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Throughput Analysis and Optimization of Wireless-Powered Multiple Antenna Full-Duplex Relay Systems

Mohammadali Mohammadi, Batu K. Chalise, Himal A. Suraweera, Caijun Zhong, Gan Zheng, Ioannis Krikidis

arXiv:1602.00145v1cs.IT

TL;DR

The paper addresses how a wireless-powered full-duplex MIMO relay can optimize beamforming and time switching for higher throughput despite loop interference. It develops optimum and suboptimum schemes, analytically characterizes outage and delay-constrained throughput, and finds that the best processing choice depends on interference cancellation and SNR.

  • Problem

    Wireless-powered full-duplex relays must balance energy harvesting, information transmission, time allocation, and loop-interference suppression to improve throughput.

  • Method

    The paper jointly designs relay receive and transmit beamformers with the time-split parameter, using optimum, SDR-based, zero-forcing, maximum-ratio combining, and maximum-ratio transmission schemes.

  • Results

    MRC/MRT can outperform RZF and TZF when loop interference is strongly canceled, whereas ZF precoders provide better outage performance at high SNRs.

  • Takeaways & Limitations

    Beamforming can improve both relay energy harvesting and loop-interference suppression, while processing scheme and time split critically determine full-duplex gains.

Abstract

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We consider a full-duplex (FD) decode-and-forward system in which the time-switching protocol is employed by the multi-antenna relay to receive energy from the source and transmit information to the destination. The instantaneous throughput is maximized by optimizing receive and transmit beamformers at the relay and the time-split parameter. We study both optimum and suboptimum schemes. The reformulated problem in the optimum scheme achieves closed-form solutions in terms of transmit beamformer for some scenarios. In other scenarios, the optimization problem is formulated as a semi-definite relaxation problem and a rank-one optimum solution is always guaranteed. In the suboptimum schemes, the beamformers are obtained using maximum ratio combining, zero-forcing, and maximum ratio transmission. When beamformers have closed-form solutions, the achievable instantaneous and delay-constrained throughput are analytically characterized. Our results reveal that, beamforming increases both the energy harvesting and loop interference suppression capabilities at the FD relay. Moreover, simulation results demonstrate that the choice of the linear processing scheme as well as the time-split plays a critical role in determining the FD gains.

I. INTRODUCTION

The paper studies a wireless-powered full-duplex MIMO decode-and-forward relay that uses multiple antennas, beamforming, and time switching to improve throughput while suppressing loop interference. It develops optimum and suboptimum processing schemes and analyzes instantaneous, outage, and delay-constrained performance.

  • System motivation and model: A multiple-antenna full-duplex relay harvests wireless energy from the source, then forwards information using a time-switching protocol.The relay uses receive and transmit antenna arrays to accumulate energy and enable spatial loop-interference cancellation.
  • Main contributions: The paper proposes optimum and suboptimum beamforming schemes to maximize instantaneous throughput while jointly selecting relay beamformers and the time-split parameter.The optimum formulation is solved efficiently despite joint optimization over the time split and transmit beamformer.
  • Optimum scheme: The optimum formulation has closed-form transmit-beamformer solutions in some scenarios and an SDR formulation whose rank-one optimum can always be guaranteed or recovered.This converts otherwise nonconvex beamformer design into a convex feasibility problem in the applicable cases.
  • Suboptimum schemes: Suboptimum designs use zero-forcing, maximum-ratio combining, and maximum-ratio transmission, with high-SNR outage expressions characterizing diversity order and array gain.The paper also derives analytical expressions useful for studying outage probability and delay-constrained throughput.
  • System assumptions: The system assumes imperfect loop-interference cancellation, Rayleigh fading, and a harvest-use relay powered entirely through wireless energy transfer.The residual interference channel is modeled statistically, while reported simulations assume estimated and actual channels coincide, providing theoretical bounds for practical design.
  • Scope and limitations: Circuit and interference-cancellation processing power are excluded, although they may significantly affect full-duplex operation and remain an open analysis direction.The paper notes that harvested power may be usable in realistic circuitry, but does not model these consumption costs.

III. JOINT RECEIVE/TRANSMIT BEAMFORMING

For fixed time splitting, the relay design maximizes the minimum two-hop SINR through optimum or suboptimum beamforming. The optimum formulation uses closed-form solutions in some cases and SDR with guaranteed rank-one optimality in the remaining cases.

  • The beamforming problem maximizes the minimum of the two-hop SINRs for a fixed time-split parameter.
  • When f1(wmSINR) ≤ f2(wmSINR), wmSINR is the optimum precoder.
  • When f2(wMRT) ≤ f1(wMRT), wMRT is the optimum transmit precoder.
  • For the remaining cases, no closed-form solution is available and the problem is solved through convex feasibility after SDR.The largest feasible auxiliary value produces the relaxed optimum Wt.
  • The optimum approach uses an SDR formulation, and its relaxation always yields a rank-one optimum solution.This makes the relaxed problem equivalent to the original beamforming problem.

