  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.
