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Throughput Optimization for Massive MIMO Systems Powered by Wireless Energy Transfer

Gang Yang, Chin Keong Ho, Rui Zhang, Yong Liang Guan

arXiv:1403.3991v2cs.IT

TL;DR

The paper addresses throughput and fairness design for wireless-powered massive MIMO when perfect CSI is unavailable. It jointly optimizes frame timing and energy allocation, derives large-M asymptotic solutions, and shows that imperfect-CSI WET-MM attains the ideal perfect-CSI MM-DoRG while asymptotically equalizing user rates.

  • Problem

    Perfect CSI is unavailable in practice, while channel-estimation time trades off against WET and WIT duration, complicating throughput optimization.

  • Method

    The paper jointly optimizes CE time, WET time, energy allocation, and CE-versus-WIT energy splitting in a three-phase WET-MM frame.

  • Results

    κWET-MM = κIdeal = 2, and all users asymptotically achieve a common rate in the large-M regime.

  • Takeaways & Limitations

    Energy beamforming enables optimal MM-DoRG relative to perfect CSI and improves fairness compared with the double near-far setting.

Abstract

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This paper studies a wireless-energy-transfer (WET) enabled massive multiple-input-multiple-output (MIMO) system (MM) consisting of a hybrid data-and-energy access point (H-AP) and multiple single-antenna users. In the WET-MM system, the H-AP is equipped with a large number $M$ of antennas and functions like a conventional AP in receiving data from users, but additionally supplies wireless power to the users. We consider frame-based transmissions. Each frame is divided into three phases: the uplink channel estimation (CE) phase, the downlink WET phase, as well as the uplink wireless information transmission (WIT) phase. Firstly, users use a fraction of the previously harvested energy to send pilots, while the H-AP estimates the uplink channels and obtains the downlink channels by exploiting channel reciprocity. Next, the H-AP utilizes the channel estimates just obtained to transfer wireless energy to all users in the downlink via energy beamforming. Finally, the users use a portion of the harvested energy to send data to the H-AP simultaneously in the uplink (reserving some harvested energy for sending pilots in the next frame). To optimize the throughput and ensure rate fairness, we consider the problem of maximizing the minimum rate among all users. In the large-$M$ regime, we obtain the asymptotically optimal solutions and some interesting insights for the optimal design of WET-MM system. We define a metric, namely, the massive MIMO degree-of-rate-gain (MM-DoRG), as the asymptotic UL rate normalized by $\log(M)$. We show that the proposed WET-MM system is optimal in terms of MM-DoRG, i.e., it achieves the same MM-DoRG as the case with ideal CE.

I. INTRODUCTION

The paper studies WET-enabled massive MIMO with imperfect channel state information, jointly optimizing frame resources to maximize minimum user throughput. It introduces MM-DoRG and shows that energy beamforming achieves the ideal perfect-CSI scaling while asymptotically equalizing user rates.

  • Research problem: Accurate CSI improves both energy-transfer efficiency and uplink rate, but longer channel estimation reduces time available for WET and WIT.This creates a throughput trade-off over channel-estimation, energy-transfer, and information-transmission durations.
  • System motivation: WET-MM combines a hybrid data-and-energy access point with a large antenna array and single-antenna users powered for uplink transmission.Each frame includes uplink channel estimation, downlink wireless energy transfer, and uplink information transmission.
  • Contributions: MM-DoRG measures the asymptotic uplink-rate scaling with respect to log M and compares beamformed WET against ideal or omnidirectional powering.The metric is motivated by the differing asymptotic rate scalings of beamformed and broadcast energy transfer.
  • Research problem: The paper jointly optimizes CE time, WET time, downlink energy weights, and each user’s CE-versus-WIT energy split.The objective is throughput optimization with rate fairness under imperfect CSI.
  • Contributions: κWET-MM = κIdeal = 2, whereas omnidirectional powering has MM-DoRG one, establishing optimal scaling for the proposed system.Energy beamforming is identified as crucial for efficient WET and optimal MM-DoRG.
  • Contributions: All users asymptotically achieve a common rate, providing the best possible fairness and addressing the double near-far effect.The paper also reports low-complexity operation and numerical accuracy for antenna counts as small as 25.

