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Dynamic User Clustering and Power Allocation for Uplink and Downlink Non-Orthogonal Multiple Access (NOMA) Systems

Md Shipon Ali, Hina Tabassum, Ekram Hossain

arXiv:1608.01081v1cs.IT

TL;DR

The paper addresses joint user clustering and power allocation for uplink and downlink NOMA under throughput, power, rate, and SIC constraints. It proposes low-complexity clustering and closed-form allocation solutions, with reported NOMA gains over OMA in both directions. The conclusion also notes ideal-SIC and interference-related scope boundaries.

  • Problem

    Existing work largely studies uplink or downlink NOMA separately, often for two users with fixed powers, leaving comprehensive multi-user treatment limited.

  • Method

    The paper uses low-complexity channel-gain-based grouping followed by KKT-derived closed-form power allocation for fixed uplink or downlink clusters.

  • Results

    NOMA achieves up to 106.3% downlink and 105.4% uplink throughput gain over OMA in the reported cases.

  • Takeaways & Limitations

    More distinctive channel gains provide impressive NOMA throughput gains over OMA, while downlink performance decreases when cluster size exceeds a threshold.

  • Takeaways & Limitations

    The analysis assumes ideal SIC, while SIC errors and inter-cell interference remain under investigation.

Abstract

from arXiv · show

In this paper, first we briefly describe the differences in the working principles of uplink and downlink NOMA transmissions. Then, for both uplink and downlink NOMA, we formulate a sum-throughput maximization problem in a cell such that the user clustering (i.e., grouping users into a single cluster or multiple clusters) and power allocations in NOMA cluster(s) can be optimized under transmission power constraints, minimum rate requirements of the users, and SIC constraints. Due to the combinatorial nature of the formulated mixed integer non-linear programming (MINLP) problem, we solve the problem in two steps, i.e., by first grouping users into clusters and then optimizing their respective power allocations. In particular, we propose a low-complexity sub-optimal user grouping scheme. The proposed scheme exploits the channel gain differences among users in a NOMA cluster and group them into a single cluster or multiple clusters in order to enhance the sum-throughput of the system. For a given set of NOMA clusters, we then derive the optimal power allocation policy that maximizes the sum throughput per NOMA cluster and in turn maximizes the overall system throughput. Using KKT optimality conditions, closed-form solutions for optimal power allocations are derived for any cluster size, considering both uplink and downlink NOMA systems. Numerical results compare the performance of NOMA over orthogonal multiple access (OMA) and illustrate the significance of NOMA in various network scenarios.

I. INTRODUCTION

NOMA multiplexes users over shared resources using power-domain signaling and SIC, motivating joint user grouping and power allocation for uplink and downlink throughput maximization.

  • NOMA Fundamentals: NOMA simultaneously serves multiple users on shared radio resources by superposing their signals in the power domain and applying SIC.This contrasts with OMA, which allocates exclusive radio resources to each user.
  • Motivation: Prior work addressed downlink and uplink NOMA separately, often with two users and fixed power allocations.The paper identifies a lack of comprehensive analysis spanning both transmission directions, user grouping, and power allocation.
  • Contributions: The paper formulates cell-throughput maximization for both uplink and downlink NOMA under transmission-power, minimum-rate, and SIC constraints.The formulation jointly optimizes user grouping and cluster power allocations.
  • Contributions: A low-complexity sub-optimal grouping scheme exploits channel-gain differences to place users into one or multiple clusters for higher sum throughput.The scheme addresses the combinatorial nature of the MINLP formulation.
  • Contributions: For fixed clusters, KKT conditions yield closed-form optimal power allocations for any cluster size in both uplink and downlink NOMA.The resulting allocation maximizes cluster sum throughput and thereby overall system throughput.
  • Downlink NOMA: Downlink SIC assigns lower power to high-channel-gain users and higher power to low-channel-gain users, allowing stronger users to cancel more interference.In a three-user example, the strongest user cancels two interfering signals, while the weakest cancels none.

B. Uplink NOMA

Uplink NOMA has multiple UEs transmit to one BS over the same channel, with SIC decoding received signals in channel-gain order under power and decoding constraints.

