## 2 RAS Selection in Conventional PSM System  
Consider an over-determined MIMO time division duplex (TDD) system with \(N_{T}\) transmit antennas and \(N_{R}(N_{R}>N_{T})\) receive antennas as shown in Figure 1. The full channel matrix is given as \(\textbf{H}\in C^{N_{R}\times N_{T}}\), which is the quasi-static channel matrix whose elements are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance. We assume that the channel side information (CSI) of **H** is perfectly known at the transmitter and receiver. To design the PSM system, RAS selection is employed to obtain the selected channel. In this work, \(N_{T}\) antennas out of \(N_{R}\) receive antennas are assumed to be selected.  
In TDD mode, CSI can be estimated by using the channel reciprocity property between uplink and downlink channels [19; 20]. Pilot symbols may be transmitted from each receive antenna through an uplink channel in a round-robin manner among all the available receive antennas. The instantaneous CSI estimated at the transmitter can be utilized for LR-aided precoding and LR-based RAS selection. The indices of the selected RAS can be transmitted through a downlink channel or the RAS selection can be conducted by exploiting the CSI acquired at the receiver side.  
Figure 1: Block diagram of an (LR-aided) PSM MIMO TDD system with (LR-based) RAS selection.  
The spatial modulated super-symbol vector is presented by \(\mathbf{x}\in C^{N_{T}\times 1}\), which can be expressed as \(\mathbf{x}=s\mathbf{e}_{r}\) where a symbol \(s\) with \(E\big{[}|s|^{2}\big{]}=1\) is selected from the \(M\)-ary quadrature amplitude modulation (QAM) or phase-shift keying (PSK) constellation set and \(\mathbf{e}_{r}\) is the \(r\)-th column of the \(N_{T}\)-dimensional unit matrix. The super-symbol \(\mathbf{x}\) is first precoded before transmission. Then, the transmit signal vector is given by \(\beta\mathbf{P}\mathbf{x}\) where \(\mathbf{P}\in C^{N_{T}\times N_{T}}\) is a precoding matrix and \(\beta\) is a power normalization factor used to ensure \(E\big{[}\|\beta\mathbf{P}\mathbf{x}\|^{2}\big{]}=1\). In [3; 6; 9; 10], the \(ZF\) precoding scheme has been applied to the PSM systems, and the precoding matrix is given as  
\[\mathbf{P}_{ZF}=\mathbf{H}_{S}^{H}\Big{(}\mathbf{H}_{S}\mathbf{H}_{S}^{H} \Big{)}^{-1} \tag{1}\]  
where \(\mathbf{H}_{S}\in C^{N_{T}\times N_{T}}\) denotes the channel matrix obtained by an RAS selection algorithm.  
The \(k\)-th received block signal at the receiver is described as  
\[\mathbf{y}=\beta_{S}\mathbf{H}_{S}\mathbf{P}\mathbf{x}+\mathbf{n}=\beta_{S} \mathbf{x}+\mathbf{n} \tag{2}\]  
where the power normalization factor related with the selected antenna subset is  
\[\beta_{S}=\sqrt{\frac{N_{T}}{Tr\big{[}\big{(}\mathbf{H}_{S}^{H}\mathbf{H}_{S} \big{)}^{-1}\big{]}}} \tag{3}\]  
and \(\mathbf{n}\in C^{N_{T}\times 1}\) is an i.i.d. additive white Gaussian noise vector whose elements are the zero-mean circular complex white Gaussian noise component of a variance of \(\sigma_{n}^{2}\). Then, the optimal RAS selection algorithm for the PSM system is expressed by [9]  
\[S^{*}=\underset{S\in\{S_{k},k=1,2,\cdots,C(N_{R},N_{T})\}}{\arg\max}\beta_{S} \tag{4}\]  
where \(S_{k}\) is the \(k\)-th enumeration of the set of all available \(C(N_{R},N_{T})\) antenna subsets. Here, \(C(N_{R},N_{T})\) is the total number of combinations of selecting \(N_{T}\) antennas among \(N_{R}\) receive antennas. In the receiver for the ZF-based PSM, the optimal maximum likelihood (ML) detector is given by  
\[\overset{\wedge}{\mathbf{x}}=\underset{\mathbf{x}}{\arg\min}\|\mathbf{y}- \beta_{S^{*}}\mathbf{x}\|^{2} \tag{5}\]