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Wireless Information and Power Transfer over an AWGN channel: Nonlinearity and Asymmetric Gaussian Signaling

Morteza Varasteh, Borzoo Rassouli, Bruno Clerckx

arXiv:1705.06350v3cs.ITmath.IT

TL;DR

The paper addresses SWIPT over a flat-fading complex AWGN channel while accounting for rectenna nonlinearity rather than the inaccurate linear harvester model. It derives delivered DC power from system baseband parameters and obtains an inner bound by restricting channel inputs to i.i.d. distributions determined by first and second moments. The studied inner bound exhibits a rate-power tradeoff, with a zero-mean Gaussian input and asymmetric power allocations across the real and imaginary dimensions.

  • Problem

    The paper addresses SWIPT over a flat-fading complex AWGN channel while accounting for rectenna nonlinearity rather than the inaccurate linear harvester model.

    Source: §I · p. 2

  • Method

    It derives delivered DC power from system baseband parameters and obtains an inner bound by restricting channel inputs to i.i.d. distributions determined by first and second moments.

    Source: §I · pp. 2–3

  • Results

    The studied inner bound exhibits a rate-power tradeoff, with a zero-mean Gaussian input and asymmetric power allocations across the real and imaginary dimensions.

    Source: §I · p. 3, §VI · p. 8

  • Takeaways & Limitations

    The optimal channel input remains zero-mean Gaussian but uses asymmetric power allocations across the real and imaginary dimensions.

    Source: §VI · p. 8

Abstract

from arXiv · show

Simultaneous transmission of information and power over a point-to-point flat-fading complex Additive White Gaussian Noise (AWGN) channel is studied. In contrast with the literature that relies on an inaccurate linear model of the energy harvester, an experimentally-validated nonlinear model is considered. A general form of the delivered Direct Current (DC) power in terms of system baseband parameters is derived, which demonstrates the dependency of the delivered DC power on higher order statistics of the channel input distribution. The optimization problem of maximizing Rate-Power (R-P) region is studied. Assuming that the Channel gain is available at both the receiver and the transmitter, and constraining to independent and identically distributed (i.i.d.) channel inputs determined only by their first and second moment statistics, an inner bound for the general problem is obtained. Notably, as a consequence of the harvester nonlinearity, the studied inner bound exhibits a tradeoff between the delivered power and the rate of received information. It is shown that the tradeoff-characterizing input distribution is with mean zero and with asymmetric power allocations to the real and imaginary dimensions.

I. INTRODUCTION

The paper studies SWIPT over a flat-fading complex AWGN channel using a nonlinear rectenna model, deriving an inner bound for the rate-power region. It finds a nontrivial rate-power tradeoff and an optimal zero-mean Gaussian input with asymmetric real and imaginary power allocations.

I. INTRODUCTION

  • The linear rectifier model used in much prior SWIPT work is inaccurate, motivating waveform and architecture design based on rectenna nonlinearity.The nonlinear output depends on higher-order statistics of the input waveform, unlike the linear model, which uses only the received signal’s second moment.Source: §I · p. 2
  • The paper derives delivered DC power in terms of system baseband parameters and studies the resulting rate-power optimization problem.The analysis assumes joint information extraction and power harvesting at the receiver.Source: §I · pp. 2–3
  • The achievable inner bound constrains channel inputs to i.i.d. distributions determined by their first and second moment statistics.This provides a tractable scheme for the general rate-power optimization problem.Source: §I · p. 3
  • For this inner bound, rectenna nonlinearity creates a tradeoff between delivered power and received-information rate.Under a linear harvester model, maximum rate and maximum energy are aligned in the flat-fading channel.Source: §I · p. 3
  • The tradeoff-achieving input remains zero-mean Gaussian but allocates unequal power to the real and imaginary dimensions.The paper contrasts this asymmetric allocation with conventional circularly symmetric Gaussian signaling.Source: §I · p. 3

II. SYSTEM MODEL

The system model considers a point-to-point flat-fading AWGN channel and introduces the operation of its transmitter and receiver.

II. SYSTEM MODEL

  • The considered communication link is a point-to-point flat-fading AWGN channel.Source: §II · p. 3

A. Transmitter

The transmitter generates a bandlimited information-power waveform from i.i.d. channel symbols, then upconverts it for transmission.

A. Transmitter

  • The transmitted waveform is generated from i.i.d. information-power symbols and is bandlimited before upconversion to the carrier frequency.The symbol at time index n is represented by Xn.Source: A. Transmitter · pp. 3–4

B. Receiver

The receiver separately describes information processing and nonlinear RF power harvesting for the received waveform.

