###### Abstract  
Accurate channel state information (CSI) is important for MIMO systems, especially in a high-speed scenario, fast time-varying CSI tends to be out of date, and a change in CSI shows complex nonlinearities. The kernel recursive least-squares (KRLS) algorithm, which offers an attractive framework to deal with nonlinear problems, can be used in predicting nonlinear time-varying CSI. However, the network structure of the traditional KRLS algorithm grows as the training sample size increases, resulting in insufficient storage space and increasing computation when dealing with incoming data, which limits the online prediction of the KRLS algorithm. This paper proposed a new sparse sliding-window KRLS (SSW-KRLS) algorithm where a candidate discard set is selected through correlation analysis between the mapping vectors in the kernel Hilbert spaces of the new input sample and the existing samples in the kernel dictionary; then, the discarded sample is determined in combination with its corresponding output to achieve dynamic sample updates. Specifically, the proposed SSW-KRLS algorithm maintains the size of the kernel dictionary within the sample budget requires a fixed amount of memory and computation per time step, incorporates regularization, and achieves online prediction. Moreover, in order to sufficiently track the strongly changeable dynamic characteristics, a forgetting factor is considered in the proposed algorithm. Numerical simulations demonstrate that, under a realistic channel model of 3 GPP in a rich scattering environment, our proposed algorithm achieved superior performance in terms of both predictive accuracy and kernel dictionary size than that of the ALD-KRLS algorithm. Our proposed SSW-KRLS algorithm with \(M=90\) achieved 2 dB NMSE less than that of the ALD-KRLS algorithm with \(v=0.001\), while the kernel dictionary was about 17% smaller when the speed of the mobile user was 120 km/h. channel prediction; time-varying channels; MIMO system; kernel methods; recursive least squares +
Footnote †: journal:  
In TDD systems, the user sends a sounding reference signal (SRS), and the base station performs channel estimation algorithms such as LS [5] and MMSE [6]. Then, the obtained CSI is used for downlink beamforming to realize cooperative signal processing. The coherence time of wireless channels is the time duration after which CSI is considered to be outdated. When the terminal is moving at a high speed, the Doppler frequency shift grows, and the time variability of the channel is severe, which leads to the shortening of channel coherence time. The measured uplink channel CSI cannot represent the real channel state of downlink slots, resulting in the mismatch between the downlink beamforming designed according to the measured CSI and the actual channel. In [1], with a typical CSI delay of 4 milliseconds, the user terminal speed at 30 km/h led to as much as 50% performance reduction versus in the low-mobility scenario at 3 km/h. In order to overcome the performance degradation caused by severe time-varying channels, [7] proposed a tractable user-centric CSI model and a robust beamforming design by taking deterministic equivalents [8] into account. However, when the user terminal is at high speed, the channel shows nonstationary characteristics, and the statistical characteristics of the channel also change with time, which cannot be used for the beamforming of high-speed time-varying channels. Another approach is to use a channel prediction algorithm [9; 10; 11; 12; 13; 14; 15; 16] to obtain more accurate CSI for beamforming. The kernel method is widely used in channel prediction algorithms due to its ability to track nonlinear channels, and its adaptation to time-varying channels [17; 18; 19; 20; 21; 22]. However, algorithms based on kernel methods face the problem in online prediction that the networks structure grows as the size of the training samples grows in time.