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Multiuser MISO Beamforming for Simultaneous Wireless Information and Power Transfer

Jie Xu, Liang Liu, Rui Zhang

arXiv:1303.1911v4cs.IT

TL;DR

The paper addresses joint information-and-energy beamforming in multiuser MISO SWIPT, maximizing weighted harvested power while meeting ID-receiver SINR constraints. It formulates separate non-convex QCQPs for ID receivers with and without energy-interference cancellation, solves them globally using SDR, and develops an equivalent uplink-downlink-duality method. The resulting structure eliminates dedicated energy beams for Type I and limits Type II to at most one.

  • Problem

    Practical SWIPT must support separately designed ID and EH receivers with substantially different received-power requirements in a multiuser MISO broadcast setting.

  • Method

    The paper formulates joint information-and-energy beamforming as two non-convex QCQPs and solves them using tight SDR and a new uplink-downlink-duality formulation.

  • Results

    For Type I ID receivers, no dedicated energy beam is needed at optimum; for Type II receivers, no more than one energy beam is optimal.

  • Takeaways & Limitations

    The derived optimal structures provide guidelines for optimizing multi-antenna SWIPT with receiver-location-based information and energy transmission.

Abstract

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Simultaneous wireless information and power transfer (SWIPT) is anticipated to have abundant applications in future machine or sensor based wireless networks by providing wireless data and energy access at the same time. In this paper, we study a multiuser multiple-input single-output (MISO) broadcast SWIPT system, where a multi-antenna access point (AP) sends information and energy simultaneously via spatial multiplexing to multiple single-antenna receivers each of which implements information decoding (ID) or energy harvesting (EH). Since EH receivers in practice operate with considerably higher received power than ID receivers, we propose a receiver location based transmission scheduling, where receivers that are close to the AP are scheduled for EH while those more distant from the AP for ID. We aim to maximize the weighted sum-power transferred to all EH receivers subject to a given set of minimum signal-to-interference-and-noise ratio (SINR) constraints at different ID receivers. In particular, we consider two types of ID receivers (referred to as Type I and Type II, respectively) without or with the capability of cancelling the interference from (a priori known) energy signals. For each type of ID receivers, we formulate the joint information and energy transmit beamforming design as a non-convex quadratically constrained quadratic program (QCQP). First, we obtain the globally optimal solutions for our formulated QCQPs by applying an optimization technique so-called semidefinite relaxation (SDR). It is shown via SDR that no dedicated energy beam is needed for Type I ID receivers to achieve the optimal solution; while for Type II ID receivers, employing no more than one energy beam is optimal. Next, we establish a new form of the celebrated uplink-downlink duality to develop alternative algorithms to obtain the same optimal solutions as SDR.

I. INTRODUCTION

The paper formulates practical multiuser MISO SWIPT beamforming for receivers dedicated to either information decoding or energy harvesting, using location-based scheduling. It develops globally optimal SDR solutions and an equivalent uplink-downlink-duality approach for maximizing harvested power under ID-receiver SINR constraints.

  • Motivation: EH receivers are scheduled near the AP and ID receivers farther away because EH operation requires substantially higher received power.A low-power EH sensor may require 0.1 mW or −10 dBm, whereas cellular and Wi-Fi ID receivers often operate below −50 dBm.
  • System model: The system uses a multi-antenna AP to transmit information and energy simultaneously to single-antenna receivers assigned exclusively to ID or EH.Each receiver is assigned one dedicated information or energy beam under linear precoding.
  • Solution approach: The resulting joint beamforming problems are non-convex QCQPs, whose globally optimal solutions are obtained efficiently through semidefinite relaxation.The formulations maximize a convex quadratic function subject to quadratic constraints.
  • Solution approach: SDR shows that Type I needs no dedicated energy beam for optimality, while Type II requires no more than one dedicated energy beam.An alternative uplink-downlink-duality formulation yields the same optimal downlink beamforming solutions as SDR.
  • Problem formulation: Two ID-receiver types are considered: Type I cannot cancel energy-signal interference, whereas Type II can cancel it perfectly because energy waveforms are known in advance.Type II receivers therefore use a different achievable SINR expression from Type I receivers.
  • Problem formulation: The design maximizes weighted sum-power transferred to EH receivers subject to individual minimum SINR constraints at ID receivers.Energy weights represent the relative priority assigned to different EH receivers.

