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Wireless Information and Power Transfer: Nonlinearity, Waveform Design and Rate-Energy Tradeoff

Bruno Clerckx

arXiv:1607.05602v3cs.ITcs.NI

TL;DR

WIPT design has relied on an inaccurate linear energy-harvester model, leaving waveform and transceiver design insufficiently grounded in rectifier behavior. The paper develops a tractable nonlinear model for multicarrier signals and optimizes a CSI-adaptive architecture that superposes unmodulated power and modulated communication waveforms. Results show that rectifier nonlinearity changes preferred waveforms, input distributions, spectrum use, and transceiver strategies, while enabling enlarged rate-energy regions.

  • Problem

    Existing WIPT designs rely on a linear rectifier model even though rectifier efficiency depends nonlinearly on the input waveform's power and shape.

  • Method

    The paper extends a tractable rectifier-nonlinearity model to multicarrier modulated signals and jointly optimizes CSI-adaptive superposed unmodulated and modulated waveforms with a power-splitter receiver.

  • Results

    Accounting for rectifier nonlinearity radically changes WIPT design, favoring different waveforms, modulation, input distributions, transceiver architectures, and RF-spectrum use.

  • Takeaways & Limitations

    A deterministic multisine power waveform can improve energy delivery without rate loss, and nonlinear design can enlarge the achievable rate-energy region.

Abstract

from arXiv · show

The design of Wireless Information and Power Transfer (WIPT) has so far relied on an oversimplified and inaccurate linear model of the energy harvester. In this paper, we depart from this linear model and design WIPT considering the rectifier nonlinearity. We develop a tractable model of the rectifier nonlinearity that is flexible enough to cope with general multicarrier modulated input waveforms. Leveraging that model, we motivate and introduce a novel WIPT architecture relying on the superposition of multi-carrier unmodulated and modulated waveforms at the transmitter. The superposed WIPT waveforms are optimized as a function of the channel state information so as to characterize the rate-energy region of the whole system. Analysis and numerical results illustrate the performance of the derived waveforms and WIPT architecture and highlight that nonlinearity radically changes the design of WIPT. We make key and refreshing observations. First, analysis (confirmed by circuit simulations) shows that modulated and unmodulated waveforms are not equally suitable for wireless power delivery, namely modulation being beneficial in single-carrier transmissions but detrimental in multi-carrier transmissions. Second, a multicarrier unmodulated waveform (superposed to a multi-carrier modulated waveform) is useful to enlarge the rate-energy region of WIPT. Third, a combination of power splitting and time sharing is in general the best strategy. Fourth, a non-zero mean Gaussian input distribution outperforms the conventional capacity-achieving zero-mean Gaussian input distribution in multi-carrier transmissions. Fifth, the rectifier nonlinearity is beneficial to system performance and is essential to efficient WIPT design.

I. INTRODUCTION

WIPT research has largely used an inaccurate linear rectifier model, motivating a nonlinear, CSI-adaptive design that jointly addresses waveform choice and the rate-energy tradeoff. The paper develops a tractable multicarrier model and a transceiver combining deterministic power and modulated communication waveforms, with analysis and simulations revealing substantially different design conclusions.

  • Motivation: WIPT jointly supports wireless information transfer and wireless power transfer, with efficient power delivery being fundamental to the overall architecture.The central WPT challenge is increasing end-to-end transfer efficiency, equivalently the rectenna's output DC power for a given transmit power.
  • Motivation: Rectifier conversion efficiency depends nonlinearly on the input waveform's power and shape, so the traditional linear model can produce inefficient waveform designs.Prior waveform-design results show that accounting for nonlinearity better exploits beamforming, channel frequency diversity, and rectifier behavior.
  • Approach: The paper revisits WIPT with a tractable rectifier-nonlinearity model extended from deterministic multisine signals to general multicarrier modulated signals.It uses a bottom-up design grounded in the underlying WPT signal model and characterizes the rate-energy tradeoff.
  • Findings: The analysis and circuit simulations show that modulation helps single-carrier power delivery but harms multicarrier power delivery, while deterministic multisine waveforms provide both energy and rate benefits.The deterministic power waveform is superior for energy delivery and does not cause rate loss at the communication receiver.
  • Approach: The proposed architecture superposes multicarrier unmodulated and modulated waveforms, jointly optimizing the waveform and power splitter according to channel state information.The resulting design characterizes the system's rate-energy region through a nonlinearity-induced posynomial maximization problem.
  • Findings: Numerical results identify multicarrier unmodulated waveforms, power splitting with time sharing, and non-zero-mean Gaussian inputs as advantageous under nonlinear rectification.The unmodulated waveform enlarges the rate-energy region when the number of subbands is sufficiently large, typically above four.

