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Joint Power Allocation and Beamforming for Non-Orthogonal Multiple Access (NOMA) in 5G Millimeter-Wave Communications

Zhenyu Xiao, Lipeng Zhu, Jinho Choi, Pengfei Xia, Xiang-Gen Xia

arXiv:1711.01380v1cs.IT

TL;DR

The paper addresses joint beamforming and power allocation for maximizing the sum rate of a two-user mmWave-NOMA system under analog beamforming constraints. It decomposes the non-convex problem into power-and-beam-gain allocation and constant-modulus beamforming sub-problems. Evaluations report close-to-bound sum-rate performance and substantially better performance than TDMA.

  • Problem

    The central problem is jointly steering an analog beam toward two users and allocating their powers to maximize sum rate under minimum-rate and constant-modulus constraints.

  • Method

    The paper decomposes joint beamforming and power allocation into power-and-beam-gain allocation and constant-modulus beamforming sub-problems.

  • Results

    The proposed suboptimal solution achieves close-to-bound sum-rate performance and is significantly better than TDMA.

  • Takeaways & Limitations

    The approach provides a suboptimal solution for two-user mmWave-NOMA and discusses extensions to systems with more users.

Abstract

from arXiv · show

In this paper we explore non-orthogonal multiple access (NOMA) in millimeter-wave (mmWave) communications (mmWave-NOMA). In particular, we consider a typical problem, i.e., maximization of the sum rate of a 2-user mmWave-NOMA system. In this problem, we need to find the beamforming vector to steer towards the two users simultaneously subject to an analog beamforming structure, while allocating appropriate power to them. As the problem is non-convex and may not be converted to a convex problem with simple manipulations, we propose a suboptimal solution to this problem. The basic idea is to decompose the original joint beamforming and power allocation problem into two sub-problems which are relatively easy to solve: one is a power and beam gain allocation problem, and the other is a beamforming problem under a constant-modulus constraint. Extension of the proposed solution from 2-user mmWave-NOMA to more-user mmWave-NOMA is also discussed. Extensive performance evaluations are conducted to verify the rational of the proposed solution, and the results also show that the proposed sub-optimal solution achieve close-to-bound sum-rate performance, which is significantly better than that of time-division multiple access (TDMA).

I. INTRODUCTION

mmWave communications can provide high capacity, but conventional orthogonal access and limited RF chains constrain service for many heterogeneous users. The paper studies mmWave-NOMA, where beamforming and power allocation are jointly designed under analog beamforming constraints.

  • mmWave communication offers abundant spectrum and higher capacity than legacy low-frequency mobile communications.
  • OMA strategies may waste resource for users with different rate requirements and face challenges supporting many future 5G users.
  • Limited RF-chain counts restrict the number of users served per resource block, making NOMA significant for increasing mmWave user support.
  • mmWave-NOMA differs from conventional NOMA because large-array beamforming intertwines power allocation with beamforming design.
  • The system uses one BS antenna array to serve single-antenna users, with constant-modulus analog beamforming vectors implemented through phase shifters.
  • The mmWave channel is directionally sparse, so beamforming can target each user's strongest multipath component and use a simpler effective channel model.

B. Decoding Order

In two-user mmWave-NOMA, decoding order depends on both channel and beamforming gains, unlike conventional single-antenna NOMA. The paper formulates the problem using Case 2 and notes its advantage under equal rate constraints.

  • Decoding order in mmWave-NOMA depends on both channel gain and beamforming gain, yielding two possible two-user cases.
  • When s1 is decoded first, User 2 decodes and subtracts s1 before decoding s2, subject to an implicit SINR constraint.
  • The paper chooses Case 2, in which s2 is decoded first, and formulates sum-rate maximization with minimum-rate and constant-modulus constraints.
  • The two decoding-order formulations are similar, but Case 2 usually achieves a better sum rate when User 1 has the better channel gain.
  • If r1 = r2, decoding User 2 first is optimal under the paper's assumed channel ordering.

III. SOLUTION OF THE PROBLEM

The formulated joint optimization is difficult because it is non-convex and high-dimensional. The paper therefore introduces a suboptimal solution based on decomposing it into two easier sub-problems.

  • Existing optimization tools cannot directly solve the problem because it is non-convex and may not become convex through simple manipulations.
  • Direct exhaustive optimization is computationally prohibitive because the problem dimension is N + 2, with N generally large.
  • The proposed suboptimal approach decomposes the joint problem into power-and-beam-gain allocation and constant-modulus beamforming sub-problems.

A. Problem Decomposition

The paper decomposes the joint power-allocation and beamforming problem into separate allocation and beamforming sub-problems, using ideal beamforming to derive gain constraints and then enforcing constant-modulus feasibility.

