# Novel Resource Allocation Techniques for Downlink Non-Orthogonal Multiple Access Systems
## 5 Results and Discussion
### Performance Evaluation  
This section assesses and validates the efficiency of the proposed techniques. Their performance is compared with the user pairing CEB method [16] and the power allocation FTPA method [11]. In the CEB method, the users in the network are paired according to their channel conditions. For instance, considering that the BS serves 10 active UEs, their channel conditions arrangement is shown in Equation (36). \[\underbrace{H_{1}\geq H_{2}\geq\cdots\cdots\geq H_{5}|}_{Group1:\text{ {Strong channel gain users}}}\geq\underbrace{H_{6}\geq\cdots\cdots\geq H_{10}}_{Group2:\text{ {Weak channel gain users}}} \tag{36}\]  
The pairing process starts by pairing the user that has strongest channel gain with the user that has weakest channel gain user; then, the user that has second strongest channel gain is paired with the user that has second weakest channel gain. The process is complete upon successfully pairing all users, (i.e., (UE\({}_{1}\), UE\({}_{6}\)), (UE\({}_{2}\), UE\({}_{7}\)), (UE\({}_{3}\), UE\({}_{8}\)), (UE\({}_{4}\), UE\({}_{9}\)) and (UE\({}_{5}\), UE\({}_{10}\))). In FTPA, the allocated power is obtained based on the condition of the user's channel. Hence, the transmission power of the \(w\)th user on the \(s\)th subchannel (UE\({}_{n,s}\)) is expressed as in Equation (37). \[P_{w\beta}=P_{s}\frac{H_{w,s}^{-\alpha}}{\sum_{l=1}^{M}H_{l,s}^{-\alpha}} \tag{37}\]  
where \(H_{w,s}\) denotes the channel gain normalized by the noise of user UE\({}_{w,s}\), \(P_{s}\) represents the subchannel's power and \(\alpha\) is the decay factor with a value between 0 and 1. Increasing \(\alpha\) indicates that high power is assigned to user with weak channel gain. The user with poorer channel gain is allocated with a greater \(\alpha\), which means a high power increase. This approach exhibits a low computational \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Complexity** \\ \hline Hungarian algorithm & \(O(N^{3})\) \\ \hline Dinkelbach algorithm & \(O(M_{1}\left(M_{2}(Q_{1}+Q_{2})+O(K)\right))\) \\ \hline Exhaustive search & \(O(2S!/2^{S})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The complexity comparison between the proposed and existing algorithms. complexity, with degraded performance as a result of a user's allocation power instability in proportion to the path loss [28]. The default simulation parameter values are given in Table 4.  
Figure 3 demonstrates the effect of the BS's transmission power on system sum rate with 10 users considered in the BS and with transmission power variation from 1 to 12 W. The sum rates of NOMA-PKKT-HNG and NOMA-DC are considerably higher than those of NOMA-FTPA and conventional OFDMA due to the power optimization. The results show that the system sum rate increases upon increasing the BS's transmission power for all applied methods. The NOMA-PKKT-HNG technique achieves a higher sum rate than NOMA-DC [28], NOMA-FTPA [11] and conventional OFDMA. Figure 4 shows the system sum rate performance with different UE numbers (10-60 users per BS that has 1 W transmission power). The results confirm that the system sum rate increases as the number of users increases. For comparison, the NOMA-PKKT-HNG is compared with NOMA-PKKT-CEB, a centralization-based approach [16]. The proposed technique outperformed other techniques (NOMA-PKKT-CEB, NOMA-FTPA and OFDMA) under the same applied method for power allocation in both proposed NOMA-PKKT-HNG and NOMA-PKKT-CEB techniques with similar sum rate for all users available at the BS. Once 20 users exist at the BS, the NOMA-PKKT-HNG has the highest performance when compared to other techniques. Figure 5 expresses the relationship between the system energy efficiency and the number of users available in a cellular system. The BS transmitting power is fixed at 1 W, and the system energy efficiency increases with the increasing number of users across all methods. The proposed NOMA-PDKL-HNG method is more energy-efficient than NOMA-DC, NOMA-FTPA and OFDMA.