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

Jaehyun Park, Bruno Clerckx

arXiv:1310.6870v3cs.IT

TL;DR

The paper studies the previously unexplored problem of joint wireless information and energy transfer in a general K-user MIMO interference channel. It analyzes three receiver-mode configurations, derives a common optimality condition, and develops rank-one beamforming and iterative optimization methods. The resulting strategies characterize achievable rate-energy tradeoffs, while SLER-based EH receiver selection further improves the achievable region.

  • Problem

    JWIET in the general K-user MIMO interference channel had not been addressed, although interference has opposite effects on information decoding and energy harvesting.

  • Method

    The paper analyzes three receiver-mode scenarios, develops rank-one energy beamforming, and jointly optimizes information covariance matrices and energy-beamforming powers iteratively.

  • Results

    The proposed strategies identify achievable rate-energy regions, and SLER-based EH receiver selection further improves those regions.

  • Takeaways & Limitations

    A common necessary condition applies across the three scenarios: energy-transferring transmitters can use rank-one energy beamforming in the K-user MIMO IFC.

Abstract

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Recently, joint wireless information and energy transfer (JWIET) methods have been proposed to relieve the battery limitation of wireless devices. However, the JWIET in a general K-user MIMO interference channel (IFC) has been unexplored so far. In this paper, we investigate for the first time the JWIET in K-user MIMO IFC, in which receivers either decode the incoming information data (information decoding, ID) or harvest the RF energy (energy harvesting, EH). In the K-user IFC, we consider three different scenarios according to the receiver mode -- i) multiple EH receivers and a single ID receiver, ii) multiple IDs and a single EH, and iii) multiple IDs and multiple EHs. For all scenarios, we have found a common necessary condition of the optimal transmission strategy and, accordingly, developed the transmission strategy that satisfies the common necessary condition, in which all the transmitters transferring energy exploit a rank-one energy beamforming. Furthermore, we have also proposed an iterative algorithm to optimize the covariance matrices of the transmitters that transfer information and the powers of the energy beamforming transmitters simultaneously, and identified the corresponding achievable rate-energy tradeoff region. Finally, we have shown that by selecting EH receivers according to their signal-to-leakage-and-harvested energy-ratio (SLER), we can improve the achievable rate-energy region further.

I. INTRODUCTION

The paper addresses JWIET in the previously unexplored general K-user MIMO interference channel, where receivers operate in information-decoding or energy-harvesting mode. It derives a common transmission condition, develops rank-one energy beamforming and optimization methods, and reports improved rate-energy regions through SLER-based receiver selection.

  • JWIET combines wireless information and energy transfer to address wireless devices' battery limitations.
  • The general K-user MIMO IFC had not previously been addressed, despite interference harming information decoding while benefiting energy harvesting.
  • The paper considers three receiver-mode scenarios: multiple EH receivers with one ID, multiple IDs with one EH, and multiple IDs with multiple EHs.
  • All energy-transferring transmitters use rank-one energy beamforming under the developed transmission strategy.
  • An iterative algorithm jointly optimizes information-transmitter covariance matrices and energy-beamforming-transmitter powers for the achievable rate-energy region.
  • SLER-based EH receiver selection further improves the achievable rate-energy region.

II. SYSTEM MODEL

The system is a K-user MIMO interference channel whose receivers switch between information decoding and energy harvesting over time. The model defines rates, harvested power, transmitter covariance constraints, and channel-information assumptions for the two receiver modes.

  • K transmitters simultaneously send signals to K receivers in a frequency-flat fading MIMO interference channel.
  • Each receiver switches between information-decoding and energy-harvesting modes at each frame or time slot because simultaneous operation is unavailable.
  • Receiver mode is communicated to all transmitters through a zero-delay, error-free feedback link at the frame beginning.
  • Transmitters know the channel state information of their associated links but do not share that information among themselves.
  • The model constrains each transmit covariance matrix by tr(Qj) ≤ P and defines harvested power using an energy-conversion efficiency constant.
  • In information-decoding mode interference is harmful, whereas in energy-harvesting mode signals from other transmitters provide useful transferred energy.

III. A NECESSARY CONDITION FOR THE OPTIMAL TRANSMISSION STRATEGY

The paper extends the two-user MIMO IFC rank-one result to K-user settings by examining receiver configurations with one or multiple information-decoding and energy-harvesting receivers.

  • The prior two-user MIMO IFC result identified rank-one energy beamforming with proper power control as a necessary optimality condition.
  • The K-user analysis covers one ID with multiple EHs, multiple IDs with one EH, and multiple IDs with multiple EHs.

A. One ID receiver and One EH receiver

For one information-decoding and one energy-harvesting receiver, the boundary of the achievable rate-energy region has an optimal energy-transmitter covariance of rank at most one in the high-SNR regime. This rank-one strategy supports energy transfer while reducing interference, with validity also reported in the low-SNR regime.

