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Relay Selection for Simultaneous Information Transmission and Wireless Energy Transfer: A Tradeoff Perspective

Diomidis S. Michalopoulos, Himal A. Suraweera, Robert Schober

arXiv:1303.1647v2cs.ITmath.IT

TL;DR

When independently fading channels favor different relays for information and energy, relay selection must trade off receiver performance against harvested energy. The paper proposes selection schemes for any number of relays and an optimal method for two; its results distinguish the schemes by capacity and outage performance.

  • Problem

    The relay best for information transmission need not provide the strongest energy-transfer channel, so choosing a relay entails a tradeoff between receiver information quality and energy delivered to the harvester.

    Source: §I · p. 3

  • Method

    The paper develops time-sharing and threshold-checking selection for N relays and an optimal capacity/outage selection policy for two relays.

    Source: §VII · p. 23

  • Results

    Threshold-checking achieves better capacity than time-sharing for a given required energy transfer, while the two-relay policy yields optimal capacity and outage probability for a given energy transfer.

    Source: §VII · p. 23

  • Takeaways & Limitations

    The two general schemes differ by performance criterion: threshold-checking favors capacity, while time-sharing can outperform it for outage probability under the stated SNR condition.

    Source: §VII · p. 23

Abstract

from arXiv · show

In certain applications, relay terminals can be employed to simultaneously deliver information and energy to a designated receiver and a radio frequency (RF) energy harvester, respectively. In such scenarios, the relay that is preferable for information transmission does not necessarily coincide with the relay with the strongest channel to the energy harvester, since the corresponding channels fade independently. Relay selection thus entails a tradeoff between the efficiency of the information transfer to the receiver and the amount of energy transferred to the energy harvester. The study of this tradeoff is the subject on which this work mainly focuses. Specifically, we investigate the behavior of the ergodic capacity and the outage probability of the information transmission to the receiver, for a given amount of energy transferred to the RF energy harvester. We propose two relay selection methods that apply to any number of available relays. Furthermore, for the case of two relays, we develop the optimal relay selection method in a maximum capacity / minimum outage probability sense, for a given energy transfer constraint. A close-to-optimal selection method that is easier to analyze than the optimal one is also examined. Closed-form expressions for the capacity-energy and the outage-energy tradeoffs of the developed schemes are provided and corroborated by simulations. Interesting insights on the aforementioned tradeoffs are obtained.

I. INTRODUCTION

Relay selection must balance information delivery against energy transfer because independently fading channels can favor different relays. The paper analyzes this tradeoff, proposes schemes for any number of relays, and develops an optimal policy for two relays.

I. INTRODUCTION

  • Relay-assisted communication is motivated by quality-of-service improvement when the source and destination are far apart, while large path loss also constrains long-distance wireless energy transfer.The paper considers relays serving a receiver and a separate RF energy harvester.Source: §I · p. 2
  • Independent fading can make the relay best for information transmission differ from the relay delivering the most energy, creating a selection tradeoff.The relay choice exchanges information quality for energy-transfer efficiency.Source: §I · p. 3
  • For N relays, the paper studies time-sharing and threshold-checking schemes; for N = 2, it develops an optimal policy and a close-to-optimal alternative.The two-relay policy optimizes capacity and/or outage probability for a given energy transfer; the alternative behaves approximately optimally in certain regions.Source: §I · p. 3

A. System Model

  • The considered system uses half-duplex decode-and-forward relays, with one relay selected per frame to carry information and energy.Terminals have single antennas; direct source-to-destination information and source-to-harvester energy transfer are assumed absent because of large path loss and/or shadowing.Source: A. System Model · pp. 4–5
  • The analysis assumes independent, identically distributed Rayleigh fading across channels, while noting that independent non-identical fading can also be accommodated.The source and relays transmit with power P.Source: A. System Model · p. 4

B. Relay Selection: General Description

A centralized unit uses channel-state information to select one relay, balancing receiver-link reliability against energy delivered to the harvester. The section also sets out how the paper characterizes tradeoffs and their efficient frontier.