B. TZF Scheme

The TZF scheme uses receive-side maximum ratio combining and transmit-side zero forcing to cancel residual loop interference. Its transmit beamformer is obtained by projecting the destination channel into the transmit-side null space of the residual loop-interference channel.

  • TZF completely cancels loop interference when the relay has more than one transmit antenna and applies MRC at its input.The feasibility condition is MT > 1.
  • The TZF transmit beamformer maximizes the destination-channel gain after projection onto the null space enforcing the zero-forcing constraint.

C. RZF Scheme

The RZF scheme uses maximum ratio transmission while selecting a receive combiner that zero-forces residual loop interference. Compared with the other schemes, MRC/MRT needs less channel information and performs well under mild loop interference.

  • RZF sets the transmit beamformer by MRT and chooses the receive combiner to satisfy the residual loop-interference zero-forcing constraint.RZF requires more than one receive antenna for feasibility.
  • MRC/MRT requires only hSR and hRD, whereas optimum, TZF, and RZF also require knowledge of HRR.MRC/MRT exhibits very good performance under mild loop-interference effects.
  • The instantaneous-throughput design balances energy-harvesting duration against information-transmission duration through the optimized time-split parameter α.Longer harvesting increases second-hop SNR but leaves less time for information transmission.

A. Optimum Scheme

For the optimum scheme, the instantaneous rate is optimized analytically over the time-split parameter α. The resulting solution uses the Lambert W function in one case and a boundary value when the rate decreases with α.

  • The optimum instantaneous rate as a function of α can be solved analytically using the Lambert W function.The function satisfies W(x)exp(W(x)) = x.
  • The optimal time portion α is obtained by differentiating ROpt(α) and applying the stated analytical procedure.
  • When the derivative of ROpt(α) is strictly negative, the optimal α is the boundary value specified by the resulting expression.

B. TZF Scheme

The TZF section derives instantaneous-throughput expressions and optimizes the time split, while the broader optimum design uses either one-dimensional search or alternating optimization with beamformer updates.

  • B. TZF Scheme: The TZF instantaneous throughput is obtained after substituting its beamformers into the general throughput expression, with the optimal time split found from a dedicated optimization problem.
  • B. TZF Scheme: The MRC/MRT time-split optimization is analytically solvable, and its throughput decreases with α in the relevant case, yielding a boundary optimum.
  • B. TZF Scheme: The optimum scheme requires joint optimization because its transmit beamformer depends on α, using either global line search or iterative alternating optimization.
  • B. TZF Scheme: The optimum algorithm updates α and the transmit beamformer iteratively, recovering the beamformer from a rank-one matrix when the SDR solution is rank one.
  • B. TZF Scheme: For line search, the algorithm evaluates candidate α values and selects the beamformer–time-split pair with the maximum objective value.

V. DELAY-CONSTRAINED THROUGHPUT

The delay-constrained analysis expresses throughput through outage probability and studies exact or numerical outage characterization alongside high-SNR approximations for diversity analysis.

  • V. DELAY-CONSTRAINED THROUGHPUT: Delay-constrained throughput depends on the outage probability that the instantaneous SINR falls below γth = 2^Rc −1.
  • V. DELAY-CONSTRAINED THROUGHPUT: Closed-form outage analysis is difficult for the optimum scheme, so its delay-constrained throughput is evaluated through simulations.
  • V. DELAY-CONSTRAINED THROUGHPUT: The TZF outage integral does not admit a closed-form expression but can be evaluated numerically.
  • V. DELAY-CONSTRAINED THROUGHPUT: For TZF, high-SNR outage approximations characterize diversity order, with one degree of freedom consumed by interference cancellation.
  • V. DELAY-CONSTRAINED THROUGHPUT: The TZF diversity order is min(MR, MT −1), and its high-SNR behavior can converge more slowly than in the constant-power case.

B. RZF Scheme

The RZF analysis derives its SINR and outage distributions, then gives a high-SNR approximation showing how receive-side interference cancellation affects diversity.

  • B. RZF Scheme: The RZF outage distribution is constructed from transformed channel variables and their chi-square and beta distributions.
  • B. RZF Scheme: The exact RZF outage expression lacks a closed-form solution but can be evaluated efficiently numerically.
  • B. RZF Scheme: The RZF high-SNR outage probability is approximated analytically for ρ1, ρ2 →∞.
  • B. RZF Scheme: The RZF diversity order is min(MR −1, MT), because one receive degree of freedom is allocated to loop-interference cancellation.

C. MRC/MRT Scheme

The MRC/MRT analysis treats special antenna configurations because arbitrary-dimension outage analysis is cumbersome, and examines high-SNR behavior and time-split optimization.

