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Joint Beamforming and Power Splitting Control in Downlink Cooperative SWIPT NOMA Systems

Yanqing Xu, Chao Shen, Zhiguo Ding, Xiaofang Sun, Shi Yan, Gang Zhu, Zhangdui Zhong

arXiv:1706.02851v1cs.IT

TL;DR

The paper addresses joint beamforming and power-splitting control for cooperative SWIPT NOMA under a nonconvex formulation. It develops SDR, exhaustive search, SCA, and GSS-based solutions for MISO and SISO cases, respectively. The proposed SWIPT-aided NOMA strategy is reported to outperform existing transmission strategies, with SISO algorithms converging to the unique global optimum.

  • Problem

    Cooperative SWIPT NOMA requires jointly optimizing beamforming and power splitting while maximizing the strong user's rate and satisfying the weak user's QoS.

  • Method

    The paper uses SDR with two-dimensional exhaustive search and rank-one optimality for MISO, SCA for efficient approximation, and GSS with closed-form updates for SISO.

  • Results

    The proposed cooperative SWIPT NOMA strategy outperforms existing transmission strategies, while both SISO algorithms converge to the unique global optimal solution.

  • Takeaways & Limitations

    The protocol uses harvested energy for relay forwarding and is presented as a promising candidate for IoT scenarios.

Abstract

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This paper investigates the application of simultaneous wireless information and power transfer (SWIPT) to cooperative non-orthogonal multiple access (NOMA). A new cooperative multiple-input single-output (MISO) SWIPT NOMA protocol is proposed, where a user with a strong channel condition acts as an energy-harvesting (EH) relay to help a user with a poor channel condition. The power splitting (PS) scheme is adopted at the EH relay. By jointly optimizing the PS ratio and the beamforming vectors, the design objective is to maximize the data rate of the "strong user" while satisfying the QoS requirement of the "weak user". It boils down to a challenging nonconvex problem. To resolve this issue, the semidefinite relaxation (SDR) technique is applied to relax the quadratic terms related with the beamformers, and then it is solved to its global optimality by two-dimensional exhaustive search. We prove the rank-one optimality, which establishes the equivalence between the relaxed problem and the original one. To further reduce the high complexity due to the exhaustive search, an iterative algorithm based on successive convex approximation (SCA) is proposed, which can at least attain its stationary point efficiently. In view of the potential application scenarios, e.g., IoT, the single-input single-output (SISO) case of the cooperative SWIPT NOMA system is also studied. The formulated problem is proved to be strictly unimodal with respect to the PS ratio. Hence, a golden section search (GSS) based algorithm with closed-form solution at each step is proposed to find the unique global optimal solution. It is worth pointing out that the SCA method can also converge to the optimal solution in SISO cases. In the numerical simulation, the proposed algorithm is numerically shown to converge within a few iterations, and the SWIPT-aided NOMA protocol outperforms the existing transmission protocols.

I. INTRODUCTION

The paper proposes cooperative SWIPT-aided NOMA in which the strong user harvests energy and relays the weak user's message, with joint beamforming and power-splitting design for MISO and SISO systems.

  • MISO design: The MISO design jointly optimizes beamforming and power splitting to maximize the strong user's rate while satisfying the weak user's QoS requirement.The formulation is nonconvex because the power-splitting ratio is coupled with quadratic beamformer terms.
  • Protocol: The proposed protocol uses the strong user as an energy-harvesting relay to improve the weak user's communication reliability.The relay forwards information using harvested energy only, avoiding battery energy consumption during information forwarding.
  • SISO design: In SISO settings, the optimal objective is strictly unimodal in the power-splitting ratio, enabling golden section search with closed-form power allocation at each iteration.The paper motivates SISO analysis by its low-power and low-cost advantages in IoT and massive machine-type communications.
  • System model: The system model uses a multi-antenna base station and two single-antenna users with different channel conditions, assigning the relay role to the stronger-channel user.The weak user is modeled as a cell-edge user and the strong user as a cell-center user.
  • Transmission protocol: The MISO transmission has two stages: the weak user receives directly while the strong user performs SWIPT, then the strong user forwards using harvested energy and the weak user applies MRC.The base station transmits x = w1x1 + w2x2 under a beamformer power constraint, while the strong user's received signal is split for information decoding and energy harvesting.

B. Cooperative Transmission Stage

In the cooperative stage, the strong user forwards the weak user's message using harvested energy, while the weak user combines both received signals and the design maximizes the strong user's rate under QoS constraints.

