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Intelligent Reflecting Surface Assisted Non-Orthogonal Multiple Access

Gang Yang, Xinyue Xu, Ying-Chang Liang

arXiv:1907.03133v2cs.ITeess.SP

TL;DR

The paper addresses fair rate optimization in IRS-assisted downlink NOMA, where user ordering depends on IRS-controlled combined channels. It jointly designs BS beamforming and IRS phase shifts using a combined-channel-strength ordering scheme and non-convex optimization algorithms, obtaining significant rate gains over the stated baselines while maintaining fairness and showing slight degradation from finite phase resolution.

  • Problem

    IRS-assisted NOMA requires fair max-min rate optimization while user ordering is coupled to the IRS-dependent combined channel strengths.

  • Method

    The paper decouples ordering through combined-channel-strength-based user ordering and jointly optimizes BS beamforming and IRS phase shifts using block coordinate descent and semidefinite relaxation.

  • Results

    IRS-assisted downlink NOMA significantly enhances rate performance versus traditional NOMA without IRS and OMA with or without IRS.

  • Takeaways & Limitations

    The numerical results indicate that good rate fairness is maintained and finite IRS phase resolution causes only slight rate degradation.

Abstract

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Intelligent reflecting surface (IRS) is a new and disruptive technology to achieve spectrum- and energy-efficient as well as cost-efficient wireless networks. This paper considers an IRS-assisted downlink non-orthogonal-multiple-access (NOMA) system. To optimize the rate performance and ensure user fairness, we maximize the minimum decoding signal-to-interference-plus-noise-ratio (i.e., equivalently the rate) of all users, by jointly optimizing the (active) transmit beamforming at the base station (BS) and the phase shifts (i.e., passive beamforming) at the IRS. A combined-channel-strength based user ordering scheme is first proposed to decouple the user-ordering design and the joint beamforming design. Efficient algorithms are further proposed to solve the formulated non-convex problem for the cases of a single-antenna BS and a multi-antenna BS, respectively, by leveraging the block coordinated decent and semidefinite relaxation (SDR) techniques. For the single-antenna BS case, the optimal solution for the power allocation at the BS and the asymptotically optimal solution for the phase shifts at the IRS are obtained in closed forms. For the multi-antenna BS case, it is shown that the rank of the SDR solution to the transmit beamforming design is upper bounded by two. Also, the convergence proof and the complexity analysis are given for the proposed algorithms. Simulation results show that the IRS-assisted downlink NOMA system can enhance the rate performance significantly, compared to traditional NOMA without IRS and traditional orthogonal multiple access with/without IRS. In addition, numerical results demonstrate that the rate degradation due to the IRS's finite phase resolution is slight, and good rate fairness among users can be always guaranteed.

I. INTRODUCTION

NOMA multiplexes users on shared resources but its gains depend on channel-strength differences, while IRS can reconfigure propagation to strengthen and realign combined channels. This motivates IRS-assisted NOMA with joint beamforming and phase-shift optimization for max-min user rate and fairness.

  • NOMA uses power-domain multiplexing and successive interference cancellation to serve multiple users on the same resource block.
  • NOMA provides significant spectral-efficiency gains over OMA only when users have sufficiently different channel strengths.
  • NOMA does not always outperform OMA, including cases where users’ channel vectors are mutually orthogonal.
  • IRS uses many low-cost reflecting elements whose phase shifts can configure propagation for power boosting, interference mitigation, and secure transmission.
  • IRS can create stronger combined channels and artificially increase channel-strength differences or realign users’ channels to obtain NOMA gains.
  • The paper targets max-min user rate by jointly optimizing BS transmit beamforming and IRS phase shifts, addressing user ordering coupled to the resulting combined channels.
  • Prior IRS-assisted NOMA work considered served-user number, BS power, or sum rate, whereas this paper focuses on max-min rate optimization with user fairness.

2) Literature on IRS-assisted Communciations:

Prior IRS-assisted communication studies examined channel modeling, beamforming, power allocation, secrecy, outage, ergodic performance, and learning-based channel estimation. This paper applies IRS assistance to downlink NOMA, formulates a fairness-oriented joint design, and develops ordering and optimization methods with favorable numerical results.

