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Optimal Resource Allocation for Power-Efficient MC-NOMA with Imperfect Channel State Information

Zhiqiang Wei, Derrick Wing Kwan Ng, Jinhong Yuan, Hui-Ming Wang

arXiv:1705.05947v1cs.IT

TL;DR

The paper addresses power-efficient MC-NOMA resource allocation under imperfect CSIT. It develops optimal and suboptimal allocation schemes, with simulations showing close-to-optimal performance and significant transmit-power savings.

  • Problem

    Power-efficient resource allocation and SIC decoding for MC-NOMA under imperfect CSIT remain unresolved.

  • Method

    The framework defines a CNR outage threshold and recasts the problem as generalized linear multiplicative programming, alongside a D.C.-programming scheme.

  • Results

    The suboptimal scheme converges rapidly to a close-to-optimal solution, while the proposed schemes provide significant transmit-power savings.

  • Takeaways & Limitations

    The optimal allocation algorithm provides a performance benchmark, while the suboptimal scheme balances performance and computational complexity.

Abstract

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In this paper, we study power-efficient resource allocation for multicarrier non-orthogonal multiple access (MC-NOMA) systems. The resource allocation algorithm design is formulated as a non-convex optimization problem which jointly designs the power allocation, rate allocation, user scheduling, and successive interference cancellation (SIC) decoding policy for minimizing the total transmit power. The proposed framework takes into account the imperfection of channel state information at transmitter (CSIT) and quality of service (QoS) requirements of users. To facilitate the design of optimal SIC decoding policy on each subcarrier, we define a channel-to-noise ratio outage threshold. Subsequently, the considered non-convex optimization problem is recast as a generalized linear multiplicative programming problem, for which a globally optimal solution is obtained via employing the branch-and-bound approach. The optimal resource allocation policy serves as a system performance benchmark due to its high computational complexity. To strike a balance between system performance and computational complexity, we propose a suboptimal iterative resource allocation algorithm based on difference of convex programming. Simulation results demonstrate that the suboptimal scheme achieves a close-to-optimal performance. Also, both proposed schemes provide significant transmit power savings than that of conventional orthogonal multiple access (OMA) schemes.

I. INTRODUCTION

NOMA enables multiple users to share frequency resources, but existing MC-NOMA resource-allocation studies leave its maximum potential gain unknown and often assume perfect CSIT. This paper addresses robust, power-efficient MC-NOMA allocation under imperfect CSIT by jointly designing key transmission and decoding decisions.

  • Motivation: NOMA multiplexes multiple users on shared frequency resources and can improve spectral efficiency, outage performance, throughput, and fairness relative to OMA.Prior studies report gains in spectral efficiency, performance outage probability, cell throughput, cell-edge throughput, sum rate, individual rates, and fairness.
  • Research gap: Existing NOMA performance analyses generally use fixed power and subcarrier allocation, leaving the maximum potential performance gain for 5G networks unknown.Resource allocation is especially important for MC-NOMA, where joint power and subcarrier allocation is generally NP-hard.
  • Research gap: Perfect CSIT assumptions are impractical, while imperfect CSIT complicates channel-gain sorting, user scheduling, SIC decoding, and resource-allocation matching.Channel-estimation error, feedback delay, and quantization error can prevent perfect CSIT and degrade system performance through allocation mismatch.
  • Proposed framework: The paper robustly designs MC-NOMA resource allocation under imperfect CSIT using a stochastic approach and users’ heterogeneous QoS requirements.The design jointly considers power allocation, rate allocation, user scheduling, and SIC decoding policy.
  • Proposed framework: A CNR outage threshold supports optimal SIC decoding, and branch-and-bound obtains an optimal allocation benchmark after reformulating the problem as generalized linear multiplicative programming.The benchmark has high computational complexity.
  • Results: An iterative D.C.-programming algorithm provides a faster close-to-optimal alternative, while both proposed schemes achieve significant transmit-power savings and robustness against channel uncertainty.The D.C. scheme has polynomial-time complexity and converges quickly.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models MC-NOMA resource allocation under imperfect CSIT as a power-efficient non-convex design problem. The system jointly represents scheduling, power and rate allocation, SIC decisions, and channel uncertainty.

