Decentralized Power Control for Over-the-Air Computation with Phase Noise

Estimation of uplink channels is required for coherent over-the-air computation (OAC). When channel estimation is done using calibrated reciprocity, the estimates are only available locally to the devices. This poses a challenge for precoding and decoding, which cannot be coordinated centrally. To this end we use truncated channel inversion (TCI) and propose an approximate closed form solution and an exact numerical solver to optimize the TCI parameters. Importantly, we prove that the proposed TCI scheme is independent of the number of receiver antennas in terms of mean-square-error (MSE). Furthermore, our analysis reveals a clear connection between the MSE and expected aggregate phase error across devices which gives insight to the scalability of OAC. Finally, simulations with comparisons to reference methods from prior work with globally available error-free channel estimates show that proposed is close, even outperforming these references in MSE under some conditions.