  This paper studies a hierarchical over-the-air computation (AirComp) network
over a large area, in which multiple relays are exploited to facilitate data
aggregation from massive WDs. We present a two-phase amplify-and-forward (AF)
relaying protocol. In the first phase, the WDs simultaneously send their data
to the relays, while in the second phase, the relays amplify the respectively
received signals and concurrently forward them to the fusion center (FC) for
aggregation. Our objective is to minimize the computational mean squared error
(MSE) at the FC, by jointly optimizing the WD transmit coefficients, the relay
AF coefficients, and the FC de-noising factor, subject to their individual
transmit power constraints. First, we consider the centralized design with
global channel state information (CSI), in which the inter-relay signals can be
exploited beneficially for data aggregation. In this case, we develop an
alternating-optimization-based algorithm to obtain a high-quality solution to
the computational MSE minimization problem. Next, to reduce the signaling
overhead caused by the centralized design, we consider an alternative
decentralized design with partial CSI, in which the relays and the FC make
their own decisions by only requiring the channel power gain information across
different relays. In this case, the relays and FC need to treat the inter-relay
signals as harmful interference or noise. Accordingly, we optimize the transmit
coefficients of the WDs associated with each relay, and the relay AF
coefficients (together with the FC de-noising factor) in an iterative manner,
which can be implemented efficiently in a decentralized way.
