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Resource Allocation for Multi-Cell IRS-Aided NOMA Networks

Wanli Ni, Xiao Liu, Yuanwei Liu, Hui Tian, Yue Chen

arXiv:2006.11811v2eess.SPcs.IT

TL;DR

The paper addresses joint resource allocation in multi-cell IRS-aided NOMA networks, where user association, subchannels, power, phase shifts, and decoding order must be coordinated. It decomposes the mixed-integer nonlinear problem into continuous optimization and matching components using iterative relaxations and swap matching. The reported results show improved throughput and energy efficiency, with IRS placement affecting the spectrum-efficiency and coverage-area trade-off.

  • Problem

    Jointly optimizing transmission power, reflection matrix, decoding order, user association, and subchannel assignment under QoS and power constraints is challenging in multi-cell IRS-aided NOMA networks.

  • Method

    The paper decomposes the mixed-integer nonlinear problem into continuous optimization and matching subproblems, using convex approximations, semidefinite relaxation, Gaussian randomization, and swap-based matching.

  • Results

    The proposed algorithms improve system throughput and energy efficiency in IRS-aided multi-cell NOMA networks.

  • Takeaways & Limitations

    IRS-assisted NOMA can enhance wireless-system performance, while IRS placement provides a way to tune the trade-off between spectrum efficiency and coverage area.

Abstract

from arXiv · show

This paper proposes a novel framework of resource allocation in multi-cell intelligent reflecting surface (IRS) aided non-orthogonal multiple access (NOMA) networks, where an IRS is deployed to enhance the wireless service. The problem of joint user association, subchannel assignment, power allocation, phase shifts design, and decoding order determination is formulated for maximizing the achievable sum rate. The challenging mixed-integer non-linear problem is decomposed into an optimization subproblem (P1) with continuous variables and a matching subproblem (P2) with integer variables. In an effort to tackle the non-convex optimization problem (P1), iterative algorithms are proposed for allocating transmission power, designing reflection matrix, and determining decoding order by invoking relaxation methods such as convex upper bound substitution, successive convex approximation, and semidefinite relaxation. In terms of the combinational problem (P2), swap matching-based algorithms are developed for achieving a two-sided exchange-stable state among users, BSs and subchannels. Numerical results demonstrate that: 1) the sum rate of multi-cell NOMA networks is capable of being increased by 35% with the aid of the IRS; 2) the proposed algorithms for multi-cell IRS-aided NOMA networks can enjoy 22% higher energy efficiency than conventional NOMA counterparts; 3) the trade-off between spectrum efficiency and coverage area can be tuned by judiciously selecting the location of the IRS.

I. INTRODUCTION

IRSs and NOMA are presented as complementary technologies for improving multi-cell wireless performance, but their integration creates a coupled resource-allocation challenge. The introduction motivates joint scheduling, power, decoding, and reflection design to address interference, coverage, and efficiency.

  • IRS advantages: IRSs use passive reflecting elements to reconfigure wireless channels while reducing hardware cost and energy consumption relative to conventional active relays.They can operate in a full-duplex, noise-free manner and redirect incident signals toward desired directions.
  • NOMA motivation: NOMA superimposes signals for multiple users on the same frequency and separates them in the power domain using successive interference cancellation.Joint power allocation and decoding-order optimization are important for improving spectrum and energy efficiency and reducing interference.
  • Problem setting: Multi-cell NOMA resource allocation is challenging because co-channel interference couples decisions across base stations and users.The introduction therefore emphasizes joint user scheduling and resource allocation for performance improvement.
  • IRS-aided NOMA benefits: IRS integration can suppress interference, enhance desired signals, reduce energy consumption, and extend service to cell-edge users with poor signal strength.IRS configuration can also tune decoding order, user pairing, and connectivity by reconfiguring the propagation environment.
  • Related work: Prior work addressed isolated aspects of NOMA or IRS-aided networks, including power allocation, matching, beamforming, throughput, energy efficiency, and fairness.The cited studies include single-cell NOMA settings and IRS analyses of outage probability, capacity, and finite-resolution phase shifts.

2) IRS-Aided Wireless Communication Networks:

The paper targets the underexplored multi-cell, multi-subchannel IRS-aided NOMA setting, where continuous and combinatorial decisions are tightly coupled. It decomposes the problem and develops relaxation-based optimization and swap-matching algorithms.

