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Wireless Information and Energy Transfer for Two-Hop Non-Regenerative MIMO-OFDM Relay Networks

Ke Xiong, Pingyi Fan, Chuang Zhang, Khaled Ben Letaief

arXiv:1412.8305v1cs.IT

TL;DR

The paper studies how an energy-constrained relay can support two-hop non-regenerative MIMO-OFDM transmission through simultaneous wireless information and energy transfer. It proposes TSR and PSR, formulates joint end-to-end rate-maximization problems, and finds distinct relay-position effects, with PSR consistently outperforming TSR.

  • Problem

    Prior work did not jointly address harvest-and-use SWIET in two-hop MIMO-OFDM relaying with an energy-harvesting relay.

  • Method

    The paper proposes TSR and PSR and jointly optimizes their resource-allocation and relay-configuration parameters to maximize end-to-end achievable information rate.

  • Results

    PSR always outperforms TSR, while PSR rate decreases monotonically with source-relay distance and TSR is relatively worst when the relay is midway.

  • Takeaways & Limitations

    Relay placement strongly affects achievable information rate, and PSR is the better-performing protocol in the studied MIMO-OFDM relaying system.

Abstract

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This paper investigates the simultaneous wireless information and energy transfer for the non-regenerative multipleinput multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) relaying system. By considering two practical receiver architectures, we present two protocols, time switchingbased relaying (TSR) and power splitting-based relaying (PSR). To explore the system performance limit, we formulate two optimization problems to maximize the end-to-end achievable information rate with the full channel state information (CSI) assumption. Since both problems are non-convex and have no known solution method, we firstly derive some explicit results by theoretical analysis and then design effective algorithms for them. Numerical results show that the performances of both protocols are greatly affected by the relay position. Specifically, PSR and TSR show very different behaviors to the variation of relay position. The achievable information rate of PSR monotonically decreases when the relay moves from the source towards the destination, but for TSR, the performance is relatively worse when the relay is placed in the middle of the source and the destination. This is the first time to observe such a phenomenon. In addition, it is also shown that PSR always outperforms TSR in such a MIMO-OFDM relaying system. Moreover, the effect of the number of antennas and the number of subcarriers are also discussed.

I. INTRODUCTION

The paper addresses harvest-and-use SWIET in two-hop MIMO-OFDM relaying, where an energy-constrained relay harvests source-transmitted RF energy and uses it for forwarding. It introduces TSR and PSR, formulates joint rate-maximization problems, and reports distinct relay-position behavior with PSR outperforming TSR.

  • Motivation: Energy harvesting can power constrained wireless devices, but external sources may require peripheral equipment and may be unavailable or uncontrollable.These constraints motivate harvesting energy from communication signals in applications such as sensor networks.
  • Related Work: Prior SWIET studies largely considered separate energy and information receivers, while earlier MIMO and OFDM studies did not jointly consider MIMO-OFDM relaying.Existing two-hop MIMO-OFDM relay work also commonly assumed energy-supplied source and relay nodes or destination-side harvesting.
  • Motivation: The proposed system uses a source with fixed energy supply and an energy-constrained relay that harvests RF energy from the source and uses it to forward information.The scenario applies to energy-constrained networks such as sensor deployments where direct source-destination transmission is unavailable.
  • Contributions: Unlike harvest-only systems, this work jointly considers energy harvesting and consumption at the relay in a single harvest-and-use system.All harvested relay energy is used to assist information transmission from source to destination.
  • Contributions: The paper proposes TSR and PSR and jointly optimizes power allocation, receiver control, subchannel pairing, and harvested-energy assignment to maximize end-to-end achievable information rate.The optimization includes time switching for TSR and power splitting for PSR under the paper’s stated formulation.
  • Results: PSR’s achievable information rate decreases monotonically as the relay moves toward the destination, whereas TSR performs relatively worst with the relay midway between source and destination.The authors identify this contrasting relay-position behavior as a first observation and report that PSR always outperforms TSR.

