# Joint Resource Allocation for Multiuser Opportunistic Beamforming Systems with OFDM-NOMA
## 5 Numerical Results  
Numerical results are provided in this section, where we assume the channel coefficients of subcarriers satisfy circular symmetric complex Gaussian random distributions with zero mean and unit variance \(\mathcal{CN}(0,1)\). During this section, binary phase shift keying (BPSK) is used. The details of simulation parameters are listed in Table 1.  
The spectrum efficiencies of different non-orthogonal multiplexing schemes with the increased \(\frac{P_{\text{total}}}{\sigma^{2}}\) are indicated in Figure 2. It is seen that the spectrum efficiency of the proposed NOMA scheme is larger than that of SCMA scheme. For example, when \(\frac{P_{\text{total}}}{\sigma^{2}}=20\) dB and \(U=20\), the spectrum efficiency of NOMA schemes is approximately \(13\)\(\text{bps}/\text{Hz}\), although the SCMA scheme achieves approximately \(11\)\(\text{bps}/\text{Hz}\). The reason \begin{table}
\begin{tabular}{c c} \hline \hline
**Simulation Parameters** & **Value** \\ \hline Number of transmit antennas \(N_{T}\) & 2 \\ AWGN variance \(\sigma_{\text{zz}}^{2}\) & \(\sigma^{2}\) \\ Ratio of transmit power to noise variance \(\frac{P_{\text{total}}}{\sigma^{2}}\) & [0–20] (dB) \\ Number of subcarriers \(N_{c}\) & 64 \\ Number of users \(U\) & [1–20] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The simulation parameters. is that resource multiplexing rate of NOMA scheme is higher than that of SCMA scheme. Moreover, comparing the curves between the different \(U\), we can draw that the curves of \(U=20\) are higher than the curves of \(U=10\), no matter of the non-orthogonal multiplexing schemes. The larger number of waiting users \(U\) causes larger equivalent channel \(h_{u}^{\text{eqf}}\), which leads to higher spectrum efficiency. Figure 3 shows the comparison between multi-subcarrier and single-subcarrier schemes in the proposed system. It is found that multi-subcarrier scheme achieves higher spectrum efficiency than single-subcarrier. Since the channel coefficients of the subcarriers are different, multi-subcarrier scheme can adaptively adjust the power and user pair on each subcarrier. Moreover, the deep fading point in the frequency band is avoided in multi-subcarrier scheme, which results in higher system performance. In Figure 4, spectrum efficiencies of single-subcarrier (SC), OBON and sparse code multiple-access (SCMA) schemes are compared with the increased number of waiting users \(U\). From Figure 4, we can draw that all curves raise with the increased \(U\). A layer number of waiting users improves both the multiuser diversity and user pairing gains [25], therefore, a larger \(U\) leads to higher spectrum efficiency. Comparing the different schemes, the proposed OBON scheme achieves the highest spectrum efficiency, followed by SCMA and SC schemes, since the proposed OBON scheme obtains the largest multiplexing gain  
Figure 3: Comparison between multi-subcarrier and single-subcarrier schemes. Figure 2: Comparison between different non-orthogonal multiplexing schemes. and adjusts the wireless resources among the different subcarriers. Obviously, the curves with the larger \(\frac{P_{\text{total}}}{\sigma^{2}}\) achieve the higher spectrum efficiencies, no matter of the schemes. Figure 5 compares the spectrum efficiencies of our proposed closed-form power allocation (PA) and fixed power allocation (FPA) schemes, where \(U=20\). For the FPA scheme, the exhaustive searching algorithm is also applied to determine subcarrier scheduling and user pairing, and the power of the 1 st and 2 nd users are respectively \(P_{1}=\zeta_{\text{fix}}P_{\text{total}}\), and \(P_{2}=(1-\zeta_{\text{fix}})P_{\text{total}}\), where \(\zeta_{\text{fix}}\in(0,0.5)\) is a parameter. It is found that the spectrum efficiency of the proposed closed-form PA scheme is always larger than that of the FPA scheme, no matter the values of \(\zeta_{\text{fix}}\). The reason is that the proposed closed-form PA scheme can adaptively allocate the power among the users to obtain the maximum spectrum efficiency.