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Energy Beamforming with One-Bit Feedback

Jie Xu, Rui Zhang

arXiv:1312.1444v3cs.IT

TL;DR

MIMO WET needs ET-side CSI for effective energy beamforming, but communication-style training and feedback may not fit hardware-limited ERs. The paper uses ACCPM with one harvested-energy comparison bit per ER and adjusts training beams to estimate multiuser MIMO channels simultaneously. Analysis and simulations report favorable convergence and energy-transfer performance, especially as the number of ERs grows, while multi-bit feedback remains an open extension.

  • Problem

    MIMO energy beamforming requires CSI at the ET, but existing communication channel-training and feedback methods may be impractical for WET receivers with limited hardware.

  • Method

    The paper applies ACCPM to one-bit-feedback channel learning, using each ER’s harvested-energy increase or decrease to guide successive ET training-beam adjustments.

  • Results

    Analysis and simulations show that ACCPM can estimate multiuser MIMO channels simultaneously and is more appealing than existing one-bit-feedback methods, especially with many ERs.

  • Takeaways & Limitations

    One-bit harvested-energy feedback can support practical simultaneous channel learning for multiuser MIMO WET without changing existing ER harvesting circuits.

Abstract

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Wireless energy transfer (WET) has attracted significant attention recently for providing energy supplies wirelessly to electrical devices without the need of wires or cables. Among different types of WET techniques, the radio frequency (RF) signal enabled far-field WET is most practically appealing to power energy constrained wireless networks in a broadcast manner. To overcome the significant path loss over wireless channels, multi-antenna or multiple-input multiple-output (MIMO) techniques have been proposed to enhance the transmission efficiency and distance for RF-based WET. However, in order to reap the large energy beamforming gain in MIMO WET, acquiring the channel state information (CSI) at the energy transmitter (ET) is an essential task. This task is particularly challenging for WET systems, since existing channel training and feedback methods used for communication receivers may not be implementable at the energy receiver (ER) due to its hardware limitation. To tackle this problem, in this paper we consider a multiuser MIMO system for WET, where a multiple-antenna ET broadcasts wireless energy to a group of multiple-antenna ERs concurrently via transmit energy beamforming. By taking into account the practical energy harvesting circuits at the ER, we propose a new channel learning method that requires only one feedback bit from each ER to the ET per feedback interval. The feedback bit indicates the increase or decrease of the harvested energy by each ER between the present and previous intervals, which can be measured without changing the existing hardware at the ER. Based on such feedback information, the ET adjusts transmit beamforming in different training intervals and at the same time obtains improved estimates of the MIMO channels to ERs by applying a new approach termed analytic center cutting plane method (ACCPM).

I. INTRODUCTION

RF far-field WET can power multiple energy-constrained devices over longer distances, but MIMO energy beamforming requires CSI that conventional communication-oriented training may not provide at hardware-limited energy receivers. The paper proposes ACCPM-based channel learning using one harvested-energy comparison bit per receiver and reports simultaneous multiuser channel estimation with favorable convergence and energy-transfer performance.

  • Motivation: RF far-field WET supports longer-range, broadcast energy delivery and can charge multiple moving devices.Inductive and magnetic-resonant coupling are efficient at shorter ranges but are difficult to use for freely located devices simultaneously.
  • Motivation: MIMO improves WET by focusing transmitted energy through ET beamforming and increasing each ER’s effective aperture area.For point-to-point MIMO WET, one energy beam is optimal for maximizing energy-transfer efficiency, unlike spatial multiplexing in communications.
  • Challenge: CSI acquisition is essential for MIMO energy beamforming but conventional communication channel-learning methods may be unsuitable for hardware-limited ERs.Reverse-link training can depend on TDD reciprocity and consume harvested ER energy, while practical ER circuits may lack baseband channel-estimation processing.
  • Approach: The proposed protocol separates channel learning from energy transmission and uses one feedback bit per ER to indicate whether harvested energy increased or decreased.The comparison can be measured without changing existing harvesting circuits, while the ET adjusts training signals and estimates multiple MIMO channels.
  • Evaluation: Simulations compare ACCPM with CJT, gradient-sign, and distributed-beamforming methods in convergence speed and energy-transfer efficiency.The paper reports greater ACCPM gains in the multiuser setting and concludes that it is especially appealing when the number of ERs is large.

III. CHANNEL LEARNING WITH ONE-BIT FEEDBACK: SINGLE-USER CASE

For the single-user MIMO WET case, the paper formulates channel learning as estimating the normalized channel from one-bit feedback during training. It bases the algorithm on ACCPM and presents convergence analysis.

