PaperlayerView exact revision on arXiv

Source-linked AI summary

Waveform Design for Wireless Power Transfer

Bruno Clerckx, Ekaterina Bayguzina

arXiv:1604.00074v2cs.ITcs.NI

TL;DR

The paper addresses waveform design for far-field WPT, where transmitter design has received less attention than rectenna design. It develops nonlinear rectenna modeling and CSI-adaptive multisine waveforms, showing advantages over linear-model and non-adaptive designs.

  • Problem

    The paper addresses how transmitter waveform design can increase rectenna DC power without increasing transmit power, complementing prior emphasis on rectenna efficiency.

  • Method

    The paper derives a tractable nonlinear rectenna model, compares it with a linear model, and uses both to design CSI-adaptive multisine waveforms.

  • Results

    Nonlinear-model designs favor mult frequency power allocation and provide significant harvested-DC-power gains over linear-model and baseline waveforms under fixed transmit power.

  • Takeaways & Limitations

    The results support accounting for rectifier nonlinearity and acquiring transmitter CSI, while motivating large-scale multisine multi-antenna WPT architectures.

  • Takeaways & Limitations

    The successive-approximation design method guarantees KKT satisfaction but not convergence to the global optimum.

Abstract

from arXiv · show

Far-field Wireless Power Transfer (WPT) has attracted significant attention in recent years. Despite the rapid progress, the emphasis of the research community in the last decade has remained largely concentrated on improving the design of energy harvester (so-called rectenna) and has left aside the effect of transmitter design. In this paper, we study the design of transmit waveform so as to enhance the DC power at the output of the rectenna. We derive a tractable model of the non-linearity of the rectenna and compare with a linear model conventionally used in the literature. We then use those models to design novel multisine waveforms that are adaptive to the channel state information (CSI). Interestingly, while the linear model favours narrowband transmission with all the power allocated to a single frequency, the non-linear model favours a power allocation over multiple frequencies. Through realistic simulations, waveforms designed based on the non-linear model are shown to provide significant gains (in terms of harvested DC power) over those designed based on the linear model and over non-adaptive waveforms. We also compute analytically the theoretical scaling laws of the harvested energy for various waveforms as a function of the number of sinewaves and transmit antennas. Those scaling laws highlight the benefits of CSI knowledge at the transmitter in WPT and of a WPT design based on a non-linear rectenna model over a linear model. Results also motivate the study of a promising architecture relying on large-scale multisine multi-antenna waveforms for WPT. As a final note, results stress the importance of modeling and accounting for the non-linearity of the rectenna in any system design involving wireless power.

I. INTRODUCTION

The paper shifts attention from rectenna design to CSI-adaptive transmit-waveform design for increasing harvested DC power. It develops nonlinear and linear models, derives waveform and scaling-law results, and evaluates their performance.

  • I. INTRODUCTION: The paper addresses waveform design for WPT, focusing on multisine signals to improve rectenna output DC power.The design targets higher DC power without increasing transmit power.
  • I. INTRODUCTION: The proposed rectenna model captures nonlinearity through second- and higher-order Taylor-expansion terms, unlike the conventional second-order linear model.The models support comparison of nonlinear and linear waveform designs.
  • I. INTRODUCTION: With perfect CSIT, the method adaptively optimizes multi-antenna multisine amplitudes and phases to maximize rectenna output DC current.Phases are obtained in closed form, while amplitudes follow a non-convex posynomial maximization under a power constraint.
  • I. INTRODUCTION: The linear model favors single-frequency narrowband allocation, whereas the nonlinear model favors distributing power across multiple frequencies.This difference reflects the distinct system designs induced by the two rectenna models.
  • I. INTRODUCTION: Under fixed transmit power, nonlinear-model DC current can increase linearly with the number of sinewaves N, while the linear-model increase is at most logarithmic.In frequency-selective channels, the linear scaling requires CSIT; the results motivate multisine transmission and transmitter CSI.
  • I. INTRODUCTION: The nonlinear-model waveforms provide significant gains over state-of-the-art, linear-model-based, and non-adaptive waveforms in realistic evaluations.The nonlinear model is validated by circuit simulations, while the linear model does not correctly predict multisine performance.

