  Over-the-air computation (AirComp) becomes a promising approach for fast
wireless data aggregation via exploiting the superposition property in a
multiple access channel. To further overcome the unfavorable signal propagation
conditions for AirComp, in this paper, we propose an intelligent reflecting
surface (IRS) aided AirComp system to build controllable wireless environments,
thereby boosting the received signal power significantly. This is achieved by
smartly tuning the phase shifts for the incoming electromagnetic waves at IRS,
resulting in reconfigurable signal propagations. Unfortunately, it turns out
that the joint design problem for AirComp transceivers and IRS phase shifts
becomes a highly intractable nonconvex bi-quadratic programming problem, for
which a novel alternating difference-of-convex (DC) programming algorithm is
developed. This is achieved by providing a novel DC function representation for
the rank-one constraint in the low-rank matrix optimization problem via matrix
lifting. Simulation results demonstrate the algorithmic advantages and
admirable performance of the proposed approaches compared with the state-of-art
solutions.