  • Single-user setup: The single-user case sets K = 1 and removes the user index from the harvested energy, feedback bit, channel gain, and normalized channel matrix.The resulting quantities are denoted Q^L_n, f_n, γ, and G for training intervals n ∈ {1, . . . , N_L}.
  • Algorithm: The proposed single-user algorithm estimates the normalized MIMO channel G from one-bit ER feedback over channel-learning training intervals.The section introduces ACCPM before presenting the algorithm and its convergence analysis.
  • Algorithm: The single-user channel-learning design is explicitly based on ACCPM in convex optimization.The stated workflow is to introduce ACCPM, present the one-bit-feedback algorithm, and analyze its convergence.

A. Introduction of ACCPM

ACCPM localizes a convex target set by iteratively querying an oracle and intersecting the current working set with returned half spaces. Here, one-bit feedback supplies cutting planes for estimating scaled channel matrices, while analytic centers generate subsequent query points.

  • ACCPM basics: ACCPM starts from a convex working set containing the target set and updates it by intersecting with oracle-returned half spaces.The nested working sets remain supersets of the target and approach it as iterations increase.
  • ACCPM basics: Neutral, deep, and shallow cutting planes are classified by the query point’s position relative to the cutting hyperplane; ACCPM requires a deep or neutral cut.A neutral cut contains the query point, whereas a deep cut places it in the interior of the retained half space.
  • Channel-learning formulation: The channel-learning target set contains all positively scaled normalized channel matrices satisfying 0 ⪯ Ḡ ⪯ I.This known semidefinite bound supplies the initial working set.
  • Channel-learning formulation: Each one-bit feedback value acts as an ACCPM oracle, producing a half-space constraint and an updated working set for the channel estimate.The analytic center of the updated set becomes the query point for the next feedback interval.
  • Probing design: For complex Hermitian channels, cvec maps matrices to real vectors while preserving the trace inner product, enabling probing matrices orthogonal to the current query.The resulting probing matrix and covariance are selected to obtain a neutral cutting plane; the paper uses ∥p∥ = P/10 in the construction.
  • Algorithm output: The single-user algorithm iterates through feedback intervals, computes analytic centers, and uses the normalized final center as the channel estimate and dominant eigenvector for energy beamforming.The covariance update may require random trials to preserve positive semidefiniteness.

C. Convergence Analysis: Single-User Case

The single-user convergence analysis gives a worst-case feedback-interval bound for ACCPM channel estimation. The bound depends on transmit-side dimension, while practical convergence may require fewer intervals.

  • Convergence bound: Under the stated target-set accuracy condition, ACCPM query points converge to a point in the target set after a bounded number of iterations.The proof relies on the number of neutral cutting planes required for semidefinite feasibility.
  • Convergence bound: The normalized channel estimate satisfies ∥G̃ − G∥F ≤ ε in at most O(...) feedback intervals.The supplied passage states the bound but does not expose its complete expression.
  • Dimension dependence: The analytic convergence speed depends on the number of transmit antennas, MT, but not on the number of receive antennas, MR.This follows because the learned composite channel matrix G = H^H H has size MT × MT.
  • Practical interpretation: The stated feedback-interval result is a worst-case upper bound, whereas numerical results are expected to use substantially fewer intervals in practice.The practical claim is framed as an observation to be shown later in the paper.

IV. CHANNEL LEARNING WITH ONE-BIT FEEDBACK: MULTIUSER CASE

The multiuser extension applies the ACCPM-based one-bit channel-learning framework to learn normalized channel matrices for multiple energy receivers and analyzes its convergence.

  • Multiuser extension: The paper extends the single-user ACCPM channel-learning algorithm to a general multiuser MIMO wireless-energy-transfer system with K > 1 energy receivers.The extension includes both the algorithm and convergence analysis.

A. ACCPM Based Multiuser Channel Learning

The multiuser method learns one normalized channel matrix per energy receiver using one-bit feedback and ACCPM cutting planes. When all receivers cannot be handled simultaneously, users and feedback intervals are partitioned into subsets.

  • Multiuser ACCPM formulation: The multiuser formulation learns the K normalized channel matrices G1, . . ., GK using feedback from the energy receivers.Each receiver has its own target set, working set, and query point.
  • Multiuser ACCPM formulation: At each feedback interval, the transmit covariance is designed so the feedback-derived cutting plane is neutral for the relevant receivers.The analytic center of each updated working set supplies the next query point.
  • User-group constraint: If K exceeds the available real degrees of freedom, satisfying all K neutrality equations simultaneously becomes infeasible.The covariance must satisfy K equations in M_T^2 real unknowns.
  • User-group constraint: To address this constraint, the receivers are grouped into subsets containing no more than M_T^2 − 1 users, and the covariance is constrained for one subset at a time.Each subset therefore receives cutting-plane updates during its assigned intervals.
  • Feedback partitioning: The feedback intervals are partitioned into corresponding subsets, during which only the assigned receivers provide one-bit feedback and receive cutting planes.The transmission protocol in Fig. 3 illustrates this interval partition.
  • Scope: The paper leaves optimal receiver grouping and feedback-interval partitioning for best channel-learning performance beyond its scope.This is an explicit scope limitation of the multiuser design.

