Degrees of Freedom of Over-the-Air Computation over a MIMO Gaussian Network with Two Transmitters and Two Receivers

2026-06-04

AI Summary

This work investigates the fundamental performance limits of over-the-air computation (AirComp) in a two-transmitter, two-receiver MIMO Gaussian network, where both receivers aim to recover the same signal aggregation. To this end, the paper introduces the AirComp degrees of freedom (ACDoF) under an asymptotic mean-square error constraint and, for the first time, derives its almost-sure closed-form expression in a general MIMO setting: ACDoF = min{M₁, M₂, N₁, N₂, (1/3)max{M₁+M₂, N₁+N₂}}. This result generalizes existing SISO findings to arbitrary antenna configurations and channel realizations. The theoretical analysis leverages information theory and random matrix theory, complemented by an iterative alternating optimization algorithm for finite signal-to-noise ratios, with numerical experiments confirming the efficacy of the proposed approach.

Abstract

The fundamental limits of over-the-air computation (AirComp) are explored in a two-transmitter, two-receiver MIMO Gaussian network, where both receivers demand the same aggregation of source symbols originating at the two transmitters. An AirComp degrees of freedom (ACDoF) metric is defined, constrained by an asymptotic mean-squared error threshold. For a generic MIMO setting where the two transmitters are equipped with $M_1, M_2$ antennas, and the two receivers with $N_1, N_2$ antennas, the AirComp DoF value is shown to be almost surely equal to $\min\{M_1,M_2,N_1,N_2,(1/3)\max\{M_1+M_2,N_1+N_2\}\}$. For SISO settings results are extended beyond generic channels to arbitrary channel realizations. For finite signal-to-noise ratio(SNR) settings, an iterative alternating optimization algorithm is explored.

Problem

• Over-the-Air Computation
• Degrees of Freedom
• MIMO Gaussian Network
• AirComp
• Aggregation Function

Innovation

• AirComp degrees of freedom
• MIMO Gaussian network
• over-the-air computation
• asymptotic mean-squared error
• iterative alternating optimization