Probability of full-diversity for simple coded and rotated multidimensional constellation systems

The case of rotation, in use together with coding in a channel with B fading blocks, is analyzed in terms of diversity properties of the coded systems with multi-dimensional rotated constellations (coded-rotated system) output. First closed-form expression obtained is related to the diversity performance of unrotated generic coding structures with given minimum distance, d, block length, and number of fading blocks. Afterwards, another closed-form expression is derived for distinguishing the effect of rotation applied on the given coding structures. This latter expression yields a strict lower bound on the probability of attaining full diversity for any coded-rotated system and gives us a tool to evaluate the improvement with respect to systems involving only coding and also systems with only rotation. The results shown summarize the benefits of utilizing rotation together with coding, even in the cases of very simple non-full-diversity rotations like DFT rotations.