  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The QoS-constrained resource allocation formulation minimizes total transmit power and is explicitly non-convex.The formulation is introduced after defining outage-probability-based QoS under the imperfect-CSIT system model.
  • A. System Model: The SIC policy lets multiplexed users either perform SIC or directly decode their own messages, with the selected user canceling the other user’s signal first.The SIC indicator is binary, and direct decoding treats the other multiplexed signal as noise.
  • A. System Model: The considered downlink MC-NOMA system has one base station, M single-antenna users, and NF orthogonal subcarriers.The primary setting is overloaded, with NF ≤ M, while the proposed scheme also applies to underloaded systems.
  • A. System Model: Each subcarrier can multiplex at most two users through power-domain NOMA, limiting receiver SIC complexity and delay.The analysis focuses on the two-user case and leaves serving multiple users per subcarrier for future work.
  • A. System Model: Binary scheduling variables determine whether user m is assigned to subcarrier i, subject to the corresponding assignment constraint.The transmitted signal on each subcarrier superimposes users’ modulated symbols with allocated powers.
  • II. SYSTEM MODEL AND PROBLEM FORMULATION: The proposed scheme is reported as more power-efficient than OMA in both overloaded and underloaded systems.The algorithms are applied to both system regimes in the simulations.
  • A. System Model: Under imperfect CSIT, the base station jointly decides SIC decoding order and positive per-subcarrier user rates because both affect system power consumption.The model captures channel estimation error and feedback delay rather than assuming perfect transmitter channel knowledge.
  • A. System Model: The channel model combines estimated coefficients with uncorrelated estimation errors, while small-scale fading follows a Rayleigh model and path loss is estimated accurately.The channel coefficient captures the joint effect of path loss and small-scale fading.

B. QoS Requirements

The QoS model uses outage probability on each subcarrier to constrain user reliability under channel uncertainty. It distinguishes SIC decoding from direct decoding and includes OMA as a special case.

  • B. QoS Requirements: The paper defines an outage probability on each subcarrier to capture unsuccessful decoding under channel uncertainty.For a user performing SIC, failure of SIC prevents decoding its own message.
  • B. QoS Requirements: For SIC users, the outage expression depends on channel uncertainty, allocated rate, and the achievable rate for decoding the other user’s interference.The relevant interference-decoding rate is denoted by CSIC_i,m.
  • B. QoS Requirements: With two users per subcarrier, SIC processing cancels interference from only one user because the scheduling constraint permits one other multiplexed user.The paper notes that the corresponding summations contain only one non-zero entry.
  • B. QoS Requirements: A non-SIC user directly decodes its own message while treating the other user’s signal as noise.Its achievable rate is denoted by C(2)_i,m.
  • B. QoS Requirements: Each user’s outage probability must satisfy δ_i,m, where δ_i,m ∈ [0,1] is the required outage probability.When the user is not assigned to the subcarrier, the inequality is automatically satisfied.
  • B. QoS Requirements: In OMA, each subcarrier is allocated exclusively to one user, and the MC-NOMA outage definition reduces to the OMA case.SIC is not required for the exclusively assigned user.

C. Optimization Problem Formulation

The paper formulates joint MC-NOMA resource allocation under imperfect CSIT as a total-transmit-power minimization problem with QoS constraints. It derives an SIC order based on CNR outage thresholds and uses that structure to address the otherwise mixed combinatorial non-convex formulation.

  • C. Optimization Problem Formulation: The optimization jointly designs power allocation, rate allocation, user scheduling, and SIC decoding for downlink MC-NOMA under imperfect CSIT.
  • C. Optimization Problem Formulation: The QoS constraints guarantee each user’s minimum total data rate, while at most two users may be multiplexed on each subcarrier.
  • C. Optimization Problem Formulation: Binary scheduling and decoding variables create combinatorial complexity, while QoS constraints introduce non-convexity and couplings with continuous variables.
  • A. Optimal SIC Policy Per Subcarrier: The CNR outage threshold incorporates estimated channel gain, channel-estimation error distribution, noise power, and required outage probability.
  • A. Optimal SIC Policy Per Subcarrier: For two multiplexed users, the user with the higher CNR outage threshold performs SIC decoding, and this order is independent of target data rates.
  • A. Optimal SIC Policy Per Subcarrier: The policy is conditioned on feasible user scheduling and minimizes total transmit power for the QoS constraint; single-user subcarriers require no SIC.
  • A. Optimal SIC Policy Per Subcarrier: Under the proposed order, the SIC process satisfies QoS whenever the lower-threshold user satisfies its own QoS, whereas reversing the order requires extra power.

B. Minimum Total Transmit Power Per Subcarrier

The paper derives minimum transmit power for each subcarrier after fixing user scheduling and rate allocation. The resulting expression covers both NOMA and OMA and enables an equivalent optimization over scheduling and rates.

  • The minimum subcarrier transmit power is derived separately for NOMA and OMA using the optimal SIC order and QoS constraint.
  • For NOMA, power allocations depend on users’ CNR outage thresholds and target data rates, with γ_i,m = 2^R_i,m − 1.
  • The general subcarrier-power expression combines scheduling variables with the NOMA and OMA cases, and total system power sums across independent subcarriers.
  • Given feasible scheduling and rate allocation, the derived SIC order and power allocation satisfy QoS with minimum power, reducing the original problem to scheduling and rate variables.
  • OMA requires doubled target data rates for equal spectral efficiency, so direct power comparisons with NOMA are not fair without this adjustment.
  • For general user counts, the paper uses simulations to demonstrate power savings over OMA because universal analytical superiority is difficult to prove.

