## 2 System Model  
### System Model  
As shown in Figure 1, we study a frame-based transmission system, with each frame containing multiple time slots. Each frame carries data from a possibly different user to the same receiver, e.g., a base station. The receiver has \(N_{R}\) antennas, and the transmitters have \(N_{T}\) antennas. The channel \(h_{l,f}\) in slot \(l=1,2,\ldots\) of frame \(f=1,2,\ldots\) is a vector with \(S=N_{R}N_{T}W\) entries, with \(W\) being the delay spread measured in number of transmission symbols within each frame \(f\), the multi-path channels \(h_{l,f}\in\mathcal{C}^{N_{R}N_{T}W\times 1}\) are characterized by fixed, frame-dependent, average path powers, path delays, Doppler spectra, and angles of arrival and departure [46]. For instance, in a frame \(f\), we may have a slow-moving user in line-of-sight condition subject to time-invariant fading, whereas in another, the channel may have significant scattering with fast temporal variations with a large Doppler frequency. In both cases, the frame is assumed to be short enough that average path powers, path delays, Doppler spectra, and angles of arrival and departure do not change within the frame [41; 42].  
As also seen in Figure 1, for each frame \(f\), we are interested in addressing the lag-\(\delta\) channel prediction problem, in which channel \(h_{l+\delta,f}\) is predicted based on the \(N\) past channels:  
\[H_{l,f}^{N}=[h_{l,f},\ldots,h_{l-N+1,f}]\in\mathcal{C}^{S\times N}. \tag{1}\]  
We adopt linear prediction with regressor \(V_{f}\in\mathcal{C}^{SN\times S}\), so that the prediction is given as  
\[\hat{h}_{l+\delta,f}=V_{f}^{\dagger}\text{vec}(H_{l,f}^{N}). \tag{2}\]  
The focus on linear prediction is justified by the optimality of linear estimation for Gaussian stationary processes [47], which provide standard models for fading channels in rich scattering environments.  
Assuming no prior knowledge of the channel model, we adopt a data-driven approach to the design of the predictor (2). Accordingly, to train the linear predictor (2), for any frame \(f\), the receiver is assumed to have available the training set  
\[\mathcal{Z}_{f}^{\text{tr}}=\{(x_{i,f},y_{i,f})\}_{i=1}^{\text{tr}}\equiv\{( \text{vec}(H_{l,f}^{N}),h_{l+\delta,f})\}_{l=N}^{\text{tr}+N-1} \tag{3}\]  
encompassing \(L^{\text{tr}}\) input-output examples. Dataset \(\mathcal{Z}_{f}^{\text{tr}}\) can be constructed from \(L^{\text{tr}}+N+\delta-1\) channels \(\{h_{1,f},\ldots,h_{L^{\text{tr}}+N+\delta-1,f}\}\) by using the lag-\(\delta\) channel \(h_{l+\delta,f}\) as label for the covariate vector \(\text{vec}(H_{l,f}^{N})\). In practice, the channel vectors \(h_{l,f}\) are estimated by using pilot symbols, and estimation noise can be easily incorporated in the model (see Section 2.5). Throughout, we implicitly assume that the channels \(h_{l,f}\) correspond to estimates available at the receiver.  
From dataset \(\mathcal{Z}_{f}^{\text{tr}}\) in (3), we write the corresponding \(L^{\text{tr}}\times SN\) input matrix \(X_{f}^{\text{tr}}=[x_{1,f}^{\dagger},\ldots,x_{L^{\text{tr}},f}^{\dagger}]^{\top}\), and the \(L^{\text{tr}}\times S\) target matrix \(\mathcal{Y}_{f}^{\text{tr}}=[y_{1,f}^{\dagger},\ldots,y_{L^{\text{tr}},f}^{ \dagger}]^{\top}\), so that the dataset can be expressed as the pair \(\mathcal{Z}_{f}^{\text{tr}}=(X_{f}^{\text{tr}},\mathcal{Y}_{f}^{\text{tr}})\).