MIMO Over-the-Air Computation: Beamforming Optimization on the Grassmann Manifold

To support future IoT networks with dense sensor connectivity, a technique called over-the-air computation (Air-Comp) was recently developed to enable a data-fusion center to receive a desired function (e.g., mean value) of sensing data from concurrent sensor transmissions. This is made possible by exploiting the superposition property of a multi-access channel. This work aims at further developing AirComp for next-generation multi-antenna multi-modal sensor networks where a multi-modal sensor monitors multiple environmental parameters such as temperature, pollution and humidity. To be specific, we design beamforming techniques for AirComp of multiple functions, each corresponding to a particular sensing-data type. Given the objective of minimizing sum mean-squared error of computed functions, the optimization of receive beamforming for multi-function AirComp is a NP-hard problem. The approximate problem based on tightening transmission-power constraints, however, is shown to be solvable using differential geometry. The solution is proved to be the weighted centroid of points on a Grassmann manifold, where each point represents the subspace spanned by the channel matrix of a sensor. Simulation results demonstrate the effectiveness of the proposed solution.