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Intelligent Reflecting Surface for Downlink Non-Orthogonal Multiple Access Networks

Min Fu, Yong Zhou, Yuanming Shi

arXiv:1906.09434v3eess.SPcs.ITcs.NI

TL;DR

The paper addresses transmit-power minimization for IRS-assisted downlink NOMA while meeting user QoS requirements. It alternates BS beamforming and IRS phase optimization, using matrix lifting and DC programming for rank-one constraints; simulations show lower power than comparison methods.

  • Problem

    The paper seeks to minimize BS transmit power in an IRS-empowered downlink NOMA network while satisfying users’ QoS requirements despite a highly non-convex bi-quadratic formulation.

  • Method

    The method alternates beamforming and IRS phase updates, lifts the resulting QCQPs into rank-one constrained matrix problems, and solves them with an exact DC representation and algorithm.

  • Results

    The proposed alternating DC method outperforms state-of-the-art methods in total transmit power and produces exact rank-one optimal solutions in the reported feasibility experiment.

  • Takeaways & Limitations

    Jointly optimizing active BS beamforming and passive IRS phase shifts decreases transmit power, while more BS antennas or IRS elements further improve performance.

Abstract

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Intelligent reflecting surface (IRS) has recently been recognized as a promising technology to enhance the energy and spectrum efficiency of wireless networks by controlling the wireless medium with the configurable electromagnetic materials. In this paper, we consider the downlink transmit power minimization problem for a IRS-empowered non-orthogonal multiple access (NOMA) network by jointly optimizing the transmit beamformers at the BS and the phase shift matrix at the IRS. However, this problem turns out to be a highly intractable non-convex bi-quadratic programming problem, for which an alternative minimization framework is proposed via solving the non-convex quadratic programs alternatively. We further develop a novel difference-of-convex (DC) programming algorithm to solve the resulting non-convex quadratic programs efficiently by lifting the quadratic programs into rank-one constrained matrix optimization problems, followed by representing the non-convex rank function as a DC function. Simulation results demonstrate the performance gains of the proposed method.

I. INTRODUCTION

The paper targets transmit-power minimization in an IRS-empowered downlink NOMA network. It alternates beamforming and IRS phase optimization, then uses a DC formulation to address the resulting non-convex rank-one problems.

  • IRS elements independently adjust reflected-signal phase shifts to shape channel responses for constructive combination and interference cancellation.
  • The paper jointly optimizes BS beamforming and IRS phase shifts under QoS requirements, but the resulting problem is a non-convex bi-quadratic program.
  • The algorithm alternates beamforming and phase-shift updates, lifting each QCQP into a rank-one constrained matrix optimization problem.
  • The proposed DC method considerably outperforms existing methods in minimizing downlink transmit power for IRS-empowered NOMA networks.
  • The method represents the rank-one constraint exactly as a difference between the trace norm and spectral norm, enabling an efficient DC algorithm.

A. System Model

The system is a single-cell downlink NOMA network in which an IRS assists an M-antenna BS serving K single-antenna users. Users are ordered by BS-channel quality for successive interference cancellation.

  • An IRS with N passive reflecting elements assists an M-antenna BS serving K single-antenna users in a dense setting with K ≥ M.
  • The IRS phase-shift matrix is diagonal with unit-modulus entries, while the direct, BS-to-IRS, and IRS-to-user channels determine each received signal.
  • Users are ordered by increasing direct BS-channel quality, and stronger users successively decode and remove weaker users’ signals before decoding their own.
  • User U1 treats signals intended for other users as noise when directly decoding its own signal.
  • After cancellation of signals for users U1 through U_{k−1}, user Uk’s achievable rate is determined by its post-cancellation SINR.

B. Problem Formulation

The formulation minimizes total BS transmit power while satisfying each user’s minimum data-rate requirement. Coupled beamforming and IRS variables make the resulting bi-quadratic constraints highly non-convex.

  • The optimization minimizes the sum of BS beamforming powers while satisfying every user’s minimum data-rate requirement.
  • Rate constraints are rewritten using SINR requirements, with each minimum rate mapped to a corresponding decoding threshold.
  • The resulting problem is equivalently reformulated before applying an alternating optimization approach to separate the coupled variables.
  • The problem is difficult because beamforming vectors and the IRS phase-shift matrix are coupled through non-convex bi-quadratic constraints.

III. ALTERNATING OPTIMIZATION FRAMEWORK

The alternating optimization framework updates the BS beamforming vectors and IRS phase shifts in turn, reformulating each resulting QCQP through matrix lifting. The lifted problems retain rank-one constraints, motivating a method beyond SDR.

  • III. ALTERNATING OPTIMIZATION FRAMEWORK: The beamforming vectors and IRS phase shift matrix are optimized alternatively until convergence.Matrix lifting converts each resulting non-convex QCQP into a rank-one constrained matrix optimization problem.
  • III. ALTERNATING OPTIMIZATION FRAMEWORK: For fixed IRS phases, lifting each beamforming vector into W_k = w_kw_k^H yields a PSD matrix problem with rank(W_k) = 1.The rank-one constraint preserves equivalence to the original beamforming formulation.
  • III. ALTERNATING OPTIMIZATION FRAMEWORK: For fixed beamformers, the phase-shift subproblem is expressed using v = [e^jθ_1, ..., e^jθ_N]^H and affine reflected-channel terms.The unit-modulus phase constraints make the resulting feasibility problem non-convex and inhomogeneous before homogenization.
  • III. ALTERNATING OPTIMIZATION FRAMEWORK: Introducing an auxiliary variable homogenizes the phase-shift QCQP, after which lifting V = ṽṽ^H produces a PSD matrix problem with V_n,n = 1 and rank(V) = 1.A feasible homogenized solution can be mapped back to the original phase vector by normalization.
  • III. ALTERNATING OPTIMIZATION FRAMEWORK: SDR relaxes the rank-one constraints but may require Gaussian randomization because rank-one solutions are unlikely in high-dimensional problems.The framework therefore motivates an exact DC representation of the rank constraint.

