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Power Efficient IRS-Assisted NOMA

Jianyue Zhu, Yongming Huang, Jiaheng Wang, Keivan Navaie, Zhiguo Ding

arXiv:1912.11768v1cs.IT

TL;DR

Conventional NOMA cannot reliably guarantee quasi-degraded channels because propagation environments determine the user channels. The paper jointly optimizes beamforming and IRS phases for MISO NOMA, reporting more attainable quasi-degradation and DPC-equivalent performance, with IRS-assisted transmission outperforming systems without IRS and OMA.

  • Problem

    Conventional NOMA cannot reliably guarantee quasi-degraded channels because propagation environments determine the user channels.

  • Method

    The paper jointly optimizes beamforming vectors and the IRS phase shift matrix in an IRS-assisted MISO NOMA system.

  • Results

    The improved quasi-degradation condition can be satisfied with greater possibility than the original condition without IRS, while NOMA matches DPC performance.

  • Takeaways & Limitations

    IRS-assisted MISO transmission outperforms the corresponding system without IRS and the conventional OMA scheme.

Abstract

from arXiv · show

In this paper, we propose a downlink multiple-input single-output (MISO) transmission scheme, which is assisted by an intelligent reflecting surface (IRS) consisting of a large number of passive reflecting elements. In the literature, it has been proved that nonorthogonal multiple access (NOMA) can achieve the capacity region when the channels are quasi-degraded. However, in a conventional communication scenario, it is difficult to guarantee the quasi-degradation, because the channels are determined by the propagation environments and cannot be reconfigured. To overcome this difficulty, we focus on an IRS-assisted MISO NOMA system, where the wireless channels can be effectively tuned. We optimize the beamforming vectors and the IRS phase shift matrix for minimizing transmission power. Furthermore, we propose an improved quasi-degradation condition by using IRS, which can ensure that NOMA achieves the capacity region with high possibility. For a comparison, we study zero-forcing beamforming (ZFBF) as well, where the beamforming vectors and the IRS phase shift matrix are also jointly optimized. Comparing NOMA with ZFBF, it is shown that, with the same IRS phase shift matrix and the improved quasi-degradation condition, NOMA always outperforms ZFBF. At the same time, we identify the condition under which ZFBF outperforms NOMA, which motivates the proposed hybrid NOMA transmission. Simulation results show that the proposed IRS-assisted MISO system outperforms the MISO case without IRS, and the hybrid NOMA transmission scheme always achieves better performance than orthogonal multiple access.

I. INTRODUCTION

The paper addresses the difficulty of guaranteeing quasi-degraded user channels for NOMA by using an IRS to reconfigure propagation conditions. It jointly optimizes beamforming and IRS phases, compares NOMA with ZFBF, and proposes hybrid transmission.

  • IRS reflecting elements can affect wireless propagation, enabling channel reconfiguration without dedicated energy for decoding, channel estimation, or transmission.
  • The paper jointly optimizes beamforming vectors and the IRS phase shift matrix for an IRS-assisted MISO NOMA system.
  • The optimization minimizes transmission power while imposing a quasi-degradation constraint so NOMA can match DPC performance.
  • An improved quasi-degradation condition is proposed for IRS-assisted NOMA and can be satisfied with higher possibility than without IRS.
  • With the same IRS phase shift matrix and improved quasi-degradation condition, NOMA outperforms ZFBF, while ZFBF can win under another condition.
  • The resulting hybrid NOMA transmission scheme is evaluated against OMA, alongside IRS-assisted ZFBF and system-level simulations.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system is a two-user downlink MISO link assisted by an IRS, with NOMA using superposition coding and SIC. The paper jointly designs beamforming and IRS phases to minimize power while enforcing QoS and quasi-degradation constraints.

  • A. System Model: The downlink system uses a base station with M antennas, two single-antenna users, and an IRS with N phase-shift elements.
  • 1) NOMA Transmission Scheme:: NOMA uses superposition coding and a fixed decoding order (2, 1): user 1 decodes user 2 first, whereas user 2 treats user 1’s signal as interference.
  • A. System Model: The IRS phase shift matrix is parameterized by element phases θ_n ∈ [0, 2π], while the BS–IRS and direct-user channels define the effective links.
  • 1) NOMA Transmission Scheme:: The proposed IRS-assisted NOMA aims to retain DPC-level performance while offering a more practical transmission approach than DPC’s high complexity.
  • 1) NOMA Transmission Scheme:: The design jointly optimizes beamforming vectors and IRS phases under perfect CSI to minimize transmission power subject to both users’ QoS requirements.
  • 1) NOMA Transmission Scheme:: The quasi-degradation constraint is included because it ensures that NOMA achieves the same performance as DPC, whereas fixed channels may not satisfy it.
  • 1) NOMA Transmission Scheme:: Using IRS phase control, the user-channel directions can be tuned so the improved condition is more attainable than in conventional NOMA.

