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Joint Transmit Beamforming and Receive Power Splitting for MISO SWIPT Systems

Qingjiang Shi, Liang Liu, Weiqiang Xu, Rui Zhang

arXiv:1304.0062v3cs.IT

TL;DR

The paper addresses minimum-power joint beamforming and receive power splitting for multi-user MISO SWIPT under SINR and harvested-power constraints. It derives feasibility conditions, solves the non-convex design through tight semidefinite relaxation, and proposes lower-complexity ZF- and SINR-optimal-based alternatives. The optimal design minimizes transmission power, while the suboptimal schemes become asymptotically equivalent to it at high SINR.

  • Problem

    The paper seeks to minimize BS transmission power while jointly meeting users’ SINR requirements for information decoding and harvested-power requirements for energy harvesting.

  • Method

    The paper jointly designs transmit beamforming and receive power-splitting ratios, solves the non-convex problem using semidefinite relaxation, and also develops ZF- and SINR-optimal-based lower-complexity designs.

  • Results

    The feasibility condition depends only on SINR constraints, while the semidefinite relaxation is proved optimal for the formulated problem.

  • Takeaways & Limitations

    The lower-complexity suboptimal designs provide alternatives to the optimal solution, with their high-SINR behavior characterized alongside the optimal design.

Abstract

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This paper studies a multi-user multiple-input single-output (MISO) downlink system for simultaneous wireless information and power transfer (SWIPT), in which a set of single-antenna mobile stations (MSs) receive information and energy simultaneously via power splitting (PS) from the signal sent by a multi-antenna base station (BS). We aim to minimize the total transmission power at BS by jointly designing transmit beamforming vectors and receive PS ratios for all MSs under their given signal-to-interference-plus-noise ratio (SINR) constraints for information decoding and harvested power constraints for energy harvesting. First, we derive the sufficient and necessary condition for the feasibility of our formulated problem. Next, we solve this non-convex problem by applying the technique of semidefinite relaxation (SDR). We prove that SDR is indeed tight for our problem and thus achieves its global optimum. Finally, we propose two suboptimal solutions of lower complexity than the optimal solution based on the principle of separating the optimization of transmit beamforming and receive PS, where the zero-forcing (ZF) and the SINR-optimal based transmit beamforming schemes are applied, respectively.

I. INTRODUCTION

The paper formulates a multi-user MISO SWIPT problem using power splitting so mobile stations can continuously decode information and harvest energy. It jointly designs BS beamforming and MS power-splitting ratios to minimize transmission power, then develops feasibility, optimal, and lower-complexity solutions.

  • SWIPT background: Power splitting generally provides better rate-energy transmission trade-offs than time switching, although it requires an RF signal splitter.Time switching uses binary power-split ratios, whereas power splitting divides received power between information decoding and energy harvesting.
  • SWIPT background: MIMO techniques address SWIPT’s transmission-distance power-transfer losses while retaining high spectral efficiency for information transmission.The introduction motivates multiple antennas because power-transfer efficiency decays substantially with propagation distance.
  • System focus: The considered system is a MISO broadcast channel with one multi-antenna BS and K ≥1 single-antenna MSs using power splitting for continuous information and energy reception.Each MS has its own SINR requirement and harvested-power requirement.
  • Problem and contributions: The paper jointly optimizes transmit beamforming and receive PS ratios to minimize total BS transmission power under SINR and harvested-power constraints.The formulation addresses both information-decoding and energy-harvesting requirements simultaneously.
  • Problem and contributions: Feasibility depends only on the SINR constraints, while semidefinite relaxation is proved tight and therefore yields an optimal beamforming solution.The paper also presents lower-complexity suboptimal designs using ZF and SINR-optimal transmit beamforming followed by PS optimization.
  • Problem and contributions: The optimal and suboptimal solutions are compared through simulations.The suboptimal schemes are designed to reduce complexity relative to the optimal solution.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system uses linear transmit beamforming from a multi-antenna BS and power splitting at each single-antenna MS. The optimization minimizes BS transmission power while satisfying per-MS SINR and harvested-power requirements, yielding a non-convex JBPS problem.

  • System model: The model contains one BS with Nt > 1 antennas and K single-antenna MSs communicating over a quasi-static flat-fading channel.Each MS is assigned one dedicated information beam.
  • System model: The BS transmits a sum of independently modulated data streams, with vk denoting the beamforming vector for MS k.The symbols are i.i.d. CSCG random variables with zero mean and unit variance.
  • Constraints: Each MS must achieve SINR at least γk and harvested power at least ek to support continuous information transfer and receiver operation.The paper considers the general case γk > 0 and ek > 0, implying 0 <ρk <1.
  • Optimization problem: The JBPS problem jointly designs {vk} and {ρk} to minimize total BS transmission power under the SINR and harvested-power constraints.Its non-convexity arises from coupled design variables and quadratic beamforming terms.
  • Optimization problem: The paper first derives feasibility conditions and then develops optimal and suboptimal solutions for the JBPS problem.The suboptimal designs separate beamforming and power-splitting optimization, while practical computation occurs at the BS.

