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Dynamic Resource Allocation for Multiple-Antenna Wireless Power Transfer

Gang Yang, Chin Keong Ho, Yong Liang Guan

arXiv:1311.4111v1cs.ITmath.IT

TL;DR

Longer channel estimation can improve CSI but leaves less time for WPT, and the paper addresses this tradeoff alongside WPT power allocation. It uses dynamic programming for dynamic-length preambles and offline optimization for fixed-length preambles; optimized fixed-length preambles with channel-power-aware allocation harvest almost as much energy as dynamic-length preambles with length-and-channel-power-aware allocation.

  • Problem

    Longer preambles improve channel-estimate accuracy but reduce WPT time, and prior work had not jointly addressed preamble overhead and WPT energy allocation.

    Source: §I · pp. 2–3

  • Method

    The paper optimizes dynamic preamble length online with dynamic programming and fixed preamble length offline, then derives power allocation schemes for both scenarios.

    Source: §I · p. 3, §VIII · p. 25

  • Results

    Optimized fixed-length preambles with channel-power-aware allocation harvest almost the same energy as dynamic-length preambles with length-and-channel-power-aware allocation.

    Source: §VII · p. 24

  • Takeaways & Limitations

    The fixed-length channel-power-aware scheme exploits channel dynamics while harvesting almost as much energy as the dynamic-length scheme.

    Source: §VII · p. 24

Abstract

from arXiv · show

We consider a point-to-point multiple-input-single-output (MISO) system where a receiver harvests energy from a wireless power transmitter to power itself for various applications. The transmitter performs energy beamforming by using an instantaneous channel state information (CSI). The CSI is estimated at the receiver by training via a preamble, and fed back to the transmitter. The channel estimate is more accurate when longer preamble is used, but less time is left for wireless power transfer before the channel changes. To maximize the harvested energy, in this paper, we address the key challenge of balancing the time resource used for channel estimation and wireless power transfer (WPT), and also investigate the allocation of energy resource used for wireless power transfer. First, we consider the general scenario where the preamble length is allowed to vary dynamically. Taking into account the effects of imperfect CSI, the optimal preamble length is obtained online by solving a dynamic programming (DP) problem. The solution is shown to be a threshold-type policy that depends only on the channel estimate power. Next, we consider the scenario in which the preamble length is fixed. The optimal preamble length is optimized offline. Furthermore, we derive the optimal power allocation schemes for both scenarios. For the scenario of dynamic-length preamble, the power is allocated according to both the optimal preamble length and the channel estimate power; while for the scenario of fixed-length preamble, the power is allocated according to only the channel estimate power. The analysis results are validated by numerical simulations. Encouragingly, with optimal power allocation, the harvested energy by using optimized fixed-length preamble is almost the same as the harvested energy by employing dynamic-length preamble, hence allowing a low-complexity WPT system to be implemented in practice.

I. INTRODUCTION

The paper balances channel-estimation time against wireless power transfer under imperfect CSI, while also allocating WPT power. It studies both dynamically chosen and fixed preamble lengths in a frame-based MISO system.

I. INTRODUCTION

  • Imperfect CSI affects energy beamforming, while longer channel estimation improves CSI but leaves less time for WPT; the paper notes prior work did not address both preamble overhead and WPT energy allocation.The paper also contrasts its harvested-energy objective with worst-case robust beamforming, which may be overly conservative.Source: §I · pp. 2–3

1) WPT with dynamic-length preamble:

  • With dynamic-length preambles, a dynamic program yields a threshold policy: continue channel estimation below a time-dependent channel-estimate-power threshold, and request WPT otherwise.The corresponding power allocation depends on both the selected preamble length and channel-estimate power.Source: §I · p. 3, 1) WPT with dynamic-length preamble · pp. 5–6

II. SYSTEM MODEL

  • The frame-based system has a multi-antenna transmitter, a single-antenna receiver, and CE, feedback, WPT, and energy-utilization phases.Feedback is assumed error-free and negligible in duration, and the energy-utilization phase duration is fixed.Source: §I · p. 3, §II · p. 4

