overdetermined MIMO setup with $N_t\leq N_r$.

As opposed to the SM–Rx scheme, the SM–Tx scheme uses some pre–computations to estimate the points that lie inside the sphere of radius $R$. These additional computations are carefully taken into account in the analysis of the computational complexity of the SM–Tx scheme and its comparison with the ML–optimum detector in section \[sec:Comp\].

$$\Pr_{e,\text{SM--SD}} \leq \left( \Pr\left((\tilde{\ell}_{\text{SM--ML}},\tilde{s}_{\text{SM--ML}})\neq \left(\ell_t,s_t\right)\right) + \Pr\left(\left(\ell_t,s_t\right) \notin \Theta _R\right) \right) \label{Eq_44}$$

Computational Complexity of SM–Rx and SM–Tx {#sec:Comp}
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In this section, we analyse the computational complexity of SM–ML, SM–Rx and SM–Tx. The complexity is computed as the number of real multiplication and division operations needed by each algorithm [@book:gl9601].

SM–ML
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The computational complexity of SM–ML receiver in , yields,

$$\mathcal{C}_{\text{SM--ML}} = 8N_r2^m, \label{Eq:MLcomp}$$

as the ML detector searches through the whole transmit and receive search spaces. Note, evaluating the Euclidean distance $\left({\left| {y_r  - h_{\ell,r} s} \right|^2 }\right)$ requires 8 real multiplications.

The computational complexity of SMX–ML receiver in is equal to

$$\mathcal{C}_{\text{SMX--ML}} = 4\left(N_t+1\right) N_r 2^m. \label{Eq:CompSMX}$$

Note, $\left(\left|\mathbf{y}-\mathbf{Hx}\right|^2\right)$ in requires $\left(N_t+1\right)$ complex multiplications.

From and , the complexity of SM does not depend on the number of transmit antennas, and it is equal to the complexity 