  • Prior IRS-assisted studies optimized transmission power, beamforming, or phase shifts for objectives including total power, sum rate, energy efficiency, and secrecy.
  • Other studies analyzed outage probability, asymptotic sum-rate distributions, ergodic spectral efficiency, and deep-learning-based IRS channel estimation.
  • IRS passively reflects signals rather than amplifying and retransmitting them, reducing energy consumption and avoiding self-interference.
  • This paper formulates max-min target-SINR optimization for IRS-assisted downlink NOMA by jointly designing BS beamforming and IRS phase shifts under system constraints.
  • A combined-channel-strength user-ordering scheme decouples ordering from joint beamforming and achieves almost the same rate performance as exhaustive ordering search.
  • BCD and SDR-based iterative algorithms address single-antenna and multi-antenna BS cases, with closed-form solutions for selected subproblems and an SDR beamforming rank bound of two.
  • Numerical results report significant gains over traditional NOMA and OMA benchmarks, slight degradation from finite phase resolution, and preserved fairness as user count increases.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system is an IRS-assisted downlink NOMA link in which a multi-antenna BS superposes user signals and an IRS adjusts reflected phases. Users apply SIC, while the paper maximizes the minimum user SINR or equivalent rate under beamforming, phase-shift, power, and decoding constraints.

  • A. System Model: The model uses a BS with N antennas, an IRS with M passive reflecting elements, and K single-antenna users receiving superposed downlink signals.
  • A. System Model: The IRS controller adjusts element phase shifts to assist NOMA transmission by providing an additional reflected propagation path.
  • A. System Model: BS-to-user channels are modeled as mutually independent Rayleigh fading, while BS-to-IRS and IRS-to-user channels use Rician models reflecting LoS and NLoS components.
  • A. System Model: The transmitted signal superposes K data flows, each assigned a dedicated BS beamforming vector, and the received signal includes IRS phase shifts and AWGN.
  • A. System Model: NOMA users decode weaker users’ signals sequentially through SIC, cancel decoded interference, and then decode their own signals while treating remaining users as interference.
  • A. System Model: For IRS phase-shift choices, the optimal user order may vary among K! possibilities, motivating the proposed efficient ordering scheme.
  • A. System Model: The optimization maximizes the minimum target SINR, equivalently the minimum achievable rate, by jointly choosing BS beamforming and IRS phase shifts.

B. Problem Formulation

The paper formulates IRS-assisted downlink NOMA as a max-min SINR problem, jointly optimizing BS beamforming and IRS phase shifts under SIC and power constraints. It separates user ordering from joint beamforming through combined channel strengths, avoiding exhaustive search over all orderings.

  • B. Problem Formulation: The objective maximizes the minimum users’ target SINR, equivalently their achievable rate, through joint BS beamforming and IRS phase-shift optimization.The slack variable Q represents the minimum target SINR.
  • B. Problem Formulation: The formulation imposes SIC decoding, combined-channel-strength ordering, BS transmission-power, and IRS phase-shift constraints.The decoding constraints require both users’ own and decoded data flows to meet Q.
  • B. Problem Formulation: The problem is difficult because beamforming, phase shifts, and Q are coupled through non-convex constraints.The paper studies single-antenna and multi-antenna BS cases separately.
  • III. CCS-BASED USER ORDERING DESIGN: Exhaustive ordering solves K! optimization problems, making its complexity high when the number of users K is large.BS-to-user ordering can also perform poorly because it ignores IRS effects.
  • III. CCS-BASED USER ORDERING DESIGN: The proposed ordering ranks users by their maximally achievable combined channel strengths, obtained by optimizing IRS phase shifts.This decouples ordering design from the joint beamforming optimization.
  • III. CCS-BASED USER ORDERING DESIGN: For each user, the combined-channel-strength problem is transformed using E = ¯e¯eH with E ⪰ 0, while relaxing the rank-one constraint through SDR.The relaxed solution can be converted to rank one using Gaussian randomization.
  • III. CCS-BASED USER ORDERING DESIGN: The SDR-plus-randomization procedure guarantees at least a π fraction of the optimal objective value of the ordering problem.The designed ordering exhibits only slight rate degradation compared with exhaustive search.

IV. OPTIMAL SOLUTION FOR SINGLE-ANTENNA BASE STATION CASE

For a single-antenna BS, the multi-antenna channel model reduces to scalar user channels and power-allocation variables. The resulting max-min problem is solved approximately by alternating optimization of power allocation and IRS phase shifts.