III. PROTOCOLS AND OPTIMIZATION PROBLEM FORMULATION

The paper presents TSR and PSR for two-hop non-regenerative MIMO-OFDM relaying and formulates an optimization problem for each protocol to explore system performance limits.

  • Protocol Design: The paper presents TSR and PSR using two practical receiver architectures for the two-hop non-regenerative MIMO-OFDM system.It also formulates one optimization problem for each protocol.

A. Protocol Description and Optimization Problem Formulation for TSR

TSR uses a time-switching receiver architecture at the relay and divides each transmission period into energy-transfer and information-relaying phases.

  • 1) TSR Protocol:: TSR employs a time-switching receiver architecture at the relay for energy-harvesting non-regenerative MIMO-OFDM relaying.The protocol is introduced as the TS-based relaying design considered in the paper.

1) TSR Protocol:

TSR divides each period into energy transfer, source-to-relay information transmission, and relay-to-destination forwarding. Its design determines harvested relay power, source and relay allocations, and subchannel pairing under power constraints.

  • 1) TSR Protocol:: TSR divides each period T into three phases: energy transfer, source-to-relay information transmission, and relay forwarding.The energy-transfer phase lasts αT, while the two information phases have equal duration (1−α)T/2.
  • 1) TSR Protocol:: During the first phase, the source transmits dedicated energy signals, whose covariance matrices determine harvested relay energy.The energy-transfer signal may differ from the information symbol vector, and tr(X_i) represents power allocated to subcarrier i.
  • 1) TSR Protocol:: All harvested energy is used for information relaying, so the relay’s forwarding power depends on the harvested energy and available relay-power constraint.The source and relay transmit over all paired subchannels during their respective information phases.
  • 1) TSR Protocol:: The source allocates normalized power factors ω_ℓ across first-hop subchannels, while the relay allocates forwarding factors ℓ′ across second-hop subchannels.These allocations satisfy aggregate power constraints over the KN subchannels.
  • 1) TSR Protocol:: The TSR end-to-end achievable information rate depends on time switching, source and relay allocations, and subchannel-pairing indicators.The pairing indicator θ_ℓ,ℓ′ equals one when first-hop subchannel ℓ is paired with second-hop subchannel ℓ′.

2) Optimization Problem Formulation for TSR:

The paper formulates TSR and PSR designs around practical receiver architectures, describing their time or power splitting operations and jointly optimizing the variables that determine end-to-end rate.

  • 2) Optimization Problem Formulation for TSR:: TSR constrains source and relay transmit powers by P_S and P_R, respectively.The source power constraint also limits energy transfer in TSR.
  • 2) Optimization Problem Formulation for TSR:: TSR requires one-to-one pairing between first-hop and second-hop subchannels.Each first-hop subchannel can be paired with only one second-hop subchannel, and vice versa.
  • 1) PSR Protocol:: PSR uses a power-splitting receiver architecture in which the period is divided into two equal parts for simultaneous transfer and forwarding.The first part transfers energy and information from S to R; the second uses harvested energy for forwarding to D.
  • 1) PSR Protocol:: At R, the received signal is transformed and split into information and energy flows using the power-splitting factor matrix ρ.The (I−ρ) flow enters the information receiver, while the ρ flow enters the EH receiver and supports amplification.
  • 1) PSR Protocol:: For each spatial subchannel, ρ_i,n specifies the fraction directed to energy harvesting, determining the harvested energy over subcarrier i.The model assumes unit energy-conversion efficiency η=1.
  • 1) PSR Protocol:: The source power-allocation factor ω_ℓ determines transmitted power over subchannel ℓ, while the remaining 1−ρ_ℓ power reaches the information receiver.The source power and power-splitting factors jointly determine the relay’s available forwarding power.
  • 1) PSR Protocol:: The received information is forwarded to D, producing an instantaneous PSR achievable rate based on the paired subchannels.The pairing indicator θ_ℓ,ℓ′ identifies the first-hop and second-hop subchannel correspondence.
  • 1) PSR Protocol:: PSR jointly optimizes power splitting ρ, source allocation ω, and subchannel pairing θ to maximize the end-to-end achievable information rate.The optimization variables cover both hops and the relay receiver architecture.

