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On Optimal Power Allocation for Downlink Non-Orthogonal Multiple Access Systems

Jianyue Zhu, Jiaheng Wang, Yongming Huang, Shiwen He, Xiaohu You, Luxi Yang

arXiv:1707.06350v1cs.IT

TL;DR

The paper addresses the limited availability of optimal multi-channel power allocations in NOMA and the high complexity of jointly optimizing power and channel assignment. It derives closed or semi-closed power solutions under several criteria while enforcing SIC order constraints, then combines them with matching-based assignment. Simulations report better performance than existing schemes and near-optimal joint optimization, with a maximum gap below 5% in the exhaustive-search comparison.

  • Problem

    Optimal power allocation was previously known mainly for single-channel special cases, while joint power and channel optimization generally required exhaustive search.

  • Method

    The paper analytically derives power allocations for multiple criteria with explicit SIC power-order constraints and combines them with matching-based channel assignment.

  • Results

    The proposed joint resource optimization outperforms existing schemes and achieves near-optimal performance, with a maximum gap below 5% from exhaustive search.

  • Takeaways & Limitations

    Closed or semi-closed power allocation can support efficient joint NOMA resource optimization across multiple channels and performance criteria.

Abstract

from arXiv · show

Non-orthogonal multiple access (NOMA) enables power-domain multiplexing via successive interference cancellation (SIC) and has been viewed as a promising technology for 5G communication. The full benefit of NOMA depends on resource allocation, including power allocation and channel assignment, for all users, which, however, leads to mixed integer programs. In the literature, the optimal power allocation has only been found in some special cases, while the joint optimization of power allocation and channel assignment generally requires exhaustive search. In this paper, we investigate resource allocation in downlink NOMA systems. As the main contribution, we analytically characterize the optimal power allocation with given channel assignment over multiple channels under different performance criteria. Specifically, we consider the maximin fairness, weighted sum rate maximization, sum rate maximization with quality of service (QoS) constraints, energy efficiency maximization with weights or QoS constraints in NOMA systems. We also take explicitly into account the order constraints on the powers of the users on each channel, which are often ignored in theexisting works, and show that they have a significant impact on SIC in NOMA systems. Then, we provide the optimal power allocation for the considered criteria in closed or semi-closed form. We also propose a low-complexity efficient method to jointly optimize channel assignment and power allocation in NOMA systems by incorporating the matching algorithm with the optimal power allocation. Simulation results show that the joint resource optimization using our optimal power allocation yields better performance than the existing schemes.

I. INTRODUCTION

NOMA’s benefits depend on jointly allocating power and channels, but existing optimal solutions are limited and joint optimization can be computationally prohibitive. This paper addresses the gap by deriving optimal power allocations under multiple criteria while preserving SIC order and combining them with matching-based channel assignment.

  • Motivation: Joint power allocation and channel assignment forms a mixed integer, NP-hard problem for NOMA systems, making exhaustive optimization impractical.Existing practice often optimizes the two components separately or alternately, potentially yielding suboptimal solutions.
  • Research gap: Prior optimal power-allocation results generally covered only single-channel or otherwise special cases, leaving the multiple-channel general case unresolved.The gap spans sum rate, fairness, and energy-efficiency objectives.
  • Scope: The paper considers maximin fairness, weighted sum rate, QoS-constrained sum rate, and energy-efficiency maximization with weights or QoS constraints.These criteria produce distinct resource-allocation formulations.
  • Model and constraints: Power-order constraints are explicitly imposed to preserve the decoding order of successive interference cancellation on each channel.NOMA assigns higher power to users with lower channel-to-noise ratios, yielding ordered user powers.
  • Contributions: Closed-form or semi-closed optimal power allocations are derived for the considered criteria, including multiple users across multiple channels.The paper also introduces SIC-stability conditions to avoid equal power allocation that can impair SIC.
  • Joint optimization: A matching-based iterative method jointly optimizes channel assignment and power allocation, while allowing the derived power allocation to work with other assignment algorithms.The method is designed to reduce the complexity associated with exhaustive joint search.

B. Problem Formulation

The formulation evaluates NOMA resource allocation under fairness, sum-rate, and energy-efficiency criteria subject to transmit-power, QoS, and SIC-order constraints. It first optimizes power for a fixed assignment and then addresses channel assignment, targeting optimal multi-channel allocations with manageable complexity.

  • Performance criteria: The paper formulates resource allocation using maximin fairness, weighted sum rate, QoS-constrained sum rate, and energy-efficiency objectives.Energy efficiency is defined as the ratio of total sum rate to whole-system power consumption.
  • Prior limitations: Existing optimal power allocations were limited to special cases, while joint resource optimization was either suboptimal or required exhaustive search.Examples include two users on one channel for sum rate and one channel for energy efficiency.
  • Constraints: The base station’s total transmit power is constrained by a power budget, while each channel imposes ordered user powers to preserve SIC decoding order.The ordering assigns higher power to the user with lower channel-to-noise ratio.
  • Optimization strategy: The proposed methodology first optimizes power allocation for a given channel assignment and then optimizes channel assignment using the resulting allocation.The authors state that optimal allocations are available in closed form or can be obtained efficiently by algorithms.