  • IV. OPTIMAL SOLUTION FOR SINGLE-ANTENNA BASE STATION CASE: With one BS antenna, the BS-to-IRS channel, IRS-to-user channels, and beamforming vectors reduce to scalar channels and power allocation coefficients.The designed CCS-based user ordering is retained in this specialization.
  • IV. OPTIMAL SOLUTION FOR SINGLE-ANTENNA BASE STATION CASE: Once the user-ordering constraint is satisfied, each user’s own-signal SINR constraint implies the corresponding decoding constraint, which can therefore be omitted.The remaining constraints include normalization and non-negativity conditions.
  • IV. OPTIMAL SOLUTION FOR SINGLE-ANTENNA BASE STATION CASE: The single-antenna max-min problem remains non-convex because the power-allocation and phase-shift blocks are coupled.Block coordinate descent alternately optimizes these two variable blocks.
  • IV. OPTIMAL SOLUTION FOR SINGLE-ANTENNA BASE STATION CASE: The alternating procedure decouples the problem into separate power-allocation and phase-shift subproblems at each iteration.The variables αn and Θn denote their values after iteration n.

A. Phase Shift Optimization

The phase-shift subproblem is handled with bisection, SDR, and Gaussian randomization, while the single-antenna power allocation admits a closed-form solution. For two users, a closed-form IRS phase-shift rule is asymptotically optimal at high transmit power but slightly degrades performance at lower powers.

  • A. Phase Shift Optimization: The IRS phase-shift design is reformulated through E = ¯e¯eH, converting channel-strength terms into trace expressions while retaining a non-convex rank-one constraint.The SDR relaxation removes the rank-one constraint during optimization.
  • A. Phase Shift Optimization: Bisection over Q decouples the target SINR from the phase shifts, and SDR solves the resulting feasibility problem for E.Gaussian randomization then produces a suboptimal feasible phase-shift solution.
  • A. Phase Shift Optimization: For sufficiently large Qmax and small Qmin, the bisection search yields a globally optimal phase-shift-related matrix En+1 for the iteration.The rank-one phase-shift realization is subsequently obtained by randomization.
  • A. Phase Shift Optimization: For K = 2, the paper proposes a closed-form IRS phase-shift solution instead of the general bisection, SDR, and randomization procedure.The general procedure is described as complicated in practice.
  • A. Phase Shift Optimization: At high transmit power P, the asymptotically optimal phase shifts are θi = ξ2 − ϕ2,i − ψi.The rule aligns the phase terms associated with the second user’s channel components.
  • A. Phase Shift Optimization: The closed-form phase-shift rule is asymptotically optimal for large P but suboptimal for small or moderate P.At lower powers, noise is not negligible relative to interference, so both users’ channel strengths should generally be enhanced.
  • A. Phase Shift Optimization: At small or moderate P, the closed-form solution incurs only slight rate degradation relative to the general algorithm.This comparison is reported by numerical results.
  • B. Power Allocation Optimization: For fixed IRS phase shifts, the optimal power allocation is available in closed form through the dominant eigenvalue and eigenvector of Π.The allocation vector uses the first K components of Π’s dominant eigenvector, scaled by its last component.

C. Overall Algorithm

The overall method alternates phase-shift and power-allocation updates, using inner bisection and SDR procedures for phase optimization. The algorithm stops when objective improvement falls below a prescribed tolerance and is guaranteed to converge.

  • C. Overall Algorithm: Algorithm 1 alternately solves the phase-shift subproblem and the closed-form power-allocation subproblem in each outer BCD iteration.The phase-shift block uses bisection and SDR, while Theorem 1 computes the power allocation.
  • C. Overall Algorithm: Algorithm 1 is guaranteed to converge.The paper states that its proof is similar to the proof for Algorithm 2.
  • C. Overall Algorithm: The phase-shift update performs bisection over Q, solves an SDR feasibility problem, and applies eigenvalue decomposition and randomization.The resulting matrix is evaluated across randomized candidates before the next block update.
  • C. Overall Algorithm: The procedure terminates when the objective increase is smaller than ε and returns α⋆, Θ⋆, and Q⋆.The algorithm’s inner and outer accuracy parameters control the stopping conditions.

V. OPTIMAL SOLUTION FOR MULTI-ANTENNA BASE STATION CASE

The multi-antenna formulation alternates optimization of IRS phase shifts and BS beamforming, using bisection, SDR, and Gaussian randomization to obtain approximate solutions. The SDR transmit-beamforming solution has rank at most two.

