## 4 Numerical Results  
In this section, we analyze the performance of the proposed two beamformer schemes and of the non-cooperative scheme. Since the sum rates of the NOMA systems depend on the resource allocation, as well, we compare in Figure 2 the outage probabilities of these three schemes with different sets of antenna configurations to compare the proposed cooperative beamformer schemes with the non-cooperative scheme, where the parameter definitions and simulation conditions used are summarized in Table 1. Here, the outage event is defined as the case when the total transmit power of the AP and the cooperative UTs required for a scheme exceeds a given power level threshold. In Figure 2, some of the curves are close-by since two curves are almost indistinguishable for the right most curves and so are the three curves in the left most curves. Since all the three curves exhibit the same slope of the outage when \(M=N=1\), it is easy to see that the power gain by beamforming is more noticeable than the diversity gain in this case. It is the case of \(M=1\) and \(N=4\), where the diversity gain from the cooperation is maximized compared to the non-cooperative scheme. On the other hand, the case of cooperative \(M=4\) and \(N=1\) exhibits the same diversity performance since the diversity gain is acquired from the sum of channel gains at UTs and all the UTs experience the sum channels of the same diversity order. Again, the power gain is more prominent when \(M=4\) and \(N=1\). Overall, the cooperative beamforming schemes provide better diversity performance than the non-cooperative one and the power gain is guaranteed even when the diversity is not available. In all the cases, the scheme with lightly power constrained cooperative UTs loses negligible amount of outage performance compared to the case absent of such constraint.  
In Figures 3 and 4, we compare the average power curves of proposed schemes to achieve the threshold (\(\gamma^{*}\)) for different antenna settings. In Figure 3, the average power curves against \(N\) with fixed \(M\)s are plotted, while those against \(M\) with fix \(N\)s appear in Figure 4. Both \(M\) and \(N\) are similarly helpful in reducing the average power to achieve the same threshold of Figure 2. Note that the average power curve of the non-cooperative scheme with \(N=3\) in Figure 4, is collapsed with that of optimal cooperative scheme with \(N=1\). Again, the cooperation schemes provide much saving in the power consumption and the one with the UT power constraint does not require noticeably more power than the one without the constraint. Note that the UT power constrained one needs little more power than the one without such constraint when either \(M\) or \(N\) is low. These results reflect that the beamforming suggested in this paper further boosts the performance gains of the cooperative NOMA system, as well as the gain from pure cooperation. Finally, the gain we observe here is acquired at the expense of additional spectral resource used for the cooperation. Therefore, a trade-off exists between the cooperation gain and the additional spectral resource.  
Figure 3: The average power of the proposed schemes against \(N\). Here, \(J=4\) and \(\gamma^{*}=10\) for all schemes; \(L=1\) and \(P_{d,1}=\frac{\gamma^{*}}{2}\) for the cooperative schemes. “coop.PC” represents the cooperative NOMA with UT power constraint.  
Figure 2: The outage probabilities of the proposed schemes against the signal-to-interference-plus-noise ratio (SINR) with different \(M\) and \(N\). Here, \(J=4\) and \(\gamma^{*}=10\) for all schemes; \(L=1\) and \(P_{d,1}=\frac{\gamma^{*}}{2}\) for the cooperative schemes. “coop.PC” represents the cooperative NOMA with UT power constraint.