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Downlink Power Allocation for CoMP-NOMA in Multi-Cell Networks

Md Shipon Ali, Ekram Hossain, Arafat Al-Dweik, Dong In Kim

arXiv:1801.04981v1eess.SP

TL;DR

The paper addresses dynamic downlink power allocation for CoMP-NOMA in multi-cell two-tier HetNets under minimum-rate constraints. It formulates joint and distributed optimization for JT-CoMP-NOMA, establishes conditions validating the distributed solution, and reports spectral- and energy-efficiency gains over CoMP-OMA.

  • Problem

    The paper investigates dynamic power allocation for sum-rate maximization in downlink CoMP-NOMA multi-cell scenarios under minimum-rate constraints.

  • Method

    The paper formulates joint power optimization for coordinating BSs and proposes a low-complexity distributed optimization whose BS-level solutions are independent.

  • Results

    Numerical evaluation reports significant spectral- and energy-efficiency gains for CoMP-NOMA compared with conventional CoMP-OMA systems.

  • Takeaways & Limitations

    The proposed dynamic CoMP-NOMA power allocation framework supports CoMP-NOMA evaluation in two-tier HetNets while reducing the complexity of joint optimization.

Abstract

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This work considers the problem of dynamic power allocation in the downlink of multi-cell networks, where each cell utilizes non-orthogonal multiple access (NOMA)-based resource allocation. Also, coordinated multi-point (CoMP) transmission is utilized among multiple cells to serve users experiencing severe inter-cell interference (ICI). More specifically, we consider a two-tier heterogeneous network (HetNet) consisting of a high-power macro cell underlaid with multiple low-power small cells each of which uses the same resource block. Under this {\em CoMP-NOMA framework}, CoMP transmission is applied to a user experiencing high channel gain with multiple base stations (BSs)/cells, while NOMA is utilized to schedule CoMP and non-CoMP users over the same transmission resources, i.e., time, spectrum and space. Different CoMP-NOMA models are discussed, but focus is primarily on the joint transmission CoMP-NOMA (JT-CoMP-NOMA) model. For the JT-CoMP-NOMA model, an optimal joint power allocation problem is formulated and the solution is derived for each CoMP-set consisting of multiple cooperating BSs (i.e., CoMP BSs). To overcome the substantial computational complexity of the joint power optimization approach, we propose a distributed power optimization problem at each cooperating BS whose optimal solution is independent of the solution of other coordinating BSs. The validity of the distributed solution for the joint power optimization problem is provided and numerical performance evaluation is carried out for the proposed CoMP-NOMA models including JT-CoMP-NOMA and coordinated scheduling CoMP-NOMA (CS-CoMP-NOMA). The obtained results reveal significant gains in spectral and energy efficiency in comparison with conventional CoMP-orthogonal multiple access (CoMP-OMA) systems.

I. INTRODUCTION

The paper combines power-domain NOMA with CoMP to address severe inter-cell interference in multi-cell HetNets, focusing on downlink transmissions over shared resources. It positions dynamic power allocation as a central requirement for improving spectral efficiency under NOMA and CoMP.

  • NOMA fundamentals: Power-domain NOMA superposes multiple users’ signals on the same transmission resources and relies on SIC for decoding.SIC follows ascending channel gains, with higher-gain users canceling lower-gain signals while higher-order signals remain as interference.
  • Multi-cell challenge: Cell-edge users in multi-cell HetNets can experience severe inter-cell interference and low received SINR, particularly from high-power macro cells affecting underlaid small cells.The paper therefore identifies advanced ICI management as crucial for multi-cell downlink NOMA.
  • CoMP-NOMA motivation: CoMP coordinates multiple cells to schedule or transmit to interference-prone users, and this paper applies CoMP to NOMA-based downlink transmissions in two-tier HetNets.The stated objective is to improve network spectral efficiency.
  • Terminology and assumptions: A CoMP-set is a group of cooperating cells or BSs serving a user, while CoMP-users receive coordinated signals and non-CoMP-users are served by one BS.CoMP-NOMA forms NOMA clusters containing both user types according to the applied CoMP scheme.
  • Terminology and assumptions: The modeled HetNet uses one macro cell underlaid by several small cells with full frequency reuse, orthogonal NOMA clusters within each cell, and common resources for CoMP transmission.The analysis assumes flat Rayleigh fading and single transmit and receive antennas.
  • Related work: Prior CoMP-NOMA studies considered settings including homogeneous networks, fixed power allocation, outage analysis, relaying, and limited user configurations.The paper reviews work on multi-cell NOMA rate analysis, spectrum allocation, power control, and CoMP-NOMA scheduling.

