## 4 Numerical Results  
This section provide numerical results of OP and EC of the proposed combining precoding and NOMA approaches for downlink MU MISO systems. The result of proposed method is compared to the conventional method to validate the derived theoretical analysis in both perfect and imperfect SICs. In most of published works, they indicated that the performance of system is deteriorated when the number of users is more than three because of remained IUI. Configurations and parameters of the system are explained as follows. The channel between the BS and each user follows the Rayleigh distribution. We perform \(10^{6}\) independent trials for each Monte-Carlo simulation. In this system, we assume that the terminals are stationary, thus only small-scale fading impacts on the system performance. The average channel gains are modeled as \(\Omega_{m,1}=\mathbb{E}\{|\mathbf{h}_{m,1}|^{2}\}\), \(\Omega_{m,2}=\mathbb{E}\{|\mathbf{h}_{m,2}|^{2}\}\) and \(\Omega_{m,3}=\mathbb{E}\{|\mathbf{h}_{m,3}|^{2}\}\). And, the average channel gains of users are chosen as \(\Omega_{m,1}=1\), \(\Omega_{m,2}=2\), \(\Omega_{m,3}=3\). Furthermore, unless otherwise stated, the data rate threshold is set at 1 bit/s/Hz. Consequently, we assume that the number of user in every cluster is three, and the precoding at the BS for each cluster is perfect. The power allocation coefficient is calculated by \(a_{n}=\frac{N-n+1}{\varphi}\), with \(\psi=\frac{N(N+1)}{2}\) to make sure that \(\sum_{n=1}^{N}\sqrt{a_{i}}=1\). In Figure 2, we investigated the total EC under the impact of the power allocation coefficient of User 1, \(a_{1}\), with different values of SNR. We can recognize that, the total EC reaches the maximal value when \(a_{1}=0.7\) in all cases of considered SNR. The User 1 is the farthest from the BS, thus the largest transmit power should be assigned to the User 1 for ensuring the fairness. From the result of this figure, \(a_{1}=0.7\), \(a_{2}=0.2\) and \(a_{3}=0.1\) are utilized for further analyzing. In Figure 3, we present the theoretical and simulation results of OP versus SNR for two cases perfect and imperfect SIC with \(a_{3}=0.1\), \(a_{2}=0.2\) and \(a_{1}=0.7\). Although the transmit power of the third user is the lowest, its OP is the lowest. It is explained as, the third user is the nearest from the BS, hence its channel gain is the largest. In addition, the third user applies the SIC processing to cancel interference of all other users, whereas, the first user detects its own signal without SIC operation, and the second user performs the SIC operation to remove the signal of user \(1\) only and considers the signal of third user as interference. Therefore, its received SNR is much higher than the received SINR of the others. The OP of the first user is the same in two cases because the first user does not employ SIC operation. Moreover, the accuracy between theoretical and simulation results verifies the proposed theoretical analysis in both perfect and imperfect SIC schemes. Figure 4 represents the OP of the first user when the required data rate is changed, i.e., \(r=1\), \(2\), and \(3\) bit per symbol, these required data rates respectively correspond to BPSK, QPSK and 8-PSK modulations. The OP of the first user is represented, the case of others is similar. As shown in Figure 4, the OP is increased when the required data rate increases, and the Monte Carlo simulation guarantees the correctness of the theoretical results. For the same value of OP, the difference of SNR of BPSK modulation from QPSK and 8-PSK modulations is respectively 3 dB and 5 dB. It means that, increasing the number of information bits in every symbol lets the potential detection of information bit of users reduce. Thus, the NOMA system should to select the reasonable modulations scheme according to the system model. Figure 2: The total EC of the system of the considered NOMA system with different SNR.