## 4 Numerical Results  
In this section, we present numerical results based on the SNR performance of the proposed optimization combinations in various parameter settings. An AP equipped with \(M\) antennas serves \(K\) multicast users, while an IRS with \(N\) reflecting elements is supporting the AP transmission. Four combination schemes are considered alongside two reference schemes denoted as SDR and LS (Algorithm 1) in the legends, where only the SDR algorithm and only Algorithm 1 are applied for the AP without the phase optimization of the reflecting elements at the IRS. The last two have the role of baseline schemes for the four combinations proposed. Figures 2 and 3 display the SINRs of the various schemes in this paper over the variation of the number of IRS reflecting elements in the under-loaded case (Figure 2) and in the overloaded case (Figure 3), respectively. Note that the transmit power is fixed at \(P=10\) with eight transmit antennas (\(M=8\)) at the AP, where the transmit power is set to the relative received power value compared to the additive noise power at the receiver, and it reflects the combination of actual transmit power and the loss through the path. The flat curves correspond to the SDR-only scheme and the LS-only scheme, while these schemes do not utilize the increasing number of reflecting elements (\(N\)) at the IRS in their performance enhancement since the diagonal elements of \(\boldsymbol{\Phi}\) matrix are set to have random phases. On the other hand, all four proposed schemes exploit the increased number of IRS reflecting antennas, but the efficiency is differentiated depending on \(N\) and the scenarios (cases). In both cases, the SDR-only and the LS-only schemes show large SNR gaps, which creates SINR disparities among the group of schemes depending on the AP pre-coder type. Comparing the performance of the IRS phase alignment schemes (Algorithms 2 and 3),  
their performances vary depending on the combined AP pre-coder type and the scenario. In the under-loaded case in Figure 2, the SDR AP pre-coder does better with Algorithm 3 than with Algorithm 2, while the LS case shows the opposite trend. In the overloaded case of Figure 3, this trend shows the reverse trend, while the curves involved with Algorithm 3 saturate with large \(N\) values. This observation indicates that the Algorithm 3 grouping of the IRS reflecting elements does not efficiently handle the cases with large numbers of users compared to Algorithm 2, where the LS-based IRS phase adjustment is applied. Overall, Algorithm 2 shows competitive performance regardless of the AP pre-coding scheme and the scenario. On the other hand, Algorithm 3 combined with LS performs well when the number of IRS reflecting elements \(N\) is low in the overloaded case, while it performs well regardless of \(N\) in the under-loaded case. Figure 2: SINR performance comparison over the number of reflecting elements \(N\) in the under-loaded case. Here, \(P=10,M=8\) and \(K=4\). In Figures 4 and 5, the SNR results of the proposed schemes over the variation of the transmit power \(P\) are compared in the under-loaded case, where the number of IRS reflecting elements is small (\(N=32\)) in Figure 4 and is large (\(N=128\)) in Figure 5. The SNR and the transmit power are expected to have linear dependence in the log scale, while the parallel gaps between different schemes reflect the power gains of the schemes. Again, all four proposed schemes exploit the increased transmit power, while the efficiency differs depending on \(N\) and the scenarios (cases). Additionally, the SDR-only scheme and the Algorithm 1-only scheme are base-line schemes for the other combination schemes. It is easy to see that Algorithm 2 widens the gap with Algorithm 3 as \(N\) increases, regardless of the AP pre-coder optimization they are combined with.