D. Motivation and Key Contributions

The paper investigates dynamic power allocation for sum-rate maximization in downlink CoMP-NOMA under minimum-rate constraints. It develops joint and distributed optimization approaches for a two-tier HetNet and evaluates their spectral- and energy-efficiency gains over CoMP-OMA.

  • Motivation: The study targets dynamic power allocation for sum-rate maximization in multi-cell CoMP-NOMA subject to users’ minimum-rate constraints.This motivation follows the importance of dynamic allocation and the limitations of fixed-power or single-cell approaches in existing work.
  • System model: Each CoMP-cell forms a NOMA cluster containing CoMP-users and non-CoMP-users, after which dynamic power allocation is performed for each cluster.The model supports one or multiple users of each type and determines CoMP-sets according to received SINR.
  • Key contributions: The paper formulates a convex single-cell NOMA power optimization problem as a basis for multi-cell and HetNet CoMP-NOMA power allocation.The formulation maximizes sum rate under constrained minimum-rate requirements.
  • Key contributions: It provides a system and signal model for downlink CoMP-NOMA and derives achievable-rate formulas for CoMP-UEs and non-CoMP-UEs in two- and three-cell CoMP-sets.The primary illustrated configuration is JT-CoMP-NOMA in a two-tier HetNet.
  • Key contributions: A joint power optimization problem is formulated and solved across coordinating BSs for CoMP-NOMA sum-rate maximization in two- and three-cell CoMP-sets.The paper also derives conditions for global optimality of the joint solution.
  • Key contributions: A low-complexity distributed power optimization approach assigns each CoMP-cell an optimization problem whose solution avoids joint optimization across all coordinating cells.The distributed solution is independent across coordinating BSs, and conditions are derived for its validity for the joint problem.
  • Performance evaluation: Numerical evaluation reports gains in spectral and energy efficiency for the proposed CoMP-NOMA model compared with traditional CoMP-OMA systems.The evaluation covers the proposed CoMP-NOMA system and its optimization approaches.

II. OPTIMAL POWER ALLOCATION FOR SINGLE-CELL DOWNLINK NOMA

The paper formulates single-cell downlink NOMA sum-rate maximization with rate, power-budget, and SIC constraints. Under ascending channel-gain SIC ordering, the problem is convex and admits a globally optimal KKT-based solution.

  • System model: The single-cell model assumes M ≥2 users with distinct channel gains and flat-fading Rayleigh channels over each NOMA resource block.Each user has a normalized-noise channel power gain and an allocated transmission power.
  • Optimization problem: The optimization maximizes sum-rate subject to a NOMA power budget, individual rate requirements, and SIC constraints.The minimum rate requirement for each NOMA user is set to its achievable rate in an equal-spectrum, equal-power OMA system.
  • Convexity: The sum-rate objective is strictly concave in user powers, while all optimization constraints are convex or concave affine functions.The formulation contains one power-budget constraint, M rate constraints, and M−1 SIC constraints.
  • Solution: The resulting problem is convex, and the closed-form KKT solution is globally optimal.Slater’s condition holds because the inequality constraints are affine, making KKT conditions necessary and sufficient.

B. JT-CoMP NOMA System Model

JT-CoMP-NOMA combines joint transmission from cooperating base stations with NOMA clusters containing CoMP and non-CoMP users. CoMP users receive signals from multiple cells, while non-CoMP users remain associated with one serving base station.

  • Network model: The HetNet contains one high-power eNB underlaid by X low-power SBSs, with each BS using NOMA for downlink scheduling.The model considers CoMP-sets with up to three coordinating base stations.
  • CoMP-sets: Two-BS CoMP-sets pair the eNB with one SBS, while three-BS CoMP-sets combine the eNB with two distinct SBSs.The corresponding CoMP-user sets are defined separately for each CoMP-set configuration.
  • User association: A JT-CoMP CoMP-UE receives multiple transmissions from all CoMP-BSs and forms distinct NOMA clusters with non-CoMP-UEs at those cells.A non-CoMP-UE receives its desired signal only from its associated BS and belongs to one NOMA cluster.
  • SIC structure: Each NOMA cluster contains at least one non-CoMP-UE, and non-CoMP users decode CoMP-user interference while CoMP users do not decode non-CoMP interference.The ordering reflects CoMP users’ cell-edge locations and reduces SIC overhead relative to the opposite ordering.
  • Achievable rates: Achievable rates account for desired signals, intra-NOMA-user interference, and inter-cell interference for non-CoMP and CoMP users.A CoMP-UE’s desired signal is jointly transmitted by both CoMP-BSs and experiences interference from both cells’ NOMA clusters.

IV. SUM-RATE MAXIMIZATION IN JT-COMP NOMA: JOINT POWER OPTIMIZATION APPROACH

The JT-CoMP-NOMA joint optimization maximizes the sum-rate of a CoMP-set while coordinating power decisions across its cooperating base stations. Its exhaustive coupling over other BS allocations and possible CoMP-user SIC orderings creates substantial computational complexity.

