${\bf{h}}_{\ell }$ is the $\ell$–th column of $\mathbf{H}$. Note, the signal-to-noise-ratio is $\text{SNR} = E_s/N_o = 1/\sigma_n^2$.

ML–Optimum Detector
-------------------

The ML optimum receiver for MIMO systems can be written as,

$$\hat{\mathbf{x}}_t^{\left(\text{ML}\right)} =  \mathop {\arg \min }\limits_{\scriptstyle \mathbf{x} \in \mathcal{Q}^m} \left\{ {\left\| {{\bf{y}} - {\bf{H}}\mathbf{x}} \right\|_{\rm{F}}^2 } \right\} \label{Eq:MLMIMO}$$

where $\mathcal{Q}^m$ is a $2^m$ space containing all possible $\left(N_t\times 1\right)$ transmitted vectors, $\left\|  \cdot  \right\|_{\rm{F}}$ is the Frobenius norm, and ${\hat  \cdot }$ denotes the estimated spatial and constellation symbols.

Note, in SM only one transmit antenna is active at a time. Therefore, the optimal receiver in can be simplified to, $$\begin{aligned}
\vspace{0.2cm}
 \left[ {\hat \ell_t^{\left( {{\rm{ML}}} \right)} ,\hat s_t^{\left( {{\rm{ML}}} \right)} } \right] &=& \mathop {\arg \min }\limits_{\scriptstyle \ell \in \left\{ {1,2, \ldots N_t } \right\} \hfill \atop
  \scriptstyle s \in \left\{ {s_1 ,s_2 , \ldots s_M } \right\} \hfill} \left\{ {\left\| {{\bf{y}} - {\bf{h}}_\ell s} \right\|_{\rm{F}}^2 } \right\} \nonumber \\
  &=& \mathop {\arg \min }\limits_{\scriptstyle \ell \in \left\{ {1,2, \ldots N_t } \right\} \hfill \atop
  \scriptstyle s \in \left\{ {s_1 ,s_2 , \ldots s_M } \right\} \hfill} \left\{ {\sum\limits_{r = 1}^{N_r } {\left| {y_r  - h_{\ell,r} s} \right|^2 } } \right\} \nonumber  \\ \label{Eq:ML}\end{aligned}$$

where $y_r$ and $h_{\ell,r}$ 