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Wireless Information and Power Transfer in Relay Systems with Multiple Antennas and Interference

Guangxu Zhu, Caijun Zhong, Himal A. Suraweera, George K. Karagiannidis, Zhaoyang Zhang, Theodoros A. Tsiftsis

arXiv:1501.05376v1cs.IT

TL;DR

This paper studies how multi-antenna energy-harvesting relays perform in dual-hop systems with and without co-channel interference. It analytically evaluates outage, capacity, diversity, and power splitting across three processing schemes, showing that interference can help or hurt depending on the scheme.

  • Problem

    The study addresses limited understanding of outage and capacity in multi-antenna energy-harvesting relays, particularly when co-channel interference supplies energy but also disrupts communication.

  • Method

    The paper analyzes dual-hop AF relaying with a power-splitting relay under no interference and single-interferer conditions using MRC/MRT, ZF/MRT, and MMSE/MRT.

  • Results

    MRC/MRT and MMSE/MRT achieve diversity order N, whereas ZF/MRT achieves N −1; MMSE/MRT benefits from stronger interference, while interference can degrade MRC/MRT and ZF/MRT.

  • Takeaways & Limitations

    Analytical outage, capacity, and power-splitting results provide design insights for selecting relay processing and power allocation under different interference conditions.

Abstract

from arXiv · show

In this paper, an energy harvesting dual-hop relaying system without/with the presence of co-channel interference (CCI) is investigated. Specifically, the energy constrained multi-antenna relay node is powered by either the information signal of the source or via the signal receiving from both the source and interferer. In particular, we first study the outage probability and ergodic capacity of an interference free system, and then extend the analysis to an interfering environment. To exploit the benefit of multiple antennas, three different linear processing schemes are investigated, namely, 1) Maximum ratio combining/maximal ratio transmission (MRC/MRT), 2) Zero-forcing/MRT (ZF/MRT) and 3) Minimum mean-square error/MRT (MMSE/MRT). For all schemes, both the systems outage probability and ergodic capacity are studied, and the achievable diversity order is also presented. In addition, the optimal power splitting ratio minimizing the outage probability is characterized. Our results show that the implementation of multiple antennas increases the energy harvesting capability, hence, significantly improves the systems performance. Moreover, it is demonstrated that the CCI could be potentially exploited to substantially boost the performance, while the choice of a linear processing scheme plays a critical role in determining how much gain could be extracted from the CCI.

I. INTRODUCTION … III. THE NOISE-LIMITED SCENARIO

The paper studies RF-powered dual-hop AF relaying with a multi-antenna energy-constrained relay, developing analytical performance results for noise-limited and interference-plus-noise settings. It models power splitting, linear relay processing, outage probability, ergodic capacity, diversity order, and the effects of CCI and system parameters.

  • I. INTRODUCTION: RF wireless energy transfer offers a controllable alternative to intermittent natural resources for powering communication devices.RF signals can carry both information and energy, motivating simultaneous wireless information and power transfer.
  • I. INTRODUCTION: The considered system is a dual-hop AF relay network with single-antenna source and destination and an N-antenna energy-constrained relay.The relay harvests ambient RF energy and uses it to forward source information under a power-splitting receiver architecture.
  • A. Noise-limited Case: For the noise-limited scenario, the paper derives an exact single-integral outage expression, a tight closed-form outage lower bound, a high-SNR approximation, and a tight closed-form ergodic-capacity upper bound.The high-SNR approximation shows diversity order N, where N is the number of relay antennas.
  • I. INTRODUCTION: For CCI, MRC/MRT and MMSE/MRT achieve diversity order N, whereas ZF/MRT achieves diversity order N −1.The paper provides tight closed-form outage lower bounds and capacity upper bounds for all three schemes and studies the outage-minimizing power-splitting ratio.
  • I. INTRODUCTION: CCI can significantly improve performance, but its gain depends heavily on linear processing: MMSE/MRT can always exploit CCI, whereas CCI is not always beneficial with MRC/MRT or ZF/MRT.The analytical expressions also enable rapid assessment of how η, N, ρ1, and ρI affect the optimal power-splitting ratio without time-consuming Monte Carlo simulations.
  • II. SYSTEM MODEL: The system model assumes no direct source-destination link, block-fading i.i.d. Rayleigh channels, no source CSI, full relay CSI, and local destination CSI.The communication uses two equal-duration time slots, and each relay antenna splits the received signal between energy harvesting and information processing.
  • B. Interference plus Noise Case: With a single dominant interferer at the relay, CCI contributes to the relay’s received signal and harvested energy, while the destination remains affected only by AWGN.The relay applies a transformation matrix W, whose choice determines system performance.
  • III. THE NOISE-LIMITED SCENARIO: In the noise-limited case, the relay uses MRC for reception and MRT for transmission, with harvested power determining its transmit-power constraint.The analysis targets outage probability, ergodic capacity, and the outage-minimizing power-splitting ratio θ.

