# Joint Resource Allocation for Multiuser Opportunistic Beamforming Systems with OFDM-NOMA
## 5 Numerical Results
In Figure 6 the spectrum efficiencies of the proposed OBON, GSP, GA and VQ scheme are compared under the limited feedback information scenario, where \(U=20\). It is found that our proposed scheme provides the largest spectrum efficiencies, followed by GA, GSP and VQ schemes, no matter of \(\frac{P_{\text{total}}}{\sigma^{2}}\). The main reason is that the proposed OBON scheme can provide extra multiuser selection gain, which equals to \(\ln U\) with many waiting users. Moreover, OBF weights can encourage the fluctuation of equivalent channel and enlarge the equivalent channel coefficients. According to [27; 28; 29], the beamforming weights of in  
Figure 4: Spectrum efficiencies of OBON, SCMA and SC versus the increased number of waiting users. Figure 5: Spectrum efficiency comparison between the proposed joint iterative algorithm and FPA, where \(U=20\). GA, GSP and VQ schemes are generated through complex matrix calculations, although a random method is applied to obtain the beamforming weights of the proposed scheme. Therefore, the complexity of the proposed scheme is lower than those of GA, GSP and VQ schemes. Comparing the spectrum efficiencies among GA, GSP and VQ schemes, it is seen that the spectrum efficiency of GA scheme is highest, since the distortion of beamforming weights are lowest in GA scheme. Figure 7 shows the BER performances of the proposed, GA, GSP and VQ schemes, where \(U=20\). The BER of the proposed scheme is lowest among these schemes, since the proposed scheme obtains both multiuser diversity and beamforming gains. Compared to the GA scheme, the proposed scheme achieves approximately an extra 5 dB gain. According to Figures 6 and 7, the proposed scheme achieves both the highest spectrum efficiency and BER performances among these beamforming schemes with the limited feedback information. In Figure 8, the influences of different user pairing schemes on spectrum efficiencies are indicated, where \(U=20\). \"Max-min pairing scheme\" means that the users with the maximum and minimum channel qualities are paired. \"Random pairing scheme\" indicates that the users are randomly paired. In \"adjacent pairing scheme\", the BS combines the users with the adjacent channel coefficients to transmit. From Figure 8, it is seen that the  
Figure 6: The comparisons of spectrum efficiencies among the proposed, GA, GSP and VQ schemes, where \(U=20\). Figure 7: Bit error ratio comparisons among the proposed, GA, GSP and VQ schemes, where \(U=20\). proposed optimal scheme achieves the maximum spectrum efficiency, followed by the max-min pairing, random pairing and adjacent pairing schemes. The optimal scheme adaptively arranges the user pairing combination according to the channel coefficients of the users, leading to the largest spectrum efficiency. Comparing the max-min pairing, random pairing and adjacent pairing schemes, the spectrum efficiency of the max-min pairing scheme is highest, and the adjacent pairing scheme provides the lowest spectrum efficiency, which have been analyzed and verified in [32]. The comparison of spectrum efficiencies between Rayleigh and Rician fading channels is shown in Figure 9, where the ratio of direct component and scattering component is set to be 10 in Rician case. It is found that the Rayleigh case always achieves the higher spectrum efficiency than the Rician cases, no matter of \(\frac{P_{\text{total}}}{\rho^{4}}\) and \(U\). The reason is that the channel fluctuation of the Rayleigh case is larger than that of the Rician case, leading to a higher multiuser diversity gain. Moreover, all curves raise with the increased \(\frac{P_{\text{total}}}{\rho^{4}}\) and \(U\). In Figure 10, we compare the OBF scheme with CBF, where the joint iterative algorithm is also used to maximize the spectrum efficiency. It is found that CBF scheme provides larger spectrum efficiencies than OBF scheme. However, the complexity of CBF scheme is higher than that of OBF scheme, since the beamforming weights are generated according to the perfect CSI in CBF scheme.