## 4 Numerical Results  
In this section, we present the analysis of performance related to the proposed channel prediction-based adaptive transmission scheme. Simulations were performed by merging typical parameters of UWAC systems concerning the transmitter in terms of power and  
Figure 2: Protocol summary for CS and EC scenarios. bandwidth, and real data coming from measurement campaigns involving both channel and interference. Specifically, real multipath channel impulse responses, taken from the Watermark database [52], were considered to model the time-varying propagation. Channels were measured in a 740 m shallow water stretch of Oslojforden, with sounding operated in a frequency range from 10 kHz to 18 kHz. The acquisition was performed by the authors of the measurement campaigns [52] by acquiring signals in raw data during a continuous time acquisition. Hence, consecutive time-variant channel impulse responses were measured and, due to the relative movement of transmitter and receiver, it was possible to infer that the average relative speed was 4 km/h. Furthermore, the interference generated by acoustic sources were taken from some recordings available in the literature [53] and directly added to the received signal. The results were strictly dependent on the communication bandwidth, chosen to be equal to 8 kHz. The other simulation parameters are summarized in Table 2.  
One of the goals of the analysis was to prove the effectiveness of a hybrid channel prediction-estimation-based approach with respect to a conventional pure estimation mechanism. So, as further detailed, the performance comparison between two such cases is also provided. In order to, firstly, evaluate the accuracy of channel estimation, we report, in Figure 3, the mean square error (MSE) regarding the channel tracking as a function of the length of the statistics acquisition phase during the CS. Overall, it can be observed that, the longer the initial acquisition was (measured in terms of number of frames \(Q_{e}\)), the lower the MSE was. In detail, the use of only a frame (\(Q_{e}=1\)) did not lead a sufficiently reliable channel estimation and tracking with Kalman filtering, while, on the other hand, spending 20 frames for statistics acquisition allowed the MSE to be reduced by one order of magnitude. Furthermore, regarding the tracking operation, we show in Figure 4 the value of \(L_{adapt}\) in Equation (18) when different, consecutive channel realizations were considered. In particular, we discuss how the switching mechanism between tracking and estimation works. We recall that the need for re-estimation is driven by the minimum between the \begin{table}
\begin{tabular}{l c} \hline Bandwidth (\(B\)) & 8 kHz \\ OFDM sub-carriers (\(N_{sc}\)) & 128 \\ Transmit power (\(P_{xx}\)) & 183 dB @ 8V \\ Noise variance (\(N_{0}\)) & \(1.2\times 10^{-23}\ \mathrm{W/Hz}\) \\ Frame duration (\(T_{f}\)) & 20 ms \\ Slot duration (\(T_{s}\)) & 125 us \\ Interference acquisition frames (\(Q_{a}\)) & 10 \\ Channel acquisition frames (\(Q_{e}\)) & 10 \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 3: MSE for different lengths of acquisition phase time. memory of the auto-regressive model and the measurement of delay spread changes. In our simulation, we set the auto-regressive model memory as equal to 8 (frames) and, from Figure 4, it was possible to appreciate how such a value resulted, in general, as being the most appropriate one. In fact, most times, channel re-estimation results after 7 or 8 frames after prediction were exploited instead. However, for some channel realizations exhibiting significant changes in propagation characteristics (e.g., delay spread), \(L_{adapt}\) = 8 was too high and the accuracy of prediction might lower. Hence, a more frequent re-estimation was required. The particular case where \(L_{adapt}\) = 0 refers to the occurrence where estimation is required on two consecutive frames.