## 2 System Modeling and Optimization Problems
### Channel Model  
Blockages are widespread in the high-speed train communication environment, where random obstacles are distributed along the railway [30]. It is assumed that the direct link between the BS and MR is obstructed by obstacles such as trees, and only the non-line-of-sight (NLOS) component is available to transmit the signal to the MR. Therefore, the direct link is modeled as Rayleigh fading, with a Doppler power spectral density following the Jakes spectrum. Assuming that both the BS and MR antennas adopt a uniform linear array (ULA) architecture, and the IRS antennas adopt a uniform planar array (UPA) architecture, the antenna response vectors at the BS and MR ends are given by [24]:  
\[\mathbf{a}(\theta)=\left[1,e^{j\frac{2\pi}{\lambda}d\sin\theta},\cdots,e^{j\frac{ 2\pi}{\lambda}d(N-1)\sin\theta}\right]^{\mathrm{T}}, \tag{3}\]  
The IRS consists of \(N\) reflecting elements, thus assuming \(\sqrt{N}\) elements in both the horizontal and vertical directions. The steering vectors of the IRS along the y-axis and z-axis are given by [31]  
\[p_{y}(\varphi,\theta)=\left[1,e^{j2\pi\frac{d_{y}}{\lambda}\sin\theta\cos \varphi},\cdots,e^{j2\pi\frac{d_{y}}{\lambda}(\sqrt{N}-1)\sin\theta\cos\varphi }\right]^{\mathrm{T}}, \tag{4}\]  
\[p_{z}(\theta)=\left[1,e^{j2\pi\frac{d_{x}}{\lambda}\cos\theta},\cdots,e^{j2\pi \frac{d_{x}}{\lambda}(\sqrt{N}-1)\cos\theta}\right]^{\mathrm{T}}, \tag{5}\]  
Since the BS and IRS locations have been deployed in advance and the BS-IRS is designated as line-of-sight transmission, the BS-IRS channel is modeled as [22]:  
\[\mathbf{G}=\alpha p_{y}\Big{(}\varphi_{IRS}^{AOA},\theta_{IRS}^{AOA}\Big{)}\otimes p _{z}\Big{(}\theta_{IRS}^{AOA}\Big{)}a^{H}\Big{(}\theta_{BS}^{AOD}\Big{)}, \tag{6}\]  
where \(\otimes\) indicates the Kronecker product, \(\alpha\) represents the complex channel gain of the BS-IRS link. \(\varphi_{IRS}^{AOA}\in[0,\pi]\) and \(\theta_{IRS}^{AOA}\in[0,\pi]\) represent the azimuth and pitch angles of the signal arriving at the IRS, respectively. \(\theta_{BS}^{AOD}\in[0,\pi]\) represents the departure angle of the BS transmitter signal. For IRS-MR links, channel \(\mathbf{h}_{r,n}\) under the nth time slot is assumed to obey Rice fading [22]:  
\[\mathbf{h}_{r,n}=\sqrt{\frac{k}{k+1}}\mathbf{\widetilde{h}}_{r,n}+\sqrt{\frac{1}{k+1}} \mathbf{\widetilde{h}}_{r,n}, \tag{7}\]  
where \(k\) is the Rician K factor, \(\mathbf{\widetilde{h}}_{r,n}\) is the NLOS component in the nth time slot of the IRS-MR link, which follows a Rayleigh distribution with a Jakes spectrum of power expansion, and \(\mathbf{\widetilde{h}}_{r,n}\) is the LOS component affected by the Doppler frequency shift. Therefore, \(\mathbf{\widetilde{h}}_{r,n}\) is:  
\[\mathbf{\widetilde{h}}_{r,n}=\beta p_{y}\Big{(}\varphi_{IRS}^{AOD},\theta_{IRS}^{ AOD}\Big{)}\otimes p_{z}\Big{(}\theta_{IRS}^{AOD}\Big{)}e^{j2\pi\mathbf{f}_{d} \mathbf{\widetilde{h}}_{r}}, \tag{8}\]where \(\beta\) denotes the complex gain of the IRS-MR link. \(\varphi_{IRS}^{AOD}\in[0,\pi]\) and \(\theta_{IRS}^{AOD}\in[0,\pi]\) denote the azimuth and pitch angles of the reflected signal leaving the IRS, respectively. \(f_{d}\) denotes the Doppler shift. The following discussion focuses on one time slot only, and \(t\) is omitted from the following discussion. The achievable rate \(R\) at MR is:  
\[R=\log_{2}\left(1+\frac{\left|\left(h_{r}^{H}\mathbf{\Theta}G+h_{d}^{H}\right) \mathbf{f}\right|^{2}}{\sigma^{2}}\right), \tag{9}\]  
The total power consumption of the IRS-assisted HST communication system includes the BS transmit signal power, the circuit consumption of the BS and MR, and the power consumption of the IRS reflection unit. Thus, the total power consumption of the system is [32]:  
\[P_{all}=\frac{1}{\xi}{\left\|\mathbf{f}\right\|}^{2}+P_{BS}+P_{IRS}+P_{MR}, \tag{10}\]  
where \(\xi\) is the power amplification efficiency of the BS. Let \(\xi=1\)[32]. \(P_{BS}\) indicates the circuit power consumption of the BS. \(P_{IRS}=NP_{R}\), where \(P_{R}\) denotes the power consumption generated by each reflective element of the IRS.