2) ZFBF Transmission Scheme:

The paper benchmarks IRS-assisted NOMA against IRS-assisted ZFBF and studies how IRS phase control can shape channel geometry. ZFBF avoids interference through null-space transmission, while the paper formulates and optimizes its power-minimization design.

  • ZFBF transmits each user’s data in the null space of the other users’ channels to mitigate multi-user interference.
  • When the two user channels are orthogonal, ZFBF avoids interference and achieves its best performance, but user channels are not always orthogonal.
  • The IRS phase shift matrix can make the user channels orthogonal under a certain condition.
  • The paper formulates an IRS-assisted ZFBF transmission-power problem with QoS constraints and jointly designs beamforming and IRS phases.
  • The analysis first examines feasibility, then optimizes beamforming for a given IRS phase matrix before optimizing the IRS matrix.

A. The Feasibility of the Quasi-degradation Constraint

The IRS-assisted system introduces an improved quasi-degradation condition and increases the likelihood that NOMA can operate in the capacity-achieving regime. Feasibility is especially favorable when the IRS is close to user 1.

  • The IRS adjusts the angle between users' channels, making the quasi-degradation condition more likely to hold than in conventional MISO NOMA.
  • The improved quasi-degradation condition provides a sufficient condition for feasibility of the quasi-degradation constraint.
  • Under the improved condition, an IRS phase shift matrix can be found that satisfies the quasi-degradation constraint, making the NOMA MTP problem feasible.
  • The IRS-assisted MISO NOMA scheme can obtain the same performance as DPC under the improved quasi-degradation condition.
  • Simulation regions show that IRS assistance enlarges the user-2 area satisfying the quasi-degradation condition compared with the system without IRS.
  • When user 1 is near the IRS, the condition is more likely to be satisfied; if user 1 and the IRS coincide, it can always be satisfied.

B. Beamforming Vectors and the IRS Phase Shift Matrix Design for MTP

The MTP design jointly optimizes beamforming vectors and IRS phase shifts under the improved quasi-degradation constraint. The nonconvex formulation is approximated through convexification, semidefinite relaxation, and iterative updates.

  • The design jointly optimizes beamforming vectors and the IRS phase shift matrix for the MTP problem.
  • For a fixed IRS phase shift matrix, the optimal beamforming vectors are characterized in closed form.
  • The quasi-degradation constraint is transformed into a convex constraint to address the nonconvex optimization problem.
  • A quadratic transform introduces auxiliary variables, which are alternately optimized with the lifted matrix variable Q.
  • Semidefinite relaxation removes the rank constraint so the relaxed problem can be solved as a standard semidefinite program.
  • If the relaxed solution is not rank one, randomization generates rank-one solutions for recovering IRS elements and beamforming vectors.

IV. BEAMFORMING AND IRS PHASE SHIFT DESIGN: IRS-ASSISTED ZFBF

The IRS-assisted ZFBF design jointly optimizes beamforming and IRS phase shifts to minimize transmission power. Its nonconvex fractional formulation is handled through parameterization, successive convex approximation, semidefinite relaxation, and iterative convergence procedures.

  • IRS-assisted ZFBF uses beamforming vectors and IRS phase shifts jointly optimized for transmission-power minimization.
  • For a fixed IRS phase shift matrix, the optimal beamforming vectors are obtained from a closed-form solution.
  • The IRS phase-shift optimization is reformulated with a lifted matrix Q and unit-diagonal constraints.
  • A fractional objective is parameterized by η, and the resulting problem is solved iteratively until the optimality condition G*(η) = 0 is reached.
  • Successive convex approximation constructs upper bounds, while semidefinite relaxation removes the rank constraint for efficient convex optimization.
  • The η-update algorithm is guaranteed to converge to the desirable η, after which IRS elements and beamforming vectors are recovered.

V. COMPARISON OF IRS-ASSISTED NOMA AND IRS-ASSISTED ZFBF

The section compares jointly optimized IRS-assisted NOMA and ZFBF, showing that NOMA is superior under the improved quasi-degradation condition while motivating hybrid transmission when that condition fails.