III. ANALYTICAL MODEL OF THE RECTENNA

The analytical rectenna model represents the antenna, diode, filtering, and load while approximating diode nonlinearity through a truncated Taylor expansion. It defines output DC current as the time-averaged diode current under waveform excitation.

  • III. ANALYTICAL MODEL OF THE RECTENNA: The tractable model relies on assumptions and truncation so waveform parameters can be optimized, with later evaluations using a more accurate circuit simulator.This separates analytical design from realistic performance evaluation.
  • III. ANALYTICAL MODEL OF THE RECTENNA: The rectenna consists of a receiving antenna, a nonlinear diode device, a low-pass filter, and a load under an assumed matched input impedance.The antenna is modeled as lossless with a 50Ω equivalent impedance, and antenna noise is neglected.
  • III. ANALYTICAL MODEL OF THE RECTENNA: The diode current is modeled by a Taylor expansion around a fixed operating voltage drop, retaining the diode’s nonlinear behavior through higher-order terms.The small-signal expansion applies only in the diode’s nonlinear operating region.
  • III. ANALYTICAL MODEL OF THE RECTENNA: The optimization objective is the steady-state output DC current, approximated as the time average of the current flowing through the diode.The ideal-rectifier assumption makes the delivered load current constant in steady state.

C. A Non-Linear Model

The nonlinear model expresses rectifier DC current using even-order waveform terms, including cross-frequency products absent from the linear approximation. This structure supports CSI-based waveform optimization.

  • C. A Non-Linear Model: The DC-current approximation is obtained by averaging nonlinear powers of the received multisine waveform over time.Only terms producing a DC component remain under the evenly spaced-frequency assumption.
  • C. A Non-Linear Model: The second-order term is linear, while even-order terms of order four and above create nonlinear interactions between different frequencies.These higher-order terms depend on products of contributions from multiple frequencies.
  • C. A Non-Linear Model: The linear model truncates the Taylor expansion at order two and therefore omits rectifier nonlinearity.Its accuracy is questioned because experiments indicate that second-order truncation does not accurately model diode rectification.
  • C. A Non-Linear Model: Assuming transmitter CSI, waveform amplitudes and phases are selected to maximize the modeled output current.The model converts the current-maximization objective into an equivalent optimization over a DC-related quantity.

A. Linear Model-based Design

The linear model reduces waveform design to matched beamforming on the strongest frequency, whereas nonlinear terms can make single-frequency transmission suboptimal for some channel states.

  • A. Linear Model-based Design: Under the linear model, the optimal design allocates all transmit power to the sinewave with the strongest channel and applies matched beamforming.This strategy is called adaptive single sinewave (ASS).
  • A. Linear Model-based Design: For perfectly flat channel magnitudes, uniformly allocating power over any non-empty subset of sinewaves achieves the same linear-model objective value.If the channel is not perfectly flat, ASS is the unique solution.
  • A. Linear Model-based Design: In the two-sinewave nonlinear example, the ASS strategy can be suboptimal because fourth-order terms include cross-products between both sinewaves.When channel magnitudes are approximately equal, the optimal strategy allocates power to both sinewaves.
  • A. Linear Model-based Design: The toy example shows that waveform allocation should adapt to CSI and that multiple-sinewave transmission benefits some channel states.This contrasts with the linear model’s single-sinewave recommendation for all channel states.
  • A. Linear Model-based Design: Generating optimized waveforms for arbitrary multipath configurations, numbers of sinewaves, and antennas is non-trivial and requires the subsequent design procedure.The section motivates a general optimization method beyond the toy example.

C. Non-Linear Model-based Design

The non-linear rectenna model yields a tractable waveform-design problem by optimizing closed-form phases and numerically computed amplitudes. Its posynomial formulation enables iterative standard geometric-program approximations, though global optimality is not guaranteed.

