Transmission Energy Allocation for Over-the-Air Computation with Energy Harvesting

Abstract

Over-the-air computation (AirComp) has recently emerged as a promising multiple-access technique for fast wireless data aggregation (WDA) from distributed wireless devices (WDs). This paper investigates an energy harvesting (EH) AirComp system, in which multiple EH-powered single-antenna WDs simultaneously send wireless signals to a single-antenna access point (AP) with conventional energy supply for WDA via AirComp. Under this setup, we minimize the average computation mean square error (MSE) over a particular time period, by jointly optimizing the transmit energy allocation at the WDs and the AirComp denoising factors at the AP over time, subject to the energy causality constraints at individual WDs. First, we consider the offline scenario by assuming that the energy state information (ESI) and channel state information (CSI) are non-causally known at the beginning of the period, in which the formulated average MSE minimization corresponds to a non-convex optimization problem. We present a high-quality converged solution by using the techniques of alternating optimization and convex optimization. It is shown that for each WD, if the EH rate is sufficiently high, then the channel inversion power allocation is adopted; while if the EH rate is low, then all the harvested energy should be used up for transmission with proper energy allocation over time. Next, we consider the online scenario with causal ESI and CSI, in which the MSE minimization becomes a stochastic optimization problem. In this scenario, we present an offline-inspired online algorithm to obtain efficient online energy allocation designs by utilizing the obtained offline solutions. Finally, numerical results show that the proposed designs significantly outperform two benchmark schemes with power-halving and full-power transmission, respectively.