III. OPTIMAL SOLUTION VIA SEMIDEFINITE RELAXATION

Semidefinite relaxation yields globally optimal solutions for both non-convex QCQPs and reveals their optimal energy-beam structures under independently distributed user channels.

  • SDR efficiently obtains globally optimal solutions for the Type I and Type II non-convex QCQPs.The relaxation exploits the specific structures of both formulated problems.
  • Under independently distributed user channels, the Type I SDR is tight and recovers a globally optimal solution of the original problem.The optimal relaxed information-beam matrices satisfy the required rank constraints.
  • For Type I receivers, no dedicated energy beam is needed to maximize weighted harvested power.Energy beams would increase uncancellable interference at ID receivers, so information-beam weighting and power allocation suffice.
  • For Type II receivers, the optimal energy covariance has rank at most one, so no more than one energy beam is required.All energy beams can be aligned with the optimal energy-beamforming direction, allowing a single beam implementation.
  • Type II receivers can achieve higher weighted harvested power than Type I receivers when the optimal Type II energy-beam power q* is positive, at the cost of interference cancellation.The Type II optimum is generally an upper bound on the Type I optimum, and the bound is tight when q*=0.
  • Even without the independent-channel assumption, both SDRs remain tight, although rank-reduction techniques may be needed to obtain rank-one solutions.The paper notes that general-channel guarantees establish existence, while independent channels provide more specific rank results without rank reduction.

IV. ALTERNATIVE SOLUTION BASED ON UPLINK-DOWNLINK DUALITY

The paper develops an alternative uplink-downlink-duality approach to provide insight into the optimal joint information and energy beamforming structure and recover the SDR solutions.

  • Uplink-downlink duality is used as an alternative approach for solving the non-convex Type I and Type II beamforming problems.The approach is motivated by recasting apparently non-convex downlink problems into convex forms when possible.

A. Algorithm for (P1) via Uplink-Downlink Duality

For Type I receivers, the optimal design eliminates dedicated energy beams and is solved through an equivalent dual formulation with strong duality and a search over the power-constraint multiplier.

  • Under the channel assumption, Type I optimality permits setting all dedicated energy beams to zero before applying uplink-downlink duality.The resulting SINR expression and reformulation involve information transmission only.
  • The dual problem uses β≥0 as the multiplier associated with the transmit sum-power constraint.The dual function and dual problem are formed from this partial Lagrangian formulation.
  • The optimal values of the primal and dual formulations are equal, establishing strong duality for the Type I reformulation.This equivalence is enabled by the tightness of the SDR.
  • The duality is represented by a MISO broadcast channel with information-only transmission and its dual SIMO multiple-access channel.The MISO-BC minimizes weighted transmit power subject to SINR constraints, while the SIMO-MAC uses conjugated and transposed channel vectors.
  • The algorithm solves the fixed-β problem and then searches over β≥0 to minimize the resulting function.This procedure obtains the optimal β* after evaluating f1(β).

1) Obtain f1(β) for given β ≥0:

For a given β ≥0, the method obtains f1(β) by solving a downlink information-beamforming problem through an equivalent uplink problem, using different procedures according to boundedness and β relative to ξE.

  • Downlink–uplink formulation: Problem (10) minimizes weighted information transmit power for a MISO broadcast channel subject to the ID receivers’ SINR constraints.The equivalent dual SIMO-MAC uses power allocation and receive beamforming under the same SINR targets.
  • Case β ≥ξE: For β ≥ξE, βI − G ⪰0, so problems (10) and (13) are equivalent and the downlink solution follows from the uplink solution.The mapping uses the uplink solution and the corresponding downlink beamforming vectors.
  • Case 0 ≤β < ξE: For 0 ≤β < ξE, the method separately handles cases where g(β) is bounded or unbounded from below.The objective becomes non-convex in this range, and g(β) may equal −∞.
  • Bounded subcase: When βI −G is indefinite but g(β)>−∞, a new uplink-downlink duality equates the optimal values of problems (10) and (13).The equivalence relies on strong duality resulting from tight semidefinite relaxation.
  • Iteration and unboundedness: The fixed-point iteration converges to the optimal uplink solution in the bounded subcase and checks unboundedness in the g(β)=−∞ subcase.The iteration requires a feasible initial point in the bounded case; Proposition 4.5 provides an efficient unboundedness check.