  • Phase optimization: Optimal sinewave phases are chosen in closed form so all sinewaves arrive in phase at the rectenna input.This choice makes the relevant cosine terms equal to 1 and leaves amplitudes as the numerical design variables.
  • Amplitude optimization: The resulting DC-current objective is a posynomial for any Taylor expansion order, with more terms at higher orders.The waveform amplitudes are optimized under a power constraint.
  • Numerical optimization: The amplitude problem is transformed into a standard geometric program by introducing an auxiliary variable and applying an AM-GM monomial lower bound.The transformation is conservative because the original reciprocal objective is not itself a posynomial.
  • Numerical optimization: An iterative successive-approximation procedure updates the monomial weights and repeatedly solves the resulting geometric program until convergence.Existing software such as CVX can solve each standard geometric-program subproblem.
  • Limitation: The successive-approximation method is not guaranteed to reach the global solution and instead guarantees only a point satisfying KKT conditions.Simulations reported in the paper indicate that the iterative algorithm often converges to the global optimum.
  • Design implication: For a single sinewave, linear and non-linear designs both reduce to a simple matched beamformer, so non-linearity matters only when N ≥2.The multi-sine case can make allocating power to multiple frequencies preferable to a single-sinewave strategy.

D. Decoupling Space and Frequency Domains

The multi-antenna multisine design can decouple spatial and frequency-domain weights without performance loss under the stated formulation. This reduces numerical optimization from an N × M amplitude matrix to an N-dimensional vector.

  • Spatial design: The optimal phase at each frequency is a matched-beamformer phase based on the dominant right singular vector of the channel.This converts the spatial part of the design into frequency-dependent beamforming.
  • Frequency design: After spatial decoupling, the effective channel gain at frequency n is ∥h_n∥ and the remaining amplitudes form an N-dimensional vector s.The amplitudes remain subject to the total transmit-power constraint.
  • Optimization: The reduced frequency-domain objective remains a posynomial and can be optimized using the same AM-GM and geometric-program methodology.The method applies the posynomial maximization procedure after substituting the matched-beamformer structure.
  • Complexity and performance: The decoupled approach achieves the same performance as the joint space-frequency design while substantially reducing computational complexity.Only an N-dimensional vector is optimized numerically instead of an N × M matrix.

E. PAPR Constraints

The waveform design is extended to account for practical peak-to-average-power-ratio constraints. Because the resulting constraints are signomial, they are handled through conservative condensation and iterative geometric-program approximations.

  • PAPR formulation: PAPR on antenna m is constrained by bounding the waveform’s maximum instantaneous power relative to its average power.The constraint is imposed as PAPR_m ≤ η.
  • PAPR formulation: With the optimized phases fixed, sampled waveform-power constraints can be written as signomials rather than posynomials because some coefficients are negative.Oversampling is used when rewriting the continuous-time PAPR constraint in sampled form.
  • Optimization: The signomial constraints are conservatively converted into standard geometric-program constraints by condensing denominator posynomials into monomial approximations.The resulting formulation introduces monomial weights for the objective and PAPR-related terms.
  • Optimization: An iterative procedure updates the condensation weights and solves the resulting geometric program at each iteration.The full procedure is summarized as the WPT waveform design with PAPR constraints.
  • Limitation: With multiple antennas, decoupling spatial and frequency domains is suboptimal compared with joint space-frequency optimization when PAPR constraints are present.The performance gap follows from the coupled per-antenna PAPR constraints.

F. Multiple Rectennas

For multiple rectennas, waveform design must balance the DC energy harvested across rectennas because a waveform effective for one rectenna may be inefficient for another. The paper formulates a weighted-sum optimization and solves it with signomial approximations.

