sphere are much simpler to compute, as there is only a single active antenna in SM. In this section, we show how to compute the set of points.

The Cholesky factorisation of the $(2N_t \times 2N_t)$ positive definite matrix ${\bf{\bar{G}}}={\bf{\bar{H}}}^T{\bf{\bar{H}}}+\varphi{\bf{\bar{I}}}_{N_t}$ is $\mathbf{\bar G} = \mathbf{\bar D}\mathbf{\bar D}^{T}$, where

$$\varphi = \left\{ \begin{array}{rl}
                    \sigma_n^2 & N_t>N_r \\
            0          & N_t\leq N_r \\
                   \end{array} \right.$$

Then the SM–Tx can be formally written as follow,

$$\left[ {\hat \ell_t^{\left( {{\rm{Tx - SD}}} \right)} ,\hat s_t^{\left( {{\rm{Tx - SD}}} \right)} } \right] = \mathop {\arg \min }\limits_{\left( {\ell,s} \right) \in \Theta _R } \left\{ {\left\| {{\bf{\bar z}} - {\bf{\bar D\bar x}}_{\ell,s} } \right\|_{\rm{F}}^2 } \right\} \label{Eq:Tx-SD}$$

where ${\Theta _R }$ is the subset of points $\left( {\ell,s} \right)$ for $\ell \in \left\{ {1,2, \ldots N_t } \right\}$ and $s \in \left\{ {s_1 ,s_2 , \ldots s_M } \right\}$ in the transmit search space that lie inside a sphere with radius $R$ and centred around the received signal ${{\bf{\bar z}}}$, $\bar{\bf z}=\bar{\bf D}\bar{\rho}$ and $\bar{\rho}=\mathbf{\bar{G}}^{-1}\mathbf{\bar{H}}^T\mathbf{\bar{y}}$.

Unlike SM–Rx, SM–Tx reduces the computational complexity of the ML receiver by reducing the transmit search space, which is done by the efficient computation of the subset ${\Theta _R }$. After 