  • Research gap: Existing IRS-aided NOMA studies largely focus on theoretical analysis or single-cell and single-carrier configurations, leaving multi-cell multi-subchannel allocation less explored.The stated gap includes joint user association, resource allocation, mutual SIC constraints, and individual QoS requirements.
  • Problem formulation: The paper formulates sum-rate maximization by jointly optimizing decoding order, transmission power, reflection matrix, user association, and subchannel assignment.The formulation includes SIC decoding conditions, QoS requirements, and maximum-power constraints.
  • Problem formulation: The resulting mixed-integer nonlinear program is NP-hard, combining nonconvex continuous optimization with combinatorial user-association and subchannel-assignment decisions.Exhaustive search has exponential complexity, making direct optimal solution difficult.
  • Continuous optimization: Convex upper-bound substitution, successive convex approximation, semidefinite relaxation, and Gaussian randomization address power, reflection-matrix, and decoding-order design.The decoding order is obtained from combined channel gains arranged in ascending order.
  • Combinatorial optimization: Swap-based matching algorithms solve the three-dimensional association problem among users, base stations, and subchannels by seeking a two-sided exchange-stable state.The paper analyzes the algorithms' stability, convergence, complexity, and optimality.
  • Reported results: The proposed resource-allocation algorithms outperform benchmarks in sum rate and energy efficiency, while IRS assistance further improves NOMA performance over conventional OMA.The comparison is reported as a numerical result of the proposed framework.

A. System Model

The system models IRS-aided multi-cell NOMA with user association, subchannel reuse, superimposed transmission, SIC decoding, and direct and reflected channels. It assumes single-antenna BSs and users, passive IRS elements, frequency-flat fading, and bounded NOMA user grouping.

  • Network and IRS model: The network contains J single-antenna BSs, I users, K reusable subchannels, and an IRS with M passive reflecting elements.Each user is associated with one BS, while subchannels can be reused among BSs.
  • Network and IRS model: The IRS reflection coefficient is represented by an amplitude and phase shift for each element, with unit amplitude adopted to simplify the rate analysis.The phase shift θ_m lies in [0, 2π], and λ_m = 1 is assumed.
  • Model assumptions and scope: The NOMA grouping limits each cell to at most Amax simultaneously paired users, with Amax ≥ 2, to reduce SIC decoding complexity.More complicated grouping schemes may improve performance at higher complexity but are outside the paper’s scope.
  • Association and transmission: Binary variables α_ij and β_jk indicate user association and BS subchannel assignment, and user i is served by BS j on subchannel k exactly when α_ijβ_jk = 1.This association structure determines which superimposed signal serves each user.
  • Signal and channel model: Each received signal includes the desired transmission, intra-cell interference, inter-cell interference, reflected and direct channels, and AWGN.The direct BS-user channel is Rayleigh fading, while BS-IRS channels are Rician and IRS-user channels are Rayleigh fading.
  • SIC and achievable rate: SIC decoding order π_jk(i) specifies the sequence in which users decode superimposed signals and cancel previously decoded interference.A later-decoded user can cancel the signal of an earlier-decoded user when the SIC condition is satisfied.
  • SIC and achievable rate: The achievable rate is determined from the received SINR, subject to successful SIC decoding and interference from both the serving and other cells.The model uses quasi-static frequency-flat fading and ignores time-delay differences between direct and reflected links.

B. Problem Formulation

The paper formulates joint sum-rate maximization over association, subchannel assignment, power, IRS phase shifts, and decoding order under QoS, SIC, and power constraints. Because the resulting problem is a coupled NP-hard MINLP, it is decomposed into tractable optimization and matching components solved alternately.

