Over-the-Air Computation over Balanced Numerals

In this study, a digital over-the-air computation (OAC) scheme for achieving continuous-valued gradient aggregation is proposed. It is shown that the average of a set of real-valued parameters can be calculated approximately by using the average of the corresponding numerals, where the numerals are obtained based on a balanced number system. By using this property, the proposed scheme encodes the local gradients into a set of numerals. It then determines the positions of the activated orthogonal frequency division multiplexing (OFDM) subcarriers by using the values of the numerals. To eliminate the need for a precise sample-level time synchronization, channel estimation overhead, and power instabilities due to the channel inversion, the proposed scheme also uses a non-coherent receiver at the edge server (ES) and does not utilize a pre-equalization at the edge devices (EDs). Finally, the theoretical mean squared error (MSE) performance of the proposed scheme is derived and its performance for federated edge learning (FEEL) is demonstrated.