B. Algorithm for (P2) via Uplink-Downlink Duality

For Type II ID receivers, optimality permits replacing all dedicated energy beams with one common beam aligned with the optimal energy beamforming direction.

  • Energy-beam reduction: For Type II ID receivers, one common energy beam wE = √qvE is sufficient without loss of optimality.This follows from the result that only one energy beam aligned with the OeBF is optimal for (P2).

2) Minimize f1(β) over β ≥0:

The Type II dual problem is minimized over β ≥0 after evaluating f2(β), but boundedness requires β ≥ξE; the resulting subproblems use the Type I procedure and one energy beam.

  • Dual formulation: Strong duality holds between (P2.1) and (P2.2), allowing (P2.1) to be solved through its dual problem.The dual approach evaluates f2(β) for fixed β and then searches for the minimizing β.
  • Dual-variable search: f2(β) is unbounded above when β <ξE, so any bounded optimum must satisfy β ≥ξE.For β ≥ξE, f2(β) decomposes into terms involving g(β) and h(β).
  • Subproblem solution: For β ≥ξE, Algorithm 1 solves the g(β) subproblem, while the h(β) subproblem determines the common Type II energy-beam component.The information-beam solution from problem (10) becomes the corresponding solution for (P2.1).
  • Solution regions: The optimal Type II solution has three regions: no energy beam, one energy beam, or the OeBF-feasible case.These regions correspond respectively to the cases β∗=β⋆>ξE, β∗=ξE>β⋆, and β∗=β⋆=ξE.

C. Solution Comparison with Type I versus Type II ID Receivers

The comparison separates three SINR regimes: equal performance at large or small constraints, and a Type II advantage at moderate constraints when one dedicated energy beam is useful.

  • Large SINR constraints: At sufficiently large SINR constraints, both receiver types use only information beams, so their optimal harvested-power values are equal.This is Region 1, where q∗=0 for Type II receivers.
  • Moderate SINR constraints: At moderate SINR constraints, Type II receivers outperform Type I receivers because one dedicated energy beam is beneficial.The comparison identifies this as Region 2, with v(P2)>v(P1) and q∗>0.
  • Evaluation setting: The numerical comparison uses M = 4, KE = 2, and 200 random channel realizations, with 30 dB EH attenuation and 70 dB ID attenuation.The channels are generated using i.i.d. Rayleigh fading under the independent-channel assumption.
  • Small SINR constraints: At sufficiently small SINR constraints, both types have identical performance because aligning all information beams with the OeBF is feasible and optimal.This is the OeBF-feasible Region 3.
  • Moderate SINR constraints: 41% average harvested power gain is achieved for Type II over Type I receivers at γ = 10 dB and KI = 4 in the stated channel setup.The result is attributed to cancellation of known energy signals at the Type II ID receivers.

B. Complexity Comparison of SDR and Uplink-Downlink Duality Based Algorithms

The uplink-downlink duality algorithms run faster than SDR-based algorithms, while separate beamforming performs differently for Type I and Type II receivers.

  • Complexity Comparison: For either (P1) or (P2), the SDR-based algorithm has a longer running time than the uplink-downlink duality-based algorithm as M varies.The comparison fixes K_I = 4, K_E = 2, and γ = 10 dB.
  • Complexity Comparison: Analytic complexity orders cannot be rigorously compared for the duality algorithms because their costs depend on inner and outer iteration parameters.
  • Complexity Comparison: The SDR method is slower because it optimizes matrices with many more unknowns than the duality method, which uses beamforming vectors.
  • Complexity Comparison: The duality algorithm for (P1) takes longer than that for (P2) because it checks positive semidefiniteness during Algorithm 2.
  • Performance Comparison: The paper compares optimal joint beamforming with separate designs for both ID receiver types.
  • Performance Comparison: Separate information and energy beamforming performs severely worse than joint design for Type I, but comparably for Type II when γ is small.Figure 6 compares average harvested power over the SINR constraint with M = 4, K_E = 2, and K_I = 2.