  • Cooperative transmission: The strong user forwards the weak user's message using the energy harvested during the first transmission stage.The forwarded signal reaches the weak user through the inter-user channel and contributes a second-stage SNR.
  • Cooperative transmission: The weak user combines its direct and relayed observations with maximal-ratio combination before decoding the weak user's message.The resulting equivalent SINR incorporates both transmission stages.
  • Problem formulation: The optimization maximizes the strong user's data rate while ensuring that both users can decode the weak user's message at the target SINR.The constraints also include the beamformer power budget and the feasible range of the power-splitting ratio.
  • Problem formulation: The resulting problem is nonconvex because the power-splitting ratio is coupled with quadratic beamformer terms in the objective and constraints.The paper proposes SDR reformulation followed by an SCA-based iterative solution.

III. SUBOPTIMAL BEAMFORMING DESIGN AND POWER SPLITTING CONTROL

The paper transforms the nonconvex design using semidefinite relaxation and then uses successive convex approximation to obtain an efficient approximate solution.

  • SDR and SCA: The SDR reformulation replaces beamformer-related quadratic terms with tractable expressions in the semidefinite cone.Further reformulations are used to handle remaining nonconvex terms before applying SCA.
  • SDR and SCA: The SCA-based iterative algorithm repeatedly solves an approximated problem and updates auxiliary variables until successive objective values differ by less than a threshold.The method is designed to efficiently obtain at least a stationary point of the reformulated problem.

A. Reformulation of P1 with SDR

The paper lifts beamforming vectors into positive semidefinite matrices, yielding an SDR formulation whose rank-one optimality establishes equivalence with the original problem. A two-dimensional exhaustive search finds the global optimum, while high complexity motivates SCA for a stationary solution.

  • Beamforming vectors are represented by positive semidefinite matrices W1 and W2 through Wi = wiwH_i.
  • Dropping rank-one constraints creates a relaxed problem whose equivalence to the original formulation is not initially guaranteed.
  • Whenever feasible, the relaxed problem has an optimal solution with rank(W1*) = 1 and rank(W2*) ≤ 1.
  • The optimal auxiliary variables can be found by two-dimensional exhaustive search, proving SDR tightness and global optimality of the reformulation.
  • If the relaxed solution is not rank-one for both matrices, eigen-decomposition gives way to Gaussian randomization for a suboptimal beamforming solution.
  • The exhaustive search is computationally costly because the reformulated problem remains nonconvex through objective curvature and coupling among β, W1, and W2.

B. Reformulation of Nonconvex Constraints in P2

The reformulation exposes the remaining nonlinear structure of the SDR problem by introducing auxiliary variables, epigraph constraints, quadratic inequalities, and linear matrix inequalities.

  • Epigraph reformulation converts the objective into a linear objective accompanied by a nonconvex quadratic inequality and a convex LMI.
  • The nonconvex constraint (14c) is decomposed into a nonconvex constraint and a further nonconvex quadratic constraint with an LMI.
  • Compared with P2, P4 explicitly identifies nonconvex constraints (22b)–(22d) as the central remaining difficulty.

C. SCA-based Algorithm for P4

An SCA algorithm repeatedly convexifies the reformulated problem and solves the resulting convex subproblems, providing an efficient stationary-point method for MISO cases and optimality in SISO cases.

  • SCA replaces nonconvex constraints with first-order Taylor approximations and an arithmetic-geometric mean approximation, producing a convex subproblem.
  • Each iteration updates β, W, and auxiliary variables from the current solution until the objective gap between successive iterations falls below a tolerance.
  • The generated rate sequence continuously decreases its successive-iteration gap and converges to at least a stationary point whenever P4 is feasible.
  • A preconditioning parameter θ balances 1 − β and Tr(H2W2) to improve problem conditioning while preserving stationary-point convergence.
  • In SISO cases, the same SCA method can attain only a stationary point directly, whereas a separate method guarantees global optimality with a closed-form optimum at each iteration.

IV. OPTIMAL TRANSMISSION PROTOCOL DESIGN FOR SISO CASES

For the single-antenna cooperative SWIPT NOMA system, beamforming reduces to power control. The resulting problem is solved globally using a golden section search with semiclosed-form updates.