  • Problem formulation: The optimization jointly determines α_ij, β_jk, Θ, p_ijk, and π_jk(i) for user association, subchannel assignment, reflection, power, and decoding order.The formulation includes all five variable classes in one objective and constraint system.
  • Problem formulation: The objective maximizes users’ sum rate subject to SIC decoding, QoS, and maximum transmission-power constraints.Rmin specifies each user’s minimum data rate, while Pmax specifies each BS’s maximum transmission power.
  • Problem formulation: Constraints enforce feasible SIC decoding, QoS, BS power limits, one-BS association, bounded user multiplexing, subchannel assignment, phase shifts, power, and decoding order.The user multiplexing constraint requires at least two and no more than Amax users per cell.
  • Problem complexity: The mixed integer-continuous variables and their non-convex coupling make the sum-rate problem an NP-hard MINLP that standard methods cannot optimally solve directly.Exhaustive search is infeasible because computational complexity grows exponentially with the number of variables.
  • Problem decomposition: The paper decomposes the original problem into a non-convex optimization problem and a combinatorial optimization problem, then addresses them using optimization methods and matching theory.This decomposition is intended to produce tractable subproblems solved separately and alternately.
  • Algorithm overview: The alternating algorithm initializes feasible variables, updates power, co-designs reflection and decoding, then updates user association and subchannel assignment.These four steps are repeated until the objective converges or the iteration limit is reached.
  • Algorithm overview: The algorithm subsequently performs user association and subchannel assignment updates before repeating the alternating cycle.The association and assignment stages are implemented by Algorithms 5 and 6.
  • Complexity and convergence: The proposed alternating algorithm has an explicit complexity expression and is guaranteed to converge when its maximum iteration number N1 is sufficiently large.The complexity includes contributions from feasibility search, power allocation, reflection design, matching, and assignment procedures.

III. JOINT OPTIMIZATION OF POWER, REFLECTION, AND DECODING ORDER

With association and subchannel assignments fixed, the paper jointly addresses power allocation, reflection-related variables, and decoding order under coupled interference constraints. It convexifies the power subproblem with auxiliary variables and upper bounds, while adding a feasibility search to obtain suitable initial points.

  • Subproblem formulation: The continuous subproblem optimizes power allocation, reflection matrix, and SIC decoding order under the original system constraints.The power profile p and decoding-order profile π are the principal optimization profiles.
  • Subproblem formulation: Intra-cell and inter-cell interference make the resulting optimization non-linear and non-convex, preventing direct solution by standard convex optimization.The difficulty appears in both the objective and constraints through coupled interference terms.
  • Power allocation: Auxiliary SINR variables are introduced to reformulate the rate-related expressions and expose non-convex product terms for approximation.The reformulated problem is equivalent to the original problem under the stated construction.
  • Power allocation: Convex upper-bound substitution replaces products such as γ_ijk P̂_ijk and γ_ijk P̄_ijk with convex upper bounds that become tight under coefficient updates.The resulting constraints are convex and can be solved iteratively using CVX.
  • Power allocation: After an additional approximate constraint replacement, the power-allocation problem becomes convex and its KKT solution is iteratively updated until convergence.Algorithm 2 uses adjustable convergence accuracy ϵ and solves the convex approximation with CVX.
  • Feasibility search: A feasibility-searching algorithm minimizes the distance from initial points to the feasible domain to reduce Algorithm 2’s sensitivity to initialization.The search problem is jointly convex, more robust to initial solutions, and can be solved by CVX.
  • Algorithm procedures: Algorithm 3 initializes power and auxiliary variables randomly, iterates the convex updates, and supplies converged feasible solutions to Algorithm 2.The stated procedures include initialization, repeated updates, and output of the converged solutions.
  • Feasibility search: The feasibility-search problem does not require initial points in the feasible domain and can generate inputs for the subsequent power-allocation algorithm.Random initialization is permitted, and when the feasibility error is zero, the output is feasible for the substituted power problem.

B. Co-design of Reflection Matrix and Decoding Order

The paper co-designs the IRS reflection matrix and NOMA decoding order by convexifying channel constraints, relaxing the reflection matrix to an SDP, and recovering candidate rank-one solutions. The final decoding order is selected by sorting IRS-tuned combined channel gains.