  • The original problem is decomposed into phase-shift and transmit-beamforming subproblems solved alternately.
  • SDR relaxes the non-convex rank-one constraints for the lifted IRS and transmit-beamforming variables.
  • Bisection search handles the remaining non-convex subproblems, while Gaussian randomization produces approximate rank-one beamforming solutions.
  • The rank of the SDR solution for the transmit-beamforming design is upper bounded by two.
  • Algorithm 2 alternates phase-shift and beamforming updates until the objective-value increase is sufficiently small.

C. Overall algorithm

The overall algorithm uses block coordinate descent with alternating phase-shift and beamforming updates, embedding bisection and SDR procedures. Its objective is non-decreasing and the algorithm is guaranteed to converge, but global optimality is not assured.

  • Overall algorithm: Algorithm 2 alternately optimizes IRS phase shifts and BS beamforming in an outer block-coordinate-descent iteration.
  • Convergence: The objective value is non-decreasing after every iteration and is upper-bounded over the compact feasible set.
  • Convergence: Algorithm 2 is guaranteed to converge.
  • Limitation: Global optimality is not assured because the joint problem is non-convex and SDR followed by Gaussian randomization is not globally optimal.
  • Overall algorithm: Each subproblem uses a bisection inner iteration containing an SDR problem, and the algorithm stops when objective improvement is sufficiently small.

VI. NUMERICAL RESULTS

Numerical evaluations compare IRS-assisted NOMA with NOMA and OMA benchmarks for single- and multi-antenna BSs. The proposed system improves rate performance, while CCS-based ordering nearly matches exhaustive search and the low-complexity phase solution substantially reduces complexity.

  • Rate performance: At P = 10 dBm, IRS-assisted NOMA improves rate performance by 53.2%, 38.5%, and 14.3% over traditional OMA, traditional NOMA, and IRS-assisted OMA, respectively.
  • Rate performance: IRS-assisted NOMA achieves significant rate gains over three benchmarks in the multi-antenna BS setup.
  • User ordering: The CCS-based ordering scheme achieves almost the same max-min rate as exhaustive search for both single- and multi-antenna BSs.
  • Low-complexity solution: For small or moderate transmission power, the closed-form low-complexity phase solution slightly degrades rate performance relative to the general algorithm.
  • Low-complexity solution: When P is higher than 16 dBm, the low-complexity phase solution outperforms the general algorithm and provides significant complexity reduction.

B. Effects of IRS’s Finite-Phase Resolution on Rate Performance

Finite IRS phase resolution causes some max-min-rate degradation relative to continuous phases, but the degradation becomes negligible quickly as quantization bits increase. Increasing reflecting elements improves max-min rate, while increasing users creates a sum-rate tradeoff.

  • Finite-phase resolution: Finite IRS phase resolution generally degrades max-min rate relative to the ideal infinite-resolution case.
  • Finite-phase resolution: The rate degradation from finite phase resolution becomes negligible very quickly as the number of quantization bits B increases.
  • Reflecting elements: The max-min rate increases with the number of reflecting elements M at the evaluated transmission powers.
  • Number of users: As the number of NOMA users K increases, sum rate first increases and then decreases, reaching its maximum at K = 8.
  • User fairness: The two-user scenario achieves almost equal user rates, and good rate fairness can be maintained as K increases.

APPENDIX

The appendix proves the optimality and uniqueness of the power-allocation solution and characterizes it through a nonnegative-matrix eigenvalue system. It also derives the phase shifts that maximize the relevant combined channel strength.

  • For sufficiently high BS transmission power, γ1→1 approaches α2, while γ2→2 increases monotonically with |h2|^2; therefore, maximizing the latter improves the minimum SINR.
  • The phase shifts maximizing |h2|^2 are given in closed form by θ_i = ξ2 − ϕ2,i − ψ_i for i = 1, …, M.
  • The proposed power allocation α* is the optimal solution, and γ* is the corresponding optimal max-min SINR.
  • The power allocation is unique because each α_k(Q) increases strictly and monotonically with Q, yielding a unique positive Q* satisfying the defining equations.
  • Q is represented as a reciprocal eigenvalue of a nonnegative matrix Π, with a nonnegative eigenvector selected to satisfy the physical constraints.
  • The optimal power-allocation vector α consists of the first K components of the dominant eigenvector after scaling its last component to one.
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