Multi-Cell Over-the-Air Computation Systems With Spectrum Sharing: A Perspective From $\alpha$-Fairness

Wireless data aggregation (WDA) is a pivotal enabling technique in the era of Internet of Things (IoT). Recently, an emerging WDA method, over-the-air computation (AirComp), has been proposed to perform fast data aggregation, with improved spectrum utilization and shortened transmission delay. In this article, we consider a multi-cell AirComp system with single-input multiple-output (SIMO) communications. All users in the system share the same wireless spectrum. The mean squared error (MSE) is used as the metric to quantify the computation accuracy of each cell. To achieve the fairness of data aggregation among different cells, we formulate a unified optimization problem to minimize the MSE-based objective function, from the perspective of
$\alpha$
-fairness. Due to the non-convexity issue, the optimization problem is first divided into four cases according to the choice of
$\alpha$
. Thereafter, an efficient solution based on the block coordinate descent (BCD) method is proposed for each case. Specifically, in the first case, the optimal solutions are obtained for the beamforming at the AP and the power allocation at the user. In the second and third cases, a tractable solution to determine the power allocation is devised by adopting the successive convex approximation (SCA) method. In the last case, the majorization minimization (MM) method is applied to tackle the power allocation problem. Furthermore, by analyzing the MSE performance, we arrive at an in-depth insight. Namely, the impact of the inter-cell interference disappears when the number of antennas at the AP is sufficiently large. The effectiveness of our proposed schemes is validated by the numerical results.