## 3 Established Connection Phase
### Channel Re-Estimation and Prediction  
The channel estimation can be performed in the discrete frequency domain. Referring to the model presented in Equation (9), by means of the DFT, the received pilot signal is:  
\[R_{k}=H_{k}X_{k}+Z_{k}\quad 0\leq k\leq N_{SC}-1 \tag{15}\]  
where \(X_{k}\) and \(Z_{k}\) are the DFT of the sampled version of \(x(t)\) and \(z(t)\), respectively, \(H_{k}\) is the channel representation in the discrete frequency domain, and the number of samples to compute DFT is exactly the number of sub-channels \(N_{SC}\). According to the orthogonal projection lemma criterion, detailed in [46], the Minimum Mean Square Error estimation of the channel can be obtained via the following relationship:  
\[\hat{H}_{k}=R_{k}\frac{X_{k}^{*}|H_{k}|^{2}}{E_{s0}|H_{k}|^{2}+Z_{k}^{2}}\quad 1 \leq k\leq N_{DFT} \tag{16}\]  
where, we recall, \(E_{s0}\) is the energy associated to the training pilot, while \(Z_{k}^{2}\) is the effect of disturbing signal \(z(t)\). By observing Equation (16), we can argue that the \(|H_{k}|^{2}\) values are not available, since \(H_{k}\) are the quantities to be estimated. This problem is solved by posing \(|H_{k}|^{2}=1\), that leads to reliable estimates if \(E_{s0}>Z_{k}^{2}\). Starting from Equation (16), the channel evolution can be described by a V-order AR model [47; 48; 49] as follows:  
\[\mathbf{h}_{k}(\ell)=\boldsymbol{\
abla}\mathbf{h}_{k}(\ell-1)+\mathbf{z}(\ell) \tag{17}\]  
where \(\mathbf{h}_{k}(\ell)\) is a vector describing the channel coefficients related to the \(k\)-th sub-carrier, and \(\boldsymbol{\
abla}\) is the \([V\times V]\) matrix related to the V-order AR model [47; 48; 49]. Based on such representation, we resort to Kalman filtering [50] to predict the channel evolution. A possible drawback of this approach concerns the system complexity. In fact, tracking the channel evolution via Kalman filtering is not so costly for a single sub-channel, while it may become significant if a huge number of samples is considered. However, it is important to note that, at this stage, once the number of sub-carriers \(N_{SC}\) is chosen, the Kalman-based prediction operates exclusively on \(N_{SC}\) samples. So, the matrix dimensions used for handling the Kalman filtering is \([VN_{SC}\times VN_{SC}]\) and, even though it may appear to lead to a huge processing cost, it is composed of all \([V\times V]\) matrices on the diagonal, hence, presenting \(V^{2}(N_{SC}^{2}-N_{SC})\) zeros. Since no matrix inversions are present in the processing, the largest part of the computational cost is related to product and sums, to be performed with a processing time in the order of frame duration (hundreds of milliseconds). Finally, we want to highlight here that channel prediction is operated both at the transmitter and receiver sides in place of channel estimation performed with overhead pilot signaling between nodes. Hence, prediction allows the reduction of latency, which represents a crucial issue in UWACs. Before proceeding, it is important to consider some key elements. First, it must be considered that the performance of the predictor (that is, the estimation accuracy) decreases in terms of prediction error variance when the coefficient to predict (in a time sense) is far from the last performed estimation. Hence, this fact suggests that, after predicting the channel for several frames, the receiver must proceed to re-estimate the channel by means of some pilot transmissions (as detailed in Figure 1). The motivation leading to requiring estimation after some prediction is twofold. First, after the CP tuning, it is expected that the delay spread may change. Second, the memory (in the sense of channel correlation) drops, thus meaning unreliability of the prediction. This is the reason for also introducing, in Figure 1, the pilot transmission during the EC stage, so as to provide the estimation to aid prediction. Therefore, a mechanism for re-estimating the channel and feeding the predictor with new estimated values is required.