  • Given the same IRS phase shift matrix and improved quasi-degradation condition, NOMA always achieves better performance than ZFBF.
  • The IRS can make the user channels orthogonal only under condition (51), which is difficult to satisfy in practice because of random channel components.
  • The improved quasi-degradation condition is more easily satisfied when the IRS is located close to user 1.
  • The resulting hybrid NOMA precoding scheme selects NOMA under the improved condition and otherwise considers ZFBF.
  • When the improved quasi-degradation condition holds, NOMA achieves optimal performance and is preferred for transmission.
  • If the condition is violated with R ≠ 0, performance loss is inevitable, motivating ZFBF when computational complexity is prioritized.

VI. SIMULATION RESULTS

The simulations evaluate transmission power across antenna count, user distance, IRS size, and IRS placement. Hybrid NOMA outperforms ZFBF and OFDMA, approaches DPC, and gains substantially over MISO NOMA without IRS.

  • H-NOMA yields a significant performance gain over ZFBF and OFDMA as the number of antennas increases.Transmission power decreases for all three schemes as antenna count increases.
  • The H-NOMA performance gain becomes significant as the distance between the two users increases.
  • Increasing the number of IRS elements improves performance, especially when the IRS is close to the BS, but yields no obvious gain when the user is far from the BS.
  • The number of IRS elements can be selected according to the IRS and user locations.
  • IRS-assisted MISO NOMA provides a significant performance gain over MISO NOMA without IRS, with performance reaching that of DPC when quasi-degradation can be guaranteed.
  • Placing the IRS very close to user 1 improves performance because the quasi-degradation condition can then be satisfied and the performance region obtained.

VII. CONCLUSION

The paper jointly optimizes beamforming and IRS phase shifts for NOMA and ZFBF, establishes improved quasi-degradation conditions, and proposes hybrid transmission based on their comparative strengths.

  • Quasi-degradation: The improved quasi-degradation condition makes NOMA achieve the same performance as DPC and is feasible with greater possibility than the original condition without IRS.This condition is presented specifically for IRS-assisted NOMA.
  • Optimization: Beamforming solutions are characterized for fixed IRS phase shifts, and algorithms optimize the IRS phase-shift matrix for both NOMA and ZFBF.The conclusion reports joint optimization procedures for both schemes.
  • NOMA versus ZFBF: With the same IRS phase-shift matrix and improved quasi-degradation condition, NOMA always outperforms ZFBF.The comparison is made under the stated shared phase-shift and channel condition.
  • Hybrid transmission: The identified condition for generating orthogonal channels motivates a hybrid NOMA transmission scheme.Orthogonalization is used as the basis for the hybrid design.
  • Simulation results: Simulation results show that IRS-assisted NOMA outperforms both NOMA without IRS and conventional OMA.The reported comparison is simulation-based.

APPENDIX

The appendix derives feasibility conditions for the IRS-assisted quasi-degradation constraint, linking feasibility to eigenvalues and to IRS placement and channel geometry.

  • Feasibility condition: If λmin(Υ2 − Υ1) > 0, no IRS phase-shift matrix can satisfy the quasi-degradation constraint.The appendix obtains this from Υ2−Υ1 being positive definite.
  • Feasibility condition: If λmin(Υ2 − Υ1) ≤ 0, equivalently λmax(Υ1 − Υ2) ≥ 0, the quasi-degradation constraint is always feasible.This is the converse feasibility condition stated in the proof.
  • IRS placement: When the IRS is placed at user 1, the quasi-degradation condition can always be satisfied under the stated boundedness argument.The proof uses the limiting behavior of the IRS-user-1 channel norm and a monotonicity argument.
  • Constraint recovery: Combining the derived inequalities recovers the quasi-degradation constraint in (20b).The appendix explicitly identifies the combined result with the constraint in (20b).

D. Proof of Proposition 3

This proof derives optimal ZFBF under fixed IRS phases and establishes that orthogonal user channels are exactly the condition for its optimal performance.

  • Optimal beamforming: Using the Moore–Penrose inverse, the optimal solution to problem (8a) is obtained for a given IRS phase-shift matrix.The channel matrix is formed from the two user channel vectors.
  • Equivalence: The proof establishes that h1^Hh2 = 0 can be satisfied if and only if R = 0.It proves necessity and sufficiency through the quadratic form v^HRv.
  • Orthogonality condition: If the two user channels are orthogonal, the ZFBF scheme obtains optimal performance.The proof identifies h1^Hh2 = 0 as the relevant orthogonality condition.
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