  • The SISO protocol equips the base station, user 1, and user 2 with single antennas.
  • In SISO, the beamforming vector w1 is replaced by power fraction α for message x1, converting beamforming design into power control.
  • The normalized channel gains h1, h2, and g parameterize the SISO formulation.
  • Golden section search obtains the global optimum through an iterative procedure with a semiclosed-form solution at each step.

B. Global Optimal Solution to P6

For SISO problem P6, the feasible PS-ratio interval is characterized, the inner power-allocation solution has a closed form, and strict unimodality enables a unique global optimum via GSS.

  • The feasible PS-ratio set is [βmin, βmax], with βmin = (γ1−h1)+.
  • The optimal α is given by a closed-form expression involving A and B, with α = min{A, 1} for β ≥ γ1/(h2g) and α = min{max{A, B}, 1} otherwise.
  • If β ≥ γ1/(h2g), constraint (32b) holds with equality; otherwise, constraint (32c) holds with equality.
  • The objective h(β) is strictly unimodal on [βmin, βmax], so GSS obtains the unique global optimum while evaluating closed-form α for each β.
  • For any feasible β, the optimal α makes at least one of constraints (32b) and (32c) hold with equality.
  • Both the GSS-based algorithm and the SCA-based algorithm converge to the unique global optimum in SISO cases whenever P6 is feasible.

V. NUMERICAL RESULTS

The numerical-results section evaluates the performance of the proposed cooperative SWIPT NOMA protocol through simulation.

  • The simulations evaluate the performance of the proposed cooperative SWIPT NOMA protocol.

A. Simulation Setup

The simulations use a two-user room-scale setup with Rician fading, specified path-loss and noise parameters, 1 MHz bandwidth, and comparisons against noncooperative NOMA and OMA strategies.

  • The two users are randomly placed in a 5-meter × 6-meter room, while the BS is fixed at coordinate (0, 2.5m).
  • The path-loss exponents are α1 = 4 for user 1 and α2 = 2 for user 2, with distance-dependent path loss PL = 10−3d−α.
  • The simulations use σ1^2 = σ2^2 = σ^2 = −90 dBm, a total bandwidth of 1 MHz, and a Rician fading channel model.
  • Results are averaged over 1,000 independent channel realizations.
  • The comparison includes noncooperative NOMA, OMA with dynamic time allocation, and OMA with fixed time allocation.

B. Sum Rate of Users

The proposed cooperative SWIPT NOMA strategy generally achieves strong sum-rate and feasibility performance, while antenna number and user-1 rate targets shape the observed tradeoffs.

  • The proposed cooperative SWIPT NOMA strategy achieves the best MISO sum-rate performance among the considered transmission strategies.It outperforms noncooperative NOMA in the low-power regime and matches it at high power.
  • In SISO cases, the proposed strategy always achieves a higher sum rate than the other considered transmission strategies.
  • The SCA-based algorithm has sum-rate performance very close to the exhaustive-search method in MISO cases.
  • Feasible probability increases with BS transmit power, while SCA remains close to exhaustive search in MISO and SISO cases.This indicates limited conservativeness of the proposed approximation in the simulations.
  • User 2's achievable rate decreases as user 1's target rate increases because beamforming prioritizes user 1 and reduces power allocated to user 2.
  • Multiple antennas improve sum rate across transmission strategies, and noncooperative MISO NOMA can outperform cooperative SWIPT NOMA with SISO at high transmission power.
  • With 90% feasible probability, cooperative SWIPT NOMA supports a 2.2 Mbps target rate for user 1, versus 0.6 Mbps for noncooperative NOMA.
  • The SCA and GSS algorithms converge to the same SISO sum rate after several iterations, supporting the global optimality of Algorithm 1 in SISO cases.

APPENDIX A PROOF OF LEMMA 1

The appendix proves that the optimal power-splitting variable must make at least one relevant constraint active, using contradiction and derivative-based monotonicity arguments.

  • For fixed β, the optimal α makes at least one of constraints (32b) and (32c) hold with equality.If both constraints were slack, reducing α would preserve feasibility while decreasing the objective, contradicting optimality.
  • The proof establishes 0 ≤ B ≤ 1 for β in [βmin, β0] by separately showing B ≤ 1 and B ≥ 0.The nonnegativity argument uses β0 ≤ γ1 h2g.
  • The appendix derives the first-order derivative of f(β) and identifies the numerator–denominator difference used to verify its monotonicity.
  • The resulting algebraic expression decomposes into squared and parameter-dependent terms in β, providing the basis for the monotonicity verification.
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