  The energy spectral efficiency maximization (ESEM) problem of a multi-user,
multi-relay, multi-cell system is considered, where all the network nodes are
equipped with multi-antenna transceivers. To deal with the potentially
excessive interference originating from a plethora of geographically
distributed transmission sources, a pair of transmission protocols based on
interference alignment (IA) are conceived. The first, termed the full-IA,
avoids all intra-cell interference (ICI) and other-cell interference by finding
the perfect interference-nulling receive beamforming matrices (RxBFMs). The
second protocol, termed partial-IA, only attempts to null the ICI. Employing
the RxBFMs computed by either of these protocols mathematically decomposes the
channel into a multiplicity of non-interfering multiple-input--single-output
channels, which we term as spatial multiplexing components (SMCs). The problem
of finding the optimal SMCs as well as their power control variables for the
ESEM problem considered is formally defined and converted into a convex
optimization form with carefully selected variable relaxations and
transformations. Thus, the optimal SMCs and power control variables can be
distributively computed using both the classic dual decomposition and
subgradient methods. Our results indicate that indeed, the ESEM algorithm
performs better than the baseline equal power allocation algorithm in terms of
its ESE. Furthermore, surprisingly the partial-IA outperforms the full-IA in
all cases considered, which is because the partial-IA is less restrictive in
terms of the number of available transmit dimensions at the transmitters. Given
the typical cell sizes considered in this paper, the path-loss sufficiently
attenuates the majority of the interference, and thus the full-IA
over-compensates, when trying to avoid all possible sources of interference.
