$\mathbf{\bar x}_{\ell_t,s_t}$ and $\mathbf{\bar x}_{\ell,s}$, and,

$$\Pr_{e,\text{SM--SD}} = \Pr\left(\left(\tilde{\ell}_{\text{SM--SD}},\tilde{s}_{\text{SM--SD}}\right)\neq \left(\ell_t,s_t\right)\right)$$

is the pairwise error probability of deciding on the point $\left(\tilde{\ell}_{\text{SM--SD}},\tilde{s}_{\text{SM--SD}}\right)$ given that the point $\left(\ell_t,s_t\right)$ is transmitted.

The probability of error $\Pr_{e,\text{SM--SD}}$ can be thought of as two mutually exclusive events depending on whether the transmitted point $\left(\ell_t,s_t\right)$ is inside the sphere. In other words, the probability of error for SM–SD can be separated in two parts, as shown in  [@jbot0901]:

-   $\Pr\left((\tilde{\ell}_{\text{SM--ML}},\tilde{s}_{\text{SM--ML}})\neq \left(\ell_t,s_t\right)\right)$: The probability of deciding on the incorrect transmitted symbol and/or used antenna combination, given that the transmitted point $\left(\ell_t,s_t\right)$ is inside the sphere.

-   $\Pr\left(\left(\ell_t,s_t\right) \notin \Theta _R\right)$: The probability that the transmitted point $\left(\ell_t,s_t\right)$ is outside the set of points $\Theta _R$ considered by the SD algorithm.

However, the probability of error for the ML decoder is, $$\Pr_{e,\text{SM--ML}} \leq \Pr\left((\tilde{\ell},\tilde{s})\neq \left(\ell_t,s_t\right)\right) \label{Eq_45}$$

Thus, SM–SD will have a near optimum performance when, $$\Pr\left(\left(\ell_t,s_t\right) \notin \Theta _R\right) << \Pr\left((\tilde{\ell},\tilde{s})\neq \left(\ell_t,s_t\right)\right) \label{Eq:NearOP}$$

The probability of *not* having the transmitted point $\left(\ell_t,s_t\right)$ inside $\Theta_R$ can be written as, $$\begin{aligned}
 \Pr\left(\left(\ell_t,s_t\right) \notin \Theta _R\right) &=& \Pr\left(\sum_{r=1}^{2N_r}\left| {\bar{y}_r  - \bar{\bf{h}}_{\ell_t,r} \bar{\bf{s}}_t} \right|^2>R^2\right) \nonumber \\
   &=& \Pr\left(\kappa > \left(\frac{R}{\sigma_n/\sqrt{2}}\right)^2\right) \nonumber \\
&=& 1 - \frac{\gamma\left(N_r,\left(\frac{R}{\sigma_n}\right)^2\right)}{\Gamma(N_r)} \label{Eq:PrNoPoint}\end{aligned}$$ where, $$\kappa = \sum^{2N_r}_{r=1} \left| 