  • Uplink Operation: In uplink NOMA, multiple UEs transmit independently to a common BS over the same radio spectrum.Each UE may use maximum or controlled transmit power, and all received signals are desired signals that interfere with one another.
  • Uplink Operation: The BS applies SIC by decoding the highest-channel-gain signal first and then successively decoding the remaining signals.The received signal from the highest-channel-gain user is likely strongest at the BS.
  • Three-User Example: In a three-user uplink cluster, the highest-gain user experiences interference from the other two, while the lowest-gain user achieves an interference-free data rate.The middle user’s rate depends on interference from the lowest-gain user.
  • SIC Constraints: Uplink SIC requires power conditions that enable decoding of the higher-gain users before the lowest-gain user.The cited conditions cover efficient decoding of UE1 and UE2 signals at the BS.
  • Power Constraints: Each uplink user’s transmit power is bounded by the per-UE maximum transmit-power budget P′_t.This bound applies to every user in the cluster.
  • SIC Constraints: For an m-user uplink cluster, the necessary SIC power constraints are expressed through received-power inequalities involving P_jγ_j and P_tol.The general condition applies for i = 1, 2, · · ·, (m − 1).

III. SYSTEM MODEL AND PROBLEM FORMULATION

The system model divides bandwidth into orthogonal resource blocks and groups users sharing blocks into NOMA clusters, then formulates constrained joint clustering and power allocation.

  • A. Network Model and Assumptions: A macro BS serves N uniformly distributed UEs in both uplink and downlink configurations using single antennas.The model applies to both transmission directions.
  • A. Network Model and Assumptions: The total bandwidth B_T is divided into resource blocks of bandwidth B, giving Ω = B_T/B available blocks.Each NOMA cluster operates on orthogonal frequency resources relative to other clusters.
  • A. Network Model and Assumptions: Users scheduled non-orthogonally on the same resource blocks form a NOMA cluster.The number of clusters can vary between 1 and N/2.
  • A. Network Model and Assumptions: The model imposes separate power budgets for the BS, each downlink NOMA cluster, and each uplink UE.These are denoted by P_T, P_t, and P′_t, respectively.
  • A. Network Model and Assumptions: Normalized channel gains γ_i incorporate distance-based path loss and shadowing, and users are ordered as γ_1 > γ_2 > · · · > γ_N.This ordering supports channel-gain-based clustering and SIC analysis.
  • A. Network Model and Assumptions: The binary variable β_i,j indicates whether user i is grouped into cluster j.The variable equals 1 for membership and 0 otherwise, with j = 1, 2, · · ·, N/2.
  • B. Problem Formulation: Downlink NOMA: The downlink problem jointly optimizes user clustering and power allocation to maximize throughput.It is formulated as a constrained optimization problem.
  • B. Problem Formulation: Downlink NOMA: The constraints enforce BS power, minimum user rates, SIC feasibility, one-cluster assignment, at least two users per downlink cluster, frequency resources, and integer variables.C6 specifies ω_j ∈ {1, 2, · · ·, Ω}, while β and ω are integer variables.

C. Problem Formulation: Uplink NOMA

The uplink NOMA throughput-maximization problem is formulated jointly over user clustering and power allocation, subject to uplink rate and SIC-related constraints.

  • The uplink problem jointly optimizes user clustering and power allocation for throughput maximization.
  • The formulation includes constraints ensuring users’ minimum uplink data rates.
  • The formulation includes an uplink SIC constraint and retains the remaining clustering and resource constraints from the downlink problem.

D. Solution Methodology

The methodology separates combinatorial user clustering from power optimization and uses channel-gain differences to construct low-complexity uplink and downlink NOMA clusters.

  • Optimal user clustering is combinatorial and may be computationally unaffordable as the number of active users grows.The paper therefore avoids exhaustive clustering search in practical large-user systems.
  • The proposed low-complexity scheme first selects a feasible number of clusters, then groups users using channel-gain differences.It is designed for both uplink and downlink NOMA and aims to enhance cell sum throughput.
  • Downlink NOMA: In downlink NOMA, the scheme pairs the highest- and lowest-gain users because strong users can retain high rates with low power while weak users receive more power.
  • Uplink NOMA: In uplink NOMA, distinct channel gains reduce interference, while high-gain users can transmit at maximum power because SIC removes their interference to weak users.
  • User classification: Users are divided into Class-A and Class-B according to a large channel-gain separation, with γ1, γ2, · · ·, γα ≫ γα+1, γα+2, · · ·, γN.
  • Clustering algorithm: The algorithm sorts users by channel gain, chooses κ based on α and N, and assigns users to downlink or uplink clusters using different ordered patterns.When α < N/2, κ = α; otherwise, κ = N/2.

V. OPTIMAL POWER ALLOCATIONS IN NOMA

For fixed NOMA clusters, the paper derives optimal power allocations for downlink clusters by solving the constrained throughput problem with KKT conditions and closed-form expressions.

