## 5 Performance Evaluation
### Simulation Results
If BMCH-only is applied instead, however, the bandwidth will be determined by the worst user on the BMCH, that is, user 3 and the required bandwidth is larger, as seen in Figure 4. Secondly, at sample index 11, user 1 (who has the worst channel quality on the SMCH) has better channel quality on the BMCH, that is, \(j^{(0)}=1\) and \(\alpha_{j^{(0)}}<0\). Therefore, the proposed algorithm enters an iteration stage. In the first iteration, \(B_{B}^{(1)}\) satisfying \(f_{1}\Big{(}B_{B}^{(1)}\Big{)}=f_{3}\Big{(}B_{B}^{(1)}\Big{)}\) is found and \(j^{(1)}\) is updated to 3. Also, because user 3 has better channel quality on the BMCH, that is, \(\alpha_{j^{(1)}}<0\), the iteration continues. In the second iteration, \(B_{B}^{(2)}\) is found to satisfy \(f_{3}\Big{(}B_{B}^{(2)}\Big{)}=f_{2}\Big{(}B_{B}^{(2)}\Big{)}\) and \(j^{(2)}\) is updated to 2. Now, user 2 has worse channel quality on the BMCH, so the algorithm stops and both the BMCH and the SMCH are allocated. Third, at sample index 1, user 4 with the worst channel quality on the SMCH has better channel quality on the BMCH; therefore, the proposed algorithm enters the iteration stage. Unlike sample 11; however, user 4 has the worst channel quality on the BMCH and on the SMCH, implying \(B_{B}^{(1)}\) satisfying \(f_{4}\Big{(}B_{B}^{(1)}\Big{)}=f_{j}\Big{(}B_{B}^{(1)}\Big{)}\) cannot be found in the range \(\Big{[}0,B_{|\Omega}\Big{)}\). Then, \(B_{B}^{(1)}\) is set to \(B_{|\Omega|}\) and only the BMCH is allocated. It is worth investigating the effect of the correlation between Rayleigh fading on the BMCH and on the SMCH, since the proposed multicast transmission scheme exploits channel diversity and hence, the correlation might degrade performance. The fading generation scheme proposed in Reference [17] was adopted, where two correlated Rayleigh fading envelopes are generated by means of a coloring matrix obtained from Cholesky decomposition of the correlation matrix. Figure 5 shows the average bandwidth under the proposed scheme by varying correlation factor \(\rho\), as well as BMCH-only for comparison. As expected, the proposed scheme requires a larger average bandwidth as \(\rho\) increases. However, the proposed scheme still outperforms BMCH-only because the variation in channel quality is still meaningfully large, despite the correlation. We observed from the simulations that \(\gamma_{j}^{S}\) is 3 dB higher than \(\gamma_{j}\) with probabilities of 70% and 58% when \(\rho=0\) and 0.5, respectively. Interestingly, even with \(\rho=0.9\), the probability is still 24%. It can be said that this amount of variation in channel quality on the BMCH and the SMCH is large enough to get the channel diversity gain. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Sample 1**} & \multicolumn{2}{c}{**Sample 11**} & \multicolumn{2}{c}{**Sample 20**} \\ \cline{2-7}  & \(\gamma_{j}\) & \(\gamma_{j}^{S}\) & \(\gamma_{j}\) & \(\gamma_{j}^{S}\) & \(\gamma_{j}\) & \(\gamma_{j}^{S}\) \\ \hline User 1 & 15.29 & 6.58 & 13.70 & \(-\)0.98 & 31.14 & 15.12 \\ \hline User 2 & 11.59 & 12.60 & \(-\)1.05 & 7.09 & 7.85 & 7.19 \\ \hline User 3 & 13.07 & 13.16 & 6.82 & 3.02 & 1.89 & 7.30 \\ \hline User 4 & \(-\)1.56 & \(-\)5.37 & 9.72 & 12.11 & 3.67 & 5.68 \\ \hline User 5 & 3.78 & 6.92 & 15.58 & 15.03 & 18.44 & 12.59 \\ \hline User 6 & 32.50 & 20.06 & 10.39 & 8.05 & 13.72 & 12.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Channel qualities of the basic multicast channel (BMCH) and the supplementary multicast channel (SMCH) for six users at three sample points (in decibels). Figure 6 investigates the impact of \(|\Omega|\) on the minimum average bandwidth. While the conventional multicast transmission scheme requires the same bandwidth, irrespective of \(|\Omega|\), the other schemes require more bandwidth as \(|\Omega|\) increases, since the worst channel among the users in \(\Omega\) has worse channel quality with a higher probability.