A. Wireless Energy Beamforming

  • The transmitter uses a beamforming vector based on the channel estimate to send signals from which the receiver harvests RF energy.The paper focuses on harvested energy and leaves specific uses such as uplink transmission for future work.Source: §I · p. 3, A. Wireless Energy Beamforming · p. 5

B. Problem Formulation

  • The objective is to maximize harvested energy by choosing CE and WPT durations and allocating transmit power for WPT.The paper compares dynamic-length preambles with fixed-length preambles optimized offline.Source: §I · p. 3, B. Problem Formulation · p. 5

2) WPT with fixed-length preamble:

For fixed-length preambles, the paper optimizes the preamble length offline and allocates WPT power according to channel-estimate power. Its beamformer is based on the conditional channel correlation given imperfect CSI.

2) WPT with fixed-length preamble:

  • To reduce implementation complexity, the fixed-length scenario uses the same preamble length across frames, optimized offline, and allocates WPT power according only to channel-estimate power.The fixed-length optimization jointly considers preamble length and beamforming when WPT transmit power is constant.Source: 2) WPT with fixed-length preamble · p. 6

A. Partial or Full Feedback

  • To reduce feedback, the receiver can send only the q largest-gain unquantized channel coefficients; q=m corresponds to full CSI feedback.The transmitter beamforms using the antennas indexed by the selected coefficients.Source: A. Partial or Full Feedback · p. 7

B. Optimal Energy Beamforming

  • The analysis models the channel and estimation error as jointly Gaussian, then derives the conditional channel correlation used to choose the beamformer.The paper considers both LS and LMMSE estimators, with fixed preamble transmit power.Source: B. Optimal Energy Beamforming · pp. 7–8

III. OPTIMAL ENERGY BEAMFORMING

  • With imperfect CSI, the optimal beamforming vector is the largest eigenmode of the channel correlation matrix conditioned on the estimate.The expected normalized harvested energy is the mean of that matrix's largest eigenvalue.Source: B. Optimal Energy Beamforming · p. 8

C. Optimal Beamformer for LS Channel Estimation

For LS channel estimation, the paper selects an orthogonal preamble and derives the optimal beamformer from the channel correlation conditioned on the estimate. The uncorrelated-channel case has a simplified matrix structure.

C. Optimal Beamformer for LS Channel Estimation

  • The LS preamble design uses an orthogonal preamble matrix to optimize estimation performance under a fixed total preamble power.The passage identifies this design as optimal for correlated and uncorrelated channels.Source: C. Optimal Beamformer for LS Channel… · p. 8

1) Correlated channel:

  • For correlated channels, the optimal beamforming vector is the largest eigenmode of the conditional channel correlation matrix.The conditional correlation matrix is derived from the channel estimate.Source: 1) Correlated channel · p. 9

2) Uncorrelated channel:

  • For uncorrelated channels, the conditional correlation matrix is a scaled identity plus a rank-one matrix, with the normalized channel estimate as the maximal-eigenvalue eigenvector.The remaining eigenvectors can be constructed orthogonally using Gram-Schmidt.Source: 2) Uncorrelated channel · p. 9

D. Optimal Beamformer for LMMSE Channel Estimation

The paper derives beamforming and preamble designs for LMMSE estimation, then uses channel-estimate statistics to formulate dynamic-length WPT as a stopping problem. The uncorrelated case simplifies the conditional correlation structure.