  • Subproblem formulation: Given converged power and SINR variables, the reflection-and-decoding subproblem retains non-convex constraints caused by coupling between the reflection matrix and decoding order.The paper therefore reformulates the constraints before optimizing the reflection matrix.
  • Convex approximation: Auxiliary variables and difference-of-convex approximations replace non-convex channel-product constraints with first-order Taylor approximations.The approximations linearize the relevant convex components at the current iteration point.
  • Semidefinite reformulation: The reflection constraint is lifted using V = ν̄ν̄^H, yielding a positive semidefinite matrix with a rank-one constraint.The lifted formulation expresses quadratic reflection terms through matrix traces.
  • Semidefinite relaxation: The approximated problem optimizes V and auxiliary variables, after which semidefinite relaxation removes the rank-one constraint to produce a standard SDP.The relaxed SDP can be solved as a convex problem with the SeDuMi solver in CVX.
  • Rank-one recovery: When the relaxed solution has rank one, the optimal reflection matrix follows directly from its eigenvalue and eigenvector; otherwise, Gaussian randomization constructs rank-one candidates.The randomization uses the eigenvalue decomposition of the higher-rank relaxed solution.
  • Rank-one recovery: Gaussian random vectors generate candidate reflection matrices, and the candidate maximizing the combined channel gains of all users is selected.The candidates are formed from UΣ^1/2r_n and evaluated through their resulting channel gains.
  • Decoding-order design: After selecting the reflection matrix, users on each BS-subchannel pair are ranked by combined channel gain, and that ranking determines the decoding order.If H_ijk ≤ H_ĩjk, the order satisfies π_jk(i) ≤ π_jk(ĩ).
  • Algorithm 4: Algorithm 4 outputs the optimized reflection matrix and decoding order after solving the relaxed SDP, performing randomization, evaluating candidates, and sorting gains.The procedure explicitly initializes the candidate-vector generation count and returns Θ* and π*_{jk}.

C. Convergence and Complexity Analysis

The iterative utility for problem (10) is non-decreasing and bounded above by the achievable sum rate, so Algorithm 2 converges when N2 is sufficiently large.

  • Convergence: Algorithm 2 repeatedly updates p(n2) and γ(n2) by resolving problem (10).The updated variables are used to obtain the next iteration's solution.
  • Convergence: The utility value of problem (10) is non-decreasing over iterations.This follows from the iteration relations combining (34), (35), and (36).
  • Convergence: The achievable sum rate has an upper bound because system bandwidth and available transmission power are limited.This upper bound supports convergence of the iterative process.
  • Convergence: Algorithm 2 is guaranteed to converge when N2 is set sufficiently large.The convergence statement follows from the bounded, non-decreasing utility sequence.
  • Convergence: Proofs for Algorithm 3 are omitted because they use similar derivations.The omission limits the detail provided for that algorithm's convergence analysis.

2) Complexity:

The association and assignment design is decomposed from an NP-hard 3D matching problem into two 2D matching problems, while interference creates peer effects and non-substitutability.

  • Matching formulation: The joint user association and subchannel assignment formulation is a 3D matching problem over users, BSs, and subchannels.The three sets are finite and disjoint.
  • Matching formulation: The 3D matching problem is NP-hard, so it is decomposed into user association and subchannel assignment problems.The decomposition yields two 2D matching problems.
  • Matching formulation: User association is many-to-one, whereas subchannel assignment is many-to-many because subchannels can be reused across BSs.A BS may receive multiple subchannels, and a subchannel may be reused by multiple BSs.
  • Preference structure: Inter-cell interference creates peer effects in subchannel assignment because a BS’s rate depends on other BSs using the same subchannel.Consequently, preference lists can vary during the matching process.
  • Preference structure: Intra-cell interference makes user preferences non-substitutable because pairing changes users’ achievable rates.A user may leave a BS’s preferred set after its paired user is unmatched.
  • Swap matching: Swap matching is introduced to handle peer effects and obtain exchange-stable outcomes.A swap exchanges matched partners while keeping other matching states unchanged, and a swap-blocking pair requires no involved utility to decrease while at least one increases.

B. Many-to-One Matching for User Association

The matching procedures construct preferences, generate an initial matching, and iteratively apply beneficial swaps for user association and subchannel assignment.

  • Many-to-One User Association: User and BS preference lists are constructed from their achievable data rates under alternative matchings.Users compare candidate BSs, while BSs compare user subsets.
  • Many-to-One User Association: Each user proposes to its most preferred unrejected BS, and each BS accepts preferred users while rejecting others.This process ends when no user remains unmatched.
  • Many-to-One User Association: After initialization, users search for swap-blocking pairs and exchange matching states until none exists.The resulting output is a stable User-BS matching with utility U1 = U(µ*).
  • Subchannel Assignment: Subchannel assignment similarly builds preferences for User-BS units and subchannels before applying swap operations.The process terminates when no swap-blocking pair remains.
  • Subchannel Assignment: Algorithm 6 outputs a stable (User,BS)-Subchannel matching after iterative swaps.Its matching state is initialized as Φ2 and updated through the swap process.

