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Reconfigurable Intelligent Surface Empowered Downlink Non-Orthogonal Multiple Access

Min Fu, Yong Zhou, Yuanming Shi, Khaled B. Letaief

arXiv:1910.07361v3eess.SPcs.ITcs.NImath.IT

TL;DR

Power-domain NOMA can have limited gains when users’ channel conditions are similar, although practical scenarios may not provide the channel differences existing schemes assume. This paper jointly designs BS beamforming and RIS phase shifts, using an alternating DC method and a user-ordering scheme. The authors report that RIS deployment is effective for NOMA energy efficiency and that low-resolution RIS phases perform almost like continuous phase shifts.

  • Problem

    Existing NOMA schemes often assume substantially different user channel conditions, an assumption that may not hold in practical propagation environments.

    Source: §I · p. 1

  • Method

    The paper jointly optimizes BS beamforming and RIS phase shifts using an alternating DC method, and proposes a closed-form user-ordering scheme.

    Source: A. Contributions · p. 2

  • Results

    The authors report that RIS deployment enhances energy efficiency in NOMA and that low-resolution RIS phases achieve almost the same performance as continuous phase shifts.

    Source: §VIII · pp. 11–12

  • Takeaways & Limitations

    The study’s initial investigation demonstrates RIS effectiveness for energy efficiency enhancement in NOMA networks.

    Source: §VIII · p. 11

    Further work is identified for channel estimation and extending the framework to multiple BSs for practical implementations.

    Source: §VIII · p. 12

Abstract

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Power-domain non-orthogonal multiple access (NOMA) has become a promising technology to exploit the new dimension of the power domain to enhance the spectral efficiency of wireless networks. However, most existing NOMA schemes rely on the strong assumption that users' channel gains are quite different, which may be invalid in practice. To unleash the potential of power-domain NOMA, we will propose a reconfigurable intelligent surface (RIS)-empowered NOMA network to introduce desirable channel gain differences among the users by adjusting the phase shifts at RIS. Our goal is to minimize the total transmit power by jointly optimizing the beamforming vectors at the base station and the phase-shift matrix at the RIS. To address the highly coupled optimization variables, we present an alternating optimization framework to decompose the non-convex bi-quadratically constrained quadratic problem into two rank-one constrained matrices optimization problems via matrix lifting. At the same time, to accurately detect the feasibility of the non-convex rank-one constraints and improve performance by avoiding early stopping in the alternating optimization procedure, we equivalently represent the rank-one constraint as the difference between nuclear norm and spectral norm. A difference-of-convex (DC) algorithm is further developed to solve the resulting DC programs via successive convex relaxation, followed by establishing the convergence of the proposed DC-based alternating optimization method. We further propose an efficient user ordering scheme with closed-form expressions, considering both the channel conditions and users' target data rates. Simulation results will validate the effectiveness of deploying an RIS and the superiority of the proposed DC-based alternating optimization method in reducing the total transmit power.

I. INTRODUCTION

The paper addresses the reliance of power-domain NOMA on channel differences that may not occur in practice, and proposes using an RIS to shape user channels while jointly optimizing transmit beamforming and RIS phases. It also introduces a DC-based alternating optimization approach and a user-ordering scheme for the resulting design.

I. INTRODUCTION

  • Similar user channel conditions can limit NOMA’s performance gain over OMA, while practical propagation may not provide the channel differences existing schemes assume.The paper motivates RIS control as a way to introduce desirable channel differences among users.Source: §I · pp. 1–2

A. Contributions

  • The paper jointly optimizes BS beamforming and RIS phase shifts to minimize transmit power under users’ target rates, despite coupled bi-quadratic QoS and unit-modulus constraints.It proposes an efficient user-ordering criterion that accounts for combined channel conditions and target data rates.Source: §I · p. 2, A. Contributions · p. 2

1) RIS-empowered OMA Networks:

  • Prior RIS-empowered OMA work spans coverage extension, energy-efficient beamforming, physical-layer security, and massive connectivity.The cited studies also report gains in energy efficiency, transmit power, rate, and secrecy-related measures for their respective settings.Source: 1) RIS-empowered OMA Networks · pp. 2–3

