  In this study, we propose a digital over-the-air computation (OAC) scheme for
achieving continuous-valued (analog) aggregation for federated edge learning
(FEEL). We show 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
exploiting this key property, the proposed scheme encodes the local stochastic
gradients into a set of numerals. Next, it determines the positions of the
activated orthogonal frequency division multiplexing (OFDM) subcarriers by
using the values of the numerals. To eliminate the need for precise
sample-level time synchronization, channel estimation overhead, and 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).
We theoretically analyze the MSE performance of the proposed scheme and the
convergence rate for a non-convex loss function. To improve the test accuracy
of FEEL with the proposed scheme, we introduce the concept of adaptive absolute
maximum (AAM). Our numerical results show that when the proposed scheme is used
with AAM for FEEL, the test accuracy can reach up to 98% for heterogeneous data
distribution.
