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Multi-Antenna Wireless Powered Communication with Energy Beamforming

Liang Liu, Rui Zhang, Kee-Chaing Chua

arXiv:1312.1450v3cs.IT

TL;DR

The paper addresses the previously unstudied joint design of downlink energy transfer and uplink information transmission in WPCNs. It develops a two-stage optimal approach whose fixed-time subproblem is transformed into spectral-radius minimization and compares lower-complexity designs with the optimal solution.

  • Problem

    The paper studies the unaddressed joint design of DL energy transfer and UL information transmission in WPCNs.

  • Method

    The approach extends WPCNs to a multi-antenna AP and solves the joint design through spectral-radius minimization with alternating optimization, followed by optimization over the remaining design variables.

  • Results

    Algorithm I converges to the globally optimal solution to problem (16), and lower-complexity beamforming designs are compared with the optimal solution.

  • Takeaways & Limitations

    The paper provides an optimal two-stage solution for jointly designing time allocation, energy beamforming, uplink power allocation, and receive beamforming under throughput fairness.

Abstract

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The newly emerging wireless powered communication networks (WPCNs) have recently drawn significant attention, where radio signals are used to power wireless terminals for information transmission. In this paper, we study a WPCN where one multi-antenna access point (AP) coordinates energy transfer and information transfer to/from a set of single-antenna users. A harvest-then-transmit protocol is assumed where the AP first broadcasts wireless power to all users via energy beamforming in the downlink (DL), and then the users send their independent information to the AP simultaneously in the uplink (UL) using their harvested energy. To optimize the users' throughput and yet guarantee their rate fairness, we maximize the minimum throughput among all users by a joint design of the DL-UL time allocation, the DL energy beamforming, and the UL transmit power allocation plus receive beamforming. We solve this non-convex problem optimally by two steps. First, we fix the DL-UL time allocation and obtain the optimal DL energy beamforming, UL power allocation and receive beamforming to maximize the minimum signal-to-interference-plus-noise ratio (SINR) of all users. This problem is shown to be in general non-convex; however, we convert it equivalently to a spectral radius minimization problem, which can be solved efficiently by applying the alternating optimization based on the non-negative matrix theory. Then, the optimal time allocation is found by a one-dimension search to maximize the minimum rate of all users. Furthermore, two suboptimal designs of lower complexity are proposed, and their throughput performance is compared against that of the optimal solution.

I. INTRODUCTION

This paper extends wireless powered communication to a multi-antenna AP that jointly coordinates downlink energy transfer and uplink information transmission while maximizing minimum user throughput.

  • Prior WPCN studies characterized DL information–energy trade-offs but did not jointly design DL energy transfer and UL information transmission.
  • The proposed multi-antenna AP controls users’ DL harvested energy through energy beamforming and supports simultaneous UL transmission via SDMA.These capabilities provide higher spectrum efficiency than orthogonal TDMA user transmissions.
  • The design maximizes the minimum UL throughput through joint DL–UL time allocation, DL energy beamforming, UL power allocation, and receive beamforming.
  • For fixed time allocation, the non-convex optimization is converted to spectral radius minimization and solved globally using alternating optimization based on non-negative matrix theory.
  • Two lower-complexity suboptimal designs using zero-forcing receive beamforming are proposed and compared with the optimal solution.

II. SYSTEM MODEL

The system is a multi-antenna WPCN in which users harvest downlink energy and simultaneously transmit uplink information using SDMA during each block.

  • The AP has M > 1 antennas, each user has one antenna, and users lack conventional energy supplies but have energy storage devices.
  • Under harvest-then-transmit, the AP broadcasts energy for τT, followed by simultaneous user UL information transmission for (1 − τ)T.
  • The AP may transmit up to l ≤ M energy beams subject to its DL sum-power constraint, and all beams’ energy can be harvested by each user.
  • Each user’s harvested energy determines its available UL transmit power, while the AP uses linear receivers to decode simultaneous UL signals amid interference and noise.
  • With multiple AP antennas, DL energy beams can schedule users’ UL transmit power, unlike the single-antenna-AP SISO setting.
  • The DL–UL time allocation has a non-trivial trade-off: more DL time increases transmit power but reduces UL information-transmission time.

III. PROBLEM FORMULATION

The paper formulates max-min throughput as a joint optimization over time allocation, DL energy beams, UL powers, and receive beamformers, recognizing the resulting non-convexity.

  • The objective is to maximize the minimum throughput of all users in each block by jointly optimizing τ, DL energy beams V, UL powers p, and receive beamformers W.
  • The number of DL energy beams l is itself a design variable and can be set to the number of columns in the obtained beamforming matrix V.
  • The formulation is non-convex because design variables are coupled in the objective and in the UL transmit-power constraints.
  • Fixing τ and V reduces the formulation to an UL SINR-balancing problem with individual user power constraints.
  • The paper proposes both optimal and suboptimal algorithms for solving the formulated problem.

