Optimal Power Control for Over-the-Air Computation

Over-the-air computation (AirComp) of a function (e.g., averaging) has recently emerged as an efficient multi-access scheme for fast aggregation of distributed data at devices (e.g., sensors) to fusion centers (FCs) over wireless channels. To realize reliable AirComp in practice, it is crucial to control the devices' transmit power for coping with channel distortion to achieve the desired magnitude alignment of simultaneous signals. % to strike a balance between enforcing signal-magnitude alignment for overcoming heterogenous channel fading and suppressing noise. In this paper, we study the power control problem for AirComp over fading channels. Our objective is to minimize the computation error by jointly optimizing the transmit power at devices and a signal scaling factor at the FC, called denoising factor, subject to the individual average power constraints at devices. The problem is generally non-convex due to the coupling of transmit power over devices and denoising factor. To optimally solve this problem, we apply the Lagrange duality method via exploiting its ''time-sharing'' property. The derived optimal power control exhibits a regularized channel inversion structure where the regularization has the function of balancing the tradeoff between the signal-magnitude alignment and noise suppression. Moreover, for the special case that only one device is power-limited, we show that the optimal power control for the power-limited device has an interesting channel-inversion water- filling structure, while those for other devices (with sufficiently large power budgets) reduce to channel-inversion power control over all fading states. Numerical results show that the optimal power control remarkably reduces the computation error as compared with other heuristic designs.