  This paper studies an multi-cluster over-the-air computation (AirComp)
system, where an intelligent reflecting surface (IRS) assists the signal
transmission from devices to an access point (AP). The clusters are activated
to compute heterogeneous functions in a time-division manner. Specifically, two
types of IRS beamforming (BF) schemes are proposed to reveal the
performancecost tradeoff. One is the cluster-adaptive BF scheme, where each BF
pattern is dedicated to one cluster, and the other is the dynamic BF scheme,
which is applied to any number of IRS BF patterns. By deeply exploiting their
inherent properties, both generic and lowcomplexity algorithms are proposed in
which the IRS BF patterns, time and power resource allocation are jointly
optimized. Numerical results show that IRS can significantly enhance the
function computation performance, and demonstrate that the dynamic IRS BF
scheme with half of the total IRS BF patterns can achieve near-optimal
performance which can be deemed as a cost-efficient approach for IRS-aided
multi-cluster AirComp systems.
