\frac{\bar{n}_r}{\sigma_n/2} \right|^2 \label{Eq:kappa}$$ is a central chi-squared random variable with $2N_r$ degree of freedom having a cumulative distribution function (CDF) equal to [@book:p0001], $$F_{\kappa}(a,b) = \frac{\gamma(b/2,a/2)}{\Gamma(b/2)} \label{Eq_50}$$ where $\gamma(c,d)$ is the lower incomplete gamma function given by, $$\gamma(c,d) = \int_0^d t^{c-1}e^{-t}dt$$ and $\Gamma(c)$ is the gamma function given by, $$\Gamma(c) = \int_0^{\infty} t^{c-1}e^{-t}dt$$

The initial sphere radius considered in SM–SD is a function of the noise variance as given in [@xhw0701], $$R^2 = 2 \alpha N_r \sigma_n^2 \label{Eq:R}$$ where $\alpha$ is a constant chosen to satisfy . This can be done by setting $\Pr\left(\left(\ell_t,s_t\right) \notin \Theta _R\right)=10^{-6}$ and back solving . For $N_r = 1,2,4$, $\alpha = 13.8,8.3,5.3$ respectively.

Finally, $\Pr_{e,\text{SM--SD}}$ can be formulated as, $$\begin{aligned}
 \Pr_{e,\text{SM--SD}} &=& \Pr\left(\left\| \mathbf{\bar{y}} - \mathbf{\bar{h}_{\ell}}\bar{\bf{s}}\right\|^2 > \left\| \mathbf{\bar{y}} - \bar{\bf{h}}_{\ell_t} \bar{\bf{s}}_t\right\|^2 \right) \nonumber \\
&=& \Pr\left(  \xi > \left\|\mathbf{\bar{h}}_{\ell_t}\bar{\bf{s}}_t-\mathbf{\bar{h}}_{\ell}\bar{\bf{s}}\right\|\right)\end{aligned}$$ where, $$\xi = 2{\rm Re}\left\{\left(\mathbf{\bar{h}}_{\ell_t}\bar{\bf{s}}_t - \mathbf{\bar{h}}_{\ell}\bar{\bf{s}}\right)^T\mathbf{\bar{n}}\right\} \sim \mathcal{N}\left(0,2\sigma_n^2\left(\left\|\mathbf{\bar{h}}_{\ell_t}\bar{\bf{s}}_t - \mathbf{\bar{h}}_{\ell}\bar{\bf{s}}\right\|\right)\right)$$

Thus, $$\Pr_{e,\text{SM--SD}}= Q\left(\sqrt{\frac{\left\|\mathbf{\bar{h}}_{\ell_t}\bar{\bf{s}}_t - \mathbf{\bar{h}}_{\ell}\bar{\bf{s}}\right\|^2}{2\sigma_n^2}}\right) \label{Eq:Q_1}$$ where $Q(x) = (1/\sqrt{2\pi})\int_{x}^{+\infty}e^{-t^2/2}dt$.

In the case of Rayleigh fading, we can derive the closed form solution for $\mathrm{E}_{\bf{H}}\left\lbrace \Pr_{e,\text{SM--SD}} \right\rbrace$ in by employing the solution from [@ag9902 eq. (62)]. Note that the argument of the $Q$-function in can be represented as the summation of $2N_r$ squared 