2) RIS-empowered NOMA Networks:

  • Related RIS-empowered NOMA studies examine reliability, fairness, power consumption, power efficiency, sum rate, and throughput objectives.The introduction distinguishes this paper’s focus on joint power minimization and user ordering from those parallel efforts.Source: A. Contributions · p. 2, 2) RIS-empowered NOMA Networks · p. 3

C. Organization

  • The paper organizes its approach across system modeling, alternating optimization, DC reformulation and convergence, user ordering, numerical results, and conclusions.The numerical results are presented in Section VII, followed by conclusions in Section VIII.Source: C. Organization · p. 3

A. System Model

  • The system is an overloaded downlink MISO NOMA network with an M-antenna BS, an RIS, and K users, where K > M.Each user’s effective channel depends on direct and reflected links and the RIS phase shifts; SIC follows the selected user ordering.Source: A. System Model · pp. 3–4

B. Problem Formulation

The problem formulation minimizes total transmit power jointly over BS beamforming vectors and the RIS phase-shift matrix while meeting users’ rate requirements. It also introduces a lower-complexity user-ordering scheme to reduce the search over possible decoding orders.

B. Problem Formulation

  • The objective is to minimize total transmit power over BS beamforming vectors and RIS phase shifts while meeting each user’s minimum data rate.The RIS elements are subject to unit-modulus constraints, and the QoS constraints couple the beamforming and phase-shift variables.Source: B. Problem Formulation · p. 4
  • The joint problem remains intractable because its bi-quadratic constraints couple the design variables and its RIS phase shifts obey non-convex unit-modulus constraints.The formulation rewrites the rate constraints to make them more tractable before applying the alternating DC framework.Source: B. Problem Formulation · p. 4
  • The proposed closed-form user-ordering criterion achieves almost the same performance as exhaustive search while reducing computational complexity.The exhaustive-search reference selects the ordering with minimum total transmit power among possible user orderings.Source: B. Problem Formulation · pp. 4–5

III. ALTERNATING OPTIMIZATION FRAMEWORK

Alternating optimization splits the joint design into separate non-convex QCQP subproblems for the BS beamformers and RIS phase shifts. Both require feasibility handling that standard relaxations may not provide reliably.

III. ALTERNATING OPTIMIZATION FRAMEWORK

  • Alternating optimization decomposes the joint design into a beamforming QCQP and a phase-shift QCQP feasibility problem, both with non-convex constraints.The beamforming subproblem optimizes transmit vectors for a fixed RIS phase-shift matrix; the phase-shift subproblem fixes the beamformers.Source: §III · p. 5

A. Transmit Beamforming Vectors Optimization

  • A phase rotation cannot turn the beamforming subproblem into an equivalent SOCP because it cannot satisfy all required real-part conditions simultaneously.An SOCP approximation is possible, but the text says it produces a suboptimal solution with performance degradation.Source: A. Transmit Beamforming Vectors… · p. 5
  • SDR drops the beamforming matrices’ rank-one constraints; if the returned matrices are not rank one, Gaussian randomization can degrade performance and often fails to meet rank constraints at high dimension.The existing methods also may not accurately detect feasibility of multiple rank-one constraints, risking early stopping in alternating optimization.Source: A. Transmit Beamforming Vectors… · p. 5

B. Phase-Shift Matrix Optimization

  • The phase-shift feasibility problem combines quadratic constraints with unit-modulus constraints on each RIS element.Lifting and SDR can fail to yield a rank-one solution, especially as the number of RIS elements grows; randomization may not satisfy the original quadratic constraints.Source: B. Phase-Shift Matrix Optimization · p. 5
  • The method’s design objective is accurate feasibility detection for both subproblems, intended to prevent early termination and improve performance over state-of-the-art methods.The paper identifies feasibility detection for beamforming rank constraints and for phase-shift quadratic constraints as the two coupled challenges.Source: B. Phase-Shift Matrix Optimization · pp. 5–6

IV. A UNIFIED DIFFERENCE-OF-CONVEX PROGRAMMING METHOD

The paper develops a unified DC approach for the non-convex beamforming and phase-shift subproblems. It uses matrix lifting and an exact nuclear-norm-minus-spectral-norm representation to enforce rank one.

