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Wirelessly Powered Data Aggregation for IoT via Over-the-Air Functional Computation: Beamforming and Power Control

Xiaoyang Li, Guangxu Zhu, Yi Gong, Kaibin Huang

arXiv:1808.04616v1cs.ITmath.IT

TL;DR

Dense IoT deployments need fast wireless data aggregation and power for energy-constrained sensors. This paper jointly optimizes WPT and AirComp in WP-AirComp, using a nested design for power control, energy beamforming, and aggregation beamforming. The reported solution structure equalizes effective channels and directs energy beams along dominant WPT-channel eigen-directions, while simulation shows WPT control can substantially reduce AirComp error.

Abstract

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As a revolution in networking, Internet of Things (IoT) aims at automating the operations of our societies by connecting and leveraging an enormous number of distributed devices (e.g., sensors and actuators). One design challenge is efficient wireless data aggregation (WDA) over tremendous IoT devices. This can enable a series of IoT applications ranging from latency-sensitive high-mobility sensing to data-intensive distributed machine learning. Over-the-air (functional) computation (AirComp) has emerged to be a promising solution that merges computing and communication by exploiting analogwave addition in the air. Another IoT design challenge is battery recharging for dense sensors which can be tackled by wireless power transfer (WPT). The coexisting of AirComp and WPT in IoT system calls for their integration to enhance the performance and efficiency of WDA. This motivates the current work on developing the wirelessly powered AirComp (WP-AirComp) framework by jointly optimizing wireless power control, energy and (data) aggregation beamforming to minimize the AirComp error. To derive a practical solution, we recast the non-convex joint optimization problem into the equivalent outer and inner sub-problems for (inner) wireless power control and energy beamforming, and (outer) the efficient aggregation beamforming, respectively. The former is solved in closed form while the latter is efficiently solved using the semidefinite relaxation technique. The results reveal that the optimal energy beams point to the dominant eigen-directions of the WPT channels, and the optimal power allocation tends to equalize the close-loop (down-link WPT and up-link AirComp) effective channels of different sensors. Simulation demonstrates that controlling WPT provides additional design dimensions for substantially reducing the AirComp error.

I. INTRODUCTION

The introduction motivates fast wireless data aggregation and sensor power delivery as complementary IoT challenges, then proposes WP-AirComp to jointly optimize them. It outlines a nested design for minimizing computation error, with distinct WPT and aggregation-beamforming solutions.

I. INTRODUCTION

  • The introduction frames WDA as fast collection and processing of distributed device data, needed for applications including environmental monitoring, federated learning, and fleet driving.It highlights ultra-dense devices and high mobility as settings where fast aggregation is needed.Source: §I · p. 2

A. WDA via Over-the-Air Computation

  • AirComp uses simultaneous analog transmission and wave addition to compute functions over the multi-access channel, whereas orthogonal data collection has latency that scales linearly with device count.The introduction identifies nomographic functions as AirComputable and cites averaging and geometric mean as examples.Source: A. WDA via Over-the-Air Computation · pp. 2–3

B. Energy Beamforming for Wirelessly Powered Communication

  • Prior WPT research includes sensor-network energy delivery, but wirelessly powered in-network computation such as WP-AirComp remains an uncharted area.Much prior work on wirelessly powered communication focuses on rate maximization through WPT-efficiency optimization.Source: B. Energy Beamforming for Wirelessly… · pp. 4–5

C. Contributions and Organization

  • WP-AirComp jointly designs wireless power control, energy beamforming, and data aggregation beamforming because harvested energy affects the sensors’ uplink superposition.The framework is formulated to minimize AirComp computation error under a WPT power constraint.Source: §I · p. 2, C. Contributions and Organization · pp. 5–6
  • The non-convex design is decomposed into nested subproblems: inner WPT optimization and outer aggregation-beamformer optimization, with an extension from MISO to MIMO.The inner problem has a closed-form solution; the outer problem uses semidefinite relaxation (SDR).Source: C. Contributions and Organization · p. 6
  • Optimal power control tends to equalize cascaded WPT-AirComp channels, while energy beams follow the WPT channel vector or dominant eigenvector, depending on the case.The contribution summary specifies channel vectors for MISO and dominant eigenvectors for MIMO.Source: C. Contributions and Organization · p. 6

II. SYSTEM MODEL

WP-AirComp alternates downlink wireless power transfer (WPT) and uplink over-the-air computation (AirComp), using harvested energy to power sensor transmissions. The server beamforms to aggregate simultaneous sensor data and measures performance by computation error.

