transmission.

  • Decoding process: Each user first decodes the common message and then decodes its private message.Private-message decoding occurs after common-message decoding at the receiver.
  • System model: The base station transmits a common message s0 and one private message sk for each user k.The common message is decoded by all users, while each private message is decoded only by its intended user.
  • Power and SIC constraints: The transmitted signal contains common-message power p0 and private-message powers pk, while SIC requires a minimum power difference for successful decoding.The total transmit power is distributed across the common and private messages.
  • Rate model: The common-message rate must support successful decoding by all users, with the weakest channel determining the limiting rate.Channel gains are ordered as h1 ≤ h2 ≤ · · · ≤ hK, and the common-message rate is selected accordingly.
  • Rate model: The total rate of user k combines its allocated common-message rate ak with its achievable private-message rate rk.The common-message allocations across users are constrained by the total common-message rate.

A. Problem Formulation

The paper formulates RSMA sum-rate maximization as a joint rate-allocation and power-control problem under total power, minimum-rate, common-decoding, and SIC constraints. Although the resulting problem is nonconcave and couples rate and power variables, the paper derives a globally optimal solution for the SISO setting and positions it as a benchmark against existing suboptimal approaches.

  • Optimization problem: The optimization jointly chooses common-message rate allocations and transmit powers to maximize the network sum-rate.The power vector includes common and private-message powers, while user rates must satisfy minimum demands.
  • Optimization problem: The formulation enforces common-message decodability, individual minimum rates, successful SIC, and a maximum BS transmit power.These requirements appear as constraints (10a)–(10d).
  • Optimization challenge: The objective is nonconcave, and rate and power variables are coupled in both the objective and constraints.These properties make problem (10) generally difficult to solve.
  • Contribution: For the SISO formulation, the paper derives a globally optimal solution that can serve as a benchmark for subsequent optimization with power control.The result addresses the coupled nonconvex problem despite its computational difficulty.
  • Research gap: Existing RSMA studies generally obtain only suboptimal resource-allocation or power-control solutions for sum-rate optimization.The paper identifies this limitation across the cited existing literature.

III. OPTIMAL RATE ALLOCATION AND POWER CONTROL

The solution procedure decomposes the nonconvex problem into sequential optimization stages. It first derives private-message power in closed form, then common-rate allocation in closed form for fixed common power, and finally searches over common-message power.

  • Solution procedure: The method first establishes optimal conditions for the original sum-rate maximization problem.These conditions provide the basis for the subsequent reductions.
  • Solution procedure: For a given rate allocation and common-message transmission power, the optimal private-message power is obtained in closed form.This eliminates the private-power variables from the next optimization stage.
  • Solution procedure: After substituting the optimal private power, the optimal common-message rate allocation is derived in closed form for fixed common-message power.The resulting allocation is used to reduce the remaining search dimension.
  • Solution procedure: A one-dimensional search over common-message transmission power obtains the optimal solution of the original problem.The complete process is summarized in Fig. 1.

A. Optimal Conditions

The optimality analysis supplies structural conditions that simplify the original problem. At an optimum, the common-message rate constraint and total-power constraint are tight, enabling an equivalent formulation with a closed-form private-power solution.

  • Optimal conditions: At the optimum, the common-message constraint holds with equality.The available common-message rate is fully accounted for by the allocated common rates.
  • Optimal conditions: At the optimum, the maximum transmit-power constraint also holds with equality.The total available BS power is exhausted by the common and private transmissions.
  • Problem reduction: Using these optimal conditions, the original problem is transformed into an equivalent optimization problem.The transformed formulation supports the subsequent derivation of private-message power.
  • Problem reduction: The reduced problem admits a closed-form solution for the optimal private-message transmission power.A maximum rate limitation 0 ≤ ak ≤ Rk is included to facilitate this derivation.

B. Optimal Private Message Transmission Power

For fixed rate allocation and common-message power, the nonconvex private-power problem admits a closed-form optimal solution that satisfies other users’ minimum-rate requirements while concentrating remaining power strategically.

  • The first objective term is constant for fixed common-message power, so it is omitted from the reduced private-power problem.The resulting problem is feasible exactly when the minimum-power requirements can be jointly satisfied.
  • The private-message power problem is nonconvex, but its optimal solution can be obtained in closed form.Theorem 2 gives the optimal power allocation and private-message sum-rate.
  • The optimal allocation gives minimum feasible power to other users and allocates additional power to the user maximizing the sum-rate.The selected user depends on channel conditions and rate requirements.
  • With equal rate allocation, all remaining power is allocated to the user with the highest channel gain.This is the special case described after Theorem 2.

C. Optimal Rate Allocation

With common-message power fixed, the paper derives optimal rate allocation by reducing the convex allocation objective to a finite set of corner-point candidates, including a closed-form equal-demand case.

