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Optimized Training Design for Wireless Energy Transfer

Yong Zeng, Rui Zhang

arXiv:1403.7870v3cs.IT

TL;DR

The paper tackles CSI acquisition for energy beamforming in MIMO WET when the ER has limited training resources. It proposes reciprocal reverse-link training and optimizes net harvested energy over Rician channels, deriving solutions for several special cases while identifying correlated fading as an open extension.

  • Problem

    CSI is needed for energy beamforming, but channel training consumes the ER's harvested energy and coherence time, creating a trade-off between estimation quality and usable energy transmission.

  • Method

    The paper uses channel reciprocity in a two-phase protocol and optimizes the trained ER-antenna subset, training time, and training power to maximize net harvested energy.

  • Results

    The paper formulates the general uncorrelated MIMO Rician training problem and derives optimal solutions for MIMO Rayleigh and MISO Rician channels, plus an approximate massive-MIMO solution.

  • Takeaways & Limitations

    Reverse-link reciprocity enables ET-side channel estimation while explicitly accounting for training costs, and the special-case analyses provide guidance on when training improves net transferred energy.

  • Takeaways & Limitations

    Correlated MIMO Rician fading remains an open case because orthogonal training may not be optimal and the training sequence structure must also be optimized.

Abstract

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Radio-frequency (RF) enabled wireless energy transfer (WET), as a promising solution to provide cost-effective and reliable power supplies for energy-constrained wireless networks, has drawn growing interests recently. To overcome the significant propagation loss over distance, employing multi-antennas at the energy transmitter (ET) to more efficiently direct wireless energy to desired energy receivers (ERs), termed \emph{energy beamforming}, is an essential technique for enabling WET. However, the achievable gain of energy beamforming crucially depends on the available channel state information (CSI) at the ET, which needs to be acquired practically. In this paper, we study the design of an efficient channel acquisition method for a point-to-point multiple-input multiple-output (MIMO) WET system by exploiting the channel reciprocity, i.e., the ET estimates the CSI via dedicated reverse-link training from the ER. Considering the limited energy availability at the ER, the training strategy should be carefully designed so that the channel can be estimated with sufficient accuracy, and yet without consuming excessive energy at the ER. To this end, we propose to maximize the \emph{net} harvested energy at the ER, which is the average harvested energy offset by that used for channel training. An optimization problem is formulated for the training design over MIMO Rician fading channels, including the subset of ER antennas to be trained, as well as the training time and power allocated. Closed-form solutions are obtained for some special scenarios, based on which useful insights are drawn on when training should be employed to improve the net transferred energy in MIMO WET systems.

I. INTRODUCTION

The paper addresses CSI acquisition for energy beamforming in point-to-point MIMO WET, where training must balance beamforming accuracy against the ER's time and energy costs. It uses channel reciprocity to formulate net-energy-maximizing training designs for Rician fading channels and derives solutions for several practical special cases.

  • I. INTRODUCTION: Multi-antenna energy beamforming can improve RF WET efficiency, but its benefit depends on CSI that must be acquired with additional time and energy costs.The paper motivates beamforming for overcoming propagation loss and notes that conventional feedback-based acquisition becomes problematic as the ET antenna count grows.
  • I. INTRODUCTION: Limited ER resources create a training trade-off: insufficient training reduces beamforming gain, whereas excessive training consumes harvested energy and shortens energy-transmission time.The proposed objective is net harvested energy: average harvested energy offset by the energy used for training.
  • I. INTRODUCTION: The proposed two-phase protocol uses reverse-link pilots for ET channel estimation, followed by forward-link energy transmission based on the estimated channel.Channel reciprocity makes the forward and reverse links usable for this training arrangement.
  • I. INTRODUCTION: For general uncorrelated MIMO Rician fading, the optimization jointly selects the trained ER-antenna subset, training duration, and training power to maximize net average harvested energy.The design explicitly accounts for the resources consumed during channel training.
  • I. INTRODUCTION: Closed-form or approximate designs are developed for MIMO Rayleigh, MISO Rician, and massive-MIMO Rician scenarios, including asymptotically optimal scaling for a rank-1 deterministic component.The special-case analyses are intended to provide practical insight when the general optimization is difficult.

A. Reverse-Link Channel Training

The ET estimates the instantaneous channel through reverse-link pilots sent by a selected subset of ER antennas, balancing measurement requirements against training cost.

  • Training-set selection: Training only a subset of ER antennas allows the design to avoid spending the entire coherence block on channel estimation.The trained set may contain between zero and all ER antennas; training all antennas can be strictly suboptimal when no time remains for energy transmission.
  • Pilot-based estimation: Orthogonal pilots are sent by the selected ER antennas, and the ET uses the received signals to form an MMSE channel estimate.The training observation contains the pilot duration, reverse-link power, orthogonal pilot matrix, and receiver noise.
  • CSI decomposition: The channel estimate separates the known CSI from estimation errors and unestimated random components associated with non-trained ER antennas.The estimated and error components are independent under the Gaussian MMSE model, enabling the channel decomposition used later for beamforming analysis.