B. Convergence Analysis: Multiuser Case

The multiuser ACCPM method learns all ER channels by partitioning users into subsets and applying simultaneous neutral cutting planes, with a worst-case convergence guarantee. When K ≤ M_T^2−1, multiple ER channels can be learned without reducing analytic convergence speed.

  • Multiuser convergence: The multiuser algorithm estimates all K MIMO channels with Frobenius-norm error at most ε within a bounded number of feedback intervals.The guarantee applies simultaneously to every ER channel.
  • User partitioning: ERs are partitioned into subsets, each containing no more than M_T^2−1 users, so neutral cutting planes can be generated simultaneously within a subset.At each feedback interval, each ER in a subset contributes one neutral cutting plane.
  • Multiuser convergence: Proposition 4.1 gives the worst-case convergence performance for arbitrary numbers of transmit antennas, receive antennas, and ERs.The bound is stated for arbitrary M_T, M_R, and K.
  • Multiuser convergence: When K ≤ M_T^2−1, the method learns more than one ER’s MIMO channel simultaneously without reducing its analytic convergence speed.This follows directly from the multiuser convergence proposition.

V. NUMERICAL RESULTS

Simulations evaluate ACCPM-based one-bit channel learning in single-user and multiuser RF-WET settings under specified Rician and array assumptions. ACCPM converges quickly and achieves higher harvested power than benchmark methods, although training length creates a harvesting tradeoff.

  • Simulation setup: The simulations use six ERs at 5 meters, 40 dB average path loss, dominant LOS propagation, and a 5 dB Rician factor.The ET transmits 1 W and each ER has 50% harvesting efficiency.
  • Single-user results: The proposed algorithm’s normalized estimation and harvested-power errors decrease exponentially with feedback intervals, indicating fast convergence.The figures vary the number of transmit antennas while fixing M_R = 2, and also examine receive-antenna effects.
  • Single-user results: ACCPM converges at nearly the same speed for different numbers of receive antennas, consistent with convergence speed being independent of M_R.This result is reported for fixed M_T = 6.
  • Harvested-power results: With N = 200, ACCPM’s average harvested power peaks at N_L = 22, whereas CJT peaks at N_L = 42.For ACCPM and CJT, increasing training length eventually reduces harvested power because channel-learning time competes with energy-transmission time.
  • Harvested-power results: ACCPM achieves higher average harvested power than CJT, gradient sign, and distributed beamforming as block length increases toward the perfect-CSI upper bound.The reported trend reflects more accurate channel estimation with a smaller training-time percentage.

B. Multiuser Setup

The multiuser setup evaluates ACCPM-based channel learning for several ERs and compares its harvested-power performance with existing one-bit-feedback algorithms.

  • Multiuser convergence: The multiuser algorithm estimates multiple ER channels simultaneously, with convergence speed analytically preserved when K ≤ M_T^2 − 1.The result is a worst-case guarantee and does not preclude slower observed convergence as K increases.
  • Multiuser convergence: As K increases, the estimated-channel and weighted-sum harvested-power errors converge more slowly in simulations.This empirical trend is explicitly distinguished from the worst-case convergence result.
  • Simulation results: After 60 feedback intervals, at least 99% of the maximum weighted-sum power is achieved for every tested K.The result is reported for the weighted-sum harvested-power metric in Fig. 12.
  • Simulation results: With M_T = 4, M_R = 2, and K = 6, ACCPM’s performance advantage over three benchmarks becomes more substantial as N increases.The compared metric is weighted sum of average harvested power per block, Q_total/T.
  • Simulation results: The advantage is attributed to ACCPM estimating more than one ER’s MIMO channel simultaneously, whereas the benchmarks learn one ER eigenvector or dominant eigenmode at a time.The paper concludes that ACCPM is more appealing for MIMO WET with multiple ERs.
  • Design scope: One-bit feedback simplifies feedback design and receiver complexity while minimizing feedback-communication energy, but richer feedback remains an open extension.The paper states that more than one feedback bit per interval should further improve channel-learning performance.

APPENDIX

The appendix reformulates the complex ACCPM channel-learning analysis in real-valued matrix form and establishes convergence through analytic-center and potential-function arguments.

