The high speed railway (HSR) channel, exhibits a significant Doppler frequency shift and an extremely short coherence time, leading difficult challenges for precoding prediction. Based on the analysis of the measured channel, it was found that the HSR channel does not exhibit significant sparsity in the time-frequency domain. However, in the Doppler domain, the Channel State Information (CSI) exhibits evident slow variations, and the Doppler spectrum obtained at the same channel location have similarities. The paper proposes the LSTM neural network precoding algorithm for joint time-frequency domain prediction based on error modeling and repetitive HSR channel. Employing the temporality inherent in time-frequency domain channel data, we apply precoding to subsequent instances of the channel across time. Compared to the AR-PS algorithm, our algorithm improvement reaches 60%. Propose the concept of the HSR network channel state transition map (HSR-CSTM), through the continuous update of the HSR-CSTM, to obtain more accurate precoding capacity. The motion of high-speed train (HST) displays a repetitive feature, and the stability of reflectors on a long time scale results in similarity in the CSI at the same location. Our algorithm enables the reuse of precoding information at the same location in subsequent train operations. This feature significantly reduces the computational complexity. LSTM networks excel in handling the temporal data, allowing them to easily provide precoding results in advance multi-steps. This approach further reduces computational complexity.

From the perspective of information theory, we investigate a model for modulation and demodulation coding scheme of an oversampled signal of given bandwidth and a limited peak of power with a finite code length under an additive white Gaussian noise channel. We extend Gallager's Theorem of finite length code and connect the problem with the volume of a smooth manifold in a complex space. Two modifications of the orthogonal frequency-division multiplexing systems are proposed based on transforms mapping the ball to the manifolds. Analysis shows relations between the proposed methods and existing PAPR reduction techniques. The result shows that the PAPR reduction ability of a proposed cascade of a clipping and a companding transform is supported by the information theory. Features of the proposed methods, such as spectral spreading, computational complexity, and code rate loss, are investigated and simulation results show that one of the proposed method is one of the best state-of-art peak-to-average power ratio reduction methods.