  • The power-allocation stage optimizes an m-user NOMA cluster after clustering, for 2 ≤ m ≤ N.
  • Downlink formulation: The downlink formulation imposes total power, minimum-rate, and SIC constraints, and is convex under these constraints.
  • KKT solution: KKT conditions are obtained by differentiating the Lagrangian with respect to powers and constraint multipliers, together with complementarity conditions.
  • KKT solution: For an m-user downlink cluster, the number of KKT multiplier combinations satisfying the conditions is 2m−1.The paper reports 2, 4, 8, and 32 combinations for 2-, 3-, 4-, and 6-user clusters, respectively.
  • Closed-form allocation: Closed-form optimal power allocations are derived for the highest-gain downlink user and for the remaining users.The corresponding necessary conditions are tabulated for 2-, 3-, and 4-user clusters.

B. Uplink NOMA

The uplink formulation optimizes per-user powers under individual budgets, rate requirements, and SIC constraints, yielding simplified KKT cases and a closed-form allocation structure.

  • The uplink power-control problem is defined for an m-user cluster with positive minimum-rate requirements and allocated resource blocks.
  • Uplink formulation: Each uplink user’s power is bounded by the maximum transmission budget P′t, while the objective accounts for inter-user interference.
  • KKT solution: The uplink optimization is convex under its power, rate, and SIC constraints, enabling KKT-based solution derivation.
  • KKT solution: Three Lagrange-multiplier combinations satisfy the KKT conditions for the examined uplink cluster sizes.The paper states this count is 3 for 2-, 3-, 4-, and 6-user cases.
  • Closed-form allocation: Closed-form optimal power allocations are derived for an m-user uplink NOMA cluster.The resulting transmission powers and necessary conditions are also provided for 2-, 3-, and 4-user clusters.
  • Power-control structure: Power control in an uplink cluster is needed only at the weakest channel-gain user; other users can transmit with full power.For four users, UE1, UE2, and UE3 transmit with full power, while UE4 may require control to satisfy rate and receive-power-difference constraints.

VI. NUMERICAL RESULTS AND DISCUSSIONS

The evaluation uses proposed NOMA grouping and power-allocation solutions, comparing NOMA with OFDMA-based LTE/LTE-Advanced systems across cluster sizes and transmission directions.

  • Simulation setup: The simulations compare downlink and uplink NOMA with OFDMA-based LTE/LTE-Advanced systems using proposed grouping and power-allocation solutions.Resource blocks are allocated according to NOMA cluster size, and downlink transmission power is uniformly allocated among available resources.
  • Compared systems: The study evaluates 2-, 3-, 4-, and 6-user NOMA clusters and reports their performance against OMA systems.The 12-user downlink performance is summarized for 2-, 3-, and 4-user clusters, while the simulation materials also include uplink cluster solutions.
  • Power allocation: Closed-form uplink power-allocation solutions are provided for 2-user, 3-user, and 4-user NOMA clusters.These solutions are part of the evaluated power-allocation framework for NOMA clusters.
  • Downlink evaluation: Downlink performance is reported for m-user NOMA systems with m = 2, 3, and 4, alongside corresponding OMA performance for 12 users.The comparison focuses on system performance across different downlink NOMA cluster sizes.

A. Downlink NOMA

Downlink NOMA generally outperforms OMA, especially with distinct user channel gains, but throughput depends strongly on the strongest user, cluster size, and minimum-rate allocation.

  • Throughput Comparison between NOMA and OMA Systems: The highest-channel-gain user receives significantly higher individual throughput in NOMA than in OMA, while the weakest user is limited by its minimum-rate requirement.The paper suggests dynamically adjusting users’ minimum-rate requirements to improve fairness.
  • Throughput Comparison between NOMA and OMA Systems: Downlink NOMA sum-throughput depends strongly on the highest-channel-gain user because that user can cancel all interfering signals before decoding.Its achievable data rate therefore does not depend on inter-user interference.
  • Throughput Comparison between NOMA and OMA Systems: The lowest-channel-gain user has minimal impact on cluster sum-throughput unless its channel is so poor that the required power becomes very large.At that point, sum-throughput sharply decays, whereas traditional OMA cannot operate at such poor channel gains.
  • Throughput Comparison between NOMA and OMA Systems: Channel variations of intermediate users do not considerably affect a downlink NOMA cluster’s sum-throughput.This observation is reported for the 4-user downlink NOMA results in Fig. 7(b) and Fig. 7(c).
  • Throughput Comparison of Various Downlink NOMA Systems: Higher cluster sizes are preferred at higher channel gains, whereas smaller cluster sizes are preferred at lower channel gains when channel gains are less distinct.For more distinct channel gains, 4-user NOMA outperforms 2-user and 3-user NOMA if power allocated to the strongest user does not decrease significantly.
  • Throughput Comparison of Various Downlink NOMA Systems: When the number of good channels equals the number of clusters, downlink NOMA achieves its maximum relative gain over OMA.The 4-user system reaches a 106.3% gain, while the 3-user and 2-user systems reach 86.6% and 52.8%, respectively, in the reported cases.
  • Throughput Comparison of Various Downlink NOMA Systems: Optimal power allocation maximizes the strongest user’s transmit power while maintaining every user’s minimum rate, but lower-gain users may experience substantial throughput differences.The paper proposes adjusting minimum data-rate requirements to balance fairness against overall system throughput.