III. WHEN IS THE JBPS PROBLEM FEASIBLE?

The JBPS problem is feasible exactly when the corresponding SINR-only problem is feasible. Thus, feasibility can be checked from the SINR targets without using the harvested-power constraints.

  • Feasibility reduction: The feasibility of the JBPS problem is equivalent to feasibility of a reduced problem.This equivalence is established by combining Lemmas 3.1 and 3.2.
  • Feasibility reduction: The feasibility condition does not depend on the harvested-power constraints.The reduced feasibility test can be obtained by letting ρk →1 for all users.
  • SINR condition: The reduced problem is the well-known SINR feasibility problem whose feasible region is characterized by the SINR targets γk.Proposition 3.1 gives a sufficient and necessary condition for this reduced problem.
  • SINR condition: For given SINR and harvested-power targets, JBPS feasibility can be verified by checking whether the SINR targets satisfy Proposition 3.1.The remainder of the paper assumes the formulated problem is feasible unless stated otherwise.

IV. OPTIMAL SOLUTION

The JBPS problem is solved optimally by applying semidefinite relaxation and proving that the relaxation is tight. The resulting solution preserves the original beamforming problem’s optimum while satisfying both SINR and harvested-power constraints.

  • IV. OPTIMAL SOLUTION: SDR converts the non-convex JBPS formulation into a convex problem after reformulation of the coupled constraints.The original SDR formulation remains non-convex until the coupled beamforming and PS constraints are reformulated.
  • IV. OPTIMAL SOLUTION: The reformulated problem is convex, enabling solution through an interior-point algorithm using software such as CVX.The paper also notes that the dual problem may be solved for better efficiency.
  • IV. OPTIMAL SOLUTION: Rank(X*_k) ≤ 1 for every user, proving that the SDR is tight for γ_k > 0 and e_k > 0.The proposition establishes the rank-one property needed to recover beamforming vectors from the relaxed solution.
  • IV. OPTIMAL SOLUTION: The rank relaxation causes no loss of optimality, so solving the SDR yields an optimal solution to the original JBPS problem.The resulting beamforming and PS variables can therefore be recovered from the SDR solution.
  • IV. OPTIMAL SOLUTION: With the optimal beamforming and PS solution, every MS’s SINR and harvested-power constraints hold with equality.This follows from the first part of Proposition 4.1.

V. SUBOPTIMAL SOLUTIONS

The paper introduces two lower-complexity alternatives to joint beamforming and PS optimization by separating the two design tasks. They use ZF-based and SINR-optimal transmit beamforming criteria, respectively.

  • V. SUBOPTIMAL SOLUTIONS: Two suboptimal algorithms reduce complexity by separately designing transmit beamforming vectors and receive PS ratios.The optimal solution instead jointly optimizes both sets of variables.
  • V. SUBOPTIMAL SOLUTIONS: The first alternative applies zero-forcing beamforming, while the second applies SINR-optimal transmit beamforming.The paper also studies the asymptotic optimality of both alternatives.

A. ZF Beamforming

The ZF-based design eliminates multiuser interference by restricting each beamformer to the other users’ channel null space. It provides a closed-form solution but requires at least as many BS antennas as users.

  • A. ZF Beamforming: ZF beamforming eliminates multiuser interference by imposing h_i^H v_k = 0 for every i ≠ k.This restriction simplifies the beamforming design and reduces the joint problem to a simpler formulation.
  • A. ZF Beamforming: The ZF formulation is feasible when N_t ≥ K and the user channels are not linearly dependent.These conditions ensure an appropriate null space exists for each beamformer.
  • A. ZF Beamforming: The optimal solution of the ZF-restricted problem is given in closed form using an orthogonal basis of each channel matrix’s null space.The basis is denoted U_k for the null space of H_k^H.
  • A. ZF Beamforming: Algorithm 2 has complexity O(K^4 + K^2N_t^2), dominated by K singular-value-decomposition operations.This is the complexity stated for the ZF beamforming-based suboptimal algorithm.
  • A. ZF Beamforming: The ZF-based design assumes σ_k^2 > 0 for every user in the stated proposition.The paper notes that this assumption concerns antenna noise at each mobile station.

B. SINR-Optimal Beamfoming

The SINR-optimal alternative first solves a power-minimization problem with only SINR constraints, then scales the beamformers and optimizes PS ratios to satisfy harvested-power constraints. Unlike ZF, it works for arbitrary N_t and K.

