  In typical sensor networks, data collection and processing are separated. A
sink collects data from all nodes sequentially, which is very time consuming.
Over-the-air computation, as a new diagram of sensor networks, integrates data
collection and processing in one slot: all nodes transmit their signals
simultaneously in the analog wave and the processing is done in the air. This
method, although efficient, requires that signals from all nodes arrive at the
sink, aligned in signal magnitude so as to enable an unbiased estimation. For
nodes far away from the sink with a low channel gain, misalignment in signal
magnitude is unavoidable. To solve this problem, in this paper, we investigate
the amplify-and-forward based relay, in which a relay node amplifies signals
from many nodes at the same time. We first discuss the general relay model and
a simple relay policy. Then, a coherent relay policy is proposed to reduce
relay transmission power. Directly minimizing the computation error tends to
over-increase node transmission power. Therefore, the two relay policies are
further refined with a new metric, and the transmission power is reduced while
the computation error is kept low. In addition, the coherent relay policy helps
to reduce the relay transmission power by half, to below the limit, which makes
it one step ahead towards practical applications.