  • A. One ID receiver and One EH receiver: rank(Q1) ≤ 1 at the achievable rate-energy boundary in the high-SNR regime.
  • A. One ID receiver and One EH receiver: The proof uses GSVD and the interlacing theorem to show that the achievable rate is maximized when rank(Q1) = 1.
  • A. One ID receiver and One EH receiver: The optimal energy transmitter either uses rank-one beamforming or remains silent, depending on the energy harvested from the information transmitter.
  • A. One ID receiver and One EH receiver: The rank-one strategy increases harvested energy at the EH receiver while simultaneously reducing interference at the ID receiver.
  • A. One ID receiver and One EH receiver: The rank-one optimality is derived in the high-SNR regime, while the paper also reports its validity in the low-SNR regime.

B. One ID and multiple EHs in a K-user IFC

For one ID receiver and multiple EH receivers, the paper characterizes boundary rate-energy solutions and establishes rank-one optimal energy transmission. The resulting beamforming direction accounts for interference from other energy transmitters.

  • The scenario places transceiver pairs (Txk, Rxk), k = 1, ..., K −1, in EH mode and the remaining pair in ID mode.
  • Proposition 2 shows that optimal covariance matrices on the boundary for the multi-EH rate-energy region also solve the corresponding effective two-user boundary problem.
  • For every energy transmitter, the boundary-optimal covariance satisfies rank(Qk) ≤1.
  • If an energy transmitter uses rank at least two, a rank-one beamformer can achieve either higher information rate or larger harvested energy.
  • The optimal beamforming direction depends on the covariance matrix of interference generated by the other energy transmitters.

C. One EH and multiple IDs in a K-user IFC

For one EH receiver and multiple ID receivers, the paper reduces the boundary analysis to an effective multi-ID problem and again obtains rank-one optimal energy transmission. The same boundary solutions extend to the multiple-EH, multiple-ID setting.

  • The configuration places (Tx1, Rx1) in EH mode and (Txk, Rxk), k = 2, ..., K, in ID mode.
  • Proposition 3 states that optimal energy-transmitter covariance matrices on the multi-ID boundary also solve the corresponding reduced boundary problem.
  • For one EH and multiple IDs, the boundary-optimal covariance satisfies rank(Q1) ≤1.
  • With multiple EHs and multiple IDs, all optimal energy-transmitter covariance matrices satisfy rank(Qk) ≤1 for k = 1, ..., K1.
  • The generalized strategy uses rank-one beamforming with proper power allocation, while beam directions depend on interference from other energy and information transmitters.
  • Global CSI and centralized optimization could further improve the rate-energy region through beam alignment, but this is outside the paper’s scope.

IV. DISTRIBUTED RANK-ONE BEAMFORMING DESIGN AND ACHIEVABLE R-E REGION

This section develops distributed rank-one beamforming methods and an iterative procedure for computing achievable rate-energy trade-off curves in the K-user MIMO IFC.

  • The paper proposes distributed rank-one beamforming methods based on the rank-one optimality result.
  • It also proposes an iterative algorithm to compute achievable rate-energy trade-off curves under different beamforming schemes.
  • The design considers K1 EH transceiver pairs and K − K1 ID transceiver pairs.

A. Distributed Rank-one Beamforming Design

The distributed design balances energy delivery to EH receivers against interference leakage to ID receivers. It uses MEB-like and MLB-like rank-one directions on effective channels, with beam tilting addressing their competing objectives.

  • Each energy transmitter steers its signal toward multiple EH receivers while accounting for the need to limit interference at ID receivers.
  • The energy-maximizing direction is analogous to maximum-energy beamforming applied to the effective channel.
  • The interference-minimizing direction is obtained from the effective channel’s singular vectors and is analogous to minimum-leakage beamforming.
  • The harvested energy from the kth transmitter is Pk∥˜H11,k[˜V21,k]M∥2 under the interference-minimizing design.
  • MEB and MLB optimize different objectives, respectively causing either larger ID interference or insufficient EH energy.

1) Energy-regularized SLER-maximizing beamforming:

The paper defines SLER to balance harvested-energy maximization against leakage minimization toward ID receivers. Its beamformer shifts between MEB and MLB as the required harvested energy changes.

  • SLER definition: SLER combines harvested-energy and leakage considerations by adding the unmet harvested-energy requirement to its denominator.The metric is motivated by the different roles of noise in SLNR and required harvested energy in EH operation.
  • Relation to SLNR: The SLER metric is comparable to SLNR, but replaces the noise-related denominator contribution with a term tied to the required harvested energy.The beamforming vector can be efficiently computed using a GSVD algorithm.
  • Beamforming regimes: When the required harvested energy is large, SLER maximizing beamforming becomes equivalent to MEB.The denominator approaches a scalar multiple of the identity matrix in this regime.
  • Beamforming regimes: When the required harvested energy is small, the SLER beamformer approaches the MLB weight vector.The beam direction is steered to reduce interference leakage into ID receivers.
  • Beamformer construction: SLER beamforming balances energy maximization at EH receivers with leakage minimization at ID receivers.The beamforming vector is obtained from the generalized eigenvector associated with the largest generalized eigenvalue.