B. Forward-Link Energy Transmission

After training, the ET beamforms energy using the available CSI, while the design evaluates harvested energy net of the ER’s pilot-training expenditure.

  • Forward transmission: The ET transmits energy-bearing signals during the remaining T −τ symbols using the estimated channel, rather than the unavailable true channel.The transmit covariance is optimized conditionally on the estimated CSI.
  • Beamforming rule: Because the CSI discrepancy is isotropic, the estimated channel can be treated as the true channel for maximum energy transfer.The resulting transmit covariance aligns energy transmission with the available channel estimate.
  • Net-energy objective: The optimization maximizes net average harvested energy by subtracting ER training energy from average harvested energy.The objective depends jointly on the trained-antenna subset, training duration, and reverse-link training power.
  • Energy assumption: The formulation assumes sufficient initial ER energy and constrains training so the net-energy objective remains strictly positive.Under this assumption, the ER can send pilots with the selected training power throughout the training phase.

IV. OPTIMAL TRAINING DESIGN

The general training-design problem is difficult because it combines antenna-subset selection with continuous training-time and power decisions, so the paper derives tractable special-case solutions and conditions for when training pays off.

  • Problem structure: The general problem is challenging because antenna selection requires up to 2^N possibilities, while the continuous optimization lacks a general closed form.The paper therefore studies MIMO Rayleigh, MISO Rician, and massive-MIMO Rician special cases.
  • MIMO Rayleigh fading channel: Training increases harvested energy through beamforming on trained channels, whereas non-trained channels receive no ET-antenna beamforming gain.The harvested-energy expression separates contributions from trained and non-trained ER antennas.
  • MIMO Rayleigh fading channel: For Rayleigh fading, the optimal training duration equals the number of trained ER antennas, reducing the optimization to antenna count and training power.The trained subset matters only through its cardinality because all ER antennas are statistically equivalent without a deterministic LOS component.
  • MIMO Rayleigh fading channel: In MIMO Rayleigh fading, training helps only when coherence time, ET antenna count, and two-way ESNR are sufficiently large.Otherwise, pilot costs outweigh beamforming gains and the optimal policy is no training with isotropic energy transmission.

B. MISO Rician Fading Channel

In MISO Rician fading, the training decision is binary: either train the single ER antenna or rely on the deterministic LOS component, with the preferred choice determined by net harvested energy.

  • Problem setup: The MISO formulation has only two training-set choices, N1 = 0 or N1 = 1, because the ER has one antenna.The channel matrix becomes a 1 × M vector, simplifying the subset decision to whether pilots are sent.
  • No-training case: Without training, the optimum is τ = 0 and Pr = 0, and energy transmission uses only the channel’s deterministic LOS component.The unestimated random component is isotropic, so it cannot provide beamforming gain at the ET.
  • Training case: The trained case has an explicit net-energy expression and an optimal training solution parameterized by the two-way ESNR Γ.The final binary decision compares the net energies for training and no training.
  • Design condition: Channel training is selected only when T, M, and Γ are sufficiently large while K is sufficiently small; otherwise LOS-based beamforming is optimal.Setting K = 0 recovers the corresponding MISO Rayleigh solution.

C. Massive MIMO Rician Fading Channel

For massive MIMO Rician WET, the paper approximates training design through a lower bound and derives antenna-ordering and training-selection rules. Training helps when the coherence block and training-energy conditions are sufficiently favorable, while optimized training matches ideal beamforming asymptotically under a rank-1 deterministic component.

  • Approximate optimization: The approximate massive-MIMO design maximizes a lower bound on net harvested energy while jointly selecting the trained-antenna subset, its ordering, and training power.The lower bound depends on both the permutation matrix and the number of trained ER antennas.
  • Approximate optimization: The optimal antenna ordering places the absolute values of the deterministic-channel vector in non-decreasing order for the training design.The resulting permutation determines the trained subset after reordering the ER antennas.
  • Approximate optimization: For fixed trained-antenna count, the optimal training power is obtained from a convex problem, and the optimal training duration equals that count.
  • Training conditions: For massive MIMO Rician channels, training helps when T and Γ are sufficiently large and the Rician factor K is sufficiently small.
  • Rank-1 deterministic channel: The rank-1 asymptotic result assumes a line-of-sight deterministic component and sufficiently large T satisfying T > KN^2 + 2N.
  • Rank-1 deterministic channel: Optimized training achieves the same asymptotic harvested-energy scaling as ideal perfect-CSI beamforming when the rank-1 channel component has sufficiently large block length T ≫ N.The result applies to MIMO Rician WET with a rank-1 deterministic channel component.

