## 4 Multi-Antenna Frequency-Selective Channels
### LSTD-Based Prediction Model  
Given the LSTD channel model (25) and (26), in this subsection we redefine the problem of predicting channel \(h_{l+\hat{s},f}=B_{f}d_{l+\hat{s},f}\) as the problem of estimating the feature matrix \(B_{f}\) and predicting the amplitude vector \(\hat{d}_{l+\hat{s},f}\) based on the available data. This will lead to a reduced-rank parametrization of the linear predictor (2).  
To start, we write the predicted channel \(\hat{h}_{l+\hat{s},f}\) as  
\[\hat{h}_{l+\hat{s},f}=\hat{\beta}_{f}\hat{d}_{l+\hat{s},f}, \tag{27}\]  
where \(\hat{\beta}_{f}\) and \(\hat{d}_{l+\hat{s},f}\) are the estimated feature matrix and the predicted amplitude vector, respectively. To define the corresponding predictor, we first observe that the input matrix \(H_{l,f}^{N}\) in (1) can be expressed by using (25) as  
\[H_{l,f}^{N}=B_{f}[d_{l,f},\ldots,d_{l-N+1,f}]. \tag{28}\]  
Assume now that we have an estimated feature matrix \(\hat{\beta}_{f}\). If this estimate is sufficiently accurate, the \(N\) past amplitudes \([d_{l,f},\ldots,d_{l-N+1,f}]\in\mathcal{C}^{K\times N}\) can be in turn estimated from \(H_{l,f}^{N}\) as  
\[[\hat{d}_{l,f},\ldots,\hat{d}_{l-N+1,f}]=\hat{\beta}_{f}^{\dagger}H_{l,f}^{N}. \tag{29}\]Consider now the prediction of the \(k\)-th amplitude \(d^{k}_{l+\delta,f}\). Generalizing (12), we adopt the linear predictor  
\[\hat{d}^{k}_{l+\delta,f}=(v^{k}_{f})^{\dagger}\text{vec}([\hat{d}^{k}_{l,f}, \ldots,\hat{d}^{k}_{l-N+1,f}]), \tag{30}\]  
where \(v^{k}_{f}\) is an \(N\times 1\) prediction vector, and  
\[[\hat{d}^{k}_{l,f},\ldots,\hat{d}^{k}_{l-N+1,f}]=(\hat{b}^{k}_{f})^{\dagger}H^{ N}_{l,f}\in\mathcal{C}^{1\times N} \tag{31}\]  
is the \(k\)-th row of the matrix (29), which represents the past \(N\) fading scalar amplitudes that correspond to the \(k\)-th feature \(b^{k}_{f}\). Plugging the prediction (30) into (27) yields the predicted channel \(\hat{h}_{l+\delta,f}\) (cf. (26))  
\[\hat{h}_{l+\delta,f}=\sum_{k=1}^{K}\hat{b}^{k}_{f}\hat{d}^{k}_{l+\delta,f}. \tag{32}\]  
As detailed in Appendix A, inserting (30) and (31) to (32), we can express the LSTD-based prediction (32) in the form (2) as  
\[\hat{h}_{l+\delta,f}=(V^{(K)}_{f})^{\dagger}\text{vec}(H^{N}_{l,f}), \tag{33}\]  
where the LSTD-based predictor matrix \(V^{(K)}_{f}\in\mathcal{C}^{5N\times S}\) is given as  
\[V^{(K)}_{f}=\sum_{k=1}^{K}v^{k}_{f}\otimes(\hat{b}^{k}_{f}(\hat{b}^{k}_{f})^{ \dagger}), \tag{34}\]  
where \(\otimes\) is the Kronecker product. Overall illustration of LSTD-based channel prediction is summarized in Figure 3.  
The LSTD prediction model (33) reduces the dimension of the learnable parameters from \(S^{2}N\) (for \(V_{f}\)) to \((S+N)K\) (for \(V^{(K)}_{f}\)). This complexity reduction comes at a minimal cost in terms of bias as long as the number of total features \(K\) is accurately chosen (a detailed discussion can be found in Section 4.7) and correlations across amplitudes \(d^{k}_{l,f}\) for different features \(k=1,\ldots,K\) are negligible.  
Figure 3: Illustration of the considered LSTD-based prediction model. (i) Estimate amplitudes \(d_{l,f}\) via the estimated long-term feature matrix \(\hat{B}_{f}\). (ii) Feature-wise short-term fading amplitude prediction \(\hat{d}^{k}_{l+\delta,f}\) based on feature-wise predictor \(v^{k}_{f}\) for \(k=1,\ldots,K\). (iii) Reconstruction of the predicted channel \(\hat{h}_{l+\delta,f}\) based on the feature matrix \(\hat{B}_{f}\) and predicted fading amplitude \(\hat{d}_{l+\delta,f}\).