B. Performance Evaluation of Uplink NOMA

Uplink NOMA throughput depends strongly on channel-gain distinctness and clustering, with gains over OMA varying by rate requirement, channel conditions, and cluster composition.

  • 2) Comparisons Among Different Uplink NOMA Systems:: The study evaluates 2-user, 3-user, 4-user, and 6-user uplink NOMA systems for 12 users under minimum-rate and SIC-related operating constraints.Users are sorted by descending channel gain, with cluster availability depending on cluster size and channel distinctness.
  • 1) Throughput Comparisons Between NOMA and OMA:: Uplink NOMA outperforms OMA more clearly when users in a cluster have distinct channel gains.With nearly equal gains, interference and power control can reduce or eliminate the advantage over OMA.
  • 1) Throughput Comparisons Between NOMA and OMA:: At a 1 Mbps minimum rate, power-controlled uplink NOMA becomes inferior to OMA when channel gains are less distinct.The comparison concerns 2-user systems and the corresponding sum throughput.
  • 1) Throughput Comparisons Between NOMA and OMA:: The sum throughput of a 4-user uplink NOMA cluster mainly depends on the highest-gain user’s channel condition.In the evaluated subfigures, one user’s gain varies while the others remain fixed.
  • 2) Comparisons Among Different Uplink NOMA Systems:: When higher-gain users equal the number of clusters, uplink NOMA achieves maximum or near-maximum relative throughput gains over OMA.Reported gains are 105.4% for 6-user, 79.9% for 4-user, 60.25% for 3-user, and 33.3% for 2-user NOMA.
  • 2) Comparisons Among Different Uplink NOMA Systems:: Higher-order uplink NOMA performs much better than OMA when one or a few users have higher channel gains, while systems become similar as more users have high gains.Properly selecting channel gains for weaker users can also provide good throughput fairness across a six-user cluster.

APPENDIX A: PROOF OF SATISFIED KKT CONDITION FOR DOWNLINK NOMA

The appendix verifies KKT satisfaction for a four-user downlink NOMA cluster by examining feasible Lagrange-multiplier combinations and positivity of the resulting solutions.

  • APPENDIX A: PROOF OF SATISFIED KKT CONDITION FOR DOWNLINK NOMA: For a four-user downlink NOMA cluster, the appendix enumerates Lagrange-multiplier sets that can satisfy the KKT conditions.The sets include multipliers for power, rate, and SIC-related constraints.
  • APPENDIX A: PROOF OF SATISFIED KKT CONDITION FOR DOWNLINK NOMA: Equations (A.1)–(A.2) provide the optimal solution, while equations (A.3)–(A.4) provide necessary conditions for those solutions.The appendix then solves the equations to obtain optimal power allocations and Lagrange multipliers.
  • APPENDIX A: PROOF OF SATISFIED KKT CONDITION FOR DOWNLINK NOMA: The candidate multiplier set S1 = {λ, µ2, µ3, µ4} satisfies the KKT conditions when the corresponding power allocations are positive.The appendix states that the remaining cases can be verified similarly.

APPENDIX-B: PROOF OF SATISFIED KKT CONDITIONS

The appendix verifies KKT conditions for a four-user uplink NOMA cluster by checking multiplier combinations and positivity of the resulting power allocations.

  • APPENDIX-B: PROOF OF SATISFIED KKT CONDITIONS: For a four-user uplink NOMA cluster, the appendix identifies Lagrange-multiplier sets that satisfy the KKT conditions.The multipliers correspond to power, rate, and SIC-related constraints.
  • APPENDIX-B: PROOF OF SATISFIED KKT CONDITIONS: Equations (B.1)–(B.2) give the optimal solution, while equations (B.3)–(B.5) give the necessary conditions.Solving the equations yields the optimal power allocations for the considered uplink cluster.
  • APPENDIX-B: PROOF OF SATISFIED KKT CONDITIONS: The multiplier set S1 = {λ1, λ2, λ3, µ3} satisfies the KKT conditions when all resulting power allocations are positive.The appendix says the other cases can be verified using a similar approach.
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