symbol emitted by the transmit antenna $\ell_t$ is denoted by $s_t$, with $s_t \in \{s_1,s_2,\dots,s_{M}\} $.

Accordingly, the $N_t \times 1 $ transmitted vector is: $${\bf{x}}_{\ell_t ,s_t }  = \left[ {{\bf{0}}_{1 \times \left( {\ell_t  - 1} \right)} ,s_t ,{\bf{0}}_{1 \times \left( {N_t  - \ell_t } \right)} } \right]^T, \label{Eq:Mod_vec}$$

where $\left[  \cdot  \right]^T$ denotes transpose operation, and ${\bf{0}}_{p \times q}$ is a $p \times q$ matrix with all–zero entries. Note, a power constraint on the average energy per transmission of unity is assumed (*i.e.* $E_s = \mathrm{E}[\mathbf{x}^{H}\mathbf{x}]=1$), where $\mathrm{E}\{\cdot\}$ is the expectation operator.

From above, the maximum achievable spectral efficiency by SM is,

$$\vspace{0.2cm}
m_{\text{SM}} = \log_2(N_t) + \log_2(M) \label{eq1}$$

However, for SMX,

$$\vspace{0.2cm}
 m_{\text{SMX}} = N_t \log_2(M) \label{eq2}$$

From and , we can see that the spectral efficiency of SM does not increase linearly with the number of transmit antennas as SMX does. Therefore, SM needs a larger number of transmit antennas/ larger constellation size to arrive at the same spectral efficiency as SMX. However, because in SM only one antenna is active:

-   The computational complexity of SM does not depend on the number of transmit antennas. Unlike SMX where the computational complexity increases linearly with the number 