  With the increase in the number of antennas at base stations (BSs),
centralized multi-antenna architectures have encountered scalability problems
from excessive interconnection bandwidth to the central processing unit (CPU),
as well as increased processing complexity. Thus, research efforts have been
directed towards finding decentralized receiver architectures where a part of
the processing is performed at the antenna end (or close to it). A recent paper
put forth an information-lossless trade-off between level of decentralization
(inputs to CPU) and decentralized processing complexity (multiplications per
antenna). This trade-off was obtained by studying a newly defined matrix
decomposition--the WAX decomposition--which is directly related to the
information-lossless processing that should to be applied in a general
framework to exploit the trade-off. {The general framework consists of three
stages: a set of decentralized filters, a linear combining module, and a
processing matrix applied at the CPU; these three stages are linear
transformations which can be identified with the three constituent matrices of
the WAX decomposition. The previous work was unable to provide explicit
constructions for linear combining modules which are valid for WAX
decomposition, while it remarked the importance of these modules being sparse
with 1s and 0s so they could be efficiently implemented using hardware
accelerators.} In this work we present a number of constructions, as well as
possible variations of them, for effectively defining linear combining modules
which can be used in the WAX decomposition. Furthermore, we show how these
structures facilitate decentralized calculation of the WAX decomposition for
applying information-lossless processing in architectures with an arbitrary
level of decentralization.