II. SYSTEM MODEL

  • The harvested-then-transmit protocol alternates fixed-duration WPT and AirComp phases, with energy harvested in one phase powering the following AirComp phase.Channels remain fixed within each phase under the block-fading model.Source: §II · p. 7

A. WPT Phase

  • In WPT, the multi-antenna server allocates power across sensors and delivers it through energy beamforming, while sensors harvest energy for transmission.The model assumes negligible harvesting from unintended beam sidelobes and a fixed energy-conversion efficiency for each sensor.Source: A. WPT Phase · pp. 7–9

B. AirComp Phase

  • In AirComp, synchronized sensors transmit preprocessed vector symbols simultaneously, and the server uses aggregation beamforming to coherently combine them into the target-function vector.The model defines computation error as the distortion between the received estimate and desired output, measured by mean-squared error.Source: B. AirComp Phase · pp. 9–10
  • Sensor transmission power is constrained by harvested energy, with only a fraction of that energy allocated to transmission.The fraction is denoted θ_k and lies in (0, 1).Source: B. AirComp Phase · p. 10

III. JOINT DESIGN OF WPT AND AIRCOMP: PROBLEM FORMULATION

The joint design minimizes AirComp computation error over wireless power control, energy beamforming, and aggregation beamforming. Because the problem is non-convex and couples beamformers across both phases, the paper develops a tractable decomposition.

III. JOINT DESIGN OF WPT AND AIRCOMP: PROBLEM FORMULATION

  • The joint design optimizes wireless power control and energy and aggregation beamforming to minimize computation error under the WPT and sensor power constraints.The formulation is non-convex because transmit and receive beamformers are coupled in both phases.Source: §III · pp. 10–11
  • For single-antenna sensors, the problem reduces to MISO channels with scalar data precoders, and only scalar-function AirComp is feasible.This simplified case removes receive energy beamformers at sensors and makes the design tractable.Source: §III · p. 11
  • The MISO solution provides insights used to develop a practical solution, while maximizing receive SNR leads to strategies fundamentally different from minimizing computation error.Source: §III · p. 11

IV. JOINT DESIGN OF WPT AND AIRCOMP: MISO CASE

  • For the MISO case, the approach decomposes the problem into an inner WPT optimization over power control and energy beamforming and an outer aggregation-beamformer optimization.The inner problem is conditioned on the aggregation beamformer.Source: §IV · p. 12, A. Decomposition Approach · p. 13

A. Decomposition Approach

  • Given an aggregation beamformer, channel-inversion sensor precoding is optimal for minimizing computation error.The sensor precoders satisfy the sensor transmission-power constraints through a scaling factor.Source: A. Decomposition Approach · p. 12
  • Substituting the optimal precoders transforms the design into minimizing the denoising factor η, which suppresses channel-noise effects as symbol energy increases.Source: A. Decomposition Approach · p. 12
  • The resulting problem is recast as equivalent outer and inner sub-problems, making the remaining non-convex design tractable through separate solutions.Source: A. Decomposition Approach · p. 13

B. Optimal Wireless Power Control and Energy Beamforming

For a fixed aggregation beamformer, the inner problem optimizes WPT energy beams and power allocation. The resulting structure steers beams along WPT channels and tends to equalize the sensors’ cascaded effective channels.

B. Optimal Wireless Power Control and Energy Beamforming

C. Optimal Aggregation Beamforming

The outer problem optimizes the aggregation beamformer after substituting the optimal WPT beamformer and power allocation. Semidefinite relaxation (SDR) converts its non-convex constraint into a convex problem, with Gaussian randomization used when needed to recover a feasible beamformer.