B. Information Decoder

The information decoder can achieve the stated rate with or without explicitly canceling the deterministic power waveform. The achievable rate is based on translated codewords when cancellation is omitted.

  • The deterministic power waveform does not change the differential entropy of the information signal, so the achievable rate remains unchanged.
  • For zero power-waveform splitting, the conventional benchmark uses maximum-ratio transmission per subband and water-filling across subbands.
  • The achievable rate is supported by receiver architectures both with and without power-waveform cancellation.
  • With cancellation, the receiver subtracts the power waveform after RF-to-baseband conversion and ADC.
  • Without cancellation, the baseband receiver decodes translated versions of the codewords.

C. Energy Harvester

The energy harvester is modeled as an antenna, a matched rectifier, and a nonlinear diode followed by low-pass filtering. A truncated Taylor expansion provides a tractable approximation of the rectifier output.

  • Under perfect matching and power splitting, the rectifier receives a fraction ρ of the available RF power.
  • The rectenna combines a nonlinear device with a low-pass filter that extracts DC power from the RF input.
  • The diode voltage is the input voltage minus the output voltage across the load resistor.
  • The diode characteristic is expanded around a quiescent operating point to obtain a tractable behavioral model.
  • Choosing the expansion point as the expected diode voltage makes the output current analyzable under steady-state ideal low-pass filtering.
  • The nonlinear model retains terms through a selected Taylor order, whereas the linear model retains only the second-order term.

2) Linear and Nonlinear Models:

The nonlinear rectifier model distinguishes deterministic multisine and modulated multicarrier waveforms through higher-order signal terms, unlike the conventional linear model. Its scope includes general multicarrier inputs but assumes ideal filtering, matching, and operation outside breakdown.

  • Linear and Nonlinear Models: The model derives DC-current approximations for both multisine and modulated multicarrier excitations, averaging over random information symbols for the latter.
  • Linear and Nonlinear Models: The modulation-related expectation factor explains the different multiplicative factors in the modulated and unmodulated current expressions.
  • Linear and Nonlinear Models: The linear model treats multisine and modulated waveforms as equally suitable because harvested energy depends only on the second-order power term.
  • Linear and Nonlinear Models: The nonlinear model reveals different multicarrier behavior because modulated and unmodulated signals have distinct second- and fourth-order terms.
  • Linear and Nonlinear Models: The conventional linear model assumes constant RF-to-DC efficiency and therefore optimizes only the input-power term.
  • Linear and Nonlinear Models: The diode model assumes ideal low-pass filtering, perfect impedance matching, and operation outside the diode breakdown region.
  • Linear and Nonlinear Models: The model accommodates a wide range of rectifier topologies and multicarrier modulated or unmodulated input signals.

III. WIPT WAVEFORM OPTIMIZATION AND RATE-ENERGY REGION CHARACTERIZATION

The paper characterizes the WIPT rate-energy region by optimizing superposed information and power waveforms with channel-state information. The optimization jointly selects waveform parameters and the power-splitting ratio.

  • The achievable rate-energy region is defined using the proposed WIPT architecture’s attainable information rate and harvested energy.
  • With channel frequency responses known at the transmitter, amplitudes, phases, and power-splitting ratio are optimized to enlarge the rate-energy region.
  • The methodology supports arbitrary Taylor-expansion truncation orders in the nonlinear rectifier model.
  • The characterization solves an optimization maximizing the rectifier output current over power and information waveform parameters and power splitting.
  • The same optimization can be expressed by maximizing the DC output metric z_DC instead of the output current.

A. WPT-only: Energy Maximization

The paper compares multicarrier unmodulated and modulated waveforms under linear and nonlinear rectifier models. Rectifier nonlinearity produces a substantially different multisine design from the linear model, while modulated waveforms retain the ASS strategy.

  • Model comparison: The nonlinear multisine formulation accounts for higher-order rectifier effects that are absent from the linear-model design.The paper contrasts the linear optimization with the nonlinear posynomial formulation for multisine transmission.
  • Model comparison: The linear model favors transmission on a single frequency, whereas the nonlinear model leads to a different multisine strategy.The nonlinear design is formulated as a posynomial maximization problem and solved as a Reversed Geometric Program.
  • Modulated waveforms: Multicarrier modulated waveforms use the adaptive single-sinewave strategy under both linear and nonlinear models.The optimal modulated-waveform beamforming weights follow the ASS design because the input symbols introduce CSCG randomness.

