  Over-the-air computation (OAC) is a promising wireless communication method
for aggregating data from many devices in dense wireless networks. The
fundamental idea of OAC is to exploit signal superposition to compute functions
of multiple simultaneously transmitted signals. However, the time- and
phase-alignment of these superimposed signals have a significant effect on the
quality of function computation. In this study, we analyze the OAC problem for
a system with unknown random time delays and phase shifts. We show that the
classical matched filter does not produce optimal results, and generates bias
in the function estimates. To counteract this, we propose a new filter design
and show that, under a bound on the maximum time delay, it is possible to
achieve unbiased function computation. Additionally, we propose a Tikhonov
regularization problem that produces an optimal filter given a tradeoff between
the bias and noise-induced variance of the function estimates. When the time
delays are long compared to the length of the transmitted pulses, our filter
vastly outperforms the matched filter both in terms of bias and mean-squared
error (MSE). For shorter time delays, our proposal yields similar MSE as the
matched filter, while reducing the bias.
