  In future Internet-of-Things networks, sensors or even access points can be
mounted on ground/aerial vehicles for smart-city surveillance or environment
monitoring. To support the high-mobility sensing with low network latency, a
technique called over-the-air-computation (AirComp) was recently developed
which enables an access-point to receive a desired function of sensing-data
from concurrent-transmissions 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. Specifically, we
design beamforming and channel-feedback techniques for multi-function AirComp.
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. As a by-product, the beamforming problem is found
to have the same form as the classic problem of multicast-beamforming,
establishing the AirComp-multicasting-duality. Its significance lies in making
the said Grassmannian-centroid solution transferable to the latter problem
which otherwise is solved using the computation-intensive
semidefinite-relaxation-technique. Last, building on the AirComp-beamforming
solution, an efficient channel-feedback technique is designed for an
access-point to receive the beamformer from distributed sensor transmissions of
designed signals that are functions of local channel-state-information.
