## 1 Introduction
The numerical results demonstrate that deploying IRS in the system increases the system's capacity while reducing its power consumption. The existing literature on IRS-assisted HST communication systems primarily focuses on the performance analysis related to the outage probability, spectrum efficiency, and channel capacity, with relatively limited research into optimizing systems' energy efficiency. Based on the aforementioned research findings, this paper considers the influence of the Doppler effect on channel modeling in the IRS-assisted HST communication scenario, taking into account the complexity reduction of algorithms and the improvement of system energy efficiency. The main contributions of this paper are as follows:  
1. Addressing the problem of decreased system energy efficiency due to line-of-sight blockages between the BS and mobile relay in the HST communication system. We analyze the Doppler spreading of non-line-of-sight (NLOS) components in the direct BS-mobile relay link and the Doppler frequency shift in the IRS-mobile relay reflected link under the movement of a high-speed train. An IRS-assisted HST communication system model is established, considering the BS transmit power constraint and the IRS unit modulus constraint, formulating a joint optimization problem of BS transmit beamforming and the IRS phase shift for energy efficiency maximization. This problem involves multi-variable-coupled non-convex fractional programming, which is challenging to solve directly. 2. To tackle this NP-hard problem, a solution approach is proposed. First, to address the issue of optimizing variable coupling, an alternating optimization strategy is employed to decouple the variables, dividing the original problem into two sub-problems: BS transmit beamforming and IRS phase-shift optimization. For the BS transmit beamforming sub-problem, the Dinkelbach method is utilized to convert the fractional objective function into a difference form, and then the successive convex approximation (SCA) algorithm is used to transform non-convex constraints into convex ones, which are solved using CVX. For the IRS phase-shift sub-problem, the Riemannian conjugate gradient (RCG) algorithm is used for optimization. Finally, an alternating iterative process is conducted to converge to a suboptimal solution. 3. Through simulations conducted under different system parameters, such as the number of IRS elements and the BS transmit power, the results demonstrate the effectiveness of the proposed algorithm in enhancing the energy efficiency of IRS-assisted HSTcommunication systems compared to baseline algorithms. It is further confirmed that increasing the number of IRS elements is more cost effective, in terms of improving HST communication systems' energy efficiency, than increasing the number of BS antennas, thus proving the necessity of employing IRS in HST communication systems. The rest of this article is organized as follows. In Section 2, we present the IRS-assisted HST communication system model. The energy efficiency optimization problem is formulated based on the proposed model. AO-EE algorithms for the optimization design are presented in Section 3. In Section 4, we provide numerical simulations and analysis. In Section 5, the conclusions of this paper and future research directions are presented.