Optimal Receive Filter Design for Misaligned Over-the-Air Computation

TL;DR

This paper addresses the challenge of designing optimal receive filters for over-the-air computation in wireless networks with unknown timing and phase misalignments, proposing new filters that improve accuracy and reduce bias.

Contribution

The authors introduce a novel filter design for OAC that accounts for unknown delays and phase shifts, outperforming classical matched filters especially with long delays.

Findings

Proposed filters achieve unbiased estimates under bounded delays.

Filter outperforms matched filter in bias and MSE with long delays.

Reduces bias while maintaining MSE with shorter delays.

Abstract

Over-the-air computation (OAC) is a promising wireless communication method for aggregating data from many devices in dense wireless networks. The fundamental idea of OAC is to exploit signal superposition to compute functions of multiple simultaneously transmitted signals. However, the time- and phase-alignment of these superimposed signals have a significant effect on the quality of function computation. In this study, we analyze the OAC problem for a system with unknown random time delays and phase shifts. We show that the classical matched filter does not produce optimal results, and generates bias in the function estimates. To counteract this, we propose a new filter design and show that, under a bound on the maximum time delay, it is possible to achieve unbiased function computation. Additionally, we propose a Tikhonov regularization problem that produces an optimal filter given a tradeoff between the bias and noise-induced variance of the function estimates. When the time delays are long compared to the length of the transmitted pulses, our filter vastly outperforms the matched filter both in terms of bias and mean-squared error (MSE). For shorter time delays, our proposal yields similar MSE as the matched filter, while reducing the bias.