## 4 Simulation Results and Diversity Analysis  
In this section, the optimal and suboptimal LR-based RAS selection algorithms proposed for the presented LR-PSM system with \(N_{T}\) transmit antennas and \(N_{R}(N_{R}>N_{T})\) receive antennas are evaluated through Monte Carlo simulations over Raleigh flat-fading channels. The signal-to-noise ratio (SNR) is defined by the symbol energy to the noise power spectral density ratio, i.e., \(\eta=1/\sigma_{n}^{2}\). The QPSK modulation is assumed and the receiver is based on ML detection. Moreover, the fcCLLL algorithm for LR operation employs \(\delta=1\) and the iteration number of \(N_{iter}=3\) for LR-RAS selection and \(N_{iter}=5\) for signal detection. For the BER performance comparison, the following five PSM systems are considered. 1. conventional ZF-based PSM without RAS selection (called ZF w/o RAS)
2. conventional ZF-based PSM with optimal RAS selection (called opt-RAS-ZF) [9]
3. LR-ZF-based PSM without RAS selection (called LR-ZF w/o RAS)
4. LR-ZF-based PSM with optimal LR-aided RAS selection (called opt-LR-RAS-LR- ZF)
5. LR-ZF-based PSM with suboptimal LR-aided RAS selection (called subopt-LR- RAS-LR-ZF)  
In addition, in the plots, the BER reference curves are given as a form of \(c/SNR^{G}\) with solid lines, where \(c\) is an appropriately selected positive constant and \(G\) denotes a diversity gain. Note that the diversity order can be employed to determine the slope of the BER curve in log-scale at high SNR ranges [26]. Figure 2 presents the simulated BER results of three antenna diversity systems of (b), (d), and (e) with \(N_{T}=2\) and \(N_{R}=4\). Thus, the spectral efficiency of 3 bps/Hz is assumed. Note that the number of selected receive antennas is equal to \(N_{T}=2\). The other cases of (a) and (c), which have no antenna selection, correspond to the scenario of \(N_{T}=2\) and \(N_{R}=2\). Note that the PSM system with no RAS selection should meet the condition of \(N_{R}\leq N_{T}\) to enable the precoding scheme. Thus, this work assumes that the number of receive antennas under the scenario without RAS selection is equal to the number of transmit antennas. It is found from BER curves of (a) and (c) that the LR-ZF-based PSM scheme outperforms the conventional ZF-based PSM. Using the similar method to [27; 28], the transmit diversity order of ZF-based PSM without RAS selection can be easily obtained as \(G=N_{T}-N_{R}+1\). It is clearly observed that ZF without RAS selection achieves the diversity order of \(G=1\). To plot the BER reference curves, the constants selected for \(G=1,2,3\), and \(6\), are \(c=0.9\), \(1.7\), \(7\), and \(8.3\times 10^{4}\), respectively. On the other hand, the LR-ZF-based PSM without RAS selection can capture the diversity order of \(G=N_{T}\). Let us approximately analyze the transmit diversity order of the LR-ZF-based PSM scheme without RAS selection. The square of \(\widetilde{\beta}_{S}\) can be re-expressed as  
\[\widetilde{\beta}_{S}^{2}=\frac{1}{E\Big{\{}diag\Big{[}\left(\widetilde{ \mathbf{H}}^{H}\widetilde{\mathbf{H}}\right)^{-1}\Big{]}\Big{\}}}=E\Big{\{} \widetilde{\beta}_{r,S}^{2}\Big{\}} \tag{20}\]  
where  
\[\widetilde{\beta}_{r,S}^{2}=\frac{1}{\Big{[}\left(\widetilde{\mathbf{H}}^{H} \widetilde{\mathbf{H}}\right)^{-1}\Big{]}_{r,r}}=\widetilde{\mathbf{h}}_{r}^{ H}\widetilde{\mathbf{h}}_{r} \tag{21}\]  
and \(\widetilde{\mathbf{h}}_{r}\) is the \(r\)-th column of \(\widetilde{\mathbf{H}}\) and \(\widetilde{\mathbf{F}}\) is an \(N_{T}\times N_{T}\) non-negative Hermitian matrix formed from \(\widetilde{\mathbf{h}}_{1},\cdots,\widetilde{\mathbf{h}}_{r-1},\widetilde{ \mathbf{h}}_{r+1},\cdots,\widetilde{\mathbf{h}}_{N_{T}}\). Then, it is assumed for analysis purpose that the LR-reduced matrix \(\widetilde{\mathbf{h}}\) consists of perfectly orthogonal column vectors. In this case, \(\widetilde{\mathbf{h}}_{r}\) is orthogonal to its projection on the subspace of \(\widetilde{\mathbf{F}}\).