B. Achievable R-E region

The achievable rate-energy problem is non-convex because information-transmitter covariance matrices are coupled through interference. The paper therefore develops iterative procedures that optimize information powers and energy-transmitter covariances while exploiting rank-one energy beamforming.

  • Transmission structure: Energy-transmitting nodes use rank-one beamforming at most, forming the structural basis for the proposed achievable-region algorithms.The energy-transmitter covariance matrices are selected from the paper’s rank-one beamforming constructions.
  • Iterative optimization: The iterative algorithm jointly optimizes information-transmitter powers and energy-transmitter powers to trace the achievable rate-energy region.The procedure adjusts energy-transmitter power downward when transferred energy exceeds the required harvested energy, reducing interference at ID receivers.
  • Problem formulation: The achievable rate-energy optimization is non-convex because interference couples the covariance matrices of information transmitters.This coupling prevents easy identification of the optimal region for the original problem.
  • Upper-bound formulation: If information-transmitter interference is sufficiently weak, the simplified problem can serve as an upper bound on the original rate-energy region.Although not tight, this upper bound provides insight for developing the iterative algorithm.
  • Convergence: For the simplified upper-bound problem, the converged solution of Algorithm 1 is globally optimal under the stated concavity and convexity conditions.The fixed-power subproblem is concave and has zero duality gap, while the relevant power sequence converges monotonically.
  • Original problem: For the original non-convex problem, the proposed iterative water-filling procedure is guaranteed to converge to a Nash equilibrium under nonsingular direct-channel matrices, not necessarily the global optimum.Global optimization methods would require centralized coordination and complete channel-response knowledge unavailable in the system model.

V. SIMULATION RESULTS

Simulations evaluate rate-energy tradeoffs among MEB, MLB, and SLER beamforming across receiver configurations and interference levels. They show that interference changes the preferred beamforming strategy, while SLER-based selection and beam tilting can further enlarge the achievable region.

  • MEB, MLB, and SLER comparisons: As the number of energy transmitters increases, total harvested energy increases, while they may remain silent below a harvested-energy threshold because the information transmitter already satisfies EH constraints.The threshold is linearly proportional to the number of EH receivers harvesting from the information transmitter.
  • MEB, MLB, and SLER comparisons: SLER’s water-filling-like approach outperforms time-sharing and covers most of the R-E regions achieved by MEB and MLB.For MEB at low required harvested energy, time-sharing can instead perform better because full-power energy beams create substantial interference.
  • Effect of network size: With more information transmitters, maximum harvested energy increases but maximum information rate does not increase drastically because interference grows proportionally.The resulting system becomes interference-limited.
  • Effect of network size: When interference dominates, MEB becomes more attractive and its R-E region nearly covers MLB’s region.This observation is consistent with the paper’s asymptotic rank-one MEB result.
  • Effect of interference: For αij = 0.6, the R-E region provides larger harvested energy but lower information rate than for αij = 0.3.The stronger cross-link interference degrades information decoding while benefiting energy harvesting.
  • SLER-based user selection: SLER-based user selection extends the achievable R-E region for both αij = 0.3 and αij = 0.6, with a slightly larger improvement under stronger interference.The simulations identify user selection as more effective when cross-link interference is strong.
  • Beamforming limitations: Independent SLER beam directions can produce suboptimal aggregate interference, allowing MLB to outperform SLER near 450µW harvested energy.The effect becomes more sensitive to beam steering and power reduction as the number of energy transmitters increases.
  • Beam tilting: Beam tilting improves SLER performance over SLER without tilting, especially when (K, K1) = (5, 4).The results indicate that jointly designing energy-beam directions and powers can further improve beamforming.

VI. CONCLUSION

The paper establishes rank-one energy beamforming as a common necessary condition across three K-user MIMO IFC receiver-mode scenarios, then develops iterative optimization and SLER-based selection to characterize and improve achievable rate-energy regions.

  • Main conclusions: The three receiver-mode scenarios share a necessary optimality condition: every energy-transferring transmitter uses rank-one energy beamforming.The scenarios are multiple EH receivers with one ID receiver, multiple IDs with one EH receiver, and multiple IDs with multiple EHs.
  • Main conclusions: Given rank-one beamforming, the paper develops an iterative algorithm for the non-convex optimization of information-transmitter covariance matrices and energy-beamforming powers.The algorithm jointly optimizes these transmission variables.
  • Rate-energy performance: SLER-maximizing beamforming balances harvested energy and information rate, producing a wider R-E region than MEB and MLB.MEB and MLB respectively emphasize harvested energy or information rate, whereas SLER maximization pursues both.
  • Rate-energy performance: As the number of information transmitters increases, the optimal energy-transmitter strategy approaches MEB in an interference-limited system.Increasing energy transmitters instead makes beam steering and power reduction important for information-rate performance, motivating beam tilting.
  • Further implications: The paper proposes SLER-based EH transceiver selection, which further improves the achievable rate-energy region.The approach is discussed as an extension toward the MIMO interference broadcast channel, while power splitting remains future work requiring joint optimization of receiver splitting ratios and transmission strategy.
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