Over-the-Air Computation in OFDM Systems with Imperfect Channel State Information

This paper studies the over-the-air computation (AirComp) in an orthogonal frequency division multiplexing (OFDM) system with imperfect channel state information (CSI), in which multiple single-antenna wireless devices (WDs) simultaneously send uncoded signals to a multi-antenna access point (AP) for distributed functional computation over multiple subcarriers. In particular, we consider two scenarios with best-effort and error-constrained computation tasks, with the objectives of minimizing the average computation mean squared error (MSE) and the computation outage probability over the multiple subcarriers, respectively. Towards this end, we jointly optimize the transmit coefficients at the WDs and the receive beamforming vectors at the AP over subcarriers, subject to the maximum transmit power constraints at individual WDs. First, for the special case with a single receive antenna at the AP, we propose the semi-closed-form globally optimal solutions to the two problems using the Lagrange-duality method. It is shown that at each subcarrier, the WDs' optimized power control policy for average MSE minimization follows a regularized channel inversion structure, while that for computation outage probability minimization follows an on-off regularized channel inversion, with the regularization dependent on the transmit power budget and channel estimation error. Next, for the general case with multiple receive antennas at the AP, we present efficient algorithms based on alternating optimization and convex optimization to find converged solutions to both problems.