2) Optimization Problem Formulation for PSR:

The PSR formulation imposes source-power and one-to-one subchannel-pairing constraints while optimizing the protocol’s resource variables.

  • 2) Optimization Problem Formulation for PSR:: The PSR source power-allocation factors satisfy a total normalized power constraint.The constraint ∑_{ℓ=1}^{KN}ω_ℓ≤1 limits source transmission.
  • 2) Optimization Problem Formulation for PSR:: PSR enforces one-to-one subchannel pairing across the two hops.Each first-hop subchannel is paired with only one second-hop subchannel, and vice versa.

IV. OPTIMAL DESIGN OF TSR

The TSR optimization is non-convex and combinatorial, but theoretical results separate energy design and channel pairing from the remaining resource allocation.

  • IV. OPTIMAL DESIGN OF TSR: TSR optimization remains non-convex even after removing its discrete subchannel-pairing variables.The presence of binary pairing variables makes the problem combinatorial as well.
  • IV. OPTIMAL DESIGN OF TSR: The optimization framework first computes optimal source energy covariance, then optimal pairing, and finally remaining power and time allocations.This sequence is summarized in Algorithm 1.
  • IV. OPTIMAL DESIGN OF TSR: For fixed α, maximizing harvested energy maximizes the relay’s available forwarding power, motivating independent optimization of the source energy covariance.The relay power increases with total harvested energy under the stated formulation.
  • IV. OPTIMAL DESIGN OF TSR: For a given tr(X_i), Lemma 1 characterizes the optimal energy-transfer covariance X♯.The lemma is used as the first theoretical step in solving the TSR design problem.
  • IV. OPTIMAL DESIGN OF TSR: For maximum energy transfer, Lemma 2 allocates all source power on each subcarrier to the spatial channel with maximum gain.The selected gain is the maximum ∥h̃^(i)_{S,1}∥² over the relevant subchannels.
  • IV. OPTIMAL DESIGN OF TSR: Theorem 1 states that optimal TSR energy allocation projects all transmit energy onto the channel-matrix eigenvector with the largest eigenvalue.The result also applies to multi-channel single-hop MIMO and OFDM energy-transfer systems.
  • B. Optimal θ∗for TSR: The TSR pairing problem is a joint power-allocation and subchannel-pairing problem that can be solved separately without losing global optimality.The separation principle pairs subchannels according to sorted channel gains.
  • B. Optimal θ∗for TSR: The optimal pairing matches the i-th largest first-hop channel gain with the i-th largest second-hop channel gain.Changing α affects relay power and information-transmission duration, but not the channel gains or this pairing rule.

C. Joint optimal ω∗, ∗and α∗

The TSR optimization separates subproblems where possible, then uses iterative or numerical procedures because joint concavity and analytic tractability are unavailable.

  • Joint optimization: The optimal channel pairing and source design can be obtained separately, without sacrificing global optimality.This reduces the coupled design burden before optimizing the remaining variables.
  • Joint optimization: The TSR objective is not jointly concave in ωℓ and ℓ, although fixing either variable makes it concave in the other.This motivates KKT-based conditional updates rather than a direct global analytic solution.
  • Joint optimization: For fixed α, alternating optimization updates ω and ℓ until convergence, producing a near-optimal solution.Each iteration improves the current objective, which is bounded by the constraint.
  • Optimization limits: Algorithm 2 depends on its initialization and cannot always guarantee global optimality.The paper also considers an asymptotically global solution for the high-SNR case.
  • Optimization over α: G(α) is monotonically increasing, while CTSR(α) is zero at the endpoints and has an interior maximum.The paper therefore searches numerically over α; simulations show a single peak with an increasing-then-decreasing shape.

V. OPTIMAL DESIGN OF PSR PROTOCOL

PSR is solved by reducing its discrete and continuous coupling: channel pairing is separated, power splitting is optimized conditionally, and source weights are then updated.