B. Relay Selection: General Description

  • Each transmission frame selects one relay; a centralized unit gathers link CSI and chooses the relay to compromise between information reliability and energy transfer.The selected relay is denoted R_s.Source: B. Relay Selection: General Description · p. 5
  • R_κ maximizes the receiver SNR, whereas R_λ maximizes energy transfer to the harvester; because they may differ, relay selection creates a tradeoff.The two criteria can point to different relays.Source: B. Relay Selection: General Description · p. 5

C. Preliminaries of Tradeoff Analysis

  • The paper defines opportunity cost as the absolute change in one metric corresponding to a given change in the other.A linear curve has equal opportunity cost, a strictly convex curve decreasing opportunity cost, and a strictly concave curve increasing opportunity cost.Source: C. Preliminaries of Tradeoff Analysis · pp. 6–7
  • A Pareto-efficient allocation cannot improve one metric without reducing the other, and the Pareto frontier gives the largest feasible value of either metric for a fixed value of the other.The paper's tradeoff factor δ specifies the operating point along the curve and the priority assigned to one metric.Source: C. Preliminaries of Tradeoff Analysis · pp. 6–7

III. TRADEOFF BETWEEN ERGODIC CAPACITY AND AVERAGE TRANSFERRED ENERGY

The section analyzes ergodic capacity against average harvested energy for relay-selection schemes with different CSI requirements. It characterizes the tradeoff boundaries and finds equal opportunity cost for time-sharing but increasing opportunity cost for threshold-checking.

III. TRADEOFF BETWEEN ERGODIC CAPACITY AND AVERAGE TRANSFERRED ENERGY

  • The paper analyzes time-sharing, threshold-checking, and weighted-difference schemes, which have different CSI requirements and therefore suit different CSI-availability scenarios.The section compares their capacity-energy tradeoffs.Source: §III · p. 7
  • The minimum and maximum achievable capacity and energy-transfer boundaries are identical for all three schemes.These boundaries are examined before the specific tradeoff curves.Source: §III · p. 7

1) Ergodic capacity:

3) Tradeoff factor:

  • The tradeoff factor δ ranges from 0 to 1 and expresses the fraction of the possible energy-transfer range that is actually transferred to the harvester.Average transferred energy is represented as ε = [1 + δ(H_N − 1)] ε̄.Source: 3) Tradeoff factor · p. 8

B. Time-Sharing Selection Scheme

  • Time-sharing selects between the information-maximizing and energy-maximizing relays with a probability that controls their long-run allocation.The scheme's average energy transfer is expressed as a function of its selection probability.Source: B. Time-Sharing Selection Scheme · pp. 8–9
  • Time-sharing produces an equal opportunity-cost tradeoff between ergodic capacity and average energy transfer.Its capacity is linear in the average energy transfer.Source: B. Time-Sharing Selection Scheme · p. 9

C. Threshold-Checking Selection Scheme

  • The threshold-checking scheme has an increasing opportunity-cost tradeoff between ergodic capacity and average energy transfer.The scheme compares the SNR of the relay maximizing information transmission with a threshold to decide whether to use it.Source: C. Threshold-Checking Selection Scheme · pp. 9–10

D. Weighted Difference Selection Scheme

For two relays, weighted-difference selection balances information and energy performance through a tunable weighting factor. Its capacity–energy tradeoff has increasing opportunity cost and is reported to lie close to the Pareto frontier, but the scheme applies only to N = 2.

1) Intuition Behind the Weighted Difference Scheme:

  • When one relay is better on both information and energy links, it is selected; otherwise, the weighting factor ν determines which link receives priority.Increasing ν can achieve a desired tradeoff factor.Source: 1) Intuition Behind the Weighted… · p. 11

2) Tradeoff Expression:

  • The weighted-difference scheme’s capacity–energy tradeoff is characterized by a proposition and has increasing opportunity cost.The supplied passage states that the capacity–energy tradeoff is given, but does not include its expression.Source: 2) Tradeoff Expression · p. 11
  • The scheme’s capacity–energy tradeoff is close to the Pareto frontier, but the method applies only when N = 2.The section notes that an extension to N > 2 is not straightforward because it would require multiple comparisons of candidate relays’ capacity and energy contributions.Source: 2) Tradeoff Expression · p. 12

E. On the CSI Requirements of the Proposed Schemes

The schemes differ in how much channel-state information they require. For outage analysis, the paper considers a fixed transmission rate and expresses outage through the selected relay’s end-to-end SNR relative to a threshold.