  • Problem Decomposition: Power allocation intertwines with beamforming under the constant-modulus constraint, motivating decomposition into separate sub-problems.The original problem is difficult because the beamforming vector must simultaneously realize user gains and satisfy constant-modulus elements.
  • Problem Decomposition: Under ideal beamforming, the two users’ beam gains satisfy a relationship determined by the antenna count N.The ideal beam pattern has significant gains along the two user directions and zero gains elsewhere, with beam width 2/N.
  • Problem Decomposition: Replacing |h_i^H w|^2 with beam gains c_i yields a power and beam gain allocation problem without the constant-modulus constraint.The constant-modulus condition is deferred to the subsequent beamforming sub-problem.
  • Problem Decomposition: When r_1 = r_2, decoding User 2 first is the optimal decoding order in the considered model.The optimal order is the order of increasing |λ_i| under identical minimum-rate constraints.
  • Problem Decomposition: The beamforming sub-problem relaxes strict gain equalities to inequalities and minimizes a surrogate objective before constant-modulus normalization.The relaxed problem is easier to solve, but normalization can affect the achieved gains and requires resetting user powers in the original problem.
  • Problem Decomposition: The complete method solves the allocation sub-problem, designs a beamformer from its gain constraints, normalizes it, and then recalculates powers.Because both sub-problems are relaxations, the resulting solution is not globally optimal.

B. Solution of the Power and Beam Gain Allocation Sub-Problem

The allocation sub-problem reduces the ideal-beamforming formulation to a two-variable optimization, whose optimum is determined by stationary-point and boundary analysis under rate, power, and decoding constraints.

  • Allocation Formulation: The allocation formulation uses p_2 = P − p_1 and expresses c_2 from c_1 under the ideal beam-gain relation.This produces a two-variable objective f(c_1, p_1) with user rates represented through the resulting gains and powers.
  • Feasible-Region Analysis: The stationary point of f(c_1, p_1) is a saddle point, so the optimum lies on a boundary of the feasible region.Function values at selected boundary intersections are both larger and smaller than the stationary-point value.
  • Feasible-Region Analysis: The feasible region is enclosed by three inequality boundaries, corresponding to c_1 = c_2, R_1 = r_1, and R_2 = r_2.The optimization therefore examines candidate optima on each boundary.
  • Boundary Cases: On the R_1 = r_1 boundary, substituting the rate constraint leaves a one-variable problem in c_1 whose positive stationary root gives the candidate optimum.The negative root is discarded because beam gain c_1 must be positive.
  • Boundary Cases: Four cases compare boundary candidates according to whether c_1,1 and c_1,2 lie below or above the stationary value c_1m.The selected solution is the candidate with the larger objective when both candidates exceed c_1m.
  • Resulting Allocation: The allocation sub-problem is solved optimally under ideal beamforming, but its optimum is an upper bound for the original constant-modulus problem.The ideal gains may not be simultaneously realizable by a constant-modulus beamforming vector.

C. Solution of the Beamforming Sub-Problem

The beamforming sub-problem constructs a relaxed solution through convex optimization, normalizes it to constant modulus, and analyzes the resulting gain distortion and performance impact.

  • Beamformer Construction: The beamforming sub-problem designs w to realize the target gains c⋆_1 and c⋆_2 under a constant-modulus constraint.The relaxed formulation is used because directly imposing the strict gain equalities is difficult.
  • Beamformer Construction: The nonconvex absolute-value constraint is handled by testing M candidate phases and solving one convex optimization problem for each phase.The solution with the smallest objective among the M optima is selected as w⋆_0.
  • Constant-Modulus Normalization: The selected beamformer is normalized to unit power and then converted to constant modulus while preserving the phases of its elements.This normalization is required because solving the relaxed problem does not guarantee the constant-modulus constraint.
  • Constant-Modulus Normalization: Constant-modulus normalization can move the achieved gains far from c⋆_i when the relaxed beamformer’s element moduli differ substantially.The paper identifies this as a potential source of unsatisfactory sum-rate performance.
  • Gain Preservation: At least N−1 elements of the optimal relaxed beamformer have equal modulus, while the remaining element has no larger modulus.This structure supports the claim that normalization has limited influence on the desired gains.
  • Gain Preservation: The influence of constant-modulus normalization on desired beam gains decreases as N increases.The stated result follows from the bound associated with the constant-modulus vector comparison.

D. Solution of the Original Problem

The proposed solution can be applied with an exact channel model because its effective-channel construction preserves feasibility in general, and simulations show close-to-bound rates.