D. Optimal Beamformer for LMMSE Channel Estimation

  • For LMMSE estimation, the preamble matrix and per-antenna transmit power are optimized using the channel's spatial correlation.The receiver is assumed to know the channel covariance matrix.Source: D. Optimal Beamformer for LMMSE… · pp. 9–10

1) Correlated channel:

  • For correlated channels, the optimal beamformer is the largest eigenmode of the conditional channel correlation matrix.The conditional correlation matrix is obtained from the channel estimate.Source: 1) Correlated channel · p. 10

2) Uncorrelated channel:

  • For uncorrelated channels, the LMMSE preamble reduces to the orthogonal preamble, and the conditional correlation has a structure similar to the LS case.The beamformer follows from the corresponding conditional correlation matrix.Source: 2) Uncorrelated channel · p. 10

IV. WIRELESS POWER TRANSFER WITH DYNAMIC-LENGTH PREAMBLE

  • For dynamic-length preambles, the paper formulates a dynamic program that reduces to an optimal stopping problem.The analysis assumes uncorrelated flat Rayleigh fading that is constant within a frame and independent across frames.Source: §IV · p. 10

A. Statistical Properties of Channel Estimates

  • The channel-estimate analysis characterizes adjacent estimates statistically and states that the most recent estimate is sufficient for estimating the channel.The channel-estimate power is defined as the squared norm of the estimate.Source: A. Statistical Properties of Channel… · p. 11

2. From Lemma 3, conditioned

The dynamic program uses the channel-estimation history to characterize the channel, but the most recent estimate alone is sufficient for the policy state. It frames each slot as a choice between continuing channel estimation and requesting wireless power transfer.

2. From Lemma 3, conditioned

  • Using all available LS channel estimates does not improve estimation accuracy over using only the most recent estimate.This observation supports the structure of the optimal dynamic-programming policy.Source: §2 · p. 12

1) Decision (or Control) Variable:

  • The receiver’s two control actions are to stop channel estimation and request wireless power transfer, or continue channel estimation.The initial decision is to continue estimation.Source: 1) Decision (or Control) Variable · p. 12

2) System State:

  • The system state records the number of channel-estimation slots used and the most recent channel estimate, which suffices to obtain channel statistics.The initial state has no available estimate.Source: 2) System State · p. 12

3) Policy:

  • A policy maps each system state at each slot to a decision for the next slot.Source: 3) Policy · p. 12

4) Reward:

The dynamic program maximizes expected harvested energy by choosing when to stop channel estimation and begin wireless power transfer. Its optimal policy has a threshold form based only on channel-estimate power and can be implemented with low complexity.

4) Reward:

5) Dynamic Program and Optimal Policy:

C. Optimal Policy

  • The optimal policy is a threshold rule depending only on channel-estimate power, and stopping channel estimation makes the receiver continue wireless power transfer thereafter.The decision sequence consists of an initial estimation period followed by a wireless power transfer period.Source: C. Optimal Policy · pp. 13–14
  • Comparing channel-estimate power with a precomputed scalar threshold at each slot avoids storing or computing the full high-dimensional policy online.The thresholds can be stored in a lookup table.Source: C. Optimal Policy · pp. 14–15

D. Optimal Thresholds

The dynamic policy’s thresholds are computed backward, and numerical results indicate one threshold per slot that decreases over time. The fixed-length scenario instead optimizes a preamble duration offline.

D. Optimal Thresholds

  • Thresholds are obtained backward from the final slot, with numerical search used when the recursive equation lacks a closed-form solution.The search discretizes channel-estimate power values.Source: D. Optimal Thresholds · pp. 15–16
  • Numerically, each slot’s threshold is unique and decreases as the slot index increases.The authors relate the decrease to having one additional slot in which the channel estimate can improve.Source: D. Optimal Thresholds · p. 16

V. WIRELESS POWER TRANSFER WITH FIXED-LENGTH PREAMBLE

  • For a fixed-length preamble, the paper derives an offline optimal duration and then an optimal power-allocation scheme based on channel-estimate power.Source: §V · p. 16

A. WPT with LS Channel Estimator

For LS channel estimation, the paper treats correlated and uncorrelated channels separately when optimizing fixed preamble length. It also introduces an LMMSE channel-estimator case.