B. WIPT: A General Approach

The general WIPT design jointly optimizes the superposed waveform and power-splitting ratio to characterize the rate-energy region. Because rectifier nonlinearity yields a nonstandard optimization problem, the method uses iterative geometric-program approximations.

  • Waveform design: Matched filtering with respect to channel phases is optimal for both rate and harvested-energy maximization.This produces the same phase decisions used in prior WPT waveform designs.
  • Optimization formulation: The achievable rate-energy region is obtained by maximizing harvested energy subject to transmit-power and rate constraints.The formulation jointly controls the information and power waveforms and the power-splitting ratio.
  • Optimization formulation: The resulting problem maximizes a posynomial but is not a standard geometric program.An auxiliary variable transforms it into an equivalent reversed geometric program.
  • Iterative solution: Arithmetic mean-geometric mean approximations replace non-posynomial constraints so standard geometric-program tools can solve the iterations.The method updates approximation weights and repeats the optimization until convergence.
  • Iterative solution: The successive-approximation procedure guarantees only a point satisfying the KKT conditions, not a global optimum.This is the principal scope limitation of the numerical optimization method.

C. WIPT: Decoupling Space and Frequency

The paper decouples spatial and frequency-domain waveform design by using channel-matched beamforming vectors. This preserves the performance of joint design while reducing the numerical optimization from matrices to frequency-domain vectors.

  • Decoupled design: The optimal phases correspond to maximum-ratio-transmission beamforming, enabling spatial-frequency design decoupling.The resulting weight vectors maximize the relevant rectifier-order terms and information rate.
  • Effective channel: After beamforming, the multi-antenna problem becomes an effective single-antenna multicarrier problem with channel gains given by the channel norms.The subband powers are represented by the squared magnitudes of the unmodulated and modulated waveform weights.
  • Complexity reduction: The decoupled formulation reduces the optimization variables from N × M matrices to N-dimensional power-allocation vectors.The spatial weights are determined from the channel, while the remaining optimization acts across subbands.
  • Complexity reduction: The decoupled approach achieves the same performance as the joint space-frequency design.Its computational advantage comes from numerically optimizing only the frequency-domain magnitude vectors.

D. WIPT: Characterizing the Twofold Benefit of using Deterministic Power Waveforms

The proposed architecture combines deterministic multisine power transmission with a communication waveform. Because the deterministic power waveform does not interfere with information decoding, it can provide both an energy benefit and a rate benefit relative to the baselines.

  • Twofold benefit: A deterministic power waveform avoids the information-receiver interference that can arise when power and communication waveforms are superposed.This absence of interference yields the architecture’s twofold energy and rate benefit.
  • Baseline comparisons: Removing the deterministic multisine waveform eliminates the twofold benefit in the simplified baseline.That baseline is obtained by forcing the power-waveform component to zero.
  • Baseline comparisons: A hypothetical baseline retains the energy benefit but loses the rate benefit when the power waveform is treated as CSCG distributed at the information receiver.The resulting interference term causes a rate loss, so its rate-energy region is a lower bound on the proposed architecture’s region.
  • Baseline optimization: For the hypothetical baseline, joint spatial-frequency optimization is used because interference prevents decoupling from being guaranteed optimal.The general design approach is therefore retained for that comparison.

IV. SCALING LAWS

The scaling laws reveal distinct nonlinear-rectifier behavior for modulated and unmodulated waveforms across carrier counts, channel selectivity, and CSIT availability. Modulation helps single-carrier WPT, whereas unmodulated multisine waveforms are advantageous for multi-carrier WPT and motivate the proposed WIPT architecture.

  • Multi-carrier transmission: For multi-carrier transmission without CSIT, unmodulated waveforms can scale linearly with N in frequency-flat channels, unlike modulated waveforms.Random symbol amplitudes and phases prevent the same periodic rectifier excitation produced by multisine waveforms.
  • Single-carrier transmission: For single-carrier transmission without CSIT, the modulated waveform outperforms the unmodulated waveform because its fourth-order term is twice as large.The gain comes from the fourth-order moment of the CSCG input symbols.
  • Channel selectivity and CSIT: With CSIT, frequency selectivity provides a frequency-diversity gain for both modulated and unmodulated waveforms.Without CSIT, frequency selectivity is detrimental to unmodulated-waveform performance but has no impact on modulated-waveform performance.
  • Nonlinear rectification: The fourth-order diode-nonlinearity term creates a clear distinction between unmodulated multisine and modulated waveforms in WPT.The comparison underlies the scaling-law differences across single-carrier and multi-carrier transmissions.
  • WIPT architecture: A multi-carrier unmodulated waveform superposed on a multi-carrier modulated waveform motivates the WIPT architecture for efficient WPT and WIT.The architecture uses the waveform complementarity identified by the scaling laws.