D. Property Analysis

The proposed matching algorithms are analyzed for stability, convergence, complexity, and optimality, with exchange stability guaranteed but not necessarily global utility optimality.

  • Stability: The final matchings produced by Algorithms 5 and 6 are two-sided exchange-stable.This is the stated stability property of the proposed outputs.
  • Convergence: Both Algorithm 5 and Algorithm 6 converge to a two-sided exchange-stable matching within a limited number of iterations.The swap process stops when no swap-blocking pair can further improve any player’s utility.
  • Complexity: Algorithm 5 has complexity O(IJ^2 + AmaxIJNit).The bound combines initial matching construction with the swap process.
  • Optimality: Local optimal utilities imply two-sided exchange-stable matchings, but the converse does not necessarily hold.Thus, exchange stability alone does not guarantee local utility optimality.

V. NUMERICAL RESULTS

The numerical evaluation compares IRS-aided NOMA with OMA and non-IRS benchmarks under varied channel, power, and user-location settings. Results show that IRS benefits depend on favorable reflective-link path loss and placement, while the proposed algorithms remain close to exhaustive search.

  • Simulation setup: The evaluation uses six users, three base stations, three subchannels, 2000 independent channel trials, and averaged results.The setup includes fixed user, base-station, and IRS locations plus specified path-loss and fading models.
  • Compared schemes: The benchmarks comprise OMA without IRS, OMA with IRS, and NOMA without IRS, alongside the proposed IRS-aided NOMA schemes.OMA allows at most one user per base station in each time slot, whereas NOMA reuses frequencies across adjacent cells and applies SIC.
  • Path-loss sensitivity: Case 1, with path-loss exponents a1 = 3.2, a2 = 2.6, and a3 = 2.2, provides the largest IRS performance gain.Larger IRS-related path-loss exponents worsen reflective-link attenuation and can make the gains from phase-shift tuning vanish.
  • Power and interference: Sum-rate growth slows at high maximum transmission power because of intra-cell and inter-cell interference, with performance reaching a peak beyond a threshold.The results also indicate that adding more reflecting elements can further eliminate interference and improve performance.

B. Performance Analysis of System Energy Efficiency

The analysis examines how transmission power, reflecting elements, IRS placement, and interference affect energy efficiency in multi-cell NOMA and OMA networks. IRS deployment improves interference management and energy efficiency, while IRS location creates a trade-off between performance and coverage.

  • Power budget: Energy efficiency decreases as maximum transmission power increases, because additional power produces less sum-rate gain and greater interference.For lower power budgets, available power can be fully utilized while interference remains weak; at larger budgets, increased interference reduces efficiency.
  • Inter-cell interference: NOMA experiences higher inter-cell interference than OMA because multiple cells reuse the same time-frequency resources.IRS deployment can effectively eliminate inter-cell interference and further improve energy efficiency.
  • Reflecting elements: IRS-aided NOMA gains more energy efficiency than IRS-aided OMA because IRS can suppress interference and enhance desired signals in NOMA.In the considered OMA schemes, IRS is used only to enhance signals.
  • IRS location: Raising the IRS y-axis coordinate reduces energy efficiency because longer BS–IRS and IRS–user distances increase path loss and reduce reflective-channel power gain.Lowering the IRS height slightly improves performance at the cost of coverage, establishing a trade-off between sum rate and coverage area.

APPENDIX A

The appendix analyzes the computational complexity and convergence of the proposed algorithms. It gives complexity expressions for the algorithmic steps and states convergence under a sufficiently large iteration limit.

  • Complexity analysis: The computational complexity of Steps 1–2–3–4 is represented by O1, O2, O3, and O4, respectively.The overall expression is O0 + N1(O1 + O2 + O3 + O4).
  • Complexity analysis: The appendix defines O0 through O4 using the algorithm dimensions and iteration parameters.The listed terms include N3(2IJK)^3, N2(2IJK)^3, (M + 4IJK)^6 + N4TGR, IJ^2 + AmaxIJNit, and J^2K^2(N̄it + 1).
  • Convergence analysis: Algorithm 1 has a non-decreasing objective value over iterations and a throughput upper bound from limited bandwidth and power budget.Therefore, the proposed algorithm is guaranteed to converge when N1 is sufficiently large.
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