  • Problem formulation: Multiple-rectenna WPT introduces a trade-off because each rectenna’s harvested DC energy can depend on the waveforms selected for the others.The setup covers both point-to-point MIMO WPT and multi-user MISO WPT.
  • Problem formulation: The optimization seeks amplitudes and phases across frequencies that maximize a weighted sum of the rectennas’ DC components.The weights can represent differing priorities among rectennas.
  • Linear-model baseline: For the linear model, the solution uses a single sinewave at the frequency with the largest relevant dominant-eigenvalue criterion and transmits along the dominant right singular vector.For one rectenna, this solution reduces to the corresponding single-rectenna strategy.
  • Phase design: Unlike the single-rectenna case, phases cannot generally make all rectenna-specific cosine terms equal to 1 simultaneously.The resulting DC-current objective is therefore a signomial with both positive and negative coefficients.
  • Optimization: The signomial maximization is converted into an iterative sequence of standard geometric programs using an auxiliary variable and monomial approximations.When the negative part vanishes, the formulation reduces to the single-rectenna geometric-program problem.
  • Phase design: A practical phase choice for the non-linear multiple-rectenna algorithm uses the phases of each frequency’s dominant right singular vector, without a claim of optimality.This heuristic is motivated by the dominant-eigenmode structure of the single-rectenna and linear-model solutions.

G. CSI Acquisition at the Transmitter

The paper proposes CSI-aware waveform design and derives scaling laws showing how nonlinear rectenna behavior, channel structure, and transmit dimensions affect harvested energy.

  • CSI acquisition: CSI can be acquired through uplink pilots and channel estimation or through CSI feedback to the power transmitter.The transmitter uses frequency-response knowledge to adapt waveform weights.
  • CSI acquisition: The waveform design does not require feedback about rectifier characteristics because their relevant parameters affect all terms equally or remain constant across designs.The diode saturation current is multiplicative, while the thermal voltage is constant irrespective of rectifier design.
  • Scaling laws: In frequency-flat channels, nonlinear-model multisine designs achieve DC-current scaling that increases linearly with N without CSI feedback.The linear increase originates from the fourth-order rectifier term.
  • Scaling laws: With multiple transmit antennas in frequency-flat channels, UPMF harvested energy scales proportionally to NM^2 and requires CSIT for spatial matched beamforming.N affects the fourth-order term, while M affects both second- and fourth-order terms.
  • Scaling laws: Adding U identically distributed rectennas in frequency-flat channels preserves each rectenna’s linear-in-N energy scaling, so the energy region is a hypercube.Each rectenna achieves the same quantity of energy as if it were alone in the system.
  • Scaling laws: The scaling laws assume diode operation in the nonlinear region; sufficiently large N can force linear-region operation and invalidate the Taylor model and scaling results.Increasing N indefinitely therefore does not imply unbounded harvested energy under the model’s assumptions.

B. Frequency-Selective Channels

In frequency-selective channels, fixed non-adaptive waveforms do not benefit from increasing the number of sinewaves, whereas CSI-adaptive strategies recover logarithmic gains and nonlinear designs can scale more favorably.

  • Analysis: For independent frequency-domain channel gains, phase-dependent remainder terms average to zero in the expected harvested-energy expression.Uniform channel phases make the relevant cosine terms have zero expectation.
  • Non-adaptive transmission: In frequency-selective Rayleigh fading without CSIT, average harvested DC energy is independent of N and waveform design.A single sinewave is sufficient in the absence of CSIT.
  • CSI-adaptive transmission: The ASS strategy increases the second-order and fourth-order terms proportionally to log N and log^2 N, respectively, through frequency selection.ASS allocates all transmit power to the sinewave with the strongest channel gain.
  • Scaling comparison: For M = 1, UPMF outperforms ASS when the fourth-order term dominates or N is sufficiently large, while ASS performs better when the second-order term dominates.The comparison is linear versus log-squared growth in N under fourth-order dominance.
  • Scaling comparison: The scaling-law summary states that linear growth in N requires no CSIT in frequency-flat channels but CSIT in frequency-selective channels.It also reports lower scaling laws for linear-model designs than for nonlinear-model designs.

C. Large-Scale Multi-Sine Multi-Antenna WPT

The scaling results motivate large-scale multisine multi-antenna WPT, where spatial matched beamforming and simple frequency allocation can support low-complexity waveform design.