  • Optimization framework: PSR contains discrete subchannel-pairing variables, making its optimization combinatorial and difficult for conventional methods.The proposed framework addresses this complexity through structural decomposition.
  • Optimization framework: The optimal TSR pairing strategy also applies to PSR, allowing subchannel pairing to be performed separately.For a given source weight vector, the framework first computes optimal power splitting and then updates the weights.
  • Optimization framework: Algorithm 3 alternates between calculating optimal ρ for a given ω and calculating optimal ω using the resulting ρ.This reduces the number of variables handled jointly.

A. Optimal θ∗for PSR

For PSR, sorted channel gains determine the optimal pairing independently of power splitting and power allocation, while conditional optimization handles the remaining variables.

  • Optimal pairing: PSR’s optimal subchannel pairing follows sorted channel gains because ρ and ω do not change the ordering across the two hops.Thus θ* can be determined independently of the other design variables.
  • Conditional optimization: For a fixed ω, the optimal power-splitting vector ρ* is obtained explicitly and substituted into the remaining optimization problem.The resulting ρ*(ω) reduces the number of variables requiring optimization.
  • Special case: For a single-carrier, single-antenna system with equal two-hop CNRs, the optimal power-splitting ratio is ρ*=0.5.This is a special case derived from the general PSR result.
  • Weight optimization: The resulting PSR rate rℓ is monotonically increasing in the source weight ωℓ, but the joint objective remains difficult to optimize analytically.Algorithm 4 uses stepwise weight updates; smaller Δω improves accuracy but increases convergence time.

VI. NUMERICAL RESULTS

The numerical study evaluates TSR and PSR in a three-node relay network with a barrier, varying transmission power and relay placement under specified MIMO-OFDM configurations.

  • Simulation setup: The simulations model a relay above a barrier between source S and destination D, with relay position controlled by φ and barrier height h.The relay lies on the direct line when h=0; the study uses dS,D=100m, B=5MHz, path-loss factor 2, and noise density −100dBm.
  • Compared schemes: Simple TSR and simple PSR serve as benchmarks, while optimized TSR-I and TSR-II use alternative iterative and high-SNR approximate designs.The benchmark schemes retain optimal subchannel pairing but simplify other allocations.

A. Performance vs PS

Performance rises with source transmission power, and optimized PSR generally outperforms optimized TSR. Relay position, antenna count, and subcarrier count substantially affect achievable information rate, with subcarrier growth exhibiting diminishing returns.

  • Higher PS increases the achievable information rate of all schemes because it produces higher system SNR.
  • Optimized PSR outperforms optimized TSR because its per-subchannel power-splitting factors provide more resource-allocation flexibility.
  • 30dBm–40dBm for N = 2, K = 4 and 20dBm–30dBm for N = 4, K = 64 are efficiently workable intervals for simple TSR.In these intervals, the simple TSR’s fixed α is close to the optimized α.
  • PSR performance decreases as the relay moves from the source toward the destination, whereas TSR first decreases and then increases, performing relatively worst near the midpoint.For h = 0, the changing balance between source-relay energy harvesting and relay-destination channel quality explains the differing trends.
  • Increasing the number of antennas or subcarriers increases achievable information rates for both protocols.More antennas provide spatial subchannels; more subcarriers provide additional subchannels and configuration flexibility, but fixed total bandwidth causes slower gains as K increases.

APPENDIX A THE PROOF OF LEMMA 1

The appendix proves Lemma 1 by applying eigenvalue decomposition and identifying the first eigenvector with the largest singular value. It then transforms the optimization problem and characterizes power-splitting maximizers through derivative and boundary analysis.

  • Applying eigenvalue decomposition to Xi identifies the relevant first column with the largest singular value of HS,i.
  • Problem (30) is equivalently transformed into an optimization over traces of X1 through XK.
  • The transformed problem’s optimal solution is obtained directly after the equivalent reformulation.
  • For 0 ≤ ρℓ ≤ 1, derivative analysis shows the objective has one maximum when ρℓ satisfies equation (53).
  • When Aℓ ≠ Qℓ, rℓ is maximized at ρℓ = 0.5 because (1 − ρℓ)ρℓ reaches its maximum there.
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