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MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer

Rui Zhang, Chin Keong Ho

arXiv:1105.4999v3cs.IT

TL;DR

The paper addresses the open information-energy tradeoff in RF-enabled wireless networks by studying simultaneous wireless information and power transfer in a MIMO broadcast system. It derives optimal rate-energy tradeoffs for separated receivers and compares time switching and power splitting with an outer bound for co-located receivers. The results characterize both the fundamental tradeoff and practical conditions under which power splitting approaches the upper bound.

  • Problem

    The fundamental information-energy transmission tradeoff in wireless networks that harvest energy from received RF signals remains an open problem.

  • Method

    The paper models a MIMO broadcast system with separate or co-located energy-harvesting and information-decoding receivers and analyzes their rate-energy regions.

  • Results

    For separated receivers, the paper derives an optimal transmission strategy and rate-energy boundary; for co-located receivers, it characterizes practical time-switching and power-splitting regions against an outer bound.

  • Takeaways & Limitations

    Power splitting approaches the tradeoff upper bound asymptotically when RF-band antenna noise becomes more dominant than baseband processing noise.

  • Takeaways & Limitations

    The analysis assumes simplified energy-conversion treatment, while practical receivers may not directly decode information and general energy-conversion constraints remain open.

Abstract

from arXiv · show

Wireless power transfer (WPT) is a promising new solution to provide convenient and perpetual energy supplies to wireless networks. In practice, WPT is implementable by various technologies such as inductive coupling, magnetic resonate coupling, and electromagnetic (EM) radiation, for short-/mid-/long-range applications, respectively. In this paper, we consider the EM or radio signal enabled WPT in particular. Since radio signals can carry energy as well as information at the same time, a unified study on simultaneous wireless information and power transfer (SWIPT) is pursued. Specifically, this paper studies a multiple-input multiple-output (MIMO) wireless broadcast system consisting of three nodes, where one receiver harvests energy and another receiver decodes information separately from the signals sent by a common transmitter, and all the transmitter and receivers may be equipped with multiple antennas. Two scenarios are examined, in which the information receiver and energy receiver are separated and see different MIMO channels from the transmitter, or co-located and see the identical MIMO channel from the transmitter. For the case of separated receivers, we derive the optimal transmission strategy to achieve different tradeoffs for maximal information rate versus energy transfer, which are characterized by the boundary of a so-called rate-energy (R-E) region. For the case of co-located receivers, we show an outer bound for the achievable R-E region due to the potential limitation that practical energy harvesting receivers are not yet able to decode information directly. Under this constraint, we investigate two practical designs for the co-located receiver case, namely time switching and power splitting, and characterize their achievable R-E regions in comparison to the outer bound.

I. INTRODUCTION

The paper develops a unified MIMO study of simultaneous wireless information and power transfer, motivated by the open information–energy tradeoff in RF-powered wireless networks. It characterizes optimal and practical rate-energy tradeoffs for separated and co-located energy-harvesting and information-decoding receivers.

  • Motivation: RF-based WPT can provide convenient energy supplies for wireless networks, including devices where replacing or recharging batteries is costly, inconvenient, hazardous, or impossible.The paper highlights environmental RF signals as a potential energy source for wireless devices.
  • Motivation: Prior research largely studied wireless power transfer and wireless information transfer separately, despite aligned objectives under a shared transmitter energy budget.The paper notes that increasing received signal power benefits both harvested energy and information capacity against receiver noise.
  • System model: The paper formulates a downlink MIMO SWIPT broadcast system in which a common transmitter sends signals to separate energy-harvesting and information-decoding receivers.The model includes co-located receivers as a special case when their MIMO channels from the transmitter are identical.
  • Separated receivers: For separated receivers, the paper designs an optimal transmission strategy and derives a semi-closed-form transmit covariance matrix for rate-energy boundary points.The covariance matrix jointly captures precoding and power allocation.
  • Co-located receivers: For co-located receivers, practical energy-harvesting circuits may be unable to decode information directly, making the optimal information-energy tradeoff more challenging to characterize.The paper therefore compares practical receiver designs with an achievable-region outer bound.
  • Co-located receivers: Time switching and power splitting yield achievable rate-energy regions for co-located receivers, with power splitting approaching the tradeoff upper bound asymptotically when RF antenna noise dominates baseband processing noise.The comparison also characterizes conditions under which practical performance gaps can be closed.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models a MIMO broadcast SWIPT system in which separate or co-located receivers harvest energy and decode information from a common transmitter. It formulates transmit-covariance designs under an average power constraint and characterizes the rate-energy tradeoff.