  • Joint optimization: The joint problem optimizes the sum-rate over the power vectors allocated by the eNB and cooperating SBS.The variables include the respective NOMA-user powers at both CoMP-BSs.
  • Problem formulation: For a two-BS CoMP-set, the formulation includes power budgets, user-rate requirements, and SIC constraints for CoMP and non-CoMP users.The constraints separately cover the eNB cluster, SBS cluster, and CoMP-user requirements.
  • SIC ordering: CoMP-user SIC ordering may not match ascending channel-gain order at both cooperating BSs simultaneously.Consequently, the joint optimization must consider all possible SIC orderings for CoMP users.
  • Cross-BS coupling: Each CoMP-BS must account for feasible power allocations chosen by the other CoMP-BSs when solving the joint problem.The eNB and SBS optimize their respective power vectors under their local budgets and shared CoMP-user constraints.
  • Complexity: The joint power optimization has substantial computational complexity, motivating a distributed power optimization method independent of other CoMP-BS allocations.The distributed method is introduced to reduce the complexity of solving the joint problem.

V. DISTRIBUTED POWER ALLOCATION IN JT-COMP NOMA

The distributed approach decomposes JT-CoMP-NOMA power allocation into independent per-BS NOMA optimizations. Its solution can be feasible for the joint problem under stated constraint-region conditions, while JT transmission still uses the resulting allocations jointly.

  • Caveat: The joint solution may be a global or local maximum depending on the SIC ordering of non-CoMP and CoMP users.This scope condition is stated for the solution obtained through the KKT-based approach.
  • Approximation: The distributed method ignores ICI and CoMP-transmission impact during local optimization, although CoMP-BSs jointly transmit the same message using the resulting allocation.This is the defining approximation of the proposed distributed solution.
  • Distributed formulation: The distributed problem at each CoMP-BS optimizes its NOMA-cluster sum-rate without considering other CoMP-BS power allocations.The local formulation retains a power budget, user-rate requirements, and SIC requirements for that BS’s cluster.
  • Feasibility: The distributed solution is feasible for the joint problem when all distributed constraints lie within the feasible solution region of the joint formulation.The same KKT optimality model can be used for both distributed and joint optimization.
  • Power allocation structure: The KKT allocation gives minimum power to non-cluster-head users while satisfying their rate and SIC requirements, then assigns remaining power to the cluster-head.This allocation structure supplies the conditions used to establish distributed-solution feasibility.

1) Power Budget Constraint:

The distributed power optimization (DPO) approach can satisfy the joint power optimization (JPO) constraints under specified ICI and SIC-ordering conditions, while simplifying optimization across cooperating BSs. In JT-CoMP-NOMA, DPO may improve CoMP-UE rates but can impose energy and interference trade-offs.

  • The JPO and DPO formulations are equivalent when the ICI component in (7) is negligible.
  • If ICI is not negligible for non-CoMP-UEs, DPO cannot meet their minimum-rate requirements except for the cluster-head without the offset-ICI adjustment.
  • Under DPO, CoMP-UE rate constraints satisfy the corresponding JPO constraints, and achievable CoMP-UE rates are higher than under JPO.The rates become equal if the noise power in the SINR expressions is divided by the number of CoMP-BSs.
  • DPO also satisfies JT-CoMP-NOMA SIC constraints for CoMP-UEs, while non-CoMP-UE SIC constraints can be met by adding offset ICI.The offset ICI is introduced into the non-CoMP-UE achievable-rate and SIC expressions.
  • The DPO problem is convex only for ascending channel-gain-based SIC ordering, and compatible SIC orderings may not be possible at every CoMP-BS simultaneously.Maximizing sum-rate across all CoMP-BSs may therefore require exhaustive checking of CoMP-UE channel-gain order combinations.
  • JT-CoMP-NOMA can require high average energy per transmitted bit, while DPS/CS-CoMP-NOMA may reduce interference to CoMP-UEs at some sum-rate cost.Controlling power at BSs transmitting only to non-CoMP-UEs can minimize their interference to CoMP-UEs.

VII. NUMERICAL ANALYSIS

The numerical analysis evaluates spectral and energy efficiency for JT-CoMP-NOMA under joint and distributed power optimization, comparing results with JT-CoMP-OMA and CS-CoMP-NOMA across three system models.