III. OPTIMAL POWER ALLOCATION FOR MAXIMIN FAIRNESS

The MMF problem is decomposed across channels and solved analytically, yielding equal user rates and absolute fairness. Under MMF, the optimal allocation is always SIC-stable.

  • Problem formulation and solution: The multi-channel MMF problem is decomposed into per-channel subproblems and solved in closed form.The resulting power-budget problem is concave and can also be solved efficiently by bisection or convex optimization.
  • Fairness result: At the optimum, UE1,m and UE2,m achieve the same rate, providing absolute fairness for the two users on each channel.Across all channels, the optimal allocation satisfies R1,m = R2,m = r for some r ≥0.
  • SIC stability: SIC-stability requires p1,m < p2,m, and the MMF optimum always satisfies this strict inequality on every channel.Thus, unlike other criteria discussed later, MMF does not produce equal power allocation that could destabilize SIC.

IV. OPTIMAL POWER ALLOCATION FOR SUM RATE

This section introduces optimal power allocation for maximizing weighted sum rate or sum rate with QoS constraints.

  • The section considers optimal power allocation for weighted sum-rate maximization and sum-rate maximization with QoS constraints.

A. Weighted SR Maximization (SR1)

Weighted sum-rate allocation is characterized in closed form under conditions that preserve SIC stability. The channel power-budget problem is then solved in waterfilling form.

  • Weighted SR maximization: The weighted sum-rate power-allocation problem is nonconvex, but its optimal solution can be characterized in closed form.The formulation accounts for interference between the two users sharing a channel.
  • Per-channel allocation: Under Γ1,m ≥ Γ2,m, 1 < W2,m/W1,m < Γ1,m/Γ2,m, and q_m > 2Ω_m, the allocation uses p⋆_2,m = q_m − p⋆_1,m.The optimal strong-user power is the unique root p1,m = Ω_m.
  • SIC stability: SIC-stability holds on channel m if and only if 1 < W2,m/W1,m < Γ1,m/Γ2,m and q_m > 2Ω_m.Violating either condition can produce equal powers q_m/2 and destabilize SIC.
  • Power-budget optimization: The channel power-budget problem is convex and has a waterfilling-form solution.The optimal allocation is jointly characterized by this result and the per-channel weighted sum-rate solution under SIC stability.

B. SR Maximization with QoS (SR2)

For sum-rate maximization with QoS constraints, the paper derives an optimal per-channel allocation and combines it with power-budget optimization. SIC stability depends on explicit QoS and power conditions.

  • Per-channel allocation: The per-channel QoS-constrained sum-rate problem is nonconvex, but its optimal solution is analytically characterized.The derivation imposes the power-order constraint and uses p2,m = q_m − p1,m.
  • Per-channel allocation: Under Γ1,m ≥ Γ2,m, A2,m ≥ 2, and q_m ≥ Υ_m, the optimal allocation is p⋆_2,m = q_m − p⋆_1,m.These conditions ensure the resulting allocation respects the relevant SIC-stability requirements.
  • SIC stability: SIC stability on channel m requires A2,m ≥ 2 and q_m ≥ Υ_m.If either condition fails, equal power allocation q_m/2 may result and SIC may fail.
  • QoS allocation: At the optimum, the lower-CNR user receives exactly the power needed to meet its QoS requirement, while remaining power serves the higher-CNR user.Specifically, R2,m(p⋆_1,m,p⋆_2,m) = Rmin 2,m.
  • Power-budget optimization: The remaining power-budget optimization is concave in q_m and has a waterfilling-form solution.The paper therefore jointly characterizes the QoS-constrained sum-rate allocation through the per-channel proposition and the power-budget theorem.

V. OPTIMAL POWER ALLOCATION FOR ENERGY EFFICIENCY

This section investigates energy-efficiency maximization in NOMA systems with weights or quality-of-service constraints.

  • The section targets energy-efficiency maximization in NOMA systems.
  • Weighted energy-efficiency optimization is considered as one problem setting.
  • Quality-of-service-constrained energy-efficiency optimization is also considered.

A. EE Maximization with Weights (EE1)

For weighted energy-efficiency maximization, the paper decomposes power allocation by channel, derives optimal solutions under SIC-stability conditions, and solves the remaining fractional problem iteratively.

