  As a revolution in networking, Internet of Things (IoT) aims at automating
the operations of our societies by connecting and leveraging an enormous number
of distributed devices (e.g., sensors and actuators). One design challenge is
efficient wireless data aggregation (WDA) over tremendous IoT devices. This can
enable a series of IoT applications ranging from latency-sensitive
high-mobility sensing to data-intensive distributed machine learning.
Over-the-air (functional) computation (AirComp) has emerged to be a promising
solution that merges computing and communication by exploiting analogwave
addition in the air. Another IoT design challenge is battery recharging for
dense sensors which can be tackled by wireless power transfer (WPT). The
coexisting of AirComp and WPT in IoT system calls for their integration to
enhance the performance and efficiency of WDA. This motivates the current work
on developing the wirelessly powered AirComp (WP-AirComp) framework by jointly
optimizing wireless power control, energy and (data) aggregation beamforming to
minimize the AirComp error. To derive a practical solution, we recast the
non-convex joint optimization problem into the equivalent outer and inner
sub-problems for (inner) wireless power control and energy beamforming, and
(outer) the efficient aggregation beamforming, respectively. The former is
solved in closed form while the latter is efficiently solved using the
semidefinite relaxation technique. The results reveal that the optimal energy
beams point to the dominant eigen-directions of the WPT channels, and the
optimal power allocation tends to equalize the close-loop (down-link WPT and
up-link AirComp) effective channels of different sensors. Simulation
demonstrates that controlling WPT provides additional design dimensions for
substantially reducing the AirComp error.