IV. OPTIMAL SOLUTION

The optimal solution uses a two-step procedure: optimize the fixed-time allocation design, then search over the downlink–uplink time split. The fixed-time problem is transformed into spectral-radius minimization and solved through alternating optimization, yielding a globally optimal solution.

  • Two-step optimal solution: Fixing the DL-UL time allocation reduces the problem to jointly optimizing DL energy beamforming, UL power allocation, and receive beamforming.The resulting formulation introduces a common SINR requirement for all users.
  • Time allocation: The optimal DL-UL time allocation is found by a one-dimensional search over 0 < τ̄ < 1 after solving the fixed-time problem.This completes the two-step procedure for the original throughput optimization problem.
  • Fixed-time optimization: The fixed-time SINR-balancing problem remains generally non-convex, even when the DL energy beamforming is fixed.Therefore, convex optimization techniques cannot generally obtain its optimal solution.
  • Spectral-radius formulation: The SINR-balancing problem is equivalently formulated as spectral-radius minimization using non-negative matrix theory.For given feasible beamformers, the optimal power solution is obtained from the dominant eigenvector of the relevant non-negative matrix.
  • Alternating optimization: Alternating optimization iteratively updates DL energy beamforming and UL receive beamforming until convergence.The DL subproblem can be transformed into a convex optimization problem and its beamforming solution recovered from the optimal covariance matrix.
  • Optimality and power control: Algorithm I converges globally to the optimal solution, although individual UL power constraints need not all be tight.The latter observation indicates that UL power control is generally needed rather than always transmitting at each user’s maximum available power.

V. SUBOPTIMAL DESIGN

The suboptimal designs assume K ≤ M, enabling ZF-based uplink receivers that eliminate inter-user interference and simplify joint optimization. They replace the optimal MMSE-based design with lower-complexity procedures for τ, p, and V.

  • V. SUBOPTIMAL DESIGN: The proposed suboptimal solutions assume K ≤ M, allowing the AP to use ZF-based receivers instead of MMSE-based receivers.ZF completely eliminates inter-user interference under this assumption.
  • V. SUBOPTIMAL DESIGN: ZF receiver vectors lie in the null space of all users’ channels except the intended user.The null-space basis is obtained from the SVD of H−k, and wk is parameterized within that basis.
  • V. SUBOPTIMAL DESIGN: Each ZF beamformer is selected within its null space and aligned with the intended user’s equivalent channel to maximize that user’s rate.This produces an equivalent uplink channel whose power determines the achievable throughput.
  • V. SUBOPTIMAL DESIGN: The ZF-based throughput expression supports two lower-complexity suboptimal solutions for jointly obtaining τ, p, and V.The two designs are developed in the following subsections after fixing the ZF receive beamformers.

A. Suboptimal Solution 1

Suboptimal Solution 1 transforms the ZF-based power-allocation problem by introducing a common throughput requirement. The resulting equivalent problem is convex and can be solved efficiently.

  • A. Suboptimal Solution 1: Introducing a common throughput requirement R̄ transforms problem (23) into an equivalent power-allocation problem.The transformed variables include ˜p = {˜p1, · · ·, ˜pK}.
  • A. Suboptimal Solution 1: Problem (24) is convex and can therefore be solved efficiently using an interior-point method.Its optimal solution determines the corresponding power-allocation solution to problem (23).
  • A. Suboptimal Solution 1: The remaining beamforming quantities are obtained from the eigenvalue decomposition of ˜S.The passage identifies EVD as the procedure for obtaining the relevant beamforming terms.

B. Suboptimal Solution 2

Suboptimal Solution 2 separates downlink energy-beamforming optimization from uplink power allocation. It uses weighted energy maximization and then solves the resulting time and power allocation efficiently.

  • B. Suboptimal Solution 2: The second design separates downlink energy beamforming from uplink power allocation to reduce optimization complexity.It first obtains the downlink energy beams and then determines τ and p.
  • B. Suboptimal Solution 2: Energy weights assign greater emphasis to weaker-channel users to support rate fairness and account for the doubly near-far effect.The proposed rule is αk = 1/(˜hk∥gk∥2).
  • B. Suboptimal Solution 2: For a specified set of energy weights, the maximum harvested energy is Emax = ψPsum and is achieved using one energy beam.The eigenvalue-based construction supplies the beamforming direction.
  • B. Suboptimal Solution 2: With ZF receivers, all users transmit at full uplink power because they cause no interference to one another.This removes the variable ˜p from the subsequent equivalent optimization problem.
  • B. Suboptimal Solution 2: The resulting time-allocation problem is convex and can be solved by an interior-point method or bisection over the common throughput.For a fixed R̄, the optimal τ follows from a convex feasibility problem.

VI. NUMERICAL RESULTS

The numerical study evaluates a multi-antenna WPCN with six AP antennas and four users under specified power, efficiency, noise, propagation-loss, reciprocity, and Rician-fading assumptions.