  • B. SINR-Optimal Beamfoming: Algorithm 3 uses SINR-optimal transmit beamforming, which is applicable for arbitrary values of N_t and K.The underlying SINR-only problem is feasible exactly when the original JBPS problem is feasible and can be solved using existing techniques.
  • B. SINR-Optimal Beamfoming: The method first obtains SINR-optimal beamformers, then scales them by a common α and jointly chooses PS ratios.This scaling is used to satisfy both SINR and harvested-power constraints while minimizing transmission power.
  • B. SINR-Optimal Beamfoming: Because harvested-power constraints require 0 < ρ_k < 1, the common scaling must satisfy α > 1.At α = 1 and ρ_k = 1, the SINR constraints hold with equality, but the additional harvested-power constraints require further scaling.
  • B. SINR-Optimal Beamfoming: The scaled formulation is feasible if and only if the original JBPS problem is feasible.The paper then characterizes the required scaling using the largest real root of a quadratic equation.
  • B. SINR-Optimal Beamfoming: The ZF-based and SINR-optimal designs are lower-complexity than the optimal solution, while ZF is lower-complexity than the SINR-optimal design.The ZF alternative remains restricted to N_t ≥ K.

C. Asymptotic Optimality

The two suboptimal solutions are generally suboptimal but become asymptotically optimal as the SINR targets grow without bound.

  • As γk’s increase, power allocation for all solutions becomes extremely large.
  • Interference terms vanish as γk’s increase to keep the SINR constraints feasible.
  • The two suboptimal solutions are asymptotically optimal when γk’s go to infinity.Their minimum transmit power matches the optimal solution in this limit.

VI. SIMULATION RESULTS

The simulations compare the optimal JBPS design with ZF- and SINR-optimal-based suboptimal schemes under varying SINR targets, harvested-power constraints, and antenna counts. The optimal solution consistently minimizes transmission power, while the suboptimal schemes converge toward it at high SINR and benefit from larger antenna arrays.

  • Simulation setup: The simulations use K = 4 MSs, Nt = 4 transmit antennas in Fig. 2, and Rician-fading channels with a dominant LOS component.The setup assumes equal 40dB signal attenuation over 5 meters, d = λ, and Rician factor KR = 5dB.
  • SINR-target comparison: Increasing the harvested-power constraint from −20dBm to 0dBm substantially increases required BS transmission power for every SINR target.This comparison is made in Fig. 2 for the optimal JBPS, ZF-based, and SINR-optimal-based schemes.
  • SINR-target comparison: The optimal JBPS solution achieves the minimum transmission power for both harvested-power constraints across all SINR values.The result holds for e = 0dBm and e = −20dBm.
  • SINR-target comparison: At low SINR, the SINR-optimal-based scheme uses less transmission power than ZF, but their gap vanishes as SINR increases.For e = 0dBm, the two suboptimal schemes converge to the optimal solution above γ > 35dB; for e = −20dBm, convergence occurs above γ > 25dB.
  • SINR-target comparison: At γ = 0dB, ZF-based transmission power is insensitive to γ because antenna noise is much smaller than ID-processing noise in the simulation setup.The stated parameters include σ^2 = −70dBm and δ^2 = −50dBm.
  • Antenna-count comparison: Increasing Nt substantially decreases BS transmission power for all proposed solutions, demonstrating a benefit of large or massive antenna arrays.Fig. 4 fixes γ = 10dB and e = −10dBm.

VII. CONCLUSION

The paper jointly designs transmit beamforming and receive power splitting for multiuser MISO SWIPT, minimizing BS transmission power under SINR and harvested-power constraints. It derives feasibility conditions, proves SDR optimality, and presents lower-complexity ZF- and SINR-optimal-based alternatives.

  • The study minimizes BS transmission power subject to SINR and harvested-power constraints in a multiuser MISO SWIPT system.
  • A sufficient and necessary feasibility condition is derived for the joint beamforming and power-splitting problem.
  • SDR is applied to the non-convex formulation, and the resulting solution is proved optimal.
  • Two lower-complexity suboptimal designs use ZF and SINR-optimal beamforming, respectively, and are compared with the optimal solution.
  • For the ZF-based design, optimal SINR and harvested-power constraints both hold with equality, and the power-splitting subproblem has a unique solution in 0 < ρ_k < 1.
  • The feasibility results establish equivalence between feasibility of the SINR-only and joint formulations in the stated problem relationships.

PROOF OF PROPOSITION 5.3

The proof examines the high-SINR behavior of feasible beamforming designs. It establishes that power allocation diverges, beam directions must satisfy zero-forcing, and all three proposed algorithms become asymptotically equivalent in transmit power.

  • High-SINR behavior: Any feasible beamforming design with γ_k → ∞ requires every user’s power allocation p_k to diverge.The proof uses bounded desired-channel gains and the coupling of power variables across SINR constraints.
  • High-SINR behavior: As γ_k → ∞, feasible normalized beam directions must satisfy the zero-forcing condition for inter-user interference.The proof derives this requirement by contradiction if an interfering inner product remains nonzero.
  • Algorithm comparison: All three algorithms asymptotically achieve the same minimum transmit power as every γ_k tends to infinity.This is the proposition’s concluding asymptotic-equivalence result.
  • Algorithm comparison: The optimal and SINR-optimal-based solutions must satisfy zero-forcing asymptotically, as does the ZF-based solution.Thus, all three designs share the same limiting beam-direction property.
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