  • Final Power Allocation: The final power allocation is obtained by substituting the normalized beamformer into the original optimization problem.The resulting single-variable power allocation uses fixed beam gains after beamformer normalization.
  • Exact-Channel Applicability: The simplified effective-channel model can be replaced by an effective channel defined from any exact channel model.The proposed approach is therefore not restricted to the simplified channel representation.
  • Exact-Channel Applicability: When the beamformer steers toward the strongest multipath component, spatial sparsity limits interference from other multipath components.The small steering-vector correlation for sufficiently separated angles supports this property.
  • Exact-Channel Applicability: The resulting solution is generally feasible for the original exact-channel problem after being obtained with the effective-channel model.The paper states that this conclusion is further verified through simulations.

E. Generalization

The proposed 2-user solution can extend across antenna-array types and more-user systems, but its allocation and phase-search complexity grows rapidly with user count.

  • Array and user extensions: The 2-user solution addresses joint power allocation and beamforming for a uniform linear array, motivating generalization to other arrays and more users.
  • Array and user extensions: For planar and circular arrays, only Lemma 1 must change because the remaining solution does not depend on antenna spacing or array shape.
  • Array and user extensions: The two-subproblem decomposition remains applicable to more-user systems, but the existing power and beam-gain allocation solution may not transfer directly.
  • Complexity boundary: A K-user system has (2K − 2) independent allocation variables, making exhaustive search complexity O((1/ǫ)^(2K−2)) and potentially prohibitive for large K.
  • Complexity boundary: The beamforming extension requires searching M^(K−1) phase combinations, while the authors suggest OMA, multiple RF chains, or grouped beams as alternatives for supporting more users.

IV. PERFORMANCE EVALUATIONS

The evaluations show that the proposed beamforming solution remains close to ideal performance after constant-modulus normalization and yields achievable rates near the performance bound. Across channel conditions, mmWave-NOMA substantially outperforms TDMA.

  • Beamforming evaluation: The designed beam patterns remain close to the ideal pattern along both user directions after constant-modulus normalization.This indicates that normalization has little influence on beam gains in the desired directions.
  • Beamforming evaluation: The designed user and sum beam gains are almost unchanged by constant-modulus normalization, with only a small gap from ideal gains.The gap is attributed to sidelobes in the designed beam pattern.
  • Beamforming evaluation: Relative gain errors for both users and the sum gain are roughly 0.1 and increase slowly for N ≤40, then nearly stabilize for N >40.The results support robustness to antenna number and the approximate constancy of sum beam gain versus N.
  • Rate and allocation evaluation: Designed achievable rates remain close to the performance bound for both users and the sum rate as the rate constraint varies.Most power or beam gain goes to User 1, while User 2 receives only what is needed to meet its rate constraint.
  • Rate and allocation evaluation: With increasing total power-to-noise ratio, designed rates remain close to their performance bounds and most power or beam gain remains allocated to User 1.User 1’s beam gain and power increase, while User 2’s beam gain decreases as P/σ2 increases.
  • Comparison with TDMA: Theoretical and practical mmWave-NOMA performance is very close, and mmWave-NOMA significantly outperforms TDMA under both LOS and NLOS channels.The comparison uses the original channel for practical performance and the effective channel for theoretical performance.

V. CONCLUSION

The paper addresses sum-rate maximization in a 2-user mmWave-NOMA system by decomposing joint beamforming and power allocation into two sub-problems. Evaluations support the suboptimal solution, which achieves close-to-bound sum-rate performance and outperforms TDMA.

  • The target problem jointly steers an analog beamforming vector toward two users while allocating their transmit power to maximize sum rate.
  • The proposed suboptimal method separates the original problem into power-and-beam-gain allocation and constant-modulus beamforming sub-problems.
  • Close-to-bound sum-rate performance is achieved, and the solution is distinctly better than TDMA.

APPENDIX A PROOF OF THEOREM 1

The appendix establishes structural properties of the optimal solution to a constrained beamforming problem and uses them to prove equivalence between two formulations. The proof proceeds by contradiction through a feasible perturbation.

  • Problem (44) is shown equivalent to Problem (39) by proving mutual feasibility of their optimal solutions.
  • The optimal vector for Problem (44) has at least N−1 elements with equal modulus, while the remaining element has smaller modulus.
  • The proof ranks the absolute weights and assumes a strict inequality after the second-ranked element to derive a contradiction.
  • A nearby feasible perturbation is constructed by scaling the free coordinates and expressing the first two coordinates as linear combinations of them.
  • Choosing a sufficiently small perturbation makes the constructed vector’s Nth element largest in modulus, contradicting optimality and forcing the remaining inequalities to be equalities.
  • Combining the resulting lemma with Lemma 2 yields Theorem 1.

APPENDIX B PROOF OF THEOREM 2

The appendix’s supplied proof text states a phase-preservation property under constant-modulus normalization and identifies the index ranges used in the argument.

  • Constant-modulus normalization preserves the phases of the elements of w⋆.
  • One indexed relation in the proof applies for i = 1, 2, ..., N.
  • Another indexed relation applies for i = 2, 3, ..., N.
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