\right] \\
\bar{\bf s} &=& \left[ {\begin{array}{*{20}c}
   \mathop{\rm Re}\{s\} \\
   \mathop{\rm Im}\{s\} \\
\end{array}} \right]\end{aligned}$$

where ${\mathop{\rm Re}\nolimits} \left\{  \cdot  \right\}$ and ${\mathop{\rm Im}\nolimits} \left\{  \cdot  \right\}$ denote real and imaginary parts respectively, and ${\bar{\bf H}}_{\ell}$ is the $\ell$–th column of $\bf{\bar H}$.

$$R_{i+1}^2 = {\left\| {{\bf{\bar z}} - {\bf{\bar D\bar x}}_{\ell,s} } \right\|_{\rm{F}}^2 }
 = (R_i^2-R'^2) + \sum_{\nu=1}^{N_t} \left( z_{\nu} - D_{\left(\nu,\ell\right)}{\rm Re}\left\lbrace s \right\rbrace - D_{\left(\nu,\ell+N_t\right)}{\rm Im}\left\lbrace s \right\rbrace \right)^2 \label{Eq_R_new}$$

SM–Rx Detector {#Rx-SD}
--------------

The SM–Rx is a reduced–complexity and close–to–optimum BER–achieving decoder, which aims at reducing the receive search space. The detector can formally be written as follows:

$$\left[ {\hat \ell_t^{\left( {{\rm{Rx}}} \right)} ,\hat s_t^{\left( {{\rm{Rx}}} \right)} } \right] = \mathop {\arg \min
}\limits_{\scriptstyle \ell \in \left\{ {1,2, \ldots N_t } \right\} \hfill \atop{
  \scriptstyle s \in \left\{ {s_1 ,s_2 , \ldots s_M } \right\} \hfill}}  \left\{ {\sum\limits_{r = 1}^{\tilde N_r \left( {\ell,s} \right)} {\left| {\bar{y}_r  - \bar{\bf{h}}_{\ell,r} \bf{\bar{s}}} \right|^2 } } \right\} \label{Eq:SM--Rx}$$

where $\bar{\bf{h}}_{\ell,r}$ is the $r$–th row of $\bar{\bf{h}}_{\ell}$, and, $$\tilde N_r \left( {\ell,s} \right) = \mathop {\max }\limits_{n \in \left\{ {1,2, \ldots 2N_r } \right\}} \left\{ { n
\left| \sum\limits_{r = 1}^n {\left| {y_r  - h_{\ell,r} s} \right|^2 }  \leq R^2 \right. 