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Joint Wireless Information and Energy Transfer in a Two-User MIMO Interference Channel

Jaehyun Park, Bruno Clerckx

arXiv:1303.1693v2cs.IT

TL;DR

The paper addresses joint information and energy transfer in a two-user MIMO interference channel, where the optimal design is unknown in several receiver-mode settings. It analyzes common and mixed modes, derives optimal or necessary beamforming conditions, and proposes SLER beamforming and scheduling. SLER provides wider R-E regions than conventional strategies, while the paper leaves receiver-condition-based switching policy as future work.

  • Problem

    The general MIMO interference-channel setup for joint wireless information and energy transfer had not been addressed, and optimal transmission remains open for several receiver modes.

  • Method

    The paper analyzes four receiver-mode scenarios, derives optimal or necessary transmission conditions, evaluates MEB and MLB, and proposes SLER beamforming with SLER-based mode scheduling.

  • Results

    SLER maximization beamforming exhibits a wider R-E region than conventional methods, while mode scheduling further extends the R-E tradeoff region.

  • Takeaways & Limitations

    The proposed SLER scheme utilizes energy harvesting without compromising information decoding, and scheduling between mixed receiver modes broadens the achievable tradeoff.

Abstract

from arXiv · show

This paper investigates joint wireless information and energy transfer in a two-user MIMO interference channel, in which each receiver either decodes the incoming information data (information decoding, ID) or harvests the RF energy (energy harvesting, EH) to operate with a potentially perpetual energy supply. In the two-user interference channel, we have four different scenarios according to the receiver mode -- ($ID_1$, $ID_2$), ($EH_1$, $EH_2$), ($EH_1$, $ID_2$), and ($ID_1$, $EH_2$). While the maximum information bit rate is unknown and finding the optimal transmission strategy is still open for ($ID_1$, $ID_2$), we have derived the optimal transmission strategy achieving the maximum harvested energy for ($EH_1$, $EH_2$). For ($EH_1$, $ID_2$), and ($ID_1$, $EH_2$), we find a necessary condition of the optimal transmission strategy and, accordingly, identify the achievable rate-energy (R-E) tradeoff region for two transmission strategies that satisfy the necessary condition - maximum energy beamforming (MEB) and minimum leakage beamforming (MLB). Furthermore, a new transmission strategy satisfying the necessary condition - signal-to-leakage-and-energy ratio (SLER) maximization beamforming - is proposed and shown to exhibit a better R-E region than the MEB and the MLB strategies. Finally, we propose a mode scheduling method to switch between ($EH_1$, $ID_2$) and ($ID_1$, $EH_2$) based on the SLER.

I. INTRODUCTION

The paper studies joint wireless information and energy transfer in a two-user MIMO interference channel, addressing four receiver-mode scenarios and unresolved transmission-design problems. It derives optimal or necessary transmission strategies, characterizes rate-energy tradeoffs, proposes SLER beamforming, and introduces SLER-based mode scheduling.

  • MIMO interference-channel models for joint information and energy transfer had not been addressed before this work.
  • The paper considers four receiver-mode scenarios: (ID1, ID2), (EH1, EH2), (EH1, ID2), and (ID1, EH2).
  • For (ID1, ID2), the maximum information bit rate and optimal transmission strategy remain open problems; the paper instead investigates achievable rates using iterative water-filling.
  • For (EH1, EH2), the paper derives an optimal strategy achieving maximum harvested energy, while mixed modes receive a necessary-condition analysis and R-E characterization for MEB and MLB.
  • SLER maximization beamforming yields a wider R-E region than MLB, MEB, and SLNR beamforming, and SLER-based mode scheduling further extends the tradeoff region.

II. SYSTEM MODEL

The system is a two-user MIMO interference channel whose receivers switch between information decoding and energy harvesting, with local CSI and no CSI sharing between transmitters. The paper models achievable rates and harvested power under separate receiver modes and treats interference as either harmful or useful depending on the mode.

  • Each receiver either decodes information or harvests RF energy, switching between ID and EH across frames or time slots because simultaneous operation is unavailable.
  • The two transmitters have perfect CSI for their associated links but do not share CSI or transmitted information.
  • The model uses equal transmitter and receiver antenna counts, Mt = Mr = M, normalized frequency-flat fading, and transmit covariance constraints tr(Qj) ≤ P.
  • Harvested power is modeled through an energy-conversion efficiency constant, with ζi = 1 and receiver noise neglected relative to transferred energy.
  • When a receiver performs ID, the other transmitter’s signal is treated as interference; during EH, that signal becomes a useful energy source.

