  In this paper, we consider a wireless multihop device-to-device (D2D) based
mobile edge computing (MEC) system, where the destination wireless device (WD)
is scheduled to compute nomographic functions. Under the MapReduce framework
and motivated by reducing communication resource overhead, we propose a new
multi-level over-the-air (OTA) aggregation scheme for the destination WD to
collect the individual partially aggregated intermediate values (IVAs) for
reduction from multiple source WDs in the data shuffling phase. For OTA
aggregation per level, the source WDs employ a channel inverse structure
multiplied by their individual transmit coefficients in transmission over the
same time frequency resource blocks, and the destination WD finally uses a
receive filtering factor to construct the aggregated IVA. Under this setup, we
develop a unified transceiver design framework that minimizes the mean squared
error (MSE) of the aggregated IVA at the destination WD subject to the source
WDs' individual power constraints, by jointly optimizing the source WDs'
individual transmit coefficients and the destination WD's receive filtering
factor. First, based on the primal decomposition method, we derive the
closed-form solution under the special case of a common transmit coefficient.
It shows that all the source WDs' common transmit is determined by the minimal
transmit power budget among the source WDs. Next, for the general case, we
transform the original problem into a quadratic fractional programming problem,
and then develop a low-complexity algorithm to obtain the (near-) optimal
solution by leveraging Dinkelbach's algorithm along with the Gaussian
randomization method.
