System model and notations

Consider a multiple-input multiple-output system with $n_t$ transmit and $n_r$ receive antennas. The channel is supposed to be frequency non-selective (no inter-symbol interference) and known to the receiver but not to the transmitter (CSI at the receiver side only). The number of independent channel realizations observed during one codeword transmission is denoted by $n_c$. The parameter $n_c$ takes values from 1 (quasi-static fading) up to the codeword time length (ergodic channel). The input-output channel model is given by

$$y = zSH + \eta$$ (1)

where $z \in (2^m-\textrm{QAM})^{sn_t}$, $S$ is the linear precoder matrix (also called full-rate space-time block code in the MIMO community) of size $sn_t \times sn_t$. The integer parameter $s$ represents the time spreading of the precoder, $1 \le s \le n_tn_c$. We restrict our study to unitary precoders, i.e., $S^{-1}=S^h$. The $sn_t \times sn_r$ MIMO channel matrix $H$ has $s$ blocks on its diagonal and zeros elsewhere. Each block is associated to the transmission over the $n_t \times n_r$ MIMO channel during one channel use. We can see $SH$ as a new correlated MIMO channel and call a precoding time period the group of $s$ channel uses associated to the matrix $SH$. Finally, $y \in \mathbb{C}^{sn_r}$ is the channel output and $\eta$ is an additive white gaussian noise with zero mean and variance $\sigma^2$ per real component.