  In this paper, we study a federated learning system at the wireless edge that
uses over-the-air computation (AirComp). In such a system, users transmit their
messages over a multi-access channel concurrently to achieve fast model
aggregation. Recently, an AirComp scheme based on digital modulation has been
proposed featuring one-bit gradient quantization and truncated channel
inversion at users and a majority-voting based decoder at the fusion center
(FC). We propose an improved digital AirComp scheme to relax its requirements
on the transmitters, where users perform phase correction and transmit with
full power. To characterize the decoding failure probability at the FC, we
introduce the normalized detection signal-to-noise ratio (SNR), which can be
interpreted as the effective participation rate of users. To mitigate wireless
fading, we further propose a cluster-based system and design the relay
selection scheme based on the normalized detection SNR. By local data fusion
within each cluster and relay selection, our scheme can fully exploit spatial
diversity to increase the effective number of voting users and accelerate model
convergence.