A. Outage Probability · B. Ergodic Capacity

The section derives exact, bounded, and high-SNR outage results for multi-antenna energy-harvesting relaying, then develops a tight ergodic-capacity upper bound. It shows diversity order N while explaining the slower outage decay caused by random harvested relay power.

  • A. Outage Probability: Outage probability is defined as the probability that instantaneous SNR falls below the threshold γth.
  • A. Outage Probability: Theorem 1 gives the exact outage probability for an arbitrary number of relay antennas.
  • A. Outage Probability: The exact result reduces to the single-relay-antenna result in [20, Proposition 3], but its integral lacks a closed-form expression and can be evaluated efficiently.
  • A. Outage Probability: A lower bound and a high-SNR approximation are derived to make outage behavior more tractable and characterize achievable diversity order.
  • A. Outage Probability: Diversity order is N, matching the conventional constant-power relay, while Pout decays as ρ−N^1 and therefore converges more slowly.
  • A. Outage Probability: Random harvested relay transmit power causes higher outage probability than the conventional constant relay-power case.
  • B. Ergodic Capacity: Because exact ergodic-capacity evaluation is generally intractable, Theorem 3 derives an upper bound that is tight across the entire SNR range and avoids Monte Carlo simulations.

C. Optimization of the Parameter θ in High SNR Value … 1 F MRC

The paper optimizes the power-splitting ratio in the high-SNR regime and analyzes MRC/MRT under co-channel interference using outage bounds and approximations. It shows that MRC/MRT retains full diversity and that interference may either help or hurt performance.

  • C. Optimization of the Parameter θ in High SNR Value: The power-splitting ratio θ balances relay transmit power against source-relay transmission quality, so its optimal value is obtained by minimizing high-SNR outage probability.The impact of θ on ergodic capacity is deferred to numerical illustration in Section V.
  • C. Optimization of the Parameter θ in High SNR Value: The optimal θ is the unique root in (0,1) of the polynomial specified in Proposition 1.The proof establishes that the objective decreases before the root and increases afterward, making the root the global minimizer.
  • IV. THE INTERFERENCE PLUS NOISE SCENARIO: With a single dominant interferer, the optimal relay processing matrix W is defined by an end-to-end SINR maximization problem, but its non-convexity prevents an easy closed-form solution.The analysis therefore considers heuristic two-stage relay processing strategies with a rank-1 structure.
  • A. MRC/MRT Scheme: For MRC/MRT, the relay combiner matches the first-hop channel, while the scaling factor ω satisfies the relay transmit-power constraint and determines the end-to-end SINR.Because exact analysis is difficult, the subsequent treatment uses an outage lower bound and a high-SNR approximation.
  • 1) Outage Probability:: The MRC/MRT outage probability is first lower bounded, with Theorem 4 providing the bound for ρ1 ≠ ρI.Theorem 4 is presented as the key result for evaluating outage probability.
  • F MRC: In the high-SNR region, Theorem 5 gives a simpler approximation of MRC/MRT outage probability that supports characterization of the achievable diversity order.The general result assumes unequal source and interference powers; the equal-power case can be derived similarly.
  • P MRC: Full diversity order N remains achievable with MRC/MRT despite CCI, whose effect can be beneficial or detrimental depending on the relationship between the interference and system parameters.The paper also establishes an ergodic-capacity upper bound using similar techniques.

2) Ergodic Capacity: · 3) Optimal θ Analysis:

The ergodic-capacity analysis gives an upper bound for MRC/MRT when ρ1 = ρI and refers to capacity expressions in (32)–(35). The optimal θ analysis uses a high-SNR outage approximation, with a closed-form N = 1 solution whose monotonic trends follow relay-energy and first-hop SINR effects.