  • C. MRC/MRT Scheme: For arbitrary MT and MR, MRC/MRT outage analysis is cumbersome, so the section focuses on cases MT = 1 or MR = 1.
  • C. MRC/MRT Scheme: With MT = 1, residual loop interference produces an outage floor and zero-diversity behavior for MRC/MRT.
  • C. MRC/MRT Scheme: For MR = 1, the section derives a high-SNR outage approximation after expressing the MRC/MRT SINR through transformed channel variables.
  • C. MRC/MRT Scheme: The MRC/MRT outage expressions may require numerical evaluation or truncated series approximations when closed forms are unavailable.
  • C. MRC/MRT Scheme: The delay-constrained throughput has an interior optimal α because increasing α lowers outage probability while reducing the Rc(1−α) throughput ceiling.

VI. NUMERICAL RESULTS AND DISCUSSION

Simulations compare optimum and suboptimum beamforming under instantaneous throughput, outage probability, and delay-constrained throughput criteria. The optimum scheme generally performs best, while the relative value of zero-forcing and MRC/MRT depends on time-split, SNR, antenna configuration, and loop-interference strength.

  • A. Instantaneous Throughput: The optimum scheme outperforms all other schemes across the evaluated time-split values for instantaneous throughput.Increasing receive antennas or source transmit power reduces the optimal energy-harvesting time because the relay harvests the required energy faster.
  • B. Outage Probability: Additional receive antennas can increase harvested energy and second-hop SNR, while equal transmit and receive antenna counts give RZF higher array gain.The comparison indicates that antenna configuration and beamforming choice must be selected jointly.
  • B. Outage Probability: At low-to-medium SNRs, MRC/MRT nearly matches the optimum outage performance, whereas ZF-based schemes approach the optimum at high SNRs.With fixed α, MRC/MRT develops a high-SNR outage floor because additional harvested energy strengthens loop interference; adjusting α can improve it.
  • C. Delay-Constrained Throughput: 0.557, 0.549, 0.453, and 0.404 are the optimized delay-constrained throughputs for optimum, RZF, MRC/MRT, and TZF, respectively.Each suboptimum scheme can surpass another for some α, exposing a performance–complexity trade-off.
  • C. Delay-Constrained Throughput: ZF-based throughput remains constant as loop-interference strength increases, whereas MRC/MRT throughput decreases.At low interference, a 2×2 MRC/MRT configuration approaches optimum performance; a 3×1 configuration performs worse than 2×2 across schemes.
  • C. Delay-Constrained Throughput: The simulations provide theoretical performance bounds because they assume estimated and actual loop-interference channels coincide and omit detailed RF circuitry effects.The paper notes that amplifier inefficiency could cause relay inactivity and high outage probability.

APPENDIX A PROOF OF PROPOSITION 2

The appendix proves that the semidefinite-relaxation formulation preserves optimality through a rank-one transmit solution. When the relaxed optimizer has higher rank, a rank-one solution can be recovered without loss of objective value.

  • APPENDIX A PROOF OF PROPOSITION 2: Complementary slackness places the optimal transmit matrix in the null-space of the dual matrix, linking its rank to the dual matrix’s nullity.The condition tr(YWt)=0 implies YWt=0 under the positive-semidefinite constraints.
  • APPENDIX A PROOF OF PROPOSITION 2: Cases a and c force the dual structure to have a one-dimensional null-space, which directly yields a rank-one transmit solution.The argument excludes multiplier choices that would make the dual matrix full rank or violate positive semidefiniteness.
  • APPENDIX A PROOF OF PROPOSITION 2: The SDR optimizer is rank one in Cases a and c, while Case b permits higher rank but still admits rank-one recovery without loss of optimality.The proof uses the KKT conditions, null-space structure, and eigenvalue decomposition of the relaxed transmit matrix.
  • APPENDIX A PROOF OF PROPOSITION 2: In Case b, the objective value is independent of the auxiliary parameter, allowing a suitable eigenvector to produce an equivalent rank-one transmit matrix.The recovered matrix selects an eigenvector and sets its corresponding coefficient to one without changing the optimum objective.

APPENDIX B PROOF OF PROPOSITION 5

The appendix derives closed-form and high-SNR outage expressions by expanding incomplete gamma functions and simplifying the resulting distributions. These expansions produce the asymptotic formulas used for outage characterization.

  • APPENDIX B PROOF OF PROPOSITION 5: High-SNR outage expressions follow by retaining leading terms after incomplete-gamma series expansions as ρ1 and ρ2 grow.Higher-order terms are omitted in the high-SNR regime to obtain the desired asymptotic result.
  • APPENDIX B PROOF OF PROPOSITION 5: Substituting the gamma-function expansions into the preceding outage expression and retaining sufficient terms yields equation (51).The derivation applies standard series identities and discards higher-order contributions.
  • APPENDIX B PROOF OF PROPOSITION 5: The outage calculation factors independent variables by deriving the CDF of Z3 = ∥hSR∥2|hRD|2 and combining it with FX2(·).Substitution of these distribution expressions produces the stated result.
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