B. Downlink Energy Transfer Phase

During downlink WET, the H-AP transmits a channel-estimate-dependent beamformed signal, and users harvest energy for later pilots and uplink data. The model accounts for beam allocation, reciprocity, storage, and detection during subsequent WIT.

  • Downlink WET: The H-AP transmits downlink WET with power pdl using a unit-norm beamformer w(bG) designed from the channel estimate.The received signal uses channel reciprocity and includes user noise.
  • Energy beamforming: The asymptotically optimal energy beamformer is a linear combination of normalized channel estimates for the users.The associated user weights determine downlink energy allocation and are optimized later.
  • Energy allocation: Users store harvested energy, reserving a fraction ρ for pilot transmission and using the remainder for uplink information transmission.Pilot energy affects CSI accuracy, which in turn affects WET and WIT performance.
  • Uplink WIT: In WIT, all users transmit simultaneously, while the H-AP uses ZF or MRC linear detection to recover their information.The received vector is processed by a detector A whose columns correspond to users.
  • Rate analysis: The achievable rate is represented by an analytically tractable lower bound obtained using Jensen’s inequality.The paper later reports that this lower bound is numerically tight for massive MIMO systems.

D. Ideal Case and OP-MM System

The paper benchmarks imperfect-CSI WET-MM against a perfect-CSI ideal system and an omnidirectional-powering massive MIMO system. It formulates max-min rate optimization because distance-dependent attenuation creates user unfairness.

  • Ideal case: The ideal benchmark removes channel estimation, sets τ = 0 and ρ = 0, and assumes perfect CSI for both WET beamforming and uplink decoding.The imperfect-CSI system cannot outperform this ideal case.
  • OP-MM system: OP-MM broadcasts energy in all directions before uplink CE and WIT, so channel estimation is used for data detection but not for WET.This provides a low-complexity non-beamforming comparison system.
  • Fairness challenge: Far users harvest less energy and require higher uplink transmit power, producing the double near-far effect and lower rates.The effect motivates explicit fairness optimization.
  • Optimization problem: The optimization maximizes the minimum achievable user rate over CE time, WET time, energy splitting, and downlink allocation.These variables are coupled through CSI accuracy, harvested energy, signal power, interference, and frame duration.
  • Optimization problem: The asymptotic analysis derives tractable solutions despite the nonlinear coupling among CE, WET, and WIT variables.The paper proceeds from achievable-rate analysis to large-M asymptotics and asymptotically optimal resource allocation.

IV. ANALYSIS ON ACHIEVABLE RATE

The analysis derives harvested-energy expressions and achievable uplink-rate bounds under imperfect CSI, using the proposed beamformer and ZF or MRC detection. It distinguishes desired beam energy from energy received from other users’ beams.

  • Harvested energy: The harvested-energy derivation begins by applying the proposed beamformer to the downlink WET signal.The resulting expression is then used in the achievable-rate analysis.
  • Harvested energy: The first harvested-energy term comes from the beam aimed at user k, while the second comes from beams aimed at other users.Both contributions can be collected by user k.
  • Channel estimation: Harvested energy increases with pilot length L or pilot power pCE_k, while LpCE_k represents the pilot-transmission energy.A fraction ρ of harvested energy is assigned to pilots, affecting estimation-error variance.
  • Channel estimation: The harvested-energy expression is obtained by solving a quadratic equation and discarding its negative solution.The resulting value determines the channel-estimation error variance.
  • Achievable uplink rate: With ZF detection, interference is due only to channel-estimation error, whereas MRC also includes multi-user interference.MRC has lower computational complexity because antennas independently apply matched filters.

C. Achievable Rate for Ideal Case and OP-MM System

The paper derives asymptotic achievable rates for the ideal WET-MM and OP-MM systems, then compares their rate-growth behavior with the proposed imperfect-CSI system. The analysis shows that energy beamforming and suitable scaling of system parameters determine the asymptotic gain.

  • Achievable rates: For sufficiently large M, the ideal-case and OP-MM uplink rates are characterized for both ZF and MRC detection.The ideal case assumes perfect CSI, while the OP-MM comparison uses the corresponding asymptotic rate expressions.
  • Detection-dependent behavior: For MRC, the asymptotic rate is independent of ρ because both signal and multiuser-interference powers are proportional to ρ.Thus, any ρ in (0, 1) can be selected, provided it does not approach one.
  • Detection-dependent behavior: ZF cancels multiuser interference asymptotically, whereas MRC retains interference and has an asymptotic SINR of order O(M).The factor M−1 in the MRC SINR arises from maximum-ratio combining.
  • MM-DoRG: The maximal asymptotic MM-DoRG is 2 for ZF and 1 for MRC in the proposed WET-MM system.The result follows from the asymptotic rate expressions and the corresponding SINR orders.

