} \right\} \label{Eq:NrTilde}$$

The idea behind SM–Rx is that it keeps combining the received signals as long as the Euclidean distance in is less or equal to the radius $R$. Whenever a point is found to be inside the sphere, the radius, $R$, is updated with the Euclidean distance of that point. The point with the minimum Euclidean distance and $\tilde N_r\left(\cdot,\cdot\right)=2N_r$ is considered to be the solution.

SM–Tx Detector {#Tx-SD}
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The conventional SD is designed for SMX, where all antennas are active at each time instance [@hv0501a; @vb9901; @ct0501; @wl0901]. However, in SM only one antenna is active at a time. Therefore, a modified SD algorithm tailored for SM, named SM–Tx, is presented. More specifically, similar to conventional SDs, the SM–Tx scheme reduces the number of points $\left( {\ell,s} \right)$ for $\ell \in \left\{ {1,2, \ldots N_t } \right\}$ and $s \in \left\{ {s_1 ,s_2 , \ldots s_M } \right\}$ to be searched through in , *i.e.*, the transmit search space, by computing the Euclidean distances only for those points that lie inside a sphere with radius $R$ and are centred around the received signal. However, unlike conventional SDs, in our scheme the set of points inside the 