## 3 LR-Based RAS Selection in LR Aided PSM System  
The ZF precoding-based SM system with \(N_{T}\geq N_{R}\) (i.e., under-determined MIMO) whose transmit diversity order is well-known to be \(N_{T}-N_{R}+1\) cannot provide the full diversity gain originating from multiple transmit antennas. This paper considers an LR-based PSM system (Figure 1) to achieve the maximum transmit diversity gain of \(N_{T}\). LR is a process to find a new basis for the given lattice with basis vectors \(\big{\{}\mathbf{h}_{1},\,\mathbf{h}_{2},\,\cdots,\,\mathbf{h}_{N_{T}}\big{\}}\), where \(\mathbf{h}_{i}\) is the \(i\)-th column vector of the full channel matrix \(\mathbf{H}\). The LR-reduced basis consists of nearly orthogonal and relatively short vectors. In this work, the LR is performed by a complex Lenstra-Lenstra-Lovasz (CLLL) algorithm [14; 15], which is the most common LR technique.  
A specific channel matrix contained in the set of all possible \(C(N_{R},N_{T})\) antenna subsets is denoted by \(\mathbf{H}_{S}\in C^{N_{T}\times N_{T}}\). Then, the CLLL algorithm on the columns of \(\mathbf{H}_{S}^{H}\) is used to obtain the newly generated channel matrix  
\[\widetilde{\mathbf{H}}_{S}=\mathbf{H}_{S}^{H}\mathbf{T}_{S} \tag{6}\]  
where \(\widetilde{\mathbf{H}}_{S}\) is a CLLL-reduced basis with approximately orthogonal columns, \(\mathbf{T}_{S}\) is an unimodular matrix, i.e., \(|\text{det}(\mathbf{T}_{S})|=1\), and all elements of \(\mathbf{T}_{S}\) are Gaussian integers.  
This work takes \(LR\)-aided \(ZF\) precoding at the transmitter for the PSM system into consideration. Then, the precoder based on \(LR\) is given by  
\[\mathbf{P}_{LR-ZF}=\widetilde{\mathbf{H}}_{S}\Big{(}\widetilde{\mathbf{H}}_{S}^{ H}\widetilde{\mathbf{H}}_{S}\Big{)}^{-1} \tag{7}\]  
The received signal vector in the LR-PSM system can be represented as  
\[\widetilde{\mathbf{y}}=\widetilde{\beta}_{S}\Big{(}\mathbf{T}_{S}^{-1}\Big{)}^ {H}\mathbf{x}+\mathbf{n} \tag{8}\]  
where \(\widetilde{\beta}_{S}\) is a power normalization factor, which is expressed as  
\[\widetilde{\beta}_{S}=\sqrt{\frac{N_{T}}{Tr\bigg{[}\left(\widetilde{\mathbf{H }}_{S}^{H}\widetilde{\mathbf{H}}_{S}\right)^{-1}\bigg{]}}} \tag{9}\]  
Note that although the spatial modulated signal, \(\mathbf{x}\), in (8) consists of only a non-zero element, \(\Big{(}\mathbf{T}_{S}^{-1}\Big{)}^{H}\mathbf{x}\in C^{N_{R}\times 1}\) given in the LR domain can contain two or more non-zero elements. Thus, the precoded signal may be received at more than one receive antenna in the LR domain.  
The optimal ML detector for the LR-PSM can be obtained as  
\[\begin{array}{rl}\overset{\star}{\mathbf{x}}&=\underset{\mathbf{x}}{\arg \min}\bigg{\|}\widetilde{\mathbf{y}}-\widetilde{\beta}_{S}\Big{(}\mathbf{T}_{S }^{-1}\Big{)}^{H}\mathbf{x}\bigg{\|}^{2}\\ &=\underset{\mathbf{x}}{\arg\min}\bigg{\|}\frac{\widetilde{\mathbf{y}}}{ \widetilde{\beta}_{S}}-\ \Big{(}\mathbf{T}_{S}^{-1}\Big{)}^{H}\mathbf{x}\bigg{\|}^{2}\end{array} \tag{10}\]  
The optimal RAS algorithm based on LR (called opt-LR-RAS) for the LR-PSM system can be straightforwardly formulated as  
\[\begin{array}{rl}S^{*}&=\underset{S\in\{S_{k},k=1,2,\cdots,C(N_{R},N_{T})\}} {\arg\max}\widetilde{\beta}_{S}\\ &=\underset{S\in\{S_{k},k=1,2,\cdots,C(N_{R},N_{T})\}}{\arg\min}Tr\bigg{[} \left(\widetilde{\mathbf{H}}_{S}^{H}\widetilde{\mathbf{H}}_{S}\right)^{-1} \bigg{]}\end{array} \tag{11}\]  
IT is pointed out that the computational complexity of (11) to find an optimal LR-based RAS is very high due to an exhaustive search and CLLL operations. In this work, the LR is performed by a hardware-friendly CLLL algorithm named a fixed-complexity CLLL (fcCLLL) [21] to limit the worst-case complexity of CLLL. Furthermore, to perform LR-based RAS selection with reduced-complexity, a suboptimal LR-RAS selection algorithm is developed as a hybrid of norm and LR.