C. Problem Transformation

The paper simplifies the original problem by eliminating QoS-related variables through the optimal SIC and power structure, then reformulates the remainder for global optimization. Continuous relaxation, penalty reformulation, and multiplicative programming expose a tractable branch-and-bound structure.

  • The original formulation is reduced by replacing rate variables with γ_i,m and safely removing the QoS constraint after deriving optimal SIC and power allocation.
  • The reduced problem remains a non-convex mixed-integer nonlinear program that is NP-hard because binary variables are coupled with continuous variables.
  • Constraint C8 preserves couplings between binary and continuous variables, while the transformed rate constraint remains equivalent through si,mlog2(1 + γ_i,m) = log2(1 + si,mγ_i,m).
  • Relaxing binary scheduling variables and augmenting coupling constraints with a penalty factor produces an equivalent formulation for sufficiently large θ.
  • The relaxed problem is equivalent to the prior formulation, and the new objective’s product terms motivate the penalty-based transformation.

B. B&B Based Optimal Resource Allocation Algorithm

The proposed branch-and-bound algorithm globally optimizes the transformed resource-allocation problem by subdividing a compact feasible region and pruning subregions with bounds. Convex relaxations provide local bounds, while the incumbent-gap test determines ε-convergence.

  • Branch-and-bound successively subdivides the feasible region and discards non-promising subregions using lower and upper bounds.
  • Convergence: Because branching is exhaustive over a finite-dimensional bounded region and convex subproblems are efficiently solvable, the procedure returns a globally optimal solution.
  • Bounding Method: The method’s direct applicability requires a tight convex objective bound and a compact feasible set, which the original problem lacks.
  • Branching Procedure: In the two-dimensional illustration, the initial rectangle is split along v_1,1, and one resulting rectangle becomes the current region for the next iteration.
  • Branching Procedure: The algorithm branches by bisecting the longest normalized edge of each hyper-rectangle, producing two subrectangles at every step.
  • Bounding Method: A convex relaxation solved within each rectangle supplies a local lower bound; infeasible rectangles are fathomed immediately.
  • Algorithm 1: The implementation initializes bounds and the unfathomed set, updates the incumbent and current rectangle, and repeats branching until the bound gap is within ε.
  • Bounding Method: Feasible points from the original problem provide local upper bounds, and rectangles whose lower bounds exceed the incumbent upper bound are deleted.

3) Overall Algorithm:

Algorithm 1 uses branching and bounding over subrectangles to obtain the globally optimal resource allocation. Its bounds converge in finite iterations, but the procedure is computationally demanding.

  • Branching and bounding: The branch-and-bound procedure partitions the feasible region into subrectangles and discards regions that cannot contain the optimum.The current subrectangle has the minimum local lower bound among unfathomed subrectangles, while incumbent and current points track feasible intermediate solutions.
  • Benchmark role: The optimal resource allocation obtained by the algorithm serves as a performance benchmark for suboptimal resource allocation methods.The paper states that the algorithm’s performance is verified through simulations.
  • Bound updates: The global lower-bound sequence is non-decreasing, whereas the global upper-bound sequence is non-increasing across iterations.For the one-dimensional illustration, UBD_k+1 ≤ UBD_k and LBD_k+1 ≥ LBD_k.
  • Convergence: The proposed branch-and-bound algorithm converges to the globally optimal solution in a finite number of iterations.The convergence claim follows from sufficient conditions cited in the paper, with the proof referenced separately.

V. SUBOPTIMAL SOLUTION

The suboptimal method reformulates the resource allocation problem as difference-of-convex programming and solves successive convex approximations. It has polynomial-time complexity and approaches the optimal value, without a global-optimality guarantee.

  • Comparison with optimal method: The optimal branch-and-bound method is guaranteed to find the optimum, but its worst-case complexity can equal that of exhaustive search.This motivates the lower-complexity suboptimal alternative.
  • Problem reformulation: The method introduces eγ_i,m = γ_i,m s_i,m to decouple binary scheduling and continuous rate variables.A big-M formulation is then used to transform the coupled problem.
  • D.C. formulation: The binary scheduling constraint is incorporated into the objective through a large penalty factor, yielding a canonical difference-of-convex formulation.The resulting problem retains convex constraints while penalizing violations of the binary structure.
  • Iterative solution: Each iteration linearizes one convex component using a first-order Taylor underestimator and solves the resulting convex optimization problem.The convex solution provides an upper bound, while successive feasible solutions tighten it.
  • Convergence and complexity: The suboptimal algorithm converges to a stationary point with polynomial-time computational complexity.The iterative procedure stops at a maximum iteration count or when the variable change falls below a predefined tolerance.
  • Performance and limitation: The algorithm has no guarantee of converging to a globally optimal solution, although simulations show close-to-optimal performance.Compared with exhaustive search, it provides substantial computational savings and is intended for practical implementation.