  • Simulation setup: The study compares joint power optimization (JPO) and distributed power optimization (DPO) for JT-CoMP-NOMA against JT-CoMP-OMA.JPO searches feasible power allocations, whereas DPO uses per-BS optimization expressions for achievable user rates.
  • Simulation setup: Simulations evaluate JT-CoMP-NOMA models 2:2:1, 3:2:1, and 2:3:2 using spectral-efficiency and energy-efficiency metrics.SE is measured in bits/sec/Hz and EE in Mb/J; CoMP-set metrics sum over users served by the set.
  • Spectral efficiency: Increasing the number of BSs in a CoMP-set reduces the CoMP-user distance range that can form a NOMA cluster with a non-CoMP user.Under JT-CoMP-NOMA, both desired signal and inter-NOMA-user interference powers increase additively with the number of cooperating BSs.
  • Optimization comparison: The JPO–DPO spectral-efficiency gap is very small around 120–160 m in Fig. 2(a) and 140–160 m in Fig. 3(a).DPO independently allocates power at each CoMP-BS and is presented as a lower-complexity alternative to JPO.

C. Energy Efficiency Performance

The energy-efficiency analysis compares CoMP-NOMA with CoMP-OMA and examines how serving-BS choice and interference affect the observed gains.

  • JT-CoMP-NOMA: JT-CoMP-NOMA shows no significant energy-efficiency improvement over JT-CoMP-OMA in the index view of Fig. 5.This behavior follows from CoMP-BSs individually satisfying CoMP-user rate requirements whose SINRs depend on desired-to-interference power ratios.
  • CS-CoMP-NOMA: CS-CoMP-NOMA exhibits a dramatic energy-efficiency gain when the CoMP-user is served by an SBS.The stated reasons include high SBS channel gain for the CoMP-user, the SBS’s low transmit-power budget, and the eNB’s high transmit-power budget.
  • CS-CoMP-NOMA: When the CoMP-user is served by an SBS, the SBS uses its full power budget while the eNB only uses the minimum power needed for non-CoMP-user rate requirements.The power asymmetry is identified as part of the explanation for the larger CS-CoMP-NOMA energy-efficiency gain.
  • CS-CoMP limitation: CS-JT-CoMP transmission may fail to meet the individual-user rate requirement obtainable with JT-CoMP-OMA because offset interference cannot be used.Under CS-CoMP, all but the CoMP-user-serving BS use power control, preventing use of the offset ICI in (15).
  • Interference effects: For CS-CoMP-NOMA, increasing the distance of an eNB-served non-CoMP-user reduces the CoMP-user’s data rate.The text attributes this to increased required eNB transmit power, which significantly increases interference to CoMP-users.

APPENDIX A PROOF OF LEMMA 1

Appendix A proves that the downlink NOMA sum-rate is strictly concave under ascending channel-gain-based SIC ordering by analyzing Hessian principal minors.

  • Three-user case: The three-user case shows negative first- and third-order principal minors and positive second-order minors.These alternating signs establish negative definiteness of the Hessian for the three-user sum-rate.
  • General M-user case: The alternating principal-minor signs prove Lemma 1 for the M-user downlink NOMA sum-rate.The appendix concludes the proof after establishing the required sign pattern.
  • Two-user case: For two-user downlink NOMA, the Hessian’s first- and second-order principal minors establish strict concavity of the sum-rate.The proof assumes γ2 > γ1 under ascending channel-gain-based SIC ordering.
  • General M-user case: The proof extends by induction to M-user downlink NOMA with ascending channel-gain-based SIC decoding order.Odd-numbered leading principal minors are negative and even-numbered minors are positive.

APPENDIX B PROOF OF LEMMA 4

The appendix proves Lemma 4 by deriving CoMP-user desired signal powers under joint and distributed optimization, then comparing them across NOMA-user indices. It shows the distributed approach achieves at least the joint approach’s normalized desired signal power and identifies when their rates coincide.

  • Joint optimization approach: It derives each CoMP-UE’s desired signal power under joint optimization from inter-NOMA-user interference, noise power, and the required data rate.The derivation uses the CoMP-UE rate requirement and OMA-based achievable-rate representation.
  • Distributed optimization approach: Under distributed optimization, each CoMP-BS independently derives its desired signal contribution using its own inter-NOMA interference, noise power, and rate requirement.The MBS and SBS expressions are combined to obtain the distributed CoMP desired signal power.
  • Joint optimization approach: The proof models each CoMP-set’s NOMA clusters using separate counts for non-CoMP and CoMP users at the eNB and SBS.The rate constraints are formulated for CoMP-UEs under the joint optimization approach.
  • Comparison between approaches: The comparison proves that distributed optimization’s desired signal power is greater than or equal to the joint optimization result, beginning with k = 1 and extending through k = 2 and any k ≥1.The proof establishes the comparison through the equality condition for the joint and distributed expressions.
  • Comparison between approaches: If a CoMP-UE’s noise power is divided by the number of CoMP-BSs, its data rates under joint and distributed power optimization are similar.This condition is the rate-level conclusion stated for Lemma 4.
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