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Reconfigurable Intelligent Surfaces Aided mmWave NOMA: Joint Power Allocation,Phase Shifts, and Hybrid Beamforming Optimization

Yue Xiu, Jun Zhao, Wei Sun, Marco Di Renzo, Guan Gui, Zhongpei Zhang, Ning Wei

arXiv:2007.05873v5eess.SPcs.SYeess.SY

TL;DR

MmWave-NOMA faces severe path loss, while OMA limits the number of users served per resource block. The paper jointly optimizes power allocation, RIS phase shifts, and hybrid beamforming through alternating optimization. Its numerical results report higher sum-rate for RIS-aided mmWave-NOMA than for a system without RIS.

  • Problem

    MmWave communications suffer severe path loss, and OMA limits the users served per resource block to no more than the number of RF chains.

    Source: §I · p. 2

  • Method

    The paper jointly optimizes power allocation, RIS phase shifts, and hybrid beamforming using an alternating optimization algorithm.

    Source: §VI · p. 24

  • Results

    The proposed RIS-aided mmWave-NOMA scheme outperforms mmWave-NOMA schemes without RIS in sum-rate.

    Source: §VI · p. 25

  • Takeaways & Limitations

    Optimizing RIS phase shifts is reported to achieve a significant performance gain.

    Source: §VI · p. 25

Abstract

from arXiv · show

In this paper, an reconfigurable intelligent surface (RIS)-aided millimeter wave (mmWave) non-orthogonal multiple access (NOMA) system is considered. In particular, we consider an RIS-aided mmWave-NOMA downlink system with a hybrid beamforming structure. To maximize the achievable sum-rate under a minimum rate constraint for the users and a minimum transmit power constraint, a joint RIS phase shifts, hybrid beamforming, and power allocation problem is formulated. To solve this non-convex optimization problem, we develop an alternating optimization algorithm. Specifically, first, the non-convex problem is transformed into three subproblems, i.e., power allocation, joint phase shifts and analog beamforming optimization, and digital beamforming design. Then, we solve the power allocation problem under fixed phase shifts of the RIS and hybrid beamforming. Finally, given the power allocation matrix, an alternating manifold optimization (AMO)-based method and a successive convex approximation (SCA)-based method are utilized to design the phase shifts, analog beamforming, and transmit beamforming, respectively. Numerical results reveal that the proposed alternating optimization algorithm outperforms state-of-the-art schemes in terms of sum-rate. Moreover, compared to a conventional mmWave-NOMA system without RIS, the proposed RIS-aided mmWave-NOMA system is capable of improving the achievable sum-rate of the system.

I. INTRODUCTION

The introduction motivates RIS-aided mmWave-NOMA as a response to mmWave path loss and OMA’s limit on users per resource block, while identifying open joint-optimization challenges. It presents an alternating optimization approach and reports higher sum-rate than mmWave-NOMA without RIS.

I. INTRODUCTION

  • OMA serves at most as many users per resource block as there are RF chains, whereas power-domain NOMA can serve more users through simultaneous transmission and SIC.The cited NOMA discussion describes superposition coding at the transmitter and successive interference cancellation at the receiver.Source: §I · p. 2
  • MmWave-NOMA faces severe path loss, blockage vulnerability, and strong user interference; RISs are proposed to address these challenges.RIS phase shifts can also make the users’ decoding order more flexible.Source: §I · pp. 2–3

A. Related Work

B. Motivations and Contributions

  • The paper formulates joint power, RIS phase-shift, and hybrid-beamforming optimization to maximize sum-rate under minimum user-rate and transmit-power constraints.The system has a hybrid beamforming structure.Source: B. Motivations and Contributions · p. 4
  • An alternating optimization algorithm combines power allocation, AMO for RIS phase shifts and analog beamforming, and SCA for digital beamforming.The phase-shift and analog-beamforming design assumes fixed transmit power and digital beamforming weights.Source: B. Motivations and Contributions · p. 4
  • The proposed RIS-aided mmWave-NOMA system yields a better sum-rate than a traditional mmWave-NOMA system without RIS.The numerical results compare the proposed scheme with a traditional scheme that does not use RISs.Source: B. Motivations and Contributions · pp. 4–5

II. SYSTEM MODEL AND PROBLEM FORMULATION

The section models a downlink multi-group, multi-user RIS-aided mmWave-NOMA system with hybrid beamforming. It formulates sum-rate optimization and identifies coupled variables and changing decoding order as challenges.

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The system uses an AP, an RIS phase-shift matrix, and hybrid analog-digital beamforming; users are grouped, with SIC applied among users in the same group.The analog and RIS phase-shift vectors have constant-modulus constraints.Source: §II · pp. 5–6
  • The NOMA decoding order is ranked by effective channel gains, which depend on both channel and beamforming gains.The order varies with the beamforming and RIS phase-shift matrices.Source: §II · pp. 7, 9
  • The model approximates inter-group interference as suppressible when it is sufficiently small, leaving same-group users as the main interference source.The resulting SINR is approximated under this assumption.Source: §II · pp. 7–8
  • The sum-rate optimization couples its variables, making the problem non-convex, while decoding-order changes with beamforming and RIS phases add another difficulty.The proposed response divides optimization into power, analog-beamforming and RIS-phase, and transmit-beamforming parts.Source: §II · pp. 8–9

III. POWER ALLOCATION OPTIMIZATION

The power-allocation section fixes the hybrid beamforming and RIS phase shifts, then optimizes power under the assumed decoding order. Its iterative procedure uses results for cases with and without a constraint.

