remainder of this paper is organised as follows: In section \[sec:SysModel\], the system model along with the ML–optimum detector is summarised. In section \[sec:SD\], the new SM–Rx and SM–Tx receivers are described. In section \[sec:Comp\], an accurate analysis of the computational complexity of both SM–Rx and SM–Tx is performed. In section \[sec:ANA\], the analytical BER performance for SM–SDs is derived along with the initial radius selection method. Numerical results are presented in section \[sec:Result\], and the paper is concluded in section \[sec:Con\].

System Model {#sec:SysModel}
============

SM Modulator
------------

In SM, the bit stream emitted by a binary source is divided into blocks containing $m=\log_2\left(N_t\right) + \log_2\left(M\right)$ bits each, where $M$ is the constellation size. Then the following mapping rule is used [@mhsay0801]:

-   The first $\log_2\left(N_t\right)$ bits are used to select the antenna which is switched on for data transmission, while the other transmit antennas are kept silent. In this paper, the actual transmit antenna which is active for transmission is denoted by $\ell_t$ with $\ell_t \in \{1,2,\dots,N_t\}$.

-   The second $\log_2\left(M\right)$ bits are used to choose a symbol in the signal–constellation diagram. Without loss of generality, Quadrature Amplitude Modulation (QAM) is considered. In this paper, the actual complex 