  • Prior work had solved weighted energy-efficiency power allocation only for one channel or suboptimally, leaving the general multichannel problem open.
  • Introducing channel power budgets q_m decomposes the allocation into independent two-user subproblems for each channel.
  • Under Γ1,m ≥ Γ2,m, 1 < W2,m/W1,m < Γ1,m/Γ2,m, and q_m > 2Ω_m, Proposition 4 provides the optimal per-channel solution.
  • The resulting SIC-stability conditions require q_m > 2Ω_m and 1 < W2,m/W1,m < Γ1,m/Γ2,m on each channel, with P > 2Σ_m=1^M Ω_m across channels.
  • The weighted energy-efficiency objective is fractional and nonconvex, so the method solves a parameterized convex problem and updates α until H⋆(α) = 0.
  • Algorithm 1 repeatedly computes q⋆ using Theorem 4, evaluates H⋆(α), and updates α; it is guaranteed to converge to the desirable α.

B. EE Maximization with QoS (EE2)

For energy-efficiency maximization with QoS constraints, the paper addresses the previously unresolved multichannel case using per-channel decomposition, closed-form optimization, and parameter updates.

  • The general multichannel QoS-constrained energy-efficiency problem was open because prior optimal solutions covered only one channel.
  • Introducing q_m with p1,m + p2,m = q_m decomposes the power allocation into channel-specific subproblems.
  • Under Γ1,m ≥ Γ2,m, A2,m ≥ 2, and q_m ≥ Υ_m, Proposition 5 gives the optimal solution for the QoS-constrained per-channel problem.
  • The final optimal allocation uses Theorem 5, Proposition 5, and Algorithm 1, while SIC stability requires q_m ≥ Υ_m and A2,m ≥ 2 on every channel.
  • The method parameterizes the fractional objective, solves a convex problem for fixed α, and updates α until Q⋆(α) = 0.
  • Theorem 5 supplies the optimal solution of the fixed-α convex problem, whose concavity enables efficient solution.

VI. CHANNEL ASSIGNMENT

The paper treats channel assignment as a two-sided matching problem and combines dynamic matching with optimal power allocation to obtain an efficient joint optimization method.

  • Joint channel assignment and power allocation is NP-hard, making exhaustive search computationally prohibitive in practice.
  • The proposed method iteratively alternates matching-based channel assignment with optimal user powers and channel power budgets.
  • Each user is matched to one channel, each channel can serve two users, and channel preferences depend on the performance of candidate user sets.
  • The deferred-acceptance procedure initializes preferences from CNRs, accepts channels directly when under capacity, and otherwise retains the user set yielding better performance.
  • Algorithm 3 initializes equal channel power budgets, updates matching and optimal power allocation iteratively, and stops after a prescribed iteration count.
  • The optimal power allocation can be paired with assignment algorithms beyond deferred acceptance and can reduce exhaustive-search complexity.

VII. NUMERICAL RESULTS

Numerical simulations compare the proposed joint resource allocation with OFDMA, DC, CUP, and exhaustive search across rate, spectral-efficiency, and energy-efficiency objectives. The proposed methods consistently outperform existing alternatives and achieve near-optimal performance with low complexity.

  • Simulation setup: The simulations use randomly distributed users in a 300m-radius cell, with minimum user and BS distances of 30m and 40m, respectively.The bandwidth is B = 5MHz, and the QoS thresholds are R_min,l,m = 2 bps/Hz for both users.
  • Sum rate: All evaluated NOMA schemes outperform OFDMA in sum rate, while SR1 JRA also outperforms the DC and CUP alternatives.SR1 JRA and SR2 JRA use the proposed joint resource allocation method for weighted sum rate and QoS-constrained sum rate, respectively.
  • Spectral efficiency: NOMA achieves higher spectral efficiency than OFDMA, and JRA exceeds both DC and CUP methods as the number of users varies.The same comparative pattern observed for sum rate also appears in spectral efficiency.
  • Energy efficiency: NOMA has significantly higher energy efficiency than OFDMA, while EE1 JRA outperforms EE1 DC across BS-power and user-count evaluations.EE1 JRA and EE2 JRA optimize energy efficiency with weights and QoS constraints, respectively.
  • Exhaustive-search comparison: The proposed methods remain within 5% of the globally optimal exhaustive-search performance for N = 6 and BS power budgets from 2W to 12W.This comparison indicates near-optimal performance while avoiding the high complexity of exhaustive search.

VIII. CONCLUSION

The paper derives optimal power allocation for multiple NOMA objectives while explicitly enforcing power-order constraints and addressing SIC stability. It combines these solutions with channel assignment to obtain joint resource optimization that simulations show is near-optimal.

  • Optimal power allocation: Closed- or semi-closed-form optimal power allocations are characterized for MMF, weighted sum rate, QoS-constrained sum rate, and weighted or QoS-constrained energy efficiency.The analysis covers multiple channels with given channel assignments.
  • SIC and ordering: Power-order constraints are explicitly included, and SIC-stability is introduced to avoid equal power allocation among users on each channel.The conclusion identifies these constraints as relevant to power allocation in NOMA systems.
  • Joint optimization: The proposed joint resource optimization method achieves near-optimal performance in simulations.The numerical results compare the method against exhaustive search and existing schemes.
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