III. TWO RECEIVERS ON A SINGLE MODE

For receivers in a common mode, the paper uses iterative water-filling for achievable rates with two IDs and derives rank-one beamforming for maximum harvested energy with two EHs. The EH solution independently maximizes each transmitter’s transferred energy.

  • A. Two IDs: maximum achievable sum rate: For (ID1, ID2), the paper seeks maximum achievable sum rate under covariance and power constraints using distributed iterative water-filling without CSI sharing.
  • B. Two EHs: maximum harvested energy: For (EH1, EH2), Proposition 1 states that each optimal transmit covariance Qj has rank one and aligns with the dominant right singular vector of the combined channel.
  • B. Two EHs: maximum harvested energy: Each transmitter can maximize its transferred energy independently, without using the other transmitter’s channel information or transmission strategy.
  • B. Two EHs: maximum harvested energy: When both receivers harvest energy and cannot decode information, the achievable information rate is zero.

IV. ONE ID RECEIVER AND ONE EH RECEIVER

The section studies the (EH1, ID2) setting, where energy harvesting and information decoding interact and make the achievable rate-energy boundary difficult to characterize. It establishes a high-SNR rank-one condition for the first transmitter and develops the supporting optimization framework.

  • The proof compares boundary points using harvested-energy and achievable-rate constraints while analyzing the eigenvalue structure induced by Q1 and Q2.
  • The (EH1, ID2) rate-energy boundary is difficult to characterize because energy harvesting and information decoding interact in the interference channel.
  • Lemma 1 uses a generalized singular value decomposition to construct an invertible transformation jointly associated with H11 and H21.
  • In the high-SNR regime, Proposition 2 states that an optimal boundary-point covariance Q1 has rank at most one.
  • When the first transmitter need not send a signal, Q1 = 0 is optimal and rank(Q1) = 0.

22. Note that the last approximation

The section explains why rank-one transmission is adopted for the first transmitter and introduces rank-one beamforming schemes to trace achievable rate-energy tradeoffs. It also records the high-SNR interpretation and the scalar-antenna special case.

  • At high SNR and large harvesting energy, the achievable rate is proportional to m2 − m and is maximized when the first-transmitter rank is m = 1.
  • Reducing the first transmitter’s signal rank can increase the degrees of freedom available at the second receiver.
  • Rank one is optimal from both power-transfer and high-SNR information-transfer perspectives because it concentrates energy while limiting interfered dimensions.
  • For single-antenna nodes, the second transmitter uses full power P at the boundary, leaving a power-allocation problem for the first transmitter.
  • The rank-one-or-no-transmission strategy depends on the energy transferred from the other transmitter, and a higher-rank alternative can increase harvested energy while reducing interference.
  • The paper states that the optimal achievable rate-energy boundary remains open, but the first transmitter will use a rank-one beamforming scheme.

B. Rank-one Beamforming Design

The section frames rank-one beamforming as a compromise between maximizing harvested energy and limiting interference to the information-decoding receiver.

  • Maximizing energy transfer to the first receiver can create considerable interference at the second receiver, which decodes information.
  • Because the first receiver operates as an energy harvester, the first transmitter’s beamforming design must account for both energy delivery and interference.

1) Maximum-energy beamforming (MEB):

The MEB construction steers the first transmitter toward energy transfer, while the resulting rate-energy region is evaluated through an iterative optimization procedure. The section also contrasts MEB with MLB in terms of harvested energy.

  • Maximum-energy beamforming (MEB): MEB selects a transmit covariance based on the SVD of H11, with V11 unitary and transmit power P1 constrained by 0 ≤ P1 ≤ P.
  • Minimum-leakage beamforming (MLB): MLB instead steers the first transmitter using the SVD of H21 to minimize interference at the second information-decoding receiver.
  • Minimum-leakage beamforming (MLB): For MLB, the harvested energy from the first transmitter is P1∥H11[V21]M∥2.
  • Achievable rate-energy region: The achievable rate-energy region is identified iteratively because E11 and the effective channel used for rate optimization depend on P1.
  • MEB versus MLB: The MEB strategy generally harvests more energy from the first transmitter than the MLB strategy.
  • Maximum transferable energy: The maximum transferable energy is attained when the second transmitter steers its signal to maximize beamforming energy over the cross-link channel H12.
  • Power adjustment: If transferred energy exceeds the required amount, the first transmitter reduces P1 to lower interference; if the requirement is already met, it may transmit nothing.
  • Achievable rate-energy region: The optimization uses conventional effective-MIMO rate maximization for lower energy requirements and a water-filling-like approach for higher requirements.