B. Receiver

  • The received RF waveform is downconverted and sampled to produce a baseband sequence used for information processing.Flat fading permits the channel-delay approximation used in sampling.Source: B. Receiver · p. 4
  • Unlike the linear model, the nonlinear delivered-power model captures dependence beyond the received RF signal’s second moment.Circuit simulations cited by the paper show that the linear model is inaccurate and inefficient for signal design.Source: B. Receiver · p. 4
  • The receiver models noise samples as i.i.d. circularly symmetric complex Gaussian variables and assumes channel knowledge at both transmitter and receiver.Source: B. Receiver · p. 4
  • The rectenna’s nonlinear output is approximated using a fourth-moment Taylor truncation of the diode characteristic.The first and third moments are zero under time averaging, while k2 and k4 parameterize the delivered-power expression.Source: B. Receiver · p. 4

III. DELIVERED POWER

The section derives delivered receiver power in baseband terms for i.i.d. channel inputs and identifies the moment statistics governing that power.

  • The formulation is designed to represent delivered power using system baseband parameters while jointly supporting information decoding and power harvesting.Source: §III · p. 5
  • Pdel is expressed in baseband parameters as αQ + ˜α ˜Q + (β + ˜β)P + γ for i.i.d. channel inputs.Source: §III · p. 5
  • The derivation assumes i.i.d. channel inputs because non-i.i.d. inputs make the received signal’s fourth-moment statistics depend on inputs at other time indices.Source: §III · p. 6

IV. PROBLEM STATEMENT

The paper formulates rate maximization under transmitter power and receiver delivered-power constraints, then contrasts the nonlinear and linear harvesting models.

  • The optimization maximizes mutual information I(X;Y) subject to a transmitter power constraint and a minimum delivered-power constraint at the receiver.Source: §IV · p. 6
  • Under the linear constraint E[|Y|2] ≥ Pd, a CSCG input achieves the maximum and no tradeoff exists between received information and delivered power.Source: §IV · p. 6
  • The nonlinear formulation instead treats rate and delivered power as competing objectives at the receiver.Source: §IV · p. 6

V. MAIN RESULT

Within i.i.d. inputs characterized by first and second moments, the paper obtains an inner bound whose nonlinear harvester model creates a rate–power tradeoff. The optimal inputs are zero-mean Gaussian with power allocation between real and imaginary dimensions determined by the required delivered power.

  • The optimization is restricted to i.i.d. inputs characterized by first and second moments, with zero-mean Gaussian real and imaginary components having variances Pr and Pi.Source: §V · pp. 6–7
  • The tradeoff spans delivered power and information rate, because the nonlinear model makes delivered power depend on fourth-order input statistics.Source: §V · pp. 6–7
  • At maximum delivered power, all available transmitter power is assigned to one real or imaginary dimension, while maximum rate uses equal allocation Pr = Pi = Pa/2.Source: §V · p. 7
  • For intermediate delivered-power requirements, the rate-maximizing allocation is asymmetric and is selected so αQ + ˜α ˜Q + (β + ˜β)Pa + γ = Pd.Source: §V · p. 7
  • The transmitter keeps the same power allocation throughout transmission because the channel is assumed fixed for the entire transmission.Source: §V · p. 7
  • The rate depends on h, whereas delivered power depends on both h and ˜h, and the resulting regions are illustrated by varying channel coefficients in Figure 1.Source: §V · p. 7, Figure 1 · p. 8

VI. CONCLUSION

The paper studies SWIPT over a point-to-point complex AWGN channel with a nonlinear receiver power harvester, deriving an inner bound for the rate-power region under stated channel and input assumptions. The inner bound exhibits a nonlinearity-induced rate-power tradeoff characterized by zero-mean Gaussian inputs with asymmetric real and imaginary power allocations.

  • The analysis assumes channel-state information at both transmitter and receiver and i.i.d. channel inputs characterized by first- and second-moment statistics.Source: §VI · p. 8
  • An inner bound for the optimal rate-power region is derived under these input restrictions.Source: §VI · p. 8
  • The nonlinear receiver power harvester creates a tradeoff between transmitted-information rate and delivered power.Source: §VI · p. 8
  • The tradeoff-characterizing input remains Gaussian with zero mean but uses asymmetric power allocations across the real and imaginary dimensions.Source: §VI · p. 8
  • The optimal input distribution for the unrestricted problem and extension to frequency-selective AWGN channels remain open problems.Source: §VI · p. 8

A. PROOF OF PROPOSITION 1

This proof derives the nonlinear received-power expression by expanding moments of the received RF signal and evaluating the resulting index cases and expectations. The derivation concludes by substituting these components into the delivered-power formula.

B. PROOF OF PROPOSITION 1

This proof reduces the constrained optimization to one power-allocation variable and compares the concavity of mutual information with the convexity of delivered power. The resulting extrema establish the proposition’s allocation behavior.

C. PROOF OF PROPOSITION 3

The proof invokes special cases of the Riemann zeta function and alternating series to establish the required series result.

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