## 2 System Model  
We considered two typical user movement scenarios: urban road where the moving speed of users is 60 km/h, and a highway where the moving speed of users is 120 km/h. As shown in Figure 1, in TDD systems, the user sends a SRS, and the base station runs channel estimation algorithms such as LS and MMSE. The channel matrix is first estimated in frequency-domain; then, the channel frequency response is transformed into the time-domain by IDFT. The noise is evenly distributed in the whole time domain and is easy to be eliminated using a raised cosine window. Therefore, the effect of noise is very small compared to the user's high mobility. Due to the channel reciprocity in TDD systems, uplink channel CSI can be directly used in the design of downlink beamforming to realize cooperative signal processing. However, nonstationary fast fading characteristics of mobile environments bring challenges to multi-input multi-output (MIMO) communications.  
When the terminal is moving at a high speed, the Doppler frequency shift grows, and the time variability of the channel is severe. The measured uplink channel CSI cannot represent the real channel state of the downlink slots, resulting in performance degradation, and a mismatch between the measured CSI and the actual channel. As shown in Figure 2, during the SRS reporting cycle, the user can be regarded as moving in a fixed direction at a fixed speed. In two adjacent time slots, due to the short moving distance of the user, the amplitude and phase of the direct and scattered components have a certain correlation. When the user speed is low, an AR-based channel prediction method can achieve good performance by utilizing the linear correlation between adjacent time slots. However, when the user is moving at a high speed, the channel correlation between adjacent time slots presents nonlinear characteristics. Kernel methods are needed to exploit the nonlinear correlation characteristics of the channel.  
Figure 1: In TDD mobile communication systems, the beamforming performance of high-speed mobile terminal worsens. The user sends a SRS, and the base station runs channel estimation algorithms and performs beamforming. However, due to the user’s movement, the terminal can only obtain sidelobe gain in the downlink slots.  
Figure 2: In TDD mobile communication systems, the interval between slots S and D is short in two adjacent SRS periods, and the LoS and scattering components are correlated.  
The user sends SRS in special time slots, and the BS performs the channel measurement and estimation to obtain the channel matrix. As shown in Figure 3, the channel matrix measured by BS in the \(i\)-th SRS period is assumed to be \(H_{i}\in C^{N_{r}\times N_{t}}\). \(N_{t}\) and \(N_{r}\) represent the number of antennas of the BS and the mobile UE, respectively. The real and imaginary parts are separately processed in the subsequent algorithm, and \(H\) represents a real matrix in the later representation. The prediction order means that each channel matrix is related to the channel matrix measured in the previous-order SRS periods, so the input vector of the prediction system is  
\[u_{i}=[H_{i},H_{i+1},\ldots,H_{i+order-1}]. \tag{1}\]  
There are some complex and unknown dependencies between \(u_{i}\) and \(H_{i}:H_{i+order-1}=f(u_{i})=f([H_{i},H_{i+1},\ldots,H_{i+order-1}])\) that need to be exploited by kernel methods.