## 4 Simulation Results and Analysis  
In this section, the energy efficiency performance of the proposed AO-EE algorithm is simulated, considering the IRS-assisted HST communication system's downlink. In the 3D Cartesian coordinate system, the coordinates of BS, IRS, and MR are set to (0 m, 0 m, 10 m), (0 m, 150 m, 2.5 m), and (10 m, 140 m, 2.5 m), respectively; the carrier frequency \(f_{C}=5\) GHz, the system bandwidth B = 100 KHz, and the train speed v = 70 m/s. The BS transmit power Pmax = 42 dBm, the Rician K-factor is set to 10, the noise power \(\sigma^{2}=-110\) dBm and \(d=\lambda/2\). \(\lambda\) is the wavelength [22], the BS static power consumption \(P_{BS}=40\) dBm, and the MR power consumption \(P_{MR}=30\) dBm [40]. The power consumption of the IRS unit \(P_{R}=10\) dBm [11], and the maximum Doppler shift \(f_{m}=\psi f_{c}/c\approx 1167\) Hz, where \(c\) is the \begin{table}
\begin{tabular}{c c} \hline \hline
**Beamforming Design Method** & **Algorithmic Complexity** \\ \hline AO-EE & \(\mathcal{O}\big{(}I_{iter}(L_{1}L_{2}M^{3}+N^{2})\log_{2}\Big{(}\frac{1}{ \varepsilon}\Big{)}\Big{)}\) \\ Baseline Scheme 1 & \(\mathcal{O}\big{(}I_{iter}(L_{1}L_{2}M^{3}+N^{3.5})\log_{2}\Big{(}\frac{1}{ \varepsilon}\Big{)}\Big{)}\) \\ Baseline Scheme 2 & \(\mathcal{O}\Big{(}I_{iter}(L_{1}L_{2}M^{3}+N)\log_{2}\Big{(}\frac{1}{ \varepsilon}\Big{)}\Big{)}\) \\ Baseline Scheme 3 & \(\mathcal{O}\Big{(}I_{iter}\big{(}M^{3.5}+N^{3.5}\big{)}\log_{2}\Big{(}\frac{1}{ \varepsilon}\Big{)}\Big{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Algorithmic complexity. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Beamforming Design Method** & \multicolumn{3}{c}{**Running Time When N = 64 (Unit: s)**} \\ \hline AO-EE & 7.4414 & 6.8477 & 7.6923 & 7.0585 & 6.9890 \\ Baseline Scheme 1 & 437.2063 & 428.4659 & 439.2299 & 424.3579 & 439.9219 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Running time of each iteration. speed of light. The time slot length is set to the correlation time \(T_{c}=0.432/f_{m}\approx 0.3\) ms. Due to the random nature of a single experiment, all numerical simulation results are averaged over 500 independent channel realizations. The simulation is performed using the MATLAB simulation platform. The running environment is Intel(R) Core (TM) i7-13700F (2.1 GHz), NVIDIA RTX4070 Ti. For large-scale fading, a distance-based path loss model is used [25]:  
\[L(d)=C_{0}\bigg{(}\frac{d}{d_{0}}\bigg{)}^{-\alpha} \tag{35}\]  
where \(C_{0}=-30\) dB, denotes the path loss at the reference distance \(d_{0}=1\) m, \(d\) denotes the link distance, \(\alpha\) denotes the path loss exponent, and the path loss coefficients for BS-MR, BS-IRS, and IRS-MR are \(\alpha_{BM}=3.5\), \(\alpha_{BI}=2\), and \(\alpha_{IM}=2.5\), respectively [25]. Figure 2 illustrates the relationship between the number of IRS reflecting elements (N) and the system energy efficiency for different algorithms when \(M=32\) and \(P_{\max}=42\) dBm. It can be observed that, in the absence of IRS (the NO-IRS case), increasing the number of IRS reflecting elements has no impact on the system's energy efficiency, resulting in a linear relationship. As the number of IRS reflecting elements increases, the system's energy efficiency for the other five IRS algorithms (Random Phase, Baseline Scheme 1, Baseline Scheme 2, and Baseline Scheme 3) shows varying degrees of improvement. This improvement is attributed to the gain achieved from both the BS transmit beamforming and the aperture gain provided by IRS, whereas the NO-IRS algorithm only benefits from the gain in the BS transmit beamforming. All four IRS optimization algorithms outperform the Random Phase algorithm, to a significant degree. This demonstrates the necessity of optimizing the IRS phase shift, as, under random phase shifts, the reflected signals may not be properly directed towards the MR, resulting in a substantial reduction in the gains offered by IRS.