  • The system contains one transmitter, one energy-harvesting receiver, and one information-decoding receiver, with multiple antennas allowed at each node.
  • The EH and ID links are represented by channel matrices G and H, respectively, and may be different or identical for separated and co-located receivers.
  • Energy harvesting is modeled through received RF-band power proportional to the received baseband signal power, with conversion-loss factor ζ assumed equal to 1 unless stated otherwise.
  • The transmitter uses covariance S with average power constraint tr(S) ≤ P to control harvested power and information rate.
  • Separate objectives: Energy beamforming maximizes harvested power with rank-one covariance SEH = Pv1v1^H, yielding Qmax = g1P.
  • Separate objectives: Rate maximization uses Gaussian signaling, spatial multiplexing, and water-filling, whereas simultaneous power and rate transfer requires continuous transmission.

I + HSHH

For intermediate harvested-power requirements, the paper maximizes information rate under a harvested-power constraint and compares the resulting covariance designs with time-sharing between separate energy and information optima.

  • For QID < Q̄ < Qmax, the corresponding optimal solutions define boundary rate points of the R-E region.
  • The jointly optimized covariance solutions generally produce larger rate-power pairs than time-sharing between separate energy-beamforming and information-maximizing covariances.
  • The optimization problem is convex because its objective is concave in S and its constraints define a convex set.
  • The harvested-power constraint has the opposite inequality from cognitive-radio interference constraints because harvesting more power is desirable.

III. SEPARATED RECEIVERS

For spatially separated EH and ID receivers, the paper derives optimal covariance structures for the general MIMO case and closed-form beamforming results for MISO special cases. It also shows that channel correlation can enlarge the achievable R-E region.

  • General MIMO: For intermediate harvested-power targets, there is one unique pair of positive dual variables λ* and μ* under the stated conditions.
  • General MIMO: The general separated-receiver solution has a semi-closed form based on dual variables λ* and μ*, with A = μ*I − λ*G^H G and precoding from the SVD of HA^-1/2.
  • MISO special cases: When the ID receiver is single-antenna, the optimal covariance is rank one, so beamforming along A^-1h is optimal.
  • MISO special cases: With MISO channels to both receivers, the optimal covariance remains beamforming and admits a closed-form unit-norm beamforming vector.
  • Correlated channels: For correlation values ρ = 0.1, 0.5, and 0.9, increasing ρ enlarges the achievable R-E region.
  • MISO special cases: If the harvested-power target is low, maximum-ratio combining toward the ID channel is optimal; higher targets require a combination of EH-aligned and orthogonal components.

IV. CO-LOCATED RECEIVERS

For co-located receivers sharing the same channel, the paper derives an outer bound on achievable rate-power pairs and evaluates time switching and power splitting as practical receiver designs.

  • Co-located EH and ID receivers have identical channels, G = H, with equal antenna counts N_EH = N_ID = N.
  • The optimal solution for the co-located case provides an outer bound for the achievable R-E region.
  • Time switching and power splitting are investigated as practical receiver designs and compared with the outer bound.

A. Performance Outer Bound

For co-located EH and ID receivers, the optimal transmission strategy spatially multiplexes over the MIMO channel’s eigenmodes while reallocating power across them to trace the R-E boundary. This boundary is generally an outer bound for practical receivers because current EH circuits cannot directly decode information.

