# Parallel Complex Quadrature Spatial Modulation
## 6 Simulation Results  
This section discusses the performance of the proposed scheme under different system configurations using Monte Carlo simulations. The obtained results are compared to the analytical results derived in Section 4. The channel is assumed to be frequency-flat Gaussian channel. The number of groups that are selected in all simulations for PCQSM is either \(P=2\) or 4. First, we investigated the performance of the PCQSM in terms of the rotation angle. For \(N_{t}=8\), \(N_{r}=4\) and a fixed SNR value of 28 dB, the BER versus the the rotation angle for six different system configurations is illustrated in Figure 2. Based on the depicted simulations, the angle that minimizes the BER is announced as the optimal rotation angle. For 8-QAM, the optimal rotation angle using \(P=2\) is either \(21^{\circ}\) or \(69^{\circ}\) as the graph is symmetrical. When PCQSM uses four groups, i.e., \(P=4\), \(\theta_{opt}\) is either \(30^{\circ}\) or \(60^{\circ}\). For \(M=16\), the optimal rotation angle is the same for both \(P=2\) and 4, with the value of \(\theta_{opt}\) being either \(26^{\circ}\) or \(64^{\circ}\). For \(M=32\), \(\theta_{opt}\) has a unique value of \(45^{\circ}\). This implies that at low values of \(M\), the optimal rotation angles are different. However, as the modulation order increases, \(\theta_{opt}\) is the same for \(P=2\) and 4. The details of the optimal rotation angles obtained for all PCQSM configurations that are selected for evaluation in this section are presented in Table 1. It is important to note that the optimal rotation angles for CQSM are determined independently using extensive simulations. The optimal rotation angles of CQSM for two different system configurations are provided in Table 2.  
As an example to validate the derived results, the analytical results of the performance of PCQSM using (14) is investigated for \(N_{t}=8\) and \(N_{r}=4,6\), and 8 with modulation order of 8-QAM. This is illustrated in Figure 3a,b. Figure 3a depicts a comparison between the simulation and analytical results for PCQSM (\(P=2\)). Figure 3b illustrates the same for PCQSM (\(P=4\)). In all system configurations, the simulation results are upper bounded by the analytical results. In Figure 4a, PCQSM is compared with the conventional SM schemes CQSM and QSM with the same modulation order and number of receive antennas, \(M=4\) and \(N_{r}=4\), respectively. While QSM and CQSM required 512 and 256 transmit antennas to achieve the transmission rate of 20 bit/s/Hz, PCQSM with \(P=2\) and \(P=4\) required 32 and 16 transmit antennas, respectively. The performances of the CQSM and PCQSM for \(P=2\) are similar, whereas PCQSM for \(P=4\) performed close to QSM. The 1 dB BER improvement in the performance of PCQSM with \(P=2\) as compared to PCQSM with \(P=4\) is due to the reduced number of antennas per subgroup. As the number of antennas reduces, the probability of transmitting the two signal symbols \(s_{a}\) and \(s_{b}\) from the same antennas increases. This leads to a slight degradation in error performance. This degradation is \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{(}N_{t};\,N_{r};\,\mathbf{M}\mathbf{)}\) & \(\theta_{opt}\) (\({}^{\circ}\)) & \(\mathbf{(}N_{t},\,N_{r},\,P,\,\mathbf{M}\mathbf{)}\) & \(\theta_{opt}\) (\({}^{\circ}\)) \\ \hline \((8,\,4,\,2,\,8)\) & & \((8,\,4,\,4,\,8)\) & \\ \((8,\,6,\,2,\,8)\) & & \((8,\,6,\,4,\,8)\) & \(30/60\) \\ \((8,\,8,\,2,\,8)\) & & \((8,\,8,\,4,\,8)\) & \\ \hline \((8,\,4,\,2,\,16)\) & \(26/64\) & \((8,\,4,\,2,\,32)\) & 45 \\ \((8,\,4,\,4,\,16)\) & & \((8,\,4,\,4,\,32)\) & \\ \hline \((32,\,4,\,2,\,4)\) & \(36/27/41\) & \((16,\,4,\,4,\,4)\) & \(31/41\) \\ \hline \((32,\,4,\,2,\,16)\) & \(12/21/43\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal rotation angle, \(\theta_{opt}\), for several PCQSM system configurations.