  • For fixed common-message power, the rate-allocation objective is convex, so its maximum occurs at a corner point of the feasible region.This reduces the search to finitely many candidate allocations.
  • The optimal allocation lies in three cases determined by which feasibility constraints are active.The cases cover corner points satisfying the common constraint alone or together with additional rate constraints.
  • The finite candidate set contains 5 × 2^(K−1) potential solutions for the general rate-allocation problem.This count follows from the candidate counts associated with the three cases.
  • For equal user rate demands, the optimal rate allocation has a closed-form expression.Theorem 4 provides the allocation, and additional power is assigned to the highest-channel-gain user.
  • For two users, it is optimal to decode user 1’s message first in the common message and allocate the remaining power to user 2.In this setting, NOMA is a special case of RSMA under a particular common/private split.

D. Optimal Rate Allocation and Power Control

The proposed algorithm alternates closed-form rate and private-power solutions with a one-dimensional search over common-message power to solve the original optimization problem.

  • Algorithm 1 obtains optimal rate allocation and power control by searching over the common-message power.Given each candidate common-message power, rate allocation and private-message power are obtained in closed form.
  • For general rate demands, solving the rate-allocation subproblem has complexity O(5 × 2^(K−1)).This complexity follows from the finite candidate set in Theorem 3.
  • For equal rate demands, the rate-allocation step has complexity O(K), giving Algorithm 1 complexity O(LK).L denotes the number of iterations used to search common-message power.
  • The method is practical for small user counts because common-message decoding and candidate enumeration become more demanding as K grows.For larger networks, users can be grouped into smaller RSMA groups on separate subchannels.

IV. SIMULATION RESULTS

Simulations compare RSMA with NOMA and OFDMA across power, rate-demand, SIC-threshold, rate-region, and user-count settings. RSMA generally performs best, with reported gains up to 15.6% over NOMA and 21.5% over OFDMA.

  • Two-user behavior: For two users, RSMA matches NOMA in sum-rate, while its rate region allows a larger maximum rate for user 2.The rate-region difference is linked to RSMA’s flexible rate splitting and NOMA’s successful-SIC power constraint.
  • Power and rate demand: RSMA achieves the best sum-rate among the compared schemes for three-user networks with equal and unequal rate demands.Its advantage is especially pronounced at low maximum BS transmit power and relative to OFDMA across the full bandwidth.
  • Power and rate demand: RSMA’s sum-rate declines with increasing minimum rate demand, but less rapidly than NOMA and OFDMA.High demands require more power for users with worse channel gains, degrading the sum-rate.
  • Power and rate demand: 15.6% and 21.5% are the reported maximum data-rate gains of RSMA over NOMA and OFDMA, respectively.The gains are reported as minimum-rate demand increases, when the other schemes decline faster.
  • SIC sensitivity: As the SIC detection threshold increases, RSMA and NOMA lose sum-rate while OFDMA remains unchanged; RSMA degrades more slowly than NOMA.RSMA particularly outperforms NOMA at high SIC detection thresholds.
  • Distribution and scalability: RSMA and NOMA substantially improve over OFDMA in the high-sum-rate CDF region, while RSMA exceeds NOMA at moderate rates of 5–15 Mbits/s.The paper attributes RSMA’s moderate-rate advantage to adjusting the common/private split to control interference.
  • Distribution and scalability: RSMA increasingly outperforms NOMA and OFDMA as the number of users grows.The paper relates this to stronger multiuser gains and lower bandwidth or SIC limitations in the comparison schemes.

V. CONCLUSIONS

The paper formulates joint common-message rate allocation and common/private-message power control for a SISO RSMA system. It derives closed-form and finite-search solutions, with simulations showing higher RSMA sum-rate than NOMA and OFDMA under several conditions.

  • The study jointly optimizes common-message rate allocation and transmit power for common and private messages in a SISO RSMA system.
  • A closed-form optimal private-message transmit power is derived before characterizing the finite solution space for optimal rate allocation.
  • A one-dimensional search algorithm obtains the optimal rate allocation and power-control solutions.
  • RSMA achieves higher sum-rate than NOMA and OFDMA, especially at low BS maximum transmit power, high user minimum-rate demand, high SIC detection threshold, and many users.
  • The optimization analysis includes feasibility and equivalence arguments, convexity of an objective function, and boundary-characterized optimal solutions.

APPENDIX F PROOF OF THEOREM 4

Theorem 4 establishes structural properties of the optimal rate allocation and then applies them to several cases involving rate demands and common-message power. The proof uses monotonicity and convexity arguments to identify boundary solutions.

  • Equal rate demand: For equal rate demand, a contradiction argument supports the stated ordered structure of the optimal rate allocation.
  • Optimal rate allocation: The objective function is convex, and it increases with each rate allocation variable except the selected index.
  • Optimal rate allocation: The optimal allocation assigns rate to the user with the lowest channel gain.
  • Common-message power: In another case, the objective increases with common-message power, so the optimal value is the maximum feasible power.
  • Common-message power: Across the considered cases, the objective can decrease with common-message power, making the minimum feasible power optimal in those cases.
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