B. Asymptotic Analysis for Ideal Case and OP-MM System

The paper establishes MM-DoRG benchmarks for the ideal WET-MM and OP-MM systems and uses them to assess the proposed system's asymptotic efficiency. Energy beamforming provides a higher rate-growth order than energy broadcasting without beamforming.

  • OP-MM comparison: The OP-MM system has maximal asymptotic MM-DoRG κOP-MM = 1 for both ZF and MRC.Its asymptotic SINR is of order O(M).
  • Ideal WET-MM: The ideal WET-MM system's maximal asymptotic MM-DoRG is obtained from its ZF and MRC SINR orders.The ideal system assumes perfect CSI at the H-AP.
  • Comparison: The proposed WET-MM system's MM-DoRG is double that of OP-MM because energy beamforming increases harvested energy, CSI accuracy, and uplink transmit power.The proposed system also asymptotically achieves a common user rate, unlike OP-MM, and requires roughly the square root as many antennas for a target common rate.

VI. ASYMPTOTICALLY OPTIMAL SOLUTION

The paper derives asymptotically optimal energy-allocation weights for maximizing the minimum user rate in the proposed WET-MM system. The resulting allocation compensates path loss and supports asymptotic rate fairness.

  • Optimal energy allocation weights: The asymptotically optimal ξ_k maximizes the minimum achievable rate when M satisfies the large-system condition.The result applies to both ZF and MRC detection.
  • Optimal energy allocation weights: ξ_k is inversely proportional to the square of user k's long-term path loss.This compensates path loss in both downlink WET and uplink WIT.
  • Rate fairness: The optimal allocation enables users to asymptotically achieve a common rate, ensuring fairness among users.The analysis is evaluated in the large-M regime, and the required condition is numerically well satisfied for M as small as 25.

2) Asymptotically Optimal Energy-Splitting:

The paper derives asymptotically optimal energy-splitting and time-allocation policies for the minimum-rate objective. The choices differ between ZF and MRC because their asymptotic rate expressions have different dependence on the energy-splitting fraction.

  • Asymptotically optimal energy-splitting: For ZF, the asymptotically optimal energy-splitting fraction ρ⋆ is obtained by maximizing a quasiconcave asymptotic-SINR function.The optimum is found by solving the function's first-order condition.
  • Asymptotically optimal energy-splitting: For ZF, the optimal ρ⋆ increases with the number of users K because larger systems are more interference-limited.More harvested energy is therefore assigned to channel estimation to reduce interference from estimation errors.
  • Asymptotically optimal time allocation: The paper derives asymptotically optimal CE and WET time allocations separately for ZF and MRC.The ZF and MRC allocations are given under their respective large-M conditions.
  • Asymptotically optimal time allocation: For MRC, the optimal WET time fraction α tends to zero as O(M^-ϕ), with 0 < ϕ < 1.Choosing ϕ close to one makes α approach zero faster with M.

4) Asymptotically Maximal Minimum Rate:

In the large-M regime, the proposed WET-MM system yields asymptotically common user rates and supports user scaling for a target common rate. Its optimized imperfect-CSI design balances time and energy resources, while pilot overhead creates a performance gap relative to ideal CSI.

  • Asymptotically maximal minimum rate: The asymptotically maximal minimum rate is common to all users under the proposed WET-MM system.This establishes best possible asymptotic rate fairness and addresses the double near-far effect affecting conventional WPCNs.
  • Asymptotically maximal minimum rate: When K scales with M at a constant ratio, the path-loss-based quantity c1 approaches a constant as K tends to infinity under uniformly distributed user distances.The stated distance model assumes dk is independent and uniformly distributed over [a, b], with path-loss exponent u > 1.
  • Asymptotically maximal minimum rate: For large M and K, a desired common rate R0 determines the asymptotically supported user scaling through the unique positive solution ζ0 of eR(ζ) = R0.The uniqueness of ζ0 makes the rate-target-to-user-scaling relation well defined.
  • Asymptotically maximal minimum rate: As the number of users grows, imperfect CSI incurs performance degradation because users spend more time and energy sending pilots than in the ideal-CSI case.Ideal CSI requires no user pilots, whereas the proposed system optimizes pilot-resource usage within the imperfect-CSI framework.
  • Asymptotically maximal minimum rate: The asymptotically optimal design varies time allocation, energy allocation, and the harvested-energy fraction used for pilot transmission.The analysis considers both ZF and MRC detection, with achievable rates depending on τ, α, and ρ.

