$<10^{-2}$ all graphs closely match the analytical results. Note, it is will–known that the union bound is loose for low SNR [@book:p0001].

Comparison of the BER performance of SM and SMX
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![BER against SNR. $m=8$, and $N_r=4$.[]{data-label="fig:BER_ANA_SM_8_4"}](BER_ANA_wordsize_8_Nr_4.eps){width="8.8cm"}

![BER against SNR. $m=6$, and $N_r=2$.[]{data-label="fig:BER_SM_6_2"}](BER_wordsize_6_Nr_2.eps){width="9.6cm"}

![BER against SNR. $m=6$, and $N_r=4$.[]{data-label="fig:BER_SM_6_4"}](BER_wordsize_6_Nr_4.eps){width="9.6cm"}

![BER against SNR. $m=8$, and $N_r=2$.[]{data-label="fig:BER_SM_8_2"}](BER_wordsize_8_Nr_2.eps){width="9.6cm"}

![BER against SNR. $m=8$, and $N_r=4$.[]{data-label="fig:BER_SM_8_4"}](BER_wordsize_8_Nr_4.eps){width="9.6cm"}

Figs. \[fig:BER\_SM\_6\_2\] and \[fig:BER\_SM\_6\_4\] show a BER comparison between all possible combinations of SM and SMX for $m=6$ and $N_r=2,4$. In Fig. \[fig:BER\_SM\_6\_2\], we can observe that the BER performance depends on the the number of transmit antennas used and, consequently, the constellation size. The smaller the constellation size, the better the performance. Another observation that can be made is that SM and SMX offer nearly the same performance when using the same constellation size. In Fig. \[fig:BER\_SM\_6\_4\], where the number of receive antennas is increased, we notice that SM performs better than SMX. In particular, BPSK–SM provides a $1$ dB better performance than BPSK–SMX. Also 8–QAM SM offers a slightly better performance ($\sim 0.5$ dB) than 8-QAM SMX.

In Figs. \[fig:BER\_SM\_8\_2\] and \[fig:BER\_SM\_8\_4\], the BER comparisons for $m=8$ and $N_r=2,4$ are shown. In Fig. \[fig:BER\_SM\_8\_2\], SM and 