$\mathcal{C}_{\text{Interval}}$ is the number of operations needed to compute the intervals in ,,

    $$\mathcal{C}_{{\rm{interval}}} = 2N_t  + (2N_t+3)N_{\eqref{Eq:int2}}$$

    where,

    -   For : $2N_t$ real divisions are needed.

    -   For : $\left(2N_t+3\right)N_{\eqref{Eq:int2}}$ real multiplications are needed, where $\left(2N_t+3\right)$ is the number of real computations needed to compute , and $N_{\eqref{Eq:int2}}$ is the number of times is computed, which is calculated by simulations. Note, i) the interval in depends on the antenna index $\ell$ and only the imaginary part of the symbol $s$, ii) some symbols share the same imaginary part. Therefore, is only calculated for those points $(\ell,s)$ which lie in the interval in and does not have the same $\ell$ and $\mathop{\rm Im}\{s\}$ as a previously calculated point.

$$\mathrm{E}_{\bf{H}}\left\lbrace \Pr_{e,\text{SM--SD}} \right\rbrace = \left[ \zeta\left(\frac{\sigma_s^2}{4\sigma_n^2}\right) \right]^{N_r} \sum_{r=0}^{N_r-1}\binom{N_r-1+r}{r}\left[1-\zeta\left(\frac{\sigma_s^2}{4\sigma_n^2}\right)\right]^r \label{Eq:EFinal}$$

Error Probability of SM–SDs and Initial Radius Selection Method {#sec:ANA}
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In this section, we derive an analytical expresion for the BER performance of SM–SD, and we show that SM–SD offers a near optimum performance. The BER for SM–SD is estimated using the union bound [@book:p0001], which can be expressed as follows,

$$\underset{\text{SM--SD}}{\text{BER}} \leq \sum_{\ell_t,s_t}\sum_{\ell,s}\frac{N\left(\mathbf{\bar x}_{\ell_t,s_t},\mathbf{\bar x}_{\ell,s}\right)}{m}\frac{\mathrm{E}_{\bf{H}}\left\lbrace \Pr_{e,\text{SM--SD}} \right\rbrace }{2^m} \label{Eq:BER_1}$$

where $N\left(\mathbf{\bar x}_{\ell_t,s_t},\mathbf{\bar x}_{\ell,s}\right)$ is the number of bits in error between 