IV. A UNIFIED DIFFERENCE-OF-CONVEX PROGRAMMING METHOD

  • The unified DC method targets accurate feasibility detection for both non-convex QCQP subproblems, avoiding early termination in alternating optimization.The authors state that this is intended to improve performance.Source: §IV · p. 6

A. DC Representation for Rank-One Constraint

  • For a positive-trace PSD matrix X, rank(X) = 1 exactly when ∥X∥∗ − ∥X∥2 = 0.The nuclear norm minus the spectral norm supplies the rank-one condition’s exact DC representation.Source: A. DC Representation for Rank-One… · p. 6

B. Proposed Unified DC Method

  • Matrix lifting turns the beamforming vectors and RIS phase variables into PSD matrix variables, whose rank-one constraints are handled through the DC representation.When the phase-shift objective reaches zero, the method obtains an exact rank-one feasible matrix and recovers a phase vector by Cholesky decomposition.Source: B. Proposed Unified DC Method · p. 6
  • The authors state that the alternating DC method converges to a stationary point and outperforms alternating SDR in numerical simulations.This section frames the method as solving the lifted subproblems while avoiding early stopping.Source: B. Proposed Unified DC Method · p. 6

V. ALTERNATING DC ALGORITHM WITH CONVERGENCE GUARANTEE

The paper solves the lifted DC subproblems with successive convex relaxations of primal and dual formulations. Its alternating algorithm updates beamformers and RIS phases and is designed with a convergence guarantee.

V. ALTERNATING DC ALGORITHM WITH CONVERGENCE GUARANTEE

  • The DC algorithm rewrites its objective as the difference of two strongly convex functions to develop high-quality solutions with a convergence guarantee.The paper states that the algorithm converges to a stationary solution in each iteration.Source: §V · p. 6

A. Difference of Strongly Convex Functions Representation

B. DC Algorithm for Problem (28)

  • The alternating DC algorithm applies the DC procedure to beamformers and RIS phases in turn, obtaining lifted solutions that satisfy rank-one constraints.It stops when total-power decrease falls below a preset threshold or the phase-shift feasibility problem becomes infeasible.Source: B. DC Algorithm for Problem (28) · pp. 7–8

C. Alternating DC Algorithm Convergence Analysis

The convergence analysis establishes stationary-point convergence for the iterated beamforming and phase-shift subproblems. It also shows that the original total-power objective decreases until the alternating algorithm converges.

C. Alternating DC Algorithm Convergence Analysis

  • The beamforming subproblem’s generated sequence converges to a stationary point from an arbitrary initial point, with objective values strictly decreasing and convergent.The stated sequence concerns the iterates for the lifted beamforming problem.Source: C. Alternating DC Algorithm… · p. 8
  • The phase-shift subproblem’s generated sequence likewise converges to a stationary point from an arbitrary initial point, with a strictly decreasing, convergent objective sequence.The analysis treats the phase-shift iterates separately from the beamforming iterates.Source: C. Alternating DC Algorithm… · p. 8
  • The objective value of the original power-minimization problem decreases with each alternating-DC iteration until convergence.Source: C. Alternating DC Algorithm… · p. 8

VI. LOW-COMPLEXITY USER ORDERING SCHEME

The scheme orders users by their estimated minimum transmit power, a closed-form criterion that accounts for both channel conditions and target data rates. It avoids exhaustive search and performs nearly as well as SDR-based ordering, with only slight degradation versus exhaustive search.