  • 2) Ergodic Capacity:: When ρ1 = ρI, Theorem 6 upper-bounds the ergodic capacity of the MRC/MRT scheme.
  • 2) Ergodic Capacity:: The ergodic-capacity results are given by expressions (32)–(35).
  • 3) Optimal θ Analysis:: The optimal θ minimizing outage probability is derived using the high-SNR approximation for P_MRC^out in (29).
  • 3) Optimal θ Analysis:: The optimal-θ result applies whether the signal power and CCI power are equal or unequal.
  • F MRC: For N = 1, the optimal solution has a closed-form expression.
  • F MRC: The optimal θ decreases with η and ρI but increases with ρ1.
  • F MRC: As η or ρI increases, smaller θ compensates for first-hop SINR limitations despite greater harvested energy.Higher η increases relay transmission power, whereas larger ρI both supplies energy and reduces first-hop SINR.
  • F MRC: With large ρ1, better first-hop quality makes allocating more energy to the relay, through larger θ, beneficial.

B. ZF/MRT Scheme · 1) Outage Probability:

The ZF/MRT relay uses multiple antennas to eliminate co-channel interference, requiring more relay antennas than interferers. Its outage probability is bounded and, at high SNR, reveals diversity order N − 1, one less than MRC/MRT because one degree of freedom removes interference.

  • B. ZF/MRT Scheme: ZF/MRT completely eliminates CCI by using multiple relay antennas, under the condition N > 1.The relay’s antenna count must exceed the number of interferers for zero forcing to be feasible.
  • B. ZF/MRT Scheme: The ZF/MRT end-to-end SINR is formulated from the relay’s zero-forcing combining and MRT transmission.The supplied derivation introduces the end-to-end SINR and the corresponding parameter γZF.
  • 1) Outage Probability:: An outage-probability lower bound is first established for the ZF/MRT scheme.The analysis separately introduces this lower bound before presenting the theorem for unequal source and interference powers.
  • 1) Outage Probability:: When ρ1 ≠ ρI, Theorem 7 lower bounds the ZF/MRT outage probability.The theorem states the bound for unequal source and interferer power parameters.
  • P LZF: In the high-SNR region, where ρ1 →∞, Theorem 8 provides an approximation for the ZF/MRT outage probability.The approximation is derived using the high-SNR expansion of the incomplete gamma function.
  • P LZF: The achievable diversity order of ZF/MRT is N −1.The result follows because the second asymptotic term is negligible relative to the first.
  • P LZF: ZF/MRT incurs a diversity loss of one relative to MRC/MRT because one degree of freedom is used to eliminate CCI.The paper describes this loss as an intuitive consequence of interference suppression.
  • P LZF: The subsequent analysis turns to the ergodic capacity of the ZF/MRT system and establishes an upper bound.This marks the transition from outage probability to capacity analysis.

2) Ergodic Capacity: … 1 F MMSE

The section derives capacity and outage characterizations for ZF/MRT and MMSE/MRT, then analyzes optimal power splitting and high-SNR behavior. MMSE/MRT balances interference suppression and noise enhancement, achieves diversity order N, and has strictly better outage performance than MRC/MRT.

  • 2) Ergodic Capacity:: Theorem 9 upper bounds the ergodic capacity of ZF/MRT when ρ1 ≠ ρI.The bound is stated under unequal source and interferer signal-to-noise ratios.
  • 2) Ergodic Capacity:: The optimal θ minimizing ZF/MRT outage probability is obtained from a high-SNR approximation and is a root of a specified polynomial.The analysis first uses the high-SNR approximation for P_ZF outage before characterizing θ.
  • C. MMSE/MRT Scheme: MMSE processing provides an optimum trade-off between incomplete CCI suppression and noise enhancement, unlike ZF, which fully eliminates CCI but can elevate noise.The MMSE scheme does not fully eliminate CCI, whereas ZF completely eliminates it at the relay.
  • 1) Outage Probability:: Theorem 10 lower bounds the outage probability of MMSE/MRT when ρ1 ≠ ρI.The result is based on the c.d.f. of γMMSE and subsequent derivation steps analogous to earlier theorems.
  • 1 F MMSE: In the high-SNR region, ρ1 →∞, Theorem 11 provides an approximation for the MMSE/MRT outage probability.The approximation is introduced to give further insight into outage behavior at high SNR.
  • P MMSE: The MMSE/MRT scheme achieves diversity order N, matching MRC/MRT.Theorem 11 explicitly identifies the diversity order as N.
  • P MMSE: MMSE/MRT always has strictly better outage performance than MRC/MRT because its first-term coefficient is strictly smaller and yields higher array gain.The schemes differ only in their first terms; aMMSE includes only the first two terms of aMRC.

