On space-time coding in the presence of spatio-temporal correlation

In this paper, we consider the problem of space-time coding over Rician channels in the presence of spatio-temporal correlation. We derive an upper bound on the pairwise word error probability (PWEP) of space-time codes and use it as a basis for a unified approach to analysis and design of space-time codes over any flat, Rician or Rayleigh, block-fading channel. Based on the statistical properties of a Rician channel, the high-signal-to-noise ratio (SNR) behavior of the bound is either exponential or rational. In the former case, design criteria for space-time codes are based on a Euclidean-distance-like measure. However, in the latter case, design criteria are based on a rank criterion together with a coding gain criterion. The analysis of the bound shows that the performance of rank-deficient codes can be highly degraded in the presence of spatio-temporal correlation, but the performance degradation is not severe for full-rank codes. These codes exhibit robustness against the channel correlation profile. Specifically, for channels with a nonsingular covariance matrix, the asymptotic performance of a full-rank code is proportional to its performance in the independent and identically distributed (i.i.d.) case. In the case that the only constraint on the codewords is a maximum energy constraint, we show that the set of rank-deficient codes forms a proper algebraic subset of measure zero in the affine space of space-time codes. Thus, any randomly selected code is full rank with probability one. Hence, in this case, the main challenge in space-time coding is not maximizing diversity but optimizing coding gain. However, optimizing the coding gain is computationally hard and there exists no satisfactory algorithm to solve this problem in general. On the other hand, when we constrain codewords to be from a finite alphabet or to satisfy particular algebraic structures, e.g., group codes, lattice codes, or coset codes, the problem might be tractable, but rank-deficient codes might be highly probable at high spectral efficiency.