III. POWER ALLOCATION OPTIMIZATION

  • With beamforming and RIS phases fixed, the group power allocation is optimized for the fixed decoding order.The formulation introduces P_n as the power allocated to group n.Source: §III · p. 9
  • For users above the first decoding position, the sum-rate maximum sets their rates to their minimum requirements, leaving as much power as possible for higher-order users once others meet theirs.The cited result refers to R_n,k = γ_k for k ≥ 2.Source: §III · p. 10
  • The method first solves a relaxed problem without constraint (12b); if its solution violates that constraint, Theorem 2 supplies the globally optimal solution for the constrained case.Theorem 1 gives the global solution when constraint (12b) is removed.Source: §III · p. 11
  • The group power-allocation procedure updates the problem and repeats until convergence.The algorithm uses Theorems 1 and 2 in that iterative procedure.Source: §III · p. 12

1 Initialization: t = 0, P (t)

The initialization excerpt lists an index set and quantities to calculate using Theorem 1.

1 Initialization: t = 0, P (t)

  • The initialization excerpt lists V as {1, 2, ..., N}.The excerpt presents this as the set V.Source: §1 · p. 12
  • Theorem 1 is used to calculate β_n, α_n, and P_n^(t).These quantities appear in the initialization instruction.Source: §1 · p. 12

8 According to Theorem 2, calculate ˆP (t)

The section reformulates the joint phase-shift and hybrid-beamforming design and addresses its non-convex constraints with auxiliary variables and optimization methods. Its outer procedure also updates the set V and limits the number of iterations.

8 According to Theorem 2, calculate ˆP (t)

  • The outer procedure updates V according to V = V/N and uses Tmax as the maximum number of iterations.Source: §8 · p. 12, §11 · p. 12

IV. PHASE SHIFTS AND HYBRID BEAMFORMING DESIGN

  • The analog-beamforming and RIS phase-shift modulus constraints make the design problem difficult, motivating a suboptimal algorithm with auxiliary variables.The auxiliary variables are {u_i}, {v_n,k,i}, and {z_n,k,i}.Source: §IV · p. 12
  • The reformulated objective is treated as a difference-of-convex program, and an exact penalty method is applied while several constraints remain non-convex.The section proposes AMO for non-convex constraints (12e), (12f), and (12h).Source: §IV · pp. 13–14

A. Phase Shift and Analog Beamforming Design Based on the AMO Algorithm

For fixed transmit beamforming and power allocation, the AMO design optimizes the RIS phase shifts and analog beamforming on manifolds. It uses gradient steps and retractions to keep updated points on the manifolds.

A. Phase Shift and Analog Beamforming Design Based on the AMO Algorithm

  • With transmit beamforming and power allocation fixed, the design problem is optimized over the phase shifts, analog beamforming, and digital beamforming variables.Source: A. Phase Shift and Analog Beamforming… · p. 15
  • The constant-modulus constraints define manifold spaces, where the algorithm uses Riemannian gradients for its search direction.The section identifies two spaces as Riemannian manifolds and the third as an oblique manifold.Source: A. Phase Shift and Analog Beamforming… · p. 15
  • After gradient steps for the phase shifts and beamforming matrices, retraction operations return updated points to the manifold.The section describes separate updates for θ, F, and D.Source: A. Phase Shift and Analog Beamforming… · p. 16

4 Calculate the Euclidean gradient and the Riemannian gradient based on (43) and (46)

The AMO procedure uses gradient updates and retractions for its design variables, while the digital-beamforming procedure uses successive convex approximation. The text states convergence properties for both methods.

4 Calculate the Euclidean gradient and the Riemannian gradient based on (43) and (46)

  • The AMO updates θ using a gradient step and retraction, then repeats until the stated accuracy condition is met.Source: §4 · p. 17

9 Calculate the Euclidean gradient and the Riemannian gradient based on (44) and (47)

  • The AMO similarly updates F using a gradient step and retraction, with an accuracy-based stopping condition.Source: §9 · p. 17

14 Calculate the Euclidean gradient and the Riemannian gradient based on (45) and (48)

  • The AMO updates D by gradient descent and continues until its stated stopping condition is met; the manifold algorithm is guaranteed to reach a point with zero objective gradient.Source: §14 · p. 17

B. Digital Beamforming Based on the SCA Algorithm

  • The SCA method transforms the non-convex objective component into a convex function and iteratively solves a convex approximation.First-order Taylor expansions provide convex bounds for the functions f({z_n,k,n}) and g({v_n,k,i}).Source: B. Digital Beamforming Based on the… · p. 18

1 Initialization: t = 0, {˜v(0)

The full alternating algorithm successively updates power allocation, RIS phase shifts and analog beamforming, then digital beamforming. It also reorders users by effective channel gain during its iterations.