2) Ergodic Capacity: … VI. CONCLUSION

The results validate tight analytical bounds, show how antennas, interference, distance, and power splitting shape performance, and establish scheme-specific diversity and CCI effects. An analytically characterized outage-optimal splitting ratio complements numerical capacity optimization.

  • 2) Ergodic Capacity:: When ρ1 = ρI, the MMSE/MRT ergodic capacity is upper bounded by the expression in Theorem 12.
  • 2) Ergodic Capacity:: The outage-optimal θ is a root of the equation in Proposition 4, with 0 < θ < 1.B is defined in (36).
  • V. NUMERICAL RESULTS AND DISCUSSION: The numerical study uses γth = 0 dB, η = 0.8, θ = 0.5, ρI = 9.5 dB, τ = 2, and d1 = d2 = dI = 1 unless specified otherwise.
  • A. Effect of Multiple Antennas: MMSE/MRT has the best outage performance; ZF/MRT outperforms MRC/MRT at low SNR, whereas MRC/MRT is better at high SNR.The proposed outage lower bounds are tight across the considered SNR range and nearly exact at high SNR.
  • A. Effect of Multiple Antennas: Increasing N improves ergodic capacity; MMSE/MRT performs best, ZF/MRT is slightly inferior, and their gap disappears as N increases.The proposed ergodic-capacity upper bounds are sufficiently tight across the considered SNR range.
  • B. Effect of CCI: CCI benefits MMSE/MRT monotonically, while ZF/MRT depends on interference power and strong CCI eventually harms MRC/MRT.
  • C. Effect of the Distance: In energy harvesting, the optimal relay location tends to be close to the source because the first-hop channel quality is more important.Increasing distance reduces received relay power and deteriorates ZF/MRT and MMSE/MRT performance.
  • D. Effect of Power Splitting Ratio θ: A unique θ optimizes outage or ergodic capacity: performance improves below it and gradually deteriorates above it; the outage-optimal θ decreases as η and N increase.Higher conversion efficiency and additional antennas reduce the signal portion needed for energy harvesting.

APPENDIX I PROOF OF COROLLARY 1 · APPENDIX II PROOF OF THEOREM 2 · APPENDIX III PROOF OF THEOREM 3

The appendices derive outage-probability bounds and asymptotic expressions from end-to-end SNR distributions, then establish an ergodic-capacity upper bound using SIMO channel capacities. The proofs rely on gamma-distribution properties, variable transformations, and incomplete-gamma-function expansions, with the high-SNR result identified as a tight approximation rather than a strict bound.

  • APPENDIX I PROOF OF COROLLARY 1: The proof first tightly upper-bounds the system end-to-end SNR and derives a corresponding lower bound on outage probability.The lower-bound calculation is then reduced to an evaluable expression.
  • APPENDIX I PROOF OF COROLLARY 1: The outage expression uses i.i.d. gamma random variables, a change of variable, and incomplete-gamma and binomial expansions to obtain the desired result.The derivation invokes standard gamma-function identities after rewriting the integral.
  • APPENDIX II PROOF OF THEOREM 2: For Theorem 2, the high-SNR end-to-end SNR is bounded and its outage analysis proceeds through the c.d.f. of an intermediate variable Y.The proof separately derives the c.d.f. of Y before computing the c.d.f. of the bounded SNR.
  • APPENDIX II PROOF OF THEOREM 2: The c.d.f. derivation is completed using incomplete-gamma-function asymptotics and additional special-function identities.These steps produce the closed-form expression used in the theorem.
  • APPENDIX II PROOF OF THEOREM 2: Because higher-order terms are omitted, the resulting outage expression is a tight asymptotic approximation rather than a bound and matches the exact value in the high-SNR region.The appendix relates the limiting outage probability to Prob(γ < γth) as ρ1 approaches infinity.
  • APPENDIX III PROOF OF THEOREM 3: Theorem 3 proves an upper bound on ergodic capacity by decomposing it into capacities associated with SIMO Rayleigh and SIMO keyhole channels.The two component capacities are taken from the cited prior results.
  • APPENDIX III PROOF OF THEOREM 3: The appendix then shows the relevant relationship among the capacity terms and combines the results to obtain the desired theorem expression.The final step is described as assembling all preceding components.