  • VI. NUMERICAL RESULTS: The simulation considers an AP with M = 6 antennas serving K = 4 users.The total AP transmit power is set to Psum = 1 Watt (30 dBm).
  • VI. NUMERICAL RESULTS: The assumed energy-conversion efficiency is ϵ = 50%, and the receiver-noise power is σ2 = −50 dBm.These values define the power and noise settings used in the numerical examples.
  • VI. NUMERICAL RESULTS: The simulations assume channel reciprocity and independently generated Rician-fading channel vectors with Rician factor KR = 3.The uplink and downlink channels satisfy hk = gk.
  • VI. NUMERICAL RESULTS: The line-of-sight component uses a far-field uniform linear-array model with antenna spacing dan = λ and user directions −45°, −15°, 15°, and 45°.The average power of each gk is normalized by Lk in (28).

A. Optimal Solution

The optimal design studies how time allocation, receiver choice, initialization, uplink power control, and AP antenna count affect max-min throughput. The results show an interior time-allocation trade-off, rapid algorithm convergence, and substantial antenna gains.

  • Time allocation: Both RMMSE(¯τ) and RZF(¯τ) first increase and then decrease as ¯τ varies.Increasing DL energy-transfer time initially raises harvested power, but eventually reduces UL transmission time.
  • Receiver comparison: MMSE receivers achieve higher throughput than ZF receivers for any given ¯τ.
  • Convergence: Algorithm I converges to the optimal solution in only 4–5 iterations from both tested initial points.The initialization using αk = 1/∥hk∥2∥gk∥2 converges faster than αk = 1.
  • Uplink power control: U1, U2, and U3 should not transmit at maximum power under the optimal DL energy beams.Thus, UL power control is needed to maximize the minimum SINR.
  • Antenna scaling: Max-min throughput increases significantly with the number of active antennas at the AP.The one-antenna case uses the TDMA-based SISO WPCN solution because spatial transmit and receive beamforming cannot be used.

B. Suboptimal Solution

The suboptimal designs reduce complexity through ZF-based receive beamforming, but their performance depends on channel conditions and how energy beams are generated. MMSE remains stronger across the tested distance range.

  • Distance impact: Throughput decays drastically as user distance d increases for all optimal and suboptimal solutions.
  • Performance comparison: MMSE throughput outperforms all three ZF-based suboptimal solutions for every tested d.When d is small, both ZF suboptimal solutions achieve throughput very close to the optimal MMSE solution.
  • Receiver behavior: ZF receiver is asymptotically optimal at high SNR when transmission is large for all users.
  • Suboptimal designs: Suboptimal Solution 2 performs very close to Suboptimal Solution 1 with ZF reception.The two designs use separate optimizations of DL energy beamforming and UL power allocation to reduce complexity.
  • Energy-beam design: Randomly generated energy beams cause a significant loss in achieved max-min throughput with ZF reception.

VII. CONCLUSION

The paper maximizes the minimum throughput in a multi-antenna WPCN through joint DL-UL design. It solves the resulting problem optimally in two stages and also develops lower-complexity ZF-based alternatives.

  • Conclusion: The minimum throughput among single-antenna users is maximized jointly over DL-UL time, DL energy beamforming, UL power allocation, and receive beamforming.
  • Optimal algorithm: The optimal solution uses a two-stage algorithm combining alternating optimization, non-negative matrix theory, and one-dimensional time search.
  • Suboptimal designs: Two lower-complexity suboptimal solutions are proposed with ZF-based receive beamforming and compared against the optimal solution.

A. Proof of Theorem 4.1

For fixed receive beamforming and energy beams, the proof characterizes the unique optimal power allocation and SINR balance through nonnegative-matrix spectral-radius properties. It establishes that all users share the balancing SINR, at least one individual power constraint is tight, and the resulting eigenvector construction is optimal.

  • Optimality conditions: All users achieve the same SINR balancing value, and at least one user reaches its individual power constraint.These are the two key structural properties established for the optimizer.
  • Unique power and SINR solution: Given fixed receive beamforming and energy beams, the optimal power allocation and SINR balancing solution satisfy a unique system of matrix equations.The proof extends the uniqueness result to individual user power constraints.
  • Spectral-radius characterization: The inverse optimal SINR balance equals the spectral radius of the matrix A_k∗(W̄,V̄).This follows by identifying the optimal extended power vector with the dominant eigenvector of A_k∗(W̄,V̄).
  • Spectral-radius characterization: For every candidate index k, the balancing SINR is at least ρ(A_k(W̄,V̄)); combining this bound with the active constraint identifies the minimizing spectral-radius formulation.The proof uses the nonnegative-matrix inequality and the index k∗ associated with a tight power constraint.
  • Optimality conclusion: The dominant-eigenvector power allocation is optimal for problem (13), and the resulting beamforming pair achieves the optimal value of problem (16).Theorem 4.1 supplies the power solution for fixed beamformers; the subsequent argument establishes optimality of the alternating beamforming solution.
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