7 Output: W ∗

  • The SCA-based digital-beamforming algorithm converges to a stationary point satisfying the KKT conditions.Source: §7 · p. 19

C. Optimization Algorithms to Solve (14) and Computational Complexity

  • The full algorithm solves power allocation first, updates RIS phase shifts and analog beamforming with Algorithm 2, then obtains digital beamforming with Algorithm 3.Power allocation is solved with the hybrid beamforming matrix and RIS phase shifts fixed.Source: C. Optimization Algorithms to Solve… · pp. 19–20

3 Using Algorithm1 to calculate {p(t)

  • After each iteration, the algorithm reorders users in each group by their effective channel gains.Source: §3 · p. 20

9 Output: {p∗

The section reports complexity bounds for the optimization algorithms and simulations evaluating convergence and sum-rate across transmission schemes and operating conditions. The proposed RIS-aided schemes outperform non-RIS comparisons in reported settings, with performance also depending on channel geometry and RIS size.

9 Output: {p∗

  • Algorithm 1 has complexity O(MKN + S1MK2), while Algorithm 4's complexity depends on its iteration count and the SCA problem's real-variable count.The text gives the Algorithm 4 bound as O(T(MKN + S1MK2 + S3(2NtNRF + 2NrK + 3NK2)3.5 log2(1 ...)); the expression is incomplete in the supplied passage.Source: §9 · p. 20

V. NUMERICAL RESULTS

  • The simulations use six users, 64 RIS elements, three RF chains, and compare RIS-aided hybrid and fully digital NOMA with non-RIS schemes.The comparisons also include RIS-aided all-digital mmWave-NOMA, non-RIS all-digital mmWave-NOMA, and non-RIS mmWave-FDMA.Source: §V · p. 21
  • Power allocation converges in about 10 iterations, while the AMO phase-shift and SCA transmit-beamforming algorithms show good convergence performance.The sum-rate increases with SNR and eventually stabilizes.Source: §V · pp. 21–22, Figure 3 · p. 22
  • At a 1 bps minimum-rate constraint, the proposed RIS-aided schemes largely outperform schemes without RIS.The proposed RIS-aided hybrid and fully digital mmWave-NOMA systems also outperform corresponding mmWave-OMA systems without RIS.Source: Figure 4 · p. 22, §V · p. 22
  • As total transmission power varies, the proposed RIS-aided NOMA schemes achieve higher sum-rate than the compared non-RIS NOMA systems, especially at low power.Fig. 5 presents sum-rate as a function of total transmission power.Source: Figure 5 · p. 23, §V · p. 23
  • The proposed RIS-based scheme outperforms the two benchmark schemes, and performance improves as the number of RIS reflection elements increases.The benchmarks use random RIS and analog-beamforming phases with ZF digital beamforming, or random analog phases with ZF digital precoding; they do not account for interference from other groups.Source: Figure 6 · p. 23, §V · p. 24
  • As the RIS-to-obstacle distance increases, sum-rate first rises and then falls because interference attenuates more sharply than the desired signal before the desired signal's decline dominates.The performance gain of the proposed RIS-aided schemes increases as this distance decreases.Source: §V · p. 24

VI. CONCLUSION

The paper jointly optimizes power allocation, RIS phase shifts, and hybrid beamforming for downlink multiuser RIS-aided mmWave-NOMA. Its alternating algorithm is reported to achieve near-optimal performance, with the RIS-aided scheme outperforming schemes without RIS.

VI. CONCLUSION

  • The design jointly optimizes RIS phase shifts, AP power allocation, and hybrid beamforming to maximize system sum-rate.Source: §VI · p. 24
  • The alternating algorithm first allocates power with phase shifts and hybrid beamforming fixed, then uses AMO and SCA to design RIS phases and hybrid beamforming.Source: §VI · p. 24
  • The proposed algorithms are reported to achieve near-optimal performance.Source: §VI · p. 24
  • Optimizing the RIS phase shifts can yield a significant performance gain.Source: §VI · p. 25
  • The proposed RIS-aided mmWave-NOMA system outperforms mmWave-NOMA schemes without RIS.Source: §VI · p. 25

APPENDIX A PROOF OF THEOREM 1

The appendices prove the stated theorems by analyzing power allocation, reformulating a constraint, and establishing convergence to a KKT solution.

APPENDIX A PROOF OF THEOREM 1

  • Under the stated assumption that inter-group interference is ignored, the proof uses the increasing objective in group power and KKT conditions to establish Theorem 1.Source: Appendix A · pp. 25–26

APPENDIX B PROOF OF THEOREM 2

  • Theorem 2 is proved by contradiction: a power-allocation perturbation is shown to produce a higher sum rate than the assumed optimum.Source: Appendix B · pp. 26–27

APPENDIX C PROOF OF THEOREM 3

  • Theorem 3 follows by using the Schur complement to replace constraint (30f) with constraints (33) and (34).Source: Appendix C · p. 27

APPENDIX D PROOF OF THEOREM 4

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