APPENDIX

The appendix derives rank properties of the SDR solutions using duality, complementary slackness, and independently distributed user channels.

  • Proof Strategy: The SDR1 formulation is a convex SDP satisfying Slater’s condition, enabling a zero duality gap and complementary-slackness analysis.
  • Type I SDR: The proof bounds the ranks of A_i and uses complementary slackness to establish rank(W_i*) = 1 for every ID receiver.
  • Type I SDR: When all λ_i* vanish, the energy covariance lies in the subspace of the dominant eigenvector v_E, corresponding to the OeBF-feasible case.
  • Type I SDR: Under independently distributed user channels, the relevant null spaces span the entire space with probability one, forcing the optimal energy covariance to be zero in the considered case.

B. Proof of Proposition 3.2

The proof of Proposition 3.2 analyzes the SDR2 dual variables and complementary-slackness conditions to show rank-one information beams and a single energy-beam direction.

  • Dual Analysis: SDR2 is a convex SDP satisfying Slater’s condition, so its primal and dual formulations have zero duality gap.
  • Case Analysis: The proof focuses on the case where at least one dual SINR multiplier λ_i* is positive and separately considers β* = ξ_E and β* > ξ_E.
  • Case Analysis: For β* > ξ_E, G − β*I is full rank, which constrains the optimal energy covariance through the complementary-slackness relation.
  • Rank Properties: The resulting SDR2 solution has rank-one information covariance for every ID receiver and an energy covariance of the form q*v_Ev_E^H.

C. Proof of Proposition 3.3

The proof establishes equivalence between the original Type I problem and its uplink-downlink-duality formulation by combining tight SDR, strong duality, and convex reformulation.

  • Value Comparison: Proposition 3.3 begins by comparing the primal and dual optimal values of (P1.1) and (P1.2).
  • Value Comparison: Because SDR1.1 has a rank-one solution, its optimal value equals that of the original problem (P1.1).
  • Strong Duality: Strong duality between SDR1.1 and SDR1.2 transfers the optimal value from the primal SDR to its dual formulation.
  • Value Comparison: The SDR relaxation of the auxiliary problem yields f_SDR,1(β) ≥ f_1(β) for every β ≥ 0, producing the required value inequality.
  • Convex Reformulation: The two compared formulations achieve the same optimum because their SINR constraints can be represented as a convex feasible set and optimized at tight constraints.
  • Proof Distinction: The proof differs from prior work by handling βI − G ≺ 0 through convex reformulation rather than solely standard-interference-function arguments.

F. Proof of Proposition 4.3

The proof establishes feasibility and convergence properties for the fixed-point solution associated with Proposition 4.3. It also identifies when the dual problem is bounded and why standard interference-function arguments do not apply directly.

  • Feasibility: The optimal uplink powers satisfy the constraints of problem (31), establishing feasibility when g(β) > −∞.The argument uses equivalence between problems (34) and (35) and strong duality.
  • Convergence: The fixed-point iteration generates an element-wise monotonically decreasing sequence bounded below by the optimal uplink powers.This establishes convergence to a stationary point for problem (13).
  • Global optimality: Because problem (13) is identical to the convex reformulation (34), every stationary point is globally optimal.Therefore, the converged fixed-point solution is the optimal solution.
  • Technical limitation: Standard and generalized interference-function techniques cannot prove the convergence result because the equivalent noise term can be negative for some ID receivers.The same difficulty also arises in establishing the associated uplink-downlink duality result.
  • Failure case: If g(β) = −∞, no fixed point exists, and the iteration drives at least one uplink power sufficiently small.This is the infeasible or unbounded case addressed in the proposition.
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