A Unified SNR-Dependent Analysis of Space–Time Codes in Fading Channels

The pairwise error probability (PEP) for multiple- input multiple-output (MIMO) radio interfaces is investigated by means of a novel formulation based on compound matrices. The proposed approach is suitable for any MIMO system having average upper-bounded PEP written as $[\det ( {\mmb I}+\gamma \, {\mmb A})]^{-\zeta }$, where $ {\mmb A}$ is a Hermitian matrix, $\zeta $ an integer number, and $\gamma $ the signal-to-noise ratio (SNR); that bound frequently results in MIMO single- and multicarrier transmissions. It is shown that the minimization of the bounded PEP should consider the whole set of nonzero compound matrices of $ {\mmb A}$. In particular, the SNR of interest marks the compound matrix that mainly drives the system performance. Both diversity advantage and coding gain are given as continuous functions of the variable $\gamma $, hence, their asymptotic behaviors are taken as important case of studies. The interaction effects between channel code and propagation environment are also discussed. It is shown how the eigenvectors and eigenvalues of the autocorrelation channel matrix may be considered for code design. It is also proved the maximization of the code rank is not always a necessary requirement for performance improvement being its optimal value fixed by the channel structure and SNR of interest. Finally, the analysis is applied to space–time trellis-coded transmissions over spatially correlated slow Rayleigh-fading channels.