  Multidimensional lattice constellations which present signal space diversity
(SSD) have been extensively studied for single-antenna transmission over fading
channels, with focus on their optimal design for achieving high diversity gain.
In this two-part series of papers we present a novel combinatorial geometrical
approach based on parallelotope geometry, for the performance evaluation of
multidimensional finite lattice constellations with arbitrary structure,
dimension and rank. In Part I, we present an analytical expression for the
exact symbol error probability (SEP) of multidimensional signal sets, and two
novel closed-form bounds, named Multiple Sphere Lower Bound (MLSB) and Multiple
Sphere Upper Bound (MSUB). Part II extends the analysis to the transmission
over fading channels, where multidimensional signal sets are commonly used to
combat fading degradation. Numerical and simulation results show that the
proposed geometrical approach leads to accurate and tight expressions, which
can be efficiently used for the performance evaluation and the design of
multidimensional lattice constellations, both in Additive White Gaussian Noise
(AWGN) and fading channels.
