Over-the-air computation (OAC) is a promising technique to realize fast model
aggregation in the uplink of federated edge learning. OAC, however, hinges on
accurate channel-gain precoding and strict synchronization among the edge
devices, which are challenging in practice. As such, how to design the maximum
likelihood (ML) estimator in the presence of residual channel-gain mismatch and
asynchronies is an open problem. To fill this gap, this paper formulates the
problem of misaligned OAC for federated edge learning and puts forth a whitened
matched filtering and sampling scheme to obtain oversampled, but independent,
samples from the misaligned and overlapped signals. Given the whitened samples,
a sum-product ML estimator and an aligned-sample estimator are devised to
estimate the arithmetic sum of the transmitted symbols. In particular, the
computational complexity of our sum-product ML estimator is linear in the
packet length and hence is significantly lower than the conventional ML
estimator. Extensive simulations on the test accuracy versus the average
received energy per symbol to noise power spectral density ratio (EsN0) yield
two main results: 1) In the low EsN0 regime, the aligned-sample estimator can
achieve superior test accuracy provided that the phase misalignment is
non-severe. In contrast, the ML estimator does not work well due to the error
propagation and noise enhancement in the estimation process. 2) In the high
EsN0 regime, the ML estimator attains the optimal learning performance
regardless of the severity of phase misalignment. On the other hand, the
aligned-sample estimator suffers from a test-accuracy loss caused by phase
misalignment.Comment: 17 pages, 11 figure