  • Large-scale architecture: Large-scale multisine multi-antenna WPT is motivated by scaling laws analogous to Massive MIMO in communication systems.The large dimension can simplify waveform design.
  • Large-scale architecture: Spatial matched beamforming can produce channel hardening on each sinewave as the number of transmit antennas grows.This follows from the law of large numbers applied to channel-vector norms.
  • Large-scale architecture: For large-scale WPT, optimal frequency-flat power allocation can be replaced by the lower-complexity UP strategy as a practical alternative.The paper describes this as a very low-complexity waveform design.
  • Large-scale architecture: Figure 3 examines zDC versus N, PAPR constraint η, and waveform amplitudes for N = 8 with one antenna and no wireless channel.The top, middle, and bottom panels correspond to these three quantities, respectively.

VI. PERFORMANCE EVALUATIONS

The evaluations compare nonlinear-model waveform designs with linear-model and non-adaptive baselines across channel bandwidths, sinewave counts, antennas, and PAPR. Results show that CSI-adaptive nonlinear designs exploit frequency selectivity and increasingly outperform linear-model designs as system scale grows.

  • Evaluation setup: The evaluations combine analytical nonlinear-model metrics with accurate PSpice rectenna simulations.The analytical evaluations use a fourth-order Taylor expansion and average zDC over many channel realizations.
  • Bandwidth and CSI: For B = 1MHz and M = 1, non-adaptive UP performs well, indicating that CSI feedback is not needed in channels with little frequency selectivity.At B = 10MHz, adaptive waveforms clearly outperform UP, highlighting the usefulness of CSI with a single transmit antenna.
  • Waveform comparisons: At B = 10MHz, ASS approaches OPT for small N but is clearly outperformed by adaptive MF and OPT as N increases.This comparison demonstrates the inefficiency of the linear-model-based ASS design in more frequency-selective settings.
  • Bandwidth and frequency selectivity: As bandwidth increases, uniform-power waveforms lose zDC while adaptive OPT and SS benefit by favoring the strongest sinewave(s).The optimized waveform allocates more power to frequencies with larger channel gains, resembling water-filling.
  • PAPR: High PAPR is not sufficient for effective WPT: MAX PAPR is inefficient because power is wasted inverting the channel to enforce the desired received waveform.For optimized waveforms, zDC and transmit PAPR show positive correlation, especially at small bandwidths, but this correlation decreases as frequency selectivity increases.
  • Large-scale multisine WPT: The nonlinear-model-based ASS comparison shows significant performance gains over the linear model as the number of sinewaves grows large.Fig. 9 compares UP, ASS, UPMF, and MF6 for M = 1 and 5 MHz bandwidth, alongside scaling laws for ASS and UPMF.

B. Accurate and Realistic Performance Evaluations

Realistic circuit evaluations validate the nonlinear rectenna model and show that adaptive, nonlinear-model waveform designs improve harvested DC power over linear-model designs. The evaluations also expose practical dependencies on bandwidth, capacitor choice, and waveform-optimization complexity.

  • PSpice evaluations use a realistic rectenna with an L-matching network to assess waveform optimization and rectenna non-linearity.The circuit is designed for an input power of -20dBm.
  • The OPT waveform is not computed for large N because of high optimization complexity, motivating alternative methods for large-scale waveform design.The paper also assumes uniformly spaced sinewaves, leaving optimal spectrum allocation for a fixed bandwidth as future work.
  • Increasing N changes capacitor charging and discharging, making larger Cout preferable for large N but potentially causing diode breakdown beyond a certain N.The SMS7630 diode breakdown voltage is 2V, beyond which efficiency sharply decreases.
  • For N = 16 and B = 10MHz, OPT achieves 6.4157µW with Cout = 100pF versus 2.3281µW for UP.The comparison uses the wireless channel of Fig. 4 and the realistic rectenna simulation shown in Fig. 12.
  • The nonlinear model agrees well with PSpice simulations, whereas the linear model fails to characterize rectenna behavior and produces inefficient multisine designs.With M = 1 in Fig. 13, the CSIT-based ASS waveform is even outperformed by non-adaptive UP.
  • 10 MHz bandwidth yields higher average DC power than 1 MHz because greater frequency selectivity and more frequent diode conduction improve harvesting.When N = 1, all waveforms achieve the same performance.
Loading 1604.00074v2…