V. PERFORMANCE EVALUATIONS

Performance is evaluated using both the simplified nonlinear rectifier model and an actual rectenna design simulated in PSpice.

  • Evaluation methodology: The study evaluates performance with a fourth-order simplified nonlinear model and with an accurate rectenna design in PSpice.The two evaluations compare analytical-model behavior with circuit-level simulation.

A. Nonlinear Model-Based Performance Evaluations

The nonlinear-model evaluations show that superposed deterministic multisine and modulated waveforms enlarge the rate-energy region, with the preferred power-splitting/time-sharing strategy depending on subcarrier count and SNR.

  • Fig. 4: Increasing N boosts harvested energy with superposed waveforms, whereas energy changes little without the multisine power waveform.This follows the predicted scaling laws: modulated-waveform energy is independent of N in frequency-flat channels and grows only logarithmically with N in frequency-selective channels.
  • Fig. 4: A deterministic multisine power waveform enlarges the rate-energy region because rectifier nonlinearity benefits deterministic inputs while modulated inputs fluctuate across subcarriers.This contrasts with the linear-model literature, where adding a power waveform was reported as useless when it was treated as interference by the receiver.
  • Fig. 5: The rate benefit of a deterministic waveform increases with SNR because the lower-bound system must reduce multisine power to avoid interference-limited communication.At very low SNR, the power waveform is largely absorbed into the noise, making the compared regions similar.
  • Fig. 5: At 10 and 20dB, time sharing outperforms power splitting between WPT-only and WIT-only points, whereas at 30 and 40dB power splitting performs better.The paper attributes this behavior to the concavity-convexity of the rate-energy region.
  • Summary: Overall, multisine power is useful for N > 4, power splitting is preferred without multisine, and the preferred strategy shifts from power splitting to combined power splitting/time sharing to time sharing as N grows.For sufficiently large N, time sharing is favored at low SNR and power splitting at high SNR; these effects are consequences of rectifier nonlinearity.
  • Model comparison: The nonlinear-model design is more spectrally efficient than the linear-model design because the latter provides no benefit from adding a multisine waveform to the modulated waveform.The linear-model rate-energy region matches the no-power-waveform strategy, which is outperformed by the superposed-waveform strategy.
  • Input distribution: With superposed waveforms, the input distribution changes from zero-mean to non-zero-mean Gaussian as more power is allocated to the multisine waveform and K-factors increase.The zero-mean CSCG distribution remains the input distribution for no-power-waveform and linear-model designs.

B. Validation of the Model and the Scaling Laws

Circuit simulations validate the rectifier model and scaling laws for modulated waveforms. They show that OFDM and multisine behave differently with subband count, exposing why PAPR and linear rectifier models are insufficient for WPT design.

  • Model validation: PSpice simulations validate the rectifier model and scaling laws for CSCG-modulated waveforms using the single-diode rectenna in Fig. 7.The circuit was designed for an average input power of -20dBm (10µW).
  • Scaling laws: OFDM DC power remains flat with N, whereas multisine DC power increases rapidly up to N = 64.For larger N, the multisine advantage reverses the small-N ordering between the two waveforms.
  • Scaling laws: For most N >2, OFDM loses to multisine because independent random symbol fluctuations disrupt the periodic rectifier excitation.For N ≤2, the CSCG fourth-order moment boosts OFDM's fourth-order term by a factor 2 relative to the unmodulated case.
  • Scaling laws: For N > 64, multisine DC power decreases because the rectenna was optimized for N =4.This identifies a circuit-design condition behind the high-N decline rather than a universal multisine scaling law.
  • PAPR comparison: Increasing OFDM PAPR with N does not explain its flat harvested-power scaling, so PAPR alone is not an accurate WPT suitability metric.The comparison uses CCDFs of maximum PAPR and instantaneous-power PAPR across subband counts.
  • WIPT implication: The validated nonlinear model supports adaptive superposition of multisine WPT and modulated WIT waveforms for rate-energy-region design.The resulting WIPT design accounts for rectifier nonlinearity and can enlarge rate-energy regions relative to designs that ignore it.
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