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Over-the-Air Computation with Spatial-and-Temporal Correlated Signals

Wanchun Liu, Xin Zang, Branka Vucetic, Yonghui Li

TL;DR

An AirComp system with spatial-and-temporal correlated sensor signals is proposed, and the optimal AirComp policy design problem for achieving the minimum computation mean-squared error (MSE) is formulated.

Abstract

Over-the-air computation (AirComp) leveraging the superposition property of wireless multiple-access channel (MAC), is a promising technique for effective data collection and computation of large-scale wireless sensor measurements in Internet of Things applications. Most existing work on AirComp only considered computation of spatial-and-temporal independent sensor signals, though in practice different sensor measurement signals are usually correlated. In this letter, we propose an AirComp system with spatial-and-temporal correlated sensor signals, and formulate the optimal AirComp policy design problem for achieving the minimum computation mean-squared error (MSE). We develop the optimal AirComp policy with the minimum computation MSE in each time step by utilizing the current and the previously received signals. We also propose and optimize a low-complexity AirComp policy in closed form with the performance approaching to the optimal policy.

Over-the-Air Computation with Spatial-and-Temporal Correlated Signals

TL;DR

An AirComp system with spatial-and-temporal correlated sensor signals is proposed, and the optimal AirComp policy design problem for achieving the minimum computation mean-squared error (MSE) is formulated.

Abstract

Over-the-air computation (AirComp) leveraging the superposition property of wireless multiple-access channel (MAC), is a promising technique for effective data collection and computation of large-scale wireless sensor measurements in Internet of Things applications. Most existing work on AirComp only considered computation of spatial-and-temporal independent sensor signals, though in practice different sensor measurement signals are usually correlated. In this letter, we propose an AirComp system with spatial-and-temporal correlated sensor signals, and formulate the optimal AirComp policy design problem for achieving the minimum computation mean-squared error (MSE). We develop the optimal AirComp policy with the minimum computation MSE in each time step by utilizing the current and the previously received signals. We also propose and optimize a low-complexity AirComp policy in closed form with the performance approaching to the optimal policy.

Paper Structure

This paper contains 11 sections, 2 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. AirComp System with Spatial-and-Temporal Correlated Signals
  3. Spatial-and-Temporal Correlated Signals
  4. AirComp System and Computation MSE
  5. Optimal AirComp Policy Design
  6. Low-Complexity AirComp Policy and Transmission Power Control
  7. Low-Complexity AirComp Policy
  8. Tx-Scaling Factor Design
  9. Joint Design of AirComp Policy and Tx-Scaling Factor
  10. Numerical Results
  11. Conclusions

Key Result

Theorem 1

Given the Tx-scaling factor vector $\mathbf{b}$, the optimal AirComp policy for achieving the minimum computation MSE is to first apply the KF defined in eq:kalman to estimate the sensor signal vector $\mathbf{x}_t$ and then calculate the sum of the estimate $\mathbf{\hat{x}}_t$. The minimum computa

Figures (2)

  • Figure 1: The computation MSE of the optimal and low-complexity AirComp policies.
  • Figure 2: The computation MSE of the low-complexity AirComp policy with different iteration rounds.

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Theorem 2