## 3 Our Contribution
### UWSN Channel Model
#### 3.1.2 Characterization of the Small Scale Acoustic Channel
Similarly, the time axis is split into multiple steps of \(\Delta t_{\rm s}\). Thus, the eventual vector derived would be:  
\[\Delta\gamma_{\rm p}[n]=\Big{[}\Delta\gamma_{\rm p}(f_{0},t_{0}+n\Delta t_{ \rm s}),\Delta\gamma_{\rm p}(f_{1},t_{0}+n\Delta t_{\rm s}),\ldots\Big{]}^{T} \tag{37}\]Defining matrix \(\mathrm{A_{P}}\), given as:  
\[\mathrm{A_{P}}=\mathrm{diag}\big{[}\alpha_{\mathrm{P}}(\mathrm{f_{k}})\big{]} \tag{38}\]  
where  
\[\alpha_{\mathrm{P}}(\mathrm{f_{k}})=\mathrm{e}^{-\pi\mathrm{B_{P}}(\mathrm{f_{k }})\Delta\mathrm{t_{s}}} \tag{39}\]  
Thus, applying the above derivations, an autoregressive process is obtained as:  
\[\Delta\gamma_{\mathrm{P}}[\mathrm{n}+1]=\mathrm{A_{P}}\Delta\gamma_{\mathrm{P} }[\mathrm{n}]+\mathrm{W_{P}}[\mathrm{n}],\ \ \mathrm{n}=0, \tag{40}\]  
where  
\[\mathrm{W_{P}}[\mathrm{n}]\sim\mathrm{C}\mathcal{N}\big{(}0,\mathrm{W_{P}} \big{)} \tag{41}\]  
and  
\[\big{[}\mathrm{W_{P}}\big{]}_{\mathrm{k,l}}=\big{[}1-\alpha_{\mathrm{P}}( \mathrm{f_{k}})\alpha_{\mathrm{P}}(\mathrm{f_{l}})\big{]}\alpha_{\mathrm{P}}( \mathrm{f_{k}}-\mathrm{f_{l}})2\alpha_{\mathrm{P}}(\mathrm{f_{k}})\alpha_{ \mathrm{P}}(\mathrm{f_{l}}) \tag{42}\]  
The optimal \(\mathrm{W_{P}}\) can ensure the expected frequency correlation as:  
\[\alpha_{\rm p}(f)={\rm e}^{-\pi{\rm B}_{\rm p}(f)\Delta t} \tag{34}\]0  
Once executing equation (40) iteratively over an expected time interval and for frequencies associated to the expected signal bandwidth, it becomes inevitable to add the mean values \(\overline{\gamma}_{\mathrm{P}}(\mathrm{f_{k}})\) to \(\Delta\gamma_{\mathrm{P}}(\mathrm{f_{k}},\mathrm{n}\Delta\mathrm{t_{s}})\). This results in a discrete time/frequency random process \(\gamma_{\mathrm{P}}^{\prime}(\mathrm{f},\mathrm{t})\), with statistical characteristics similar to the sampled process \(\gamma_{\mathrm{P}}(\mathrm{f},\mathrm{t})\). We ensure that the channel exhibits a precise replica of the actual channel, and it is further strengthened because of the statistical equivalence to the autoregressive process \(\gamma_{\mathrm{P}}(\mathrm{f},\mathrm{t})\). Moreover, our model considers dynamic and rough surface scattering, and hence, it does not consider any specific surface shape and applies the associated Doppler factor as \(\mathrm{a_{dp}}=\mathrm{v_{dp}}/\mathrm{C}\). In case drifting occurs in different directions and with a different value for each transmitter and receiver, then \(\mathrm{E}\Big{\{}\mathrm{e}^{\mathrm{i}2\pi\mathrm{f}\mathrm{a_{dp}}\Delta t }\Big{\}}\) considers averaging over the parameters \(\mathrm{\theta_{td}}\) and \(\mathrm{\theta_{rd}}\). With similar drifting in all directions, we obtain  
\[\alpha_{\rm p}(f)={\rm e}^{-\pi{\rm B}_{\rm p}(f)\Delta t} \tag{34}\]1  
where \(\mathrm{\theta_{td}^{\prime}}=\mathrm{\theta_{p}}-\mathrm{\theta_{td}}\) and \(\mathrm{\theta_{rd}^{\prime}}=\pi+\mathrm{\theta_{p}}+\mathrm{\theta_{rd}}\) are homogeneously distributed in the range of \([-\pi,\pi]\), and \(\mathrm{J_{0}}(. )\) refers the Bessel function of the zero order. With \(\mathrm{v_{td}}=\mathrm{v_{rd}}=\mathrm{v_{d}}\) and \(\mathrm{a_{d}}=\mathrm{v_{d}}/\mathrm{c}\), (44) minimized to \(\mathrm{J_{0}^{2}}(2\pi\mathrm{a_{d}}\mathrm{f}\mathrm{d}\mathrm{t})\). The moving component of the Doppler-effect can be obtained in the same way as discussed above, except for the motion components that can be compensated for by means of synchronization. Once performing initial synchronization, \(\mathrm{a_{vp}}=\mathrm{v_{vp}}/\mathrm{c}\) can be stated as the residual Doppler factor. If synchronization balances the Doppler factor (DF) in conjunction with the projection of the transceiver velocity onto the reference path \(\mathrm{p}=0\), the resulting DF will be (45):  
\[