V. NUMERICAL RESULTS

The numerical section evaluates the proposed training design under specified transmit-power, attenuation, noise, efficiency, and array assumptions. It includes Rayleigh and Rician WET settings used to validate the analytical results.

  • Simulation setup: The numerical examples assume ET transmit power Pf = 1 Watt, 60 dB attenuation, received training noise power σ^2_r = −90 dBm, and harvesting efficiency η = 0.5.The channel model uses a rank-1 deterministic component with specified angles and half-wavelength-spaced uniform linear arrays.
  • Simulation setup: The simulations use a rank-1 deterministic Rician component with fixed angles and uniform linear arrays whose adjacent elements are separated by half a wavelength.

A. MIMO Rayleigh Fading Channel

In MIMO Rayleigh fading, numerical results validate the analytical expressions and show that optimal training grows with block length. Optimized training outperforms isotropic transmission and approaches perfect-CSI beamforming for sufficiently long blocks.

  • Analytical validation: The analytical and simulation results match perfectly for M = 5 and N = 10, validating the derived net harvested-energy expressions.The comparison averages over 10000 random channel realizations.
  • Training allocation: The optimal number of trained ER antennas is 2 for T = 25 and 5 for T = 50, illustrating the training-size trade-off.
  • Training allocation: For M = 1, no training is optimal, whereas with M = 2 or M = 5 the optimal trained-antenna count increases with T and eventually includes all ER antennas.All ER antennas should be trained for T ≥ 150 when M = 2 and T ≥ 80 when M = 5.
  • Training energy: The optimal training energy increases with T in a square-root relationship for sufficiently large blocks, indicating diminishing marginal gains from training.
  • Benchmark comparison: Optimized training significantly outperforms isotropic transmission, improves with T, and approaches ideal perfect-CSI energy beamforming for sufficiently large T.

B. MISO Rician Fading Channel

In MISO Rician fading, training improves CSI-based beamforming relative to LOS-only beamforming when the random component matters. The benefit decreases as the Rician factor grows, and training is favored for sufficiently large systems and coherence blocks.

  • Rician-factor effects: As K increases, the performance gap between trained beamforming and LOS-only beamforming diminishes, with the proposed scheme converging to LOS-based beamforming for sufficiently large K.The LOS-based scheme approaches ideal perfect-CSI beamforming in that limit.
  • Rician-factor effects: With M = 5 and T = 200, channel training improves CSI accuracy and makes energy beamforming more effective than beamforming based only on the LOS component.
  • Training region: For fixed K, training helps if and only if the block length T and ET antenna count M are sufficiently large.The training region expands as K decreases.

C. Massive MIMO Rician Fading Channel

The massive-MIMO analysis studies how net harvested energy varies with ET antenna count and identifies asymptotic training behavior. It also outlines limitations involving correlated fading, near-far multiuser setups, and energy-outage objectives.

  • C. Massive MIMO Rician Fading Channel: The massive-MIMO analysis evaluates net average harvested energy versus ET antennas with T = 1000, K = 1, and N = 5.Figure 9 varies M from 5 to 300 and compares benchmark schemes.
  • C. Massive MIMO Rician Fading Channel: A high-quality approximate training solution achieves the optimal asymptotic scaling as the number of ET antennas increases.
  • VI. CONCLUSION AND FUTURE WORK: Future work should address correlated fading, near-far multiuser training, and energy-outage probability rather than only average harvested energy.The current discussion specifically identifies correlated channels, doubly near-far multiuser effects, and outage-based objectives as open directions.

APPENDIX A PROOF OF LEMMA 1

The appendix proves structural properties of the training optimization and derives closed-form or reduced solutions under specified channel assumptions. The arguments use time-overhead reduction, antenna-selection optimization, and massive-MIMO approximations.

  • APPENDIX A PROOF OF LEMMA 1: Reducing training duration while preserving trained antennas and total training energy strictly improves the objective, so optimal training cannot use unnecessary duration.The constructed design remains feasible and increases net harvested energy because it reduces time overhead.
  • APPENDIX A PROOF OF LEMMA 1: In the massive-MIMO regime, dominant-term approximations unify the net-energy expressions and reduce the antenna-selection problem to a relaxed optimization followed by integer selection.The derivation assumes M ≫ N with moderate K and Γ, then obtains the integer solution from the relaxed solution.
  • APPENDIX A PROOF OF LEMMA 1: For sufficiently large channel block length T > KN^2 + 2N, the optimal design trains all ER antennas.This follows from the closed-form solution for the antenna-selection problem under the stated condition.
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