2) OP-MM System:

Numerical results validate the asymptotically optimized resource allocation and show that the proposed WET-MM system improves rate scaling, antenna requirements, and fairness over OP-MM.

  • Resource allocation: At M = 200, the optimized CE time is τ⋆ = 0.00825, WET time is α⋆ = 0.0760, and energy-splitting fraction is ρ⋆ = 0.5961.The resulting asymptotic maximal minimum rate is 16.0096 bps/Hz.
  • Resource allocation: 15.9540 bps/Hz is the numerically achieved maximal minimum rate, closely approximating the asymptotic value 16.0096 bps/Hz.At the selected allocation, user 1 and user 2 achieve 16.3491 and 15.9540 bps/Hz, respectively.
  • Resource allocation: The data rate is quasi-concave in ρ, with a unique optimum near ρ⋆ ≈ 0.6.This numerical optimum coincides with the analytic result in Lemma 7.
  • Resource allocation: As M increases, the optimal CE time tends quickly to zero, while the optimal WET time tends to zero more slowly.The numerical behavior agrees with the asymptotic results in Lemmas 7 and 8.
  • Rate comparison: The proposed system achieves more than 80% of the perfect-CSI rate limit and higher rates than OP-MM, with analytical rates matching simulations even at M = 10.ZF detection also outperforms MRC detection in the simulations.
  • Rate comparison: κWET−MM = κIdeal = 2, double the MM-DoRG of OP-MM.The proposed system is therefore numerically verified as optimal in MM-DoRG.
  • Rate comparison: For max-min rates of 10, 8, and 6.4 bps/Hz at d2 = 12 m, WET-MM requires about 21, 10, and 5 antennas, versus 400, 100, and 25 for OP-MM.The WET-MM antenna requirements are roughly the square root of OP-MM requirements.
  • Fairness: The proposed WET-MM system asymptotically gives both users the same rate, whereas OP-MM leaves the far user with a much lower rate.This establishes better user fairness for the proposed system in the massive-MIMO regime.

APPENDIX A PROOF OF LEMMA 1

The proof establishes the asymptotically optimal structure of the energy beamformer by comparing a general beamformer with a structured construction based on channel estimates.

  • Beamformer construction: A general beamformer can be written using an orthonormal basis for the complement of the estimated-channel subspace.
  • Channel decomposition: The MMSE channel estimate is independent of its estimation error, enabling conditional analysis of harvested energy.
  • Harvested-energy analysis: The harvested-energy expression is decomposed into four terms and evaluated using independence and conditional-distribution properties.
  • Optimality argument: A beamformer with the structured form in (8) can be constructed to asymptotically harvest more energy than any beamformer of the general form.Therefore, the asymptotically optimal beamformer has the structure in (8).
  • Harvested-energy analysis: Substitution of the channel-estimation error variance yields the harvested-energy expression in (16).

APPENDIX D PROOF FOR LEMMA 6

The proof optimizes the energy allocation weights by equalizing users’ rates, showing that any unequal-rate allocation can improve the minimum rate.

  • ZF rate optimization: The asymptotic ZF rate is rewritten so that each user’s energy allocation weight appears through a coefficient Ck(τ, α, ρ).
  • ZF rate optimization: If one user has a higher rate than the others, shifting energy allocation toward the lower-rate users increases the minimum rate.
  • ZF rate optimization: The minimum rate is maximized when all users achieve the same rate.The corresponding energy allocation weight ξk⋆ is given by (36).
  • ZF rate optimization: The resulting optimal allocation depends only on the users’ long-term path losses βi.
  • MRC rate optimization: For MRC detection, the minimum-rate optimization can be characterized using the Karush-Kuhn-Tucker conditions.
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