VI. SIMULATION RESULTS

Simulations evaluate convergence, SIC decisions, and power consumption under varied system settings and compare the proposed methods with OMA and other MC-NOMA baselines. The suboptimal method is much faster while retaining the optimal value in tested cases.

  • Convergence: For NF = 4 and M = 7, the optimal solution is found when the bounds meet after 600 iterations on average, while the suboptimal method reaches the optimal value within 80 iterations.The optimal algorithm produces non-increasing upper bounds and non-decreasing lower bounds.
  • Scalability: The proposed algorithms are computationally efficient relative to the optimal method and can apply to scenarios with more users.The paper adopts small M and NF values for comparisons because branch-and-bound complexity is high.
  • Convergence explanation: The suboptimal algorithm converges faster with more subcarriers because time sharing tends to convexify the optimization problem.Successive convex approximation exploits this large-scale structure, unlike feasible-set partitioning in the optimal method.
  • SIC and allocation: Users with higher CNR outage thresholds are selected for SIC and receive only a fraction of power because of better channels or less stringent QoS requirements.The CNR outage threshold serves as a metric for determining the SIC decoding policy and resource allocation under imperfect CSIT.

B. Power Consumption versus Target Data Rate

Power consumption increases with target data rate and channel-estimation uncertainty, while the proposed MC-NOMA schemes consistently outperform the OMA and baseline MC-NOMA schemes. Their advantage grows in demanding rate regimes.

  • Target data rate: Power consumption increases monotonically with the target data rate for all evaluated schemes.Higher transmit power is required to satisfy more stringent data-rate requirements.
  • Target data rate: NOMA’s power-consumption advantage over OMA becomes larger as the target data rate increases because NOMA multiplexes users across subcarriers.OMA power consumption increases exponentially in the overloaded scenario.
  • Scheduling: Random user scheduling can make MC-NOMA consume more power than OMA, showing that scheduling strategy strongly affects NOMA performance.The paper therefore identifies careful user scheduling as important for MC-NOMA power efficiency.
  • Rate allocation: The proposed schemes consume less power than equal-rate allocation at high target data rates by assigning higher rates to better subcarriers.Equal-rate allocation captures most NOMA gains at low target data rates, but frequency diversity benefits the proposed schemes at higher rates.
  • Channel estimation error: Power consumption increases monotonically as channel-estimation error variance increases because more power is needed to satisfy the outage-probability requirement.The proposed schemes remain the most power-efficient among the evaluated schemes, with a 6 dB extra-power change reported as the error variance increases from 0 to 0.5.

D. Power Consumption versus Number of Users

Power consumption increases with the number of users, but the proposed MC-NOMA resource allocation remains the most power-efficient and increasingly outperforms OMA as users increase.

  • Power consumption increases with the number of users for all considered schemes.The section attributes this to more users requiring stringent QoS requirements.
  • The proposed scheme is the most power-efficient among all schemes and remains applicable to both overloaded and underloaded systems.It is more power-efficient than OMA in both settings.
  • The proposed suboptimal scheme’s power saving over baseline scheme 1 increases with the number of users.The paper attributes this to NOMA’s spectral-efficiency and multiuser-diversity gains under exclusive subcarrier allocation in the baseline.
  • NOMA reduces required transmit power through multiuser multiplexing, which provides higher spectral efficiency than OMA.Power-domain multiplexing leaves relatively more spectrum available per user in the proposed scheme as the user count grows.
  • The proposed NOMA scheme exploits multiuser diversity to reduce total transmit power, with gains increasing as CNR outage thresholds become more heterogeneous.The paper links this heterogeneity to the increasing number of users.
  • The proposed resource allocation guarantees the required outage probability under channel uncertainty, at slightly higher transmit power than with perfect CSIT.The naive scheme instead produces significantly higher-than-required outage probability.

APPENDIX

The appendix compares SIC decoding cases and establishes which ordering minimizes total transmit power under the users’ outage-threshold conditions.

  • Four SIC decoding cases represent selecting user m, user n, both users, or neither user to perform SIC.Case I selects user m for SIC while user n directly decodes its own message.
  • Case I requires a feasible SIC condition; otherwise SIC cannot succeed.The stated prerequisite is p_i,n − p_i,mγ_i,n > 0.
  • Case I is optimal for minimizing total transmit power.The appendix derives and compares total transmit power across the four SIC policies.
  • When both users have the same CNR outage threshold, Cases I and II consume the same total transmit power.
  • The relaxed optimization problem is equivalent to the original problem through the stated mapping relationship, preserving an optimal solution.
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