C. Optimal Aggregation Beamforming

  • Substituting the optimal energy beamformer and power allocation simplifies the outer problem of optimizing the aggregation beamformer.Source: C. Optimal Aggregation Beamforming · p. 15
  • SDR relaxes the non-convex norm constraint using its convex hull, yielding a convex optimization problem.The relaxed matrix is defined as F̂ = ffᴴ.Source: C. Optimal Aggregation Beamforming · p. 15
  • If the SDR solution has rank greater than one, Gaussian randomization extracts a feasible aggregation beamformer by selecting the best candidate among random samples.Source: C. Optimal Aggregation Beamforming · p. 16
  • When the SDR solution is rank one, its dominant eigenvector gives the global optimal normalized aggregation beamformer.The paper reports that the SDR solution has a high probability of being rank one in simulation.Source: C. Optimal Aggregation Beamforming · p. 16

V. JOINT DESIGN FOR WPT AND AIRCOMP: MIMO CASE

The MIMO extension solves the vector-function WP-AirComp problem using the outer–inner framework. It simplifies the design through an optimal zero-forcing precoder and separates WPT optimization from aggregation-beamformer design.

V. JOINT DESIGN FOR WPT AND AIRCOMP: MIMO CASE

  • The MIMO vector-function WP-AirComp problem is addressed through the outer–inner framework, separating WPT optimization from aggregation-beamformer design.The inner sub-problem jointly optimizes power control and energy beamforming; the outer sub-problem designs the MIMO aggregation beamformer.Source: §V · pp. 16, 18
  • Given an aggregation beamformer, zero-forcing precoders minimize the computation error.This optimality result simplifies the non-convex Problem P1.Source: §V · pp. 16–17
  • The design is reduced by normalizing the aggregation beamformer as F, with A = √ηF and tr(FFH) = 1.The reduction substitutes the optimal precoder into Problem P1.Source: §V · p. 17

A. Optimal Wireless Power Control and Energy Beamforming

For a fixed normalized aggregation beamformer, the MIMO inner problem yields optimal energy beams and power allocation. The beams follow dominant WPT-channel directions, while sensor power depends inversely on the effective closed-loop channel gain.

A. Optimal Wireless Power Control and Energy Beamforming

  • The inner sub-problem derives optimal energy transmit and receive beamformers and power control as functions of normalized aggregation beamformer F.The energy beams are chosen first; the resulting inner solution is then substituted into the outer aggregation-beamformer problem.Source: A. Optimal Wireless Power Control and… · pp. 18–19
  • Optimal energy transmit and receive beams point along the left and right dominant singular vectors of each WPT channel matrix.The dominant singular vectors maximize the WPT objective for each sensor.Source: A. Optimal Wireless Power Control and… · p. 18
  • The optimal MIMO power allocation is inversely proportional to each sensor’s effective closed-loop channel gain.That gain combines the dominant singular value of the WPT channel matrix with the beamed AirComp channel after aggregation beamforming.Source: A. Optimal Wireless Power Control and… · p. 19

B. Optimal Aggregation Beamformer

The outer problem designs the MIMO aggregation beamformer after substituting the optimal WPT strategies. Its non-convex objective is relaxed to enable convex optimization, then Gaussian randomization yields a feasible approximate solution.

B. Optimal Aggregation Beamformer

  • The outer aggregation-beamformer problem is formed by substituting the optimal WPT strategies into the original design.The normalized beamformer is constrained by tr(FFH) = 1.Source: B. Optimal Aggregation Beamformer · p. 20
  • The aggregation-beamformer objective is non-convex, prompting an inequality relaxation that is exact when each sensor channel is well-conditioned.The stated condition is that the singular values of Hk are identical.Source: B. Optimal Aggregation Beamformer · p. 20
  • Removing the rank constraint produces a convex relaxation, whose solution is converted into a feasible rank-L beamformer using Gaussian randomization.The algorithm samples candidate matrices, extracts and normalizes feasible beamformers, then selects the one with the minimum objective.Source: B. Optimal Aggregation Beamformer · pp. 20–21

VI. FURTHER DISCUSSION

The discussion contrasts computation-error minimization with receive-SNR maximization and examines how wireless power control affects AirComp error. These criteria yield different resource-allocation strategies.