E. On the CSI Requirements of the Proposed Schemes

  • Time-sharing has the lowest CSI requirement, threshold-checking has a medium requirement, and weighted-difference selection has a high requirement.Time-sharing needs CSI for either the S-R-D or R-H links; weighted-difference selection needs continuous CSI for all S-R-D and R-H links.Source: E. On the CSI Requirements of the… · p. 12

A. Outage Probability

  • For fixed-rate transmission, outage occurs when the active relay’s end-to-end SNR falls below the threshold γth = 2^r − 1.The analysis gives outage probabilities for the three schemes at a prescribed energy transfer to H.Source: A. Outage Probability · p. 13

1) Outage Probability of the Time-Sharing Scheme:

2) Outage Probability of the Threshold-Checking Scheme:

  • For threshold-checking, when τ ≤ γth outage depends on γκ; when τ > γth, outage occurs if the selected relay’s SNR is below γth while all other relays’ SNRs are below τ.The resulting outage expression can also be written as a function of the tradeoff factor.Source: 2) Outage Probability of the… · pp. 13–14

3) Outage Probability of the Weighted Difference Scheme:

The paper gives an outage-probability expression for weighted-difference selection and compares the schemes’ high-SNR behavior. The asymptotic outage ranking differs from their ergodic-capacity comparison.

3) Outage Probability of the Weighted Difference Scheme:

  • The paper provides an expression for the weighted-difference scheme’s outage probability, including a form in terms of the tradeoff factor.The supplied passages do not state the expression’s values.Source: 3) Outage Probability of the Weighted… · p. 14

B. Asymptotic Analysis

  • For N = 2 and any tradeoff factor, weighted-difference selection has the best asymptotic outage performance, time-sharing is intermediate, and threshold-checking is worst.For ergodic capacity, threshold-checking outperforms time-sharing.Source: B. Asymptotic Analysis · p. 15
  • The diversity order of all three schemes is one unless the tradeoff factor is zero; any energy transfer above its lower boundary causes this diversity gain to be lost.The stated loss applies even when the required increase above the lower boundary is infinitesimally small.Source: B. Asymptotic Analysis · p. 15

V. PARETO EFFICIENCY FOR N = 2

For two relays, the paper derives a Pareto-efficient selection policy for long-term information metrics and energy transfer. The capacity frontier requires numerical methods, while the outage tradeoff is Pareto efficient under the stated selection policy.

V. PARETO EFFICIENCY FOR N = 2

  • A selection policy based on a non-decreasing instantaneous metric F(γi) achieves the Pareto frontier between long-term information performance and average energy transfer.F may represent metrics such as ergodic capacity, average SNR, probability of no-outage, or probability of correct bit detection.Source: §V · pp. 15–16

A. Optimal Ergodic Capacity for a Given Energy Transfer

  • For ergodic capacity, the Pareto-frontier curve is obtainable only through numerical methods because its derivation involves a two-fold integral.The passage states that the integral is solvable only by numerical methods, to the authors’ knowledge.Source: A. Optimal Ergodic Capacity for a… · p. 16

B. Optimal Outage Probability for a Given Energy Transfer

VI. NUMERICAL EXAMPLES

The numerical examples examine ergodic-capacity and outage tradeoffs for a required average energy transfer. They compare relay-selection schemes for two relays and also report outage behavior for three relays.

A. Ergodic Capacity vs. Energy Transfer

B. Outage and No-Outage Probability vs. Energy Transfer

  • In the no-outage-versus-energy tradeoff, weighted difference outperforms threshold-checking and time-sharing except at small δ; threshold-checking is Pareto efficient at small δ but degrades at large δ.The Pareto-efficient scheme attains maximum feasible no-outage probability at its lower boundary; a small increase in ζ can add transferred energy with no outage cost.Source: B. Outage and No-Outage Probability… · p. 20, Figure 6 · p. 22
  • At medium and high normalized SNR (¯γ/γth), threshold-checking is inferior to all counterparts; for ¯γ/γth < 5 dB, it outperforms time-sharing and weighted-difference and approaches Pareto-efficient behavior.Source: B. Outage and No-Outage Probability… · p. 22
  • For N = 3, in the reported outage behavior, time-sharing outperforms threshold-checking at high values of normalized SNR (γ̄/γth), while threshold-checking performs slightly better at low values.Setting δ = 0.01, reflecting even a slight increase in required energy transfer to H, severely deteriorates asymptotic outage performance.Source: B. Outage and No-Outage Probability… · p. 22
  • For N = 2 and δ ≠ 0, time-sharing, threshold-checking, and weighted difference have unit diversity order, while the Pareto-efficient scheme has diversity order 2 except at δ = 1.Fig. 7 plots outage probability against normalized average SNR per link.Source: B. Outage and No-Outage Probability… · p. 21, Figure 7 · p. 23