  • Real counterpart: A real counterpart maps the complex MIMO channel matrix G into a real matrix and seeks a feasible point in the corresponding target set.The mapping separates real and imaginary parts and supports a real-valued convergence proof.
  • Real counterpart: Each iteration adds a cutting-plane half-space to the working set, which is initialized by the semidefinite constraints 0 ⪯ Ḡ ⪯ I.The analytic center of the updated working set becomes the next query point.
  • Convergence proof: The analytic center is obtained by solving a real-matrix optimization problem, and the proof uses potential functions associated with the working sets.The argument relates the real counterpart’s analytic centers to those of the original complex algorithm.
  • Convergence proof: Once the iteration index satisfies the stated inequality, the ACCPM estimates converge to a point in the target set X.The appendix completes this implication using the relationship between the complex and real algorithms.
  • Convergence proof: The proof shows that contradictory upper and lower potential-function bounds force the analytic centers into the target set.This implication is transferred from the real counterpart to the original ACCPM algorithm.
  • Technical lemmas: The supporting lemmas rely on properties of matrix mappings, second-order derivatives, and the svec isometry for real symmetric matrices.Several lemma proofs are omitted because they follow directly from prior results or simple matrix manipulations.

B. Proof of Proposition 3.2

The proof of Proposition 3.2 connects target-set accuracy to channel-matrix accuracy and derives the feedback-interval complexity needed for ACCPM convergence.

  • Accuracy transfer: If the estimate at N_L lies within ε/8 of a scaled channel matrix βG in Frobenius norm, then the channel estimate is within ε of G.The proof introduces the Frobenius error of the estimate and uses this implication to establish the proposition.
  • Feedback complexity: The proposed algorithm therefore converges to an estimate satisfying the required channel accuracy after a bounded number of feedback intervals.This conclusion follows by combining the accuracy implication with Proposition 3.1’s convergence result.
  • Feedback complexity: The required interval count is expressed through an ACCPM convergence bound involving M_T, T, ε, and logarithmic terms.The displayed bound is summarized in the paper by ignoring lower-order terms after combining the preceding inequalities.
  • Conclusion: The proposition is completed by combining the two accuracy arguments and the resulting interval bound.The proof explicitly states that these combined arguments establish Proposition 3.2.

C. Other Algorithms for Comparison

The comparison section adapts CJT, gradient sign, and distributed beamforming to multiuser MIMO WET, highlighting their sequential single-channel learning structure and implementation trade-offs.

  • Benchmark setup: The three benchmark algorithms were originally designed to learn one MIMO or MISO channel and are extended here to K ≥ 1 ERs.The benchmarks come from application scenarios other than WET.
  • CJT: CJT estimates each ER’s eigenvector matrix through channel-learning slots that use one-bit feedback to update training signals.The estimate Ṽ_k is obtained at the end of the kth slot.
  • CJT: CJT performs successive two-dimensional unitary rotations that eliminate paired off-diagonal entries of the channel matrix.A complete sweep contains M_T(M_T − 1)/2 rotations, after which the eigenvector estimate is improved.
  • Energy beamforming: Because the benchmarks estimate eigenvectors but not eigenvalues, their energy beamforming design uses the dominant eigenvector of an aggregate estimate as the beamforming vector.This design is suboptimal in general but becomes optimal when K = 1 or M_R = 1 as N_L approaches infinity.
  • CJT: CJT accuracy depends on the number of sweeps and line-search tolerance η, with greater accuracy requiring more feedback intervals.The paper identifies this as an explicit performance-versus-feedback trade-off.

2) Gradient Sign [17]:

The gradient sign method probes beamforming vectors with paired random perturbations and uses one-bit harvested-energy feedback to retain the better vector. Repeated updates approach each ER’s dominant channel eigenvector, which is then used for energy beamforming.

  • Gradient Sign: The ET defines reference beamforming vectors and probes each one by adding and then subtracting the same random perturbation.Odd intervals use an added perturbation, while the following even intervals use the corresponding subtraction.
  • Gradient Sign: Each ER’s one-bit feedback identifies which probed beam achieves higher transferred energy, enabling the ET to update its reference vector.The better beam becomes the reference for the next iteration.
  • Gradient Sign: Repeated gradient-sign updates approach the dominant eigenvector direction, yielding a finite-interval estimate for each ER.The estimate is denoted by ˜v_k,1.
  • Energy Transmission: During energy transmission, the ET applies the dominant eigenvector of the estimated aggregate matrix as its energy beamforming vector.The aggregate matrix is formed from the estimated dominant eigenvectors of the ERs.
  • Gradient Sign: The step size ξ trades convergence speed against estimation error: larger values accelerate convergence but increase estimation errors.ξ is the norm of the added or subtracted random perturbation vector.
  • Distributed Beamforming: For distributed beamforming, the ET randomly perturbs antenna phases and keeps an updated beam only when feedback indicates improved harvested energy.The phase perturbations are uniformly generated over [−χ/2, χ/2], with χ > 0 controlling the interval.
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