A. Computation Error Minimization versus SNR Maximization

  • Receive-SNR maximization is an alternative design criterion, but it leads to a different resource-allocation strategy from computation-error minimization.The discussion uses the MISO case to illustrate this difference.Source: A. Computation Error Minimization… · pp. 21–22
  • Computation-error minimization equalizes effective channel gains across sensors, whereas receive-SNR maximization favors sensors with better channel conditions.The latter strategy aims to maximize the sum effective channel gain.Source: A. Computation Error Minimization… · p. 22

B. How does WPT Help AirComp?

  • Wireless power control reduces the computation error compared with the equal-power benchmark.The benchmark equally allocates wireless power; under equal allocation, the aggregation-beamformer problem is NP-hard.Source: B. How does WPT Help AirComp? · p. 22

VII. SIMULATION

The simulations evaluate normalized computation error and report that WP-AirComp outperforms the baselines across tested transmission-power settings, while antenna and sensor counts also affect error.

VII. SIMULATION

  • The simulations use five sensors by default, with server antenna counts of 20 for MISO and 30 for MIMO, and normalized computation error as the metric.The setup assumes i.i.d. Rician fading channels, maximum transmission power P0 = 1, and noise variance 1.Source: §VII · p. 23

A. Baseline Schemes

  • The three baselines use antenna selection, eigenmode beamforming, or equal power allocation, with channel-inversion data precoding and optimal energy beamforming.The aggregation beamformers are scaled to have the same norm for fair comparison.Source: A. Baseline Schemes · p. 23

B. Optimality of the SDR Solution

  • The SDR solution is globally optimal with probability above 85% across server antenna settings, and that probability increases with antenna count.The probability is evaluated for the MISO case and is defined by rank(F̂*) = 1.Source: Table II · p. 24, B. Optimality of the SDR Solution · p. 24

C. Computation Performance of WP-AirComp

D. Effects of Different System Parameters on WP-AirComp Performance

  • Normalized computation error increases with sensor count but decreases with server antenna count under the evaluated settings.The authors relate the sensor-count trend to the difficulty of designing one aggregation beamformer to equalize different sensor channels.Source: D. Effects of Different System… · p. 25, Figure 6 · p. 26

VIII. CONCLUDING REMARKS

  • The conclusion identifies joint wireless-power allocation and energy and aggregation beamforming as the WP-AirComp design, with sensor scheduling, clustering, and multi-server cooperation as future topics.The authors state that the decomposition approaches the global optimum with high probability and that wireless power control can boost AirComp accuracy.Source: §VIII · p. 26

APPENDIX

The appendix proofs establish an equality condition for Lemma 1, use a power-constraint argument for Lemma 2, and show convexity for Problem P10.

A. Proof of Lemma 1

B. Proof of Lemma 2

  • The Lemma 2 proof uses a contradiction argument that adjusts the maximum-power term without violating the power constraint.Source: B. Proof of Lemma 2 · p. 27

C. Proof of Lemma 4

  • The Lemma 4 proof establishes that Problem P10 has a convex objective and convex constraints.It treats the objective as convex in F̂ and combines this with the convex constraints.Source: C. Proof of Lemma 4 · p. 27

D. Proof of Lemma 6

These appendix proofs use a power-constraint argument to establish the Lemma 6 strategy as optimal and show that Problem P19 is convex.

D. Proof of Lemma 6

  • The Lemma 6 proof uses a contradiction argument that preserves the power constraint to establish the stated strategy as optimal.Source: D. Proof of Lemma 6 · p. 28

E. Proof of Lemma 7

  • The Lemma 7 proof establishes that Problem P19 has a convex objective and convex constraints.The objective is treated as convex in F̂ before the convex constraints are combined with it.Source: E. Proof of Lemma 7 · p. 28
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