  • Optimal transmission strategy: The optimal strategy spatially multiplexes over the eigenmodes of H and varies their allocated power to achieve information-energy tradeoffs.When the harvested-power constraint is active, the modified water-filling policy has a non-decreasing water level as channel gains increase.
  • Boundary points: The maximum harvested power Qmax = Ph1 is achieved by beamforming all transmit power through the largest-gain channel.Conversely, conventional water-filling with λ*=0 yields the maximum transmission rate Rmax.
  • Practical achievability: The co-located R-E boundary is generally only an outer bound for practical receiver designs.All boundary pairs except (Rmax, 0) and (0, Qmax) require simultaneous full signal harvesting and decoding up to MIMO capacity, which existing EH circuits cannot generally provide.
  • Practical achievability: Achieving the remaining co-located boundary rate-power pairs remains an open problem because EH receivers cannot directly decode RF-band information.This limitation applies even in the SISO case according to the supplied discussion.

B. Time Switching

Time switching separates energy harvesting and information decoding into orthogonal transmission slots, with either fixed or flexible transmitter power constraints. Flexible power can characterize a broader tradeoff, but its ideal boundary relies on unbounded EH-slot power unless peak power is constrained.

  • Scheme: Time switching periodically alternates the co-located receiver between energy harvesting and information decoding.The transmitter and receiver synchronize their operation switching, while separate EH and ID signals remain subject to total power constraints.
  • Fixed power constraint: Under the fixed power constraint, the achievable R-E region is the straight line connecting (Rmax, 0) and (0, Qmax).Sweeping the EH-slot fraction α from 0 to 1 traces this line.
  • Flexible power constraint: With flexible power, all boundary points except (Rmax, 0) and (0, Qmax) are achieved asymptotically as the EH-slot fraction α approaches zero.Allocating O(log n) EH symbols in an n-symbol block gives α = log n/n → 0.
  • Peak-power limitation: The flexible-power boundary assumes infinite EH-slot power and therefore cannot be implemented with practical power amplifiers.Adding a peak constraint Ppeak yields α = Q/(h1Ppeak) for the achievable boundary.

C. Power Splitting

Power splitting divides each received signal between energy harvesting and information decoding, with the split also affecting receiver noise. In ideal noise-free processing it reaches the outer bound, whereas practical processing noise generally prevents full outer-bound achievability.

  • Receiver design: Power splitting sends a fraction ρ of received signal power to EH and 1−ρ to ID before RF-to-baseband processing.The ID branch incurs additional processing noise independent of antenna noise.
  • SISO regimes: For SISO channels, the PS R-E region depends on antenna and processing noise, with three regimes ranging from ideal antenna noise to ideal processing.The paper analyzes the achievable region separately for these noise conditions.
  • SISO regimes: Only noise-free RF-to-baseband processing allows PS to achieve the entire R-E outer bound.In that case, setting ρ toward one preserves the ID SNR while maximizing harvested power.
  • MIMO extension: For MIMO channels, the outer bound is achievable by PS when every receiving antenna satisfies the noise-free processing condition.The paper otherwise studies a worst-case PS region as a lower performance bound for practical receiver circuits.
  • MIMO extension: The general PS region is formed by optimizing over antenna-specific splitting ratios, including uniform and on-off splitting as special cases.On-off splitting assigns antennas wholly to EH or ID, while uniform splitting uses the same ρ at every antenna.
  • MIMO extension: For a co-located SIMO channel, the achievable PS region satisfies R ≤ log(1 + (∥h∥2P − Q)) with 0 ≤ Q ≤ ∥h∥2P.This gives the stated rate-energy tradeoff for any P > 0.

D. Performance Comparison

The comparison shows that fixed-power time switching is dominated by uniform power splitting, while flexible-power time switching can be better unless a stated channel-power condition makes them identical. In the illustrated symmetric MIMO channels, the gap between uniform splitting and the outer bound shrinks as channel symmetry increases.