VII. CONCLUSIONS

The paper analyzes the information–energy tradeoff for i.i.d. Rayleigh fading and develops relay-selection schemes for general relay counts and for two relays. For two relays, the Pareto-efficient policy is optimal under the stated energy or information-performance constraints, while the schemes’ relative performance depends on the metric and operating conditions.

VII. CONCLUSIONS

  • For N candidate relays, the paper develops time-sharing and threshold-checking selection schemes and analyzes the tradeoff for i.i.d. Rayleigh fading channels.Source: §VII · p. 23
  • Threshold-checking achieves better capacity than time-sharing for a given required energy transfer, while time-sharing outperforms threshold-checking in outage probability when normalized average SNR exceeds 5 dB; the outage ranking reverses at low SNR.Source: §VII · p. 23
  • For N = 2, the Pareto-efficient policy yields optimum capacity and outage probability for a given energy transfer, and maximum energy transfer under a capacity or outage constraint.The weighted-difference scheme performs similarly and is easier to analyze.Source: §VII · pp. 23–24
  • Including energy-transfer links in relay selection (δ > 0) loses diversity gain unless the Pareto-efficient policy also includes information links (δ < 1).Source: §VII · p. 24
  • Near δ ≈ 0 and δ ≈ 1, the Pareto-efficient and weighted-difference schemes achieve substantial capacity and/or outage improvement with relatively little energy-transfer cost.Source: §VII · p. 24

APPENDIX A

Appendices A–C derive capacity and energy-transfer expressions and detail the weighted-difference selection cases. They also delimit the capacity minimum to opportunistic selection policies.

APPENDIX A

  • For decode-and-forward relaying, the information rate is limited by the weaker source–relay or relay–destination link; the maximum capacity uses the strongest candidate end-to-end SNR, while the minimum considered capacity does not exploit link CSI for selection.The maximum-capacity calculation uses the distribution of the maximum of exponentially distributed SNRs; the minimum reduces to the single-relay expression.Source: Appendix A · pp. 24–25

APPENDIX B

  • The analysis excludes deliberately selecting a relay known to have weak channels, because that is not opportunistic relay selection.Source: DERIVATION OF (17) · p. 25

APPENDIX C

  • The weighted-difference scheme’s two-relay policy selects between four cases based on the ordering of the relays’ information SNRs and harvested energies, then derives average energy transfer and capacity.The appendix uses the case policy to obtain the scheme’s average energy-transfer function and capacity expression.Source: Appendix C · p. 26

APPENDIX D

Appendix D derives the weighted-difference scheme’s outage probability from the four selection cases used in Appendix C. It expresses the result in terms of the scheme’s energy threshold and the average transferred energy.

APPENDIX D

  • The weighted-difference scheme’s outage probability is constructed from the four relay-selection cases developed in Appendix C.Source: Appendix D · p. 28
  • The derivation simplifies the case-based outage expression before substituting the energy-threshold relation.Source: Appendix D · pp. 28–29
  • The resulting outage probability is expressed as a function of the weighted-difference energy threshold and average transferred energy.Source: Appendix D · p. 29

APPENDIX E

Appendix E formulates two-relay selection as maximizing an average performance metric under an energy-transfer constraint. A binary relay choice and nonnegative multipliers yield the per-session selection rule.

APPENDIX E

  • The optimization chooses between R1 and R2 in each transmission session using a binary variable, and maximizes average performance subject to energy-transfer constraints.Source: Appendix E · p. 29
  • The formulation uses each relay’s SNR and harvested energy for the session and introduces nonnegative Lagrange multipliers for the constraints.Source: Appendix E · p. 29
  • Setting the derivative of the Lagrangian to zero yields a binary relay-selection rule based on the differences in performance and harvested energy.Source: Appendix E · pp. 29–30
  • Because performance is non-decreasing in SNR, the derived policy also maximizes average energy transfer for a given performance metric.Source: Appendix E · p. 30

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