are the $r$–th entries of ${\bf{y}}$ and ${\bf{h}}_{\ell}$ respectively.

Sphere Decoders for SM {#sec:SD}
======================

In this section we introduce two SDs tailored for SM, SM–Rx and SM–Tx. SM–Rx aims at reducing the number of summations over $N_r$ required by the ML receiver in . SM–Tx aims at reducing the number of points $(\ell,s)$ the ML receiver searches over.

First, for ease of derivation, we introduce the real–valued equivalent of the complex–valued model in following [@book:th01],

$$\begin{aligned}
{\bf{\bar y}} &=& {\bf{\bar H\bar x}}_{\ell_t ,s_t }  + {\bf{\bar n}} \nonumber \\
              &=& {\bf{\bar h}}_{\ell_t}{\bf{\bar s}}_t + \bf{\bar n}\end{aligned}$$

where,

$$\begin{aligned}
 {\bf{\bar y}} &=& \left[ {{\mathop{\rm Re}\nolimits} \left\{ {{\bf{y}}^T } \right\},{\mathop{\rm Im}\nolimits} \left\{ {{\bf{y}}^T } \right\}} \right]^T   \\
 {\bf{\bar H}} &=& \left[ {\begin{array}{*{20}c}
   {{\mathop{\rm Re}\nolimits} \left\{ {\bf{H}} \right\}} & {{\mathop{\rm Im}\nolimits} \left\{ {\bf{H}} \right\}}  \\
   { - {\mathop{\rm Im}\nolimits} \left\{ {\bf{H}} \right\}} & {{\mathop{\rm Re}\nolimits} \left\{ {\bf{H}} \right\}}  \\
\end{array}} \right]  \\
  {\bf{\bar x}}_{\ell ,s }  &=& \left[ {{\mathop{\rm Re}\nolimits} \left\{ {{\bf{x}}_{\ell ,s }^T } \right\},{\mathop{\rm Im}\nolimits} \left\{ {{\bf{x}}_{\ell ,s }^T } \right\}} \right]^T \label{Eq_9}  \\
 {\bf{\bar n}} &=& \left[ {{\mathop{\rm Re}\nolimits} \left\{ {{\bf{n}}^T } \right\},{\mathop{\rm Im}\nolimits} \left\{ {{\bf{n}}^T } \right\}} \right]^T  \\
 \bar{\bf h}_{\ell} &=& \left[ \bar{\bf H}_{\ell} , \bar{\bf H}_{\ell+N_t} 