APPENDIX IV PROOF OF THEOREM 4 · P LMRC

The appendix proves Theorem 4 by evaluating an outage lower bound and deriving probabilities for I1 and I2. For P LMRC, it uses distributional calculations and a Monte Carlo-supported approximation when ρI is close to ρ1.

  • APPENDIX IV PROOF OF THEOREM 4: The proof begins by evaluating the outage lower bound from (26).This establishes the starting expression for the Theorem 4 derivation.
  • P LMRC: When ρI is close to ρ1, Monte Carlo simulations support approximating the relevant probability term across the whole SNR region.The same approximation was previously adopted in.
  • P LMRC: The approximation is illustrated in Fig. 7(a), which plots probability versus ρ1 in dB.Fig. 7 documents the approximations used in the proofs of Theorems 4 and 5.
  • P LMRC: For P LMRC, I1 is expressed using an exponential random variable.The derivation then applies the corresponding exponential distribution result.
  • P LMRC: For I1, binomial expansion and the cited equation are applied after establishing its exponential representation.These steps provide the probability expression needed in the proof.
  • P LMRC: I2 is expressed as a sum of two independent gamma random variables when ρ1 ≠ ρI.Its probability density function is then obtained from the cited gamma-variable result.
  • P LMRC: For I2, algebraic manipulation and [24, Eq. (8.432.7)] yield equation (81).Substituting equations (78) and (81) into (75) produces the desired result.

APPENDIX V PROOF OF THEOREM 5 · P MRC

The appendix derives a high-SNR approximation for the MRC outage probability as ρ1 →∞. The derivation neglects a comparatively negligible probability term and evaluates the resulting integrals using asymptotic expansions and variable transformations.

  • APPENDIX V PROOF OF THEOREM 5: As ρ1 →∞, the system outage probability is approximated in the high-SNR regime.The approximation is stated as the starting result for Theorem 5.
  • P MRC: The high-SNR approximation follows by treating one probability term as negligible compared with another as ρ1 increases.This comparison is justified through Fig 7(b).
  • P MRC: The resulting high-SNR outage-probability approximation is then given explicitly.The passage identifies this expression as the high-SNR approximation for outage probability.
  • P MRC: Conditioning on y1 = ρ1, the derivation applies the incomplete-gamma asymptotic expansion to obtain I2(γth).The conditioned expression is subsequently averaged over yI.
  • P MRC: After changing variables with y1 +x = t, the expression is rewritten using the binomial expansion and further averaged over y1.These transformations reduce the remaining derivation to integral evaluation.
  • P MRC: The integral I1 is solved using [24, Eq. (6.455.1)] and [24, Eq. (9.131.1)], yielding the desired result after substitution.Expressions (84) and (90) are substituted into (83).

APPENDIX VI PROOF OF THEOREM 6

The appendix derives the ergodic-capacity upper bound for the MRC scheme by evaluating the relevant terms involving the c.d.f. of γ_MRC and integral formulas. Following analogous appendix procedures, the resulting expression for I2 is obtained in closed form.

  • The ergodic-capacity upper bound is formulated similarly to the proof of Theorem 3.
  • The remaining derivation evaluates C_γMRC through terms I1 and I2, using the c.d.f. of γ_MRC and an integral formula.The I1 calculation follows the method of, while I2 uses integral formula [36, Eq. (2.6.2)].
  • I2 is ultimately expressed in closed form by following the approach used in Appendix III.

C. Calculation of E · D. Calculation of E · m m X

The section derives expectation terms through alternative expressions, substitutions, probability-density functions, and standard integral identities. It separately addresses the calculation of E and the remaining expectation involving ln Z.

  • C. Calculation of E: I1 is introduced as a quantity to be computed in the calculation of E.
  • C. Calculation of E: An identity from [24, Eq. (9.211.4)] is used in the derivation.
  • C. Calculation of E: Expression (99) is adopted as a convenient alternative form of (98).
  • C. Calculation of E: The calculation of E continues by expressing I1 in computable form and establishing an intermediate result.
  • C. Calculation of E: After substitution into (100), the expectation of ln γMRC is evaluated to obtain I1.
  • D. Calculation of E: The remaining task in the second calculation is to determine E(ln Z), using the p.d.f. of Z given in (80) to formulate (104).
  • D. Calculation of E: The integral in (104) is solved by invoking [24, Eq. (4.352.1)].
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