1) Correlated channel:

  • For a general correlated channel, the optimal preamble length has no apparent closed-form expression and is found by one-dimensional numerical search.Source: 1) Correlated channel · p. 17

2) Uncorrelated channel:

  • For an uncorrelated channel with LS estimation, the optimal preamble length is unique, and the maximal harvested energy is evaluated at that length.The quantity G_m,q captures the effect of q-dimensional feedback on harvested energy.Source: 2) Uncorrelated channel · p. 18

B. WPT with LMMSE Channel Estimator

1) Correlated channel:

The analysis derives energy beamforming and preamble-length results for correlated and uncorrelated channels, then develops dynamic-preamble power allocation based on channel conditions and preamble length.

1) Correlated channel:

  • The optimal beamforming vector for a correlated channel is the largest eigenmode of Rhq|bhq, while its optimal preamble length is obtained numerically.For the uncorrelated channel, the paper notes an analogous result to Theorem 2, also validated numerically.Source: 1) Correlated channel · p. 19, 2) Uncorrelated channel · p. 19

A. Dynamic-Length-Preamble Based Power Allocation

1) Optimal Length-and-Channel-Power Aware Power Allocation:

  • LCPA allocates power according to both optimal preamble length κ and channel estimate power eVκ.Its optimal rule gives up to P1 to frames in descending order of transfer efficiency η(eVκ, κ), until the average energy constraint P2 is met.Source: 1) Optimal Length-and-Channel-Power… · p. 20

2) Optimal Length-Aware Power Allocation:

The section compares simpler power-allocation schemes with dynamic allocation and reports that optimized fixed-length preambles with channel-aware allocation nearly match the dynamic scheme.

2) Optimal Length-Aware Power Allocation:

  • LPA restricts power allocation to depend only on optimal preamble length κ, not on channel estimate power.Its optimal rule allocates power in descending order of η(κ) until the average energy constraint is met.Source: 2) Optimal Length-Aware Power Allocation · pp. 21–22

B. Fixed-Length-Preamble Based Power Allocation

  • With fixed preamble length, CPA allocates power according to channel estimate power, using the offline-optimal preamble length for every frame.The optimal rule allocates up to P1 to frames in descending order of channel estimate power until the average constraint is met.Source: B. Fixed-Length-Preamble Based Power… · p. 22

VII. NUMERICAL RESULTS

  • The numerical evaluation uses 5 ms for the CE and WPT phases, T = 126 symbol periods, m = 3, and N = 42.It assumes a 10 m transmitter–receiver distance and a 20 dB path loss at a 1 m reference distance.Source: §VII · p. 22
  • 75% more harvested power is achieved by LCPA than FwoPA when P1 = 8P0, under the same average WPT energy consumption.The comparison is between dynamic-length preambles with LCPA and optimized fixed-length preambles without adaptive power allocation.Source: §VII · p. 24, Figure 5 · p. 25
  • CPA harvests almost as much energy as LCPA, while LPA harvests less than both schemes.The paper attributes CPA's result to exploiting channel dynamics after optimizing the preamble length over channel realizations; LPA only partially exploits those dynamics.Source: §VII · p. 24, Figure 6 · p. 25

VIII. CONCLUSION

  • The conclusion identifies optimal beamforming, dynamic and offline preamble optimization, and power allocation for both preamble scenarios as the paper's contributions.Applying the WPT system to specific applications is left for future work.Source: §VIII · pp. 25–26

APPENDIX A

The appendices support the channel-estimation and policy arguments, and establish how the continuous preamble-length optimum is converted to an allowed discrete choice.

APPENDIX A

  • For the uncorrelated-channel proof, the channel conditioned on successive estimates depends on the latest estimate, and the conditional distribution is Gaussian and independent of the previous estimate.The proof also uses independence between channel-vector elements.Source: Appendix A · p. 26

APPENDIX B

  • Policy 2, which performs channel estimation before wireless power transfer in the next slot, has strictly higher expected harvested energy than Policy 1.The argument extends to intervening estimation slots and to subsequences ending in channel estimation.Source: Appendix B · p. 27

APPENDIX C

  • For β ≥ 0, bE′′(β) < 0, so β1 is the unique value maximizing bE(β).Because the preamble length must be a multiple of m, the discrete optimum is selected from the floor and ceiling of τ1.Source: Appendix C · p. 29

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