Gaussian random variables, with zero mean and variance equal to 1. This means that the argument in the $Q$–function can be described by a central chi–squared distribution with $2N_r$ degrees of freedom.

The result for $\mathrm{E}_{\bf{H}}\left\lbrace \Pr_{e,\text{SM--SD}} \right\rbrace$ is as given in , where $\sigma_s^2 = \| \bar{\bf{x}}_{\ell_t,s_t} - \bar{\bf{x}}_{\ell,s}\|^2_{\rm F}$ and,

$$\zeta(c) = \frac{1}{2}\left( 1 - \sqrt{ \frac{c}{1+c} } \right)$$

Plugging into gives a closed form expression for the BER of SM–SD. In the next section, we show that gives a tight approximation of the BER of SM–SD, and that SM–SD offers a near optimum performance.

![BER against SNR. $m=6$, and $N_r=4$.[]{data-label="fig:BER_ANA_SM_6_4"}](BER_ANA_wordsize_6_Nr_4.eps){width="9.1cm"}

Results {#sec:Result}
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In the following, Monte Carlo simulation results for at least $10^6$ Rayleigh fading channel realisations are shown to compare the performance and computational complexity of large scale MIMO, SM–ML, SM–SD and SMX–SD.

Analytical performance of SM–SD
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Figs. \[fig:BER\_ANA\_SM\_6\_4\]-\[fig:BER\_ANA\_SM\_8\_4\] show the BER simulation results for SM–ML, SM–Rx and SM–Tx compared with the analytical bound derived in section \[sec:ANA\], where $m=6,8$ and $N_r=4$. From the figures we can see that both SM–Tx and SM–Rx offer a near optimum performance, where the results overlap with SM–ML. Furthermore, Figs. \[fig:BER\_ANA\_SM\_6\_4\]-\[fig:BER\_ANA\_SM\_8\_4\] validate our analytical bound as for BER 