some algebraic manipulations as shown in Appendix \[App:Int\], the subset of points ${\Theta _R }$ lie in the intervals:

$$\label{Eq:int1}   \frac{{ - R_i + \bar z_{\ell+N_t} }}{{\bar D_{(\ell+N_t,\ell+N_t)} }} \le {\rm Im}\left\lbrace s \right\rbrace  \le \frac{{R_i  + \bar z_{\ell+N_t} }}{{\bar D_{(\ell+N_t,\ell+N_t)} }}$$

$$\label{Eq:int2} \frac{{ - R'  + \bar z_{\ell|\ell + N_t } }}{{\bar D_{\ell,\ell} }} \le {\rm Re}\left\lbrace s \right\rbrace \le \frac{{R'  + \bar z_{\ell|\ell + N_t} }}{{\bar D_{\ell,\ell} }}$$

where $$\begin{aligned}
\bar z_{a|b }  &=& \bar z_{a}  - \bar D_{\left(a,b\right) }{\rm Im}\left\lbrace s \right\rbrace \\
R'^2 &=& R^2 - \sum_{\nu=N_t+1}^{2N_t} \bar z^2_{\nu|\ell+N_t} \label{Eq_R_dash}\end{aligned}$$

Note, every time a point is found inside the sphere, the radius $R$ is updated as shown in , with the Euclidean distance of that point. Moreover, (\[Eq:int2\]) needs to be computed only for those points that lie inside the interval in (\[Eq:int1\]), for the reason that (\[Eq:int2\]) depends implicitly on (\[Eq:int1\]).

Because of the unique properties of SM the intervals in (\[Eq:int1\]) and (\[Eq:int2\]) needs to be calculated only once for each possible transmit point, unlike conventional SDs where the intervals have to be calculate $N_t$ times for each transmit point. Furthermore, we note that SM–Tx works for both underdetermined MIMO setup with $N_t>N_r$, and 