V. ENERGY-REGULARIZED SLER-MAXIMIZING BEAMFORMING

The paper defines SLER to jointly promote energy transfer to an EH receiver and reduce leakage to an ID receiver. The resulting beamformer interpolates between MEB and MLB as the required harvested energy changes, supporting mode scheduling based on SLER.

  • SLER measures energy transfer to the EH receiver against leakage to the ID receiver.The metric incorporates the required harvested energy in its denominator.
  • The SLER-maximizing beamformer is obtained from a generalized eigenvector and can be computed using GSVD.Its implementation uses one QR decomposition, one SVD, and a matrix-vector multiplication.
  • When required harvested energy is large, SLER beamforming becomes equivalent to MEB.The denominator approaches a scaled identity matrix in this regime.
  • When required harvested energy is small, the beamformer approaches MLB by steering to reduce interference leakage.

A. The rank-one optimality in the low SNR regime for one ID receiver and one EH receiver

In the low-SNR regime, the paper establishes rank-one optimality at the boundary of the achievable R-E region for one ID receiver and one EH receiver. The result follows because the achievable rate becomes independent of interference from the first transmitter.

  • The optimal Q1 at the boundary of the achievable R-E region has rank one in the low-SNR regime.This rank-one optimality is stated for the (EH1, ID2) scenario without loss of generality.
  • At low SNR, the achievable rate is independent of interference from the first transmitter.The system is therefore noise-limited in this regime.
  • Because interference does not affect the low-SNR rate, Q1 can be designed to maximize harvested energy.
  • The low-SNR optimal information-transfer strategy is also rank-one beamforming.This is consistent with the rank-one boundary design for the energy-transfer transmitter.

B. Asymptotic behavior for a large M

The paper analyzes how low SNR and large antenna arrays simplify energy-transfer design and reports simulation results for MEB, MLB, SLER, and scheduling. Larger arrays and low SNR favor own-link energy maximization, while SLER scheduling broadens the R-E region under interference.

  • Asymptotic behavior for a large M: As M grows, det(R) approaches 1 + P independently of the beamforming vector, making the energy beamformer independent of the interference channel.
  • Asymptotic behavior for a large M: MEB with power control becomes optimal for large M because it maximizes energy transferred over the transmitter’s own link.
  • Asymptotic behavior for a large M: When SNR decreases or antenna count increases, energy-transfer design increasingly focuses on the transmitter’s own EH link rather than leakage to the ID receiver.
  • Simulation results: For MEB, time-sharing outperforms the alternative approach below 120 Joule/sec because MEB creates substantial interference at the ID receiver.
  • Simulation results: Rank-one MEB has superior R-E boundary points to rank-two MEB, although the exact optimal R-E boundary remains unidentified.
  • Simulation results: At low SNR, MEB provides higher harvested energy than MLB without degrading achievable rate.
  • Simulation results: The SLER R-E region covers most regions achieved by MEB and MLB, whereas SLNR beamforming does not cover the MEB region.

VIII. CONCLUSION

The paper characterizes transmission strategies across receiver-mode configurations in a two-user MIMO interference channel, identifying optimal or necessary conditions and achievable R-E tradeoffs. It proposes SLER maximization beamforming, which yields a wider R-E region while using interference for EH without compromising ID, though optimal boundary identification remains open.

  • For single-operation modes, iterative water-filling and energy-maximizing beamforming maximize information rate and harvested energy, respectively.
  • For mixed EH-ID modes, an optimal strategy must include rank-one beamforming by one transmitter with power control.
  • The MEB and MLB strategies satisfy this condition and have identified achievable R-E tradeoff regions.
  • MEB provides larger harvested energy, whereas MLB provides a larger achievable rate.
  • When SNR decreases or antenna count increases, joint information and energy transfer can be split into disjoint non-interfering links.
  • SLER maximization beamforming yields a wider R-E region and uses interference for EH without compromising ID.
  • Identifying the optimal R-E boundary remains challenging, and partial or erroneous CSI degrades achievable rate and harvested energy.
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