On diversity and coding gains and optimal matrix constellations for space-time block codes

The design of optimal space-time (matrix) constellations for multiple-input multiple-output (MIMO) wireless systems with quasistatic Rayleigh flat-fading channel and coherent maximum likelihood (ML) decoder is an open problem of great interest. In 1998, Tarokh et al. proposed rank and determinant criteria as guidelines for designing space-time codes. In this paper, we revisit these criteria and explore their fundamental limitations. In particular, we investigate how they characterize the role of the rank of the code difference matrix in the error probability of the coherent ML decoder and study the shortcomings of such a characterization. Motivated by the results of our study of the rank and determinant criteria, we then address the problem of finding optimal space-time constellations in the sense of minimizing the exact average pairwise error probability under the average energy constraint. We show that for any fixed space-time constellation size and block length, orthogonal space-time block codes (OSTBCs) are optimal in that sense among all space-time codes if the constellation size is not greater than 2K+1, where K is the number of the complex variables of the OSTBC. We also demonstrate that for such constellation sizes and when the number of receivers is large, the optimal constellations are regular simplex. Furthermore, we show that for any constellation size, OSTBCs are optimal among all linear dispersion (LD) codes with the same number of complex variables.