IV. PROPOSED ALTERNATING DC METHOD

The proposed alternating DC method represents the rank function exactly through the difference between trace and spectral norms, then alternates DC updates for the lifted subproblems.

  • IV. PROPOSED ALTERNATING DC METHOD: The method uses an exact DC representation of the rank constraint and alternates DC optimization over the lifted matrix problems.This approach targets the fixed-rank constraints created by matrix lifting.

A. Proposed Alternating DC Programming

The alternating DC programming formulation enforces rank-one solutions by penalizing the trace-minus-spectral-norm difference and tests feasibility through the same quantity.

  • A. Proposed Alternating DC Programming: The fixed-rank constraint is represented using the difference between the trace norm and spectral norm of a PSD matrix.For PSD matrices, this difference characterizes whether the matrix has rank one.
  • A. Proposed Alternating DC Programming: With fixed phase shifts, a DC program finds K PSD matrices whose penalty term is driven to zero to induce rank-one beamforming matrices.The recovered beamforming vectors are obtained through Cholesky decomposition.
  • A. Proposed Alternating DC Programming: With fixed beamformers, minimizing the trace-minus-spectral-norm difference detects feasibility of the lifted phase-shift problem.A zero objective yields an exact rank-one solution, while a nonzero objective indicates infeasibility of the original phase problem.

B. DC Algorithm for Problems (16) and (17)

The DC subproblems are solved by successive convex approximations using primal-dual updates and spectral-norm subgradients. Alternating these updates produces beamformers and phase shifts until convergence or infeasibility.

  • B. DC Algorithm for Problems (16) and (17): The DC iterations update primal and dual variables through successive convex approximation based on Fenchel duality and subgradients.The method uses conjugate functions and a subgradient of the concave component at the current iterate.
  • B. DC Algorithm for Problems (16) and (17): The convex subproblems for the two DC programs can be solved efficiently with CVX.The spectral-norm subgradient is computed from an eigenvector associated with the largest eigenvalue.
  • B. DC Algorithm for Problems (16) and (17): The alternating DC algorithm iteratively solves the beamforming and phase-shift DC programs until convergence.Each DC program is handled through successive convex relaxation steps.
  • B. DC Algorithm for Problems (16) and (17): The alternating DC method can guarantee feasibility of the rank-one constraint and is intended to yield good network performance.The paper compares it with alternating SDR and random phase shifts.
  • B. DC Algorithm for Problems (16) and (17): The algorithm recovers beamforming vectors and phase variables by Cholesky decomposition and stops when power reduction is below a threshold or the phase problem is infeasible.It initializes the IRS phase matrix and repeats the two lifted updates in an outer alternating loop.

V. SIMULATION RESULTS

Simulations evaluate the alternating DC method against alternating SDR and random phase shifts under IRS-assisted downlink NOMA settings. The proposed method achieves feasible rank-one solutions and lower transmit power, with performance improving as BS antennas or IRS reflecting elements increase.

  • Simulation setup: The simulations use a 3D deployment with Rayleigh fading and path-loss exponents of 3.5, 2.2, and 2.8 for the BS-user, BS-IRS, and IRS-user links.The BS, IRS, and users are placed at the coordinates and region specified in the simulation setup.
  • Convergence behavior: The proposed alternating DC method induces exact rank-one optimal solutions and accurately detects feasibility, whereas alternating SDR with Gaussian randomization terminates after failing to find a feasible solution.This comparison uses K = 6, M = 5, and N = 20 in Fig. 2(a).
  • Impact of BS antennas: The alternating DC method consumes much less total transmit power than alternating SDR and random phase shifts as the number of BS antennas increases.The comparison fixes N = 15 and K = 6; both optimized methods outperform random phase shifts.
  • Impact of IRS elements: Total transmit power decreases quickly as the number of IRS passive reflecting elements increases, indicating better performance with larger IRS surfaces.Fig. 2(c) uses M = 5 and K = 6.
  • Effect of IRS deployment: IRS-empowered networks outperform networks without IRS in total transmit power, demonstrating the importance of deploying IRS in cellular networks.The comparison uses M = 5 and N = 10.

VI. CONCLUSIONS

The paper concludes that alternating DC jointly designs BS beamforming and IRS phase shifts for transmit-power minimization in IRS-empowered NOMA networks. Simulations show that it outperforms state-of-the-art methods in total transmit power.

  • VI. CONCLUSIONS: The method alternates between two non-convex QCQP subproblems, using matrix lifting and an exact DC representation of the rank-one constraint.This design jointly optimizes BS beamforming vectors and the IRS phase shift matrix.
  • VI. CONCLUSIONS: The alternating DC method outperforms state-of-the-art methods in total transmit power for IRS-empowered NOMA networks.The conclusion refers to performance comparisons under different network settings.
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