Optimization of Adaptive Channel Estimation in Over-the-Air Computing Systems

Over-the-air computing (AirComp) has recently attracted significant attention as an efficient way of data fusion by integrating uncoded communication transmissions with computation thanks to the signal superposition offered by the multiple access channels. However, appropriate processing is required to neutralize the wireless channel effect. Since, internet-of-things (IoT) applications through low-cost devices are the main target of AirComp, perfect channel state information (CSI) availability is not always practical, there exists the need to investigate the effect of imperfect CSI on AirComp. Specifically, we present novel closed-form expressions for tight approximations that can be used to design and evaluate AirComp systems. Moreover, we design a general optimization framework that takes into account both magnitude and phase errors in CSI. Finally, a pilot retransmission policy is designed that offers a trade-off between a cost in resources and gain in the accuracy of the computations. To validate its use, a utility function of the cost of retransmission is introduced, namely, the Retransmission Policy Cost (RPC), which can incorporate the power or throughput cost opposing to the expected gain of the selected policy. Simulation results show the deterioration caused by the imperfect CSI, but, also, highlight the value of the proposed policy under various system conditions.