## 4 Simulation Results and Discussion  
In this section, the performances of the proposed resource allocation algorithm ACO-SPA are shown by contrast with root-finding and linear methods. In all simulation results, the wireless channel is defined as the frequency selective multipath channel. It contains six independent Rayleigh multipaths. The channel attenuation is an exponentially decay with e\({}^{-2l}\), where \(l\) is the multipath index. Therefore, the six multipaths of relative channel attenuation are [0, \(-\)8.69, \(-\)17.37, \(-\)26.06, \(-\)37.74, \(-\)43.43] dB. The total power is 1 W. The total bandwidth and total subcarriers are assumed as 1 MHz and 64, respectively. A set of proportionality constants are generated by the following function  
\[p\left( \begin{array}{l}A_{k}\end{array} \right)= \begin{cases}0.5,\ A_{k}=1\\ 0.3,\ A_{k}=2\\ 0.2,\ A_{k}=4\end{cases}  \tag{31}\]  
where \(p(A_{k})\) is the probability mass function (PMF). \[A_{k}=\frac{\varphi_{k}}{\min\varphi_{k}} \tag{32}\]  
In the ACO algorithm for subcarrier allocation, the number of ants is set to 10, and maximum iterations is 100, \(q_{0}=0.9\), the initial pheromone in each subchannel of each user is set to 1. The pheromone volatilization coefficient \(\rho\) ensures the random search capability. The weight factor \(\alpha\) represents the importance of the pheromone while \(\beta\) represents the importance of the heuristic information. Table 1 shows the influence of the pheromone volatilization coefficient \(\rho\) on the ACO algorithm for the subcarrier allocation. In order to analyze the influence the pheromone volatilization coefficient \(\rho\) on the ACO algorithm for subcarrier allocation, the weight factors \(\alpha=1\) and \(\beta=1\). According to Table 1, the needed iterations of convergence increases, with \(\rho\) increasing from 0.1 to 0.4, since the pheromone volatilization coefficient \(\rho\) has an impact on the random search capability of the ACO algorithm. With the impacts on the solution and the needed iterations of convergence taken into consideration, the \(\rho\) is set to 0.1 in the ACO algorithm in the paper. Table 2 shows the influence of the weight factors \(\alpha\) and \(\beta\) on the ACO algorithm for the subcarrier allocation. The weight factor \(\alpha\) represents the importance of the pheromone in the ACO algorithm, and the weight factor \(\beta\) represents the importance of the heuristic information. If \(\alpha\) is comparatively high, the algorithm would have an over-reliance on the pheromone and does not take full account of heuristic information, \begin{table}
\begin{tabular}{c c c} \hline \hline \(\rho\) & **Output Sum Rate (bit/s/Hz)** & **Iterations of Convergence** \\ \hline
0.1 & 3.882 & 43 \\
0.2 & 3.805 & 48 \\
0.3 & 3.847 & 51 \\
0.4 & 3.796 & 57 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The influence of the pheromone volatilization coefficient \(\rho\). \begin{table}
\begin{tabular}{c c c} \hline \hline \(\alpha\) & \(\beta\) & **Output Sum Rate (bit/s/Hz)** & **Iterations of Convergence** \\ \hline
0.5 & 1 & 3.768 & 47 \\
1 & 0.5 & 3.623 & 40 \\
1 & 1 & 3.882 & 43 \\
1 & 2 & 4.017 & 38 \\
2 & 1 & 3.675 & 31 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The influence of the weight factors \(\alpha\) and \(\beta\). which in turn affects the ability of solving the objective function. If \(\beta\) is comparatively high, the algorithm would lead to premature convergence and fall into the local optimal solution. As is seen from Table 2, when \(\alpha=1\) and \(\beta=2\), the algorithm has the preferable performance. Therefore, in the following simulation, the weight factor \(\alpha\) is set to 1 and the value of \(\beta\) is 2. _A. The ACO Algorithm for Subcarrier Allocation_  
Figure 4 shows the sum rate _versus_ the number of iterations. The AWGN noise power density is set to \(-80\) dB\(\cdot\)W/Hz, BER = \(10^{-6}\), the number of users is 16. In Figure 4, the blue line represents the maximum sum rate achieved by ACO algorithm up to the current iteration.