VI. LOW-COMPLEXITY USER ORDERING SCHEME

  • Users are ordered by estimated minimum transmit power, combining channel conditions and target data rates to determine decoding order.The user with the largest estimate decodes its own signal first; the user with the smallest estimate decodes the others first.Source: §VI · pp. 8–9
  • For a given RIS phase-shift matrix, maximum-ratio transmission is optimal, and required power depends on target rate and combined channel gain.The phase-shift design is simplified by relaxing unit-modulus constraints to a norm constraint, yielding a closed-form expression.Source: §VI · p. 9
  • The closed-form power estimate is σ1(Qk)(N+1), where σ1(Qk) is the largest eigenvalue of Qk.The paper obtains the largest-eigenvalue solution for the related optimization problem and uses its objective value as an approximation.Source: §VI · p. 9
  • The closed-form ordering rule has lower computational complexity than SDR-based ordering, which requires solving K SDP problems.The proposed scheme performs almost as well as SDR-based ordering and suffers only slight performance degradation relative to exhaustive search.Source: §VI · pp. 8–9

VII. NUMERICAL RESULTS

Simulations average results over 100 channel realizations and compare alternating DC with alternating SDR and random phase shifts. The reported results favor alternating DC, show lower transmit power with more BS antennas or RIS elements, and find RIS-empowered NOMA outperforms the no-RIS network.

VII. NUMERICAL RESULTS

  • The numerical setup places the BS at (0, 0, 10) m and the RIS at (50, 50, 15) m, assumes Rayleigh fading, and averages results over 100 channel realizations.Unless otherwise specified, each user's target rate is 1.5 bits per channel use and noise power is −80 dBm.Source: §VII · p. 9

A. Performance Comparison of Different Methods

B. Performance Comparison of Different User Ordering Schemes

The proposed user ordering approaches exhaustive search while avoiding its high complexity, and three-bit RIS phase shifters nearly match continuous shifts. The paper concludes that its alternating DC method reduces total transmit power versus state-of-the-art methods, while identifying several directions for further work.

B. Performance Comparison of Different User Ordering Schemes

  • At K = 5, direct-link ordering is 2.04 dBm above optimal, while proposed and SDR-based ordering are each 0.72 dBm above optimal.The proposed scheme achieves almost the same performance as SDR-based ordering; exhaustive search over K! orders has extremely high complexity, while SDR-based ordering requires K SDP problems.Source: Figure 6 · p. 11, B. Performance Comparison of Different… · p. 11

C. Performance Comparison of Discrete and Continuous Phase Shifts

  • 3-bit phase shifters are practically sufficient to achieve almost the same performance as continuous phase shifts.Discrete-phase RIS consumes more transmit power than continuous-phase RIS, and that power decreases as the quantization-bit count increases.Source: Figure 7 · p. 11, C. Performance Comparison of Discrete… · p. 11

VIII. CONCLUSIONS

  • The alternating DC method outperforms state-of-the-art methods in total transmit power minimization.The method uses an exact DC representation of lifted rank-one constraints to detect feasibility and avoid early stopping in alternating optimization.Source: §VIII · p. 11
  • The conclusion identifies theoretical analysis, channel estimation, and large-scale optimization as needed to further exploit RIS-empowered NOMA.Future work is also proposed for multiple-BS scenarios while accounting for channel estimation.Source: §VIII · p. 12

A. Proof of Proposition 3

The proof handles Proposition 3’s properties separately, showing a non-increasing sequence that converges at a limit point and establishing a critical-point condition.

A. Proof of Proposition 3

  • The proof first treats property (i), then addresses property (ii).The proof states that the other properties follow similarly.Source: A. Proof of Proposition 3 · p. 12
  • The sequence under consideration is non-increasing and converges at a limit point.The proof derives the non-increasing property before concluding convergence.Source: A. Proof of Proposition 3 · p. 12
  • The proof concludes that the stated point is a critical point of f1 = g1 − h1.It invokes the closed and convex property of h1 in the preceding argument.Source: A. Proof of Proposition 3 · p. 12

B. Proof of Proposition 4

The proof establishes decreasing objective values through the alternating procedure and concludes that the proposed alternating DC algorithm converges.

B. Proof of Proposition 4

  • For a given RIS phase-shift matrix, the DC algorithm solves problem P3 and supplies the next beamforming-vector iterate.The proof uses rank-one feasibility to obtain the next vectors via Cholesky decomposition.Source: B. Proof of Proposition 4 · p. 12
  • The proof states that a feasible solution to problem P4 is also feasible for the original problem (13).Source: B. Proof of Proposition 4 · p. 13
  • The objective value of problem P2 decreases over iterations, so the proposed alternating DC algorithm converges.Source: B. Proof of Proposition 4 · p. 13
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