  • Scheme comparison: For any P > 0, fixed-power time switching is contained within the achievable region of uniform power splitting.Thus, uniform power splitting performs at least as well in achievable rate-energy pairs.
  • Scheme comparison: Flexible-power time switching generally outperforms uniform power splitting, and they coincide iff P ≤ (1/h2 − 1/h1).The equality condition can also hold when h2 = 0, corresponding to MISO or SIMO channels.
  • Numerical comparison: As θ increases from 0.5 to 0.8, the performance gap between the UPS region and the outer bound is reduced.For this channel, h1 = (1 + θ)2 and h2 = (1 − θ)2, so the equality condition becomes satisfied as θ approaches one.
  • Practical regime: Under high SNR, practical co-located SWIPT receivers typically operate in the high-energy regime, with α and ρ approaching one.This regime corresponds to assigning very large time-switching or power-splitting coefficients.

V. CONCLUDING REMARKS

The paper identifies fundamental information–energy tradeoffs in wireless MIMO systems and outlines practical limits, assumptions, and open extensions.

  • The study reveals fundamental tradeoffs in designing wireless MIMO systems for simultaneous information and energy transmission.
  • Future work includes characterizing rate-energy regions for more general systems with more than two receivers and varied receiver configurations.The optimal solutions for these settings are described as challenging to obtain.
  • For co-located receivers, the performance bound generally cannot be achieved by practical receivers.Power splitting is discussed as one practical design for approaching this bound.
  • Further research is required to reduce or close the gap between practical receiver designs and the performance bound, even for the SISO AWGN channel.
  • The analysis assumes energy conversion efficiency is independent of the instantaneous amplitude of the received radio signal, unlike practical RF harvesting circuits.
  • Designing broadcast waveforms to maximize energy transfer under practical conversion constraints remains an open problem.

PROOF OF THEOREM 3.1

The proof establishes the dual formulation and boundedness conditions for the optimization problem, then derives the optimal transmit covariance through singular-value decompositions.

  • The dual problem is obtained by maximizing the Lagrangian over the transmit covariance and minimizing the resulting dual function over λ ≥ 0 and µ ≥ 0.
  • The optimization has a bounded optimal value only when µ > λg1.Here, g1 is the largest eigenvalue of GHG.
  • When µ > λg1, A = µI − λGHG is positive definite, so A^-1 exists and the problem can be reformulated accordingly.
  • The optimal solution is characterized using the reduced SVD of HA^-1/2 and singular-value-based power allocation.The allocation uses ˜pi = (1 − 1/˜hi)+.
  • A subgradient-based method such as the ellipsoid method finds the optimal dual variables and converges to the primal-optimal transmit covariance.
  • For the co-located case, the SVD expressions simplify because G = H, yielding specialized forms for the covariance and maximum achievable rate.

APPENDIX E

The appendix derives rate-energy boundaries for time switching and establishes a closed-form co-located SIMO rate-energy region under power splitting.

  • The time-switching rate-energy boundary is obtained by varying Q over 0 < Q < Qmax and solving the corresponding rate-maximization problem.
  • In time switching, the energy needed to harvest power Q is Q/h1, regardless of the energy-slot fraction α.
  • The information transmission rate is maximized as α → 0, with remaining transmit energy constrained by P − Q/h1.
  • For the SIMO co-located channel, the achievable region is R−E(P) = {(R, Q) : R ≤ log(1 + (∥h∥2P − Q)), 0 ≤ Q ≤ ∥h∥2P}.
  • The same SIMO rate-energy region follows for arbitrary power-splitting factors ρi with 0 ≤ ρi ≤ 1.

APPENDIX G

The appendix compares time switching and power splitting, showing when their rate-energy regions coincide and when time switching is strictly better.

  • The two schemes coincide when the optimal information covariance is rank-one, including the low-power regime characterized by P ≤ 1/h2 − 1/h1.
  • Power splitting matches time switching at the boundary points (0, Qmax) and (Rmax, 0).
  • For any harvested power Q, time switching achieves at least the power-splitting rate, so RTS ≥ RUPS.
  • When P > 1/h2 − 1/h1, time switching is strictly better for some harvested-power values because its optimal information covariance can have rank greater than one.
  • The contradiction argument shows equality cannot hold when the same solution would need to be both rank-one and higher-rank.
  • Thus, the claimed equality of the two rate-energy regions fails in the higher-power regime.
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