## 3 Our Contribution
### UWSN Channel Model
#### 3.1.2 Characterization of the Small Scale Acoustic Channel
Let the relative intra-path delays be \(\delta\mathrm{T_{p,i}}\) with variance \(\sigma_{\mathrm{s_{p}}}^{2}\). Typically, Gaussian-distributed delays come into existence due to the height of the Gaussian-distributed surface or bottom and, hence, in that case, represent the variance on both the surface as well as bottom by \(\sigma_{\mathrm{s}}^{2}\) and \(\sigma_{\mathrm{b}}^{2}\), correspondingly. Therefore the following could be derived:  
\[\sigma_{\mathrm{s_{p}}}^{2}=\frac{1}{\mathrm{c}^{2}}\big{(}2\sin\theta_{ \mathrm{p}}\big{)}^{2}\Big{[}\mathrm{n_{sp}}\sigma_{\mathrm{s}}^{2}+\mathrm{n_ {bp}}\sigma_{\mathrm{b}}^{2}\Big{]} \tag{20}\]  
Thus, the mean and variance parameters of the scattering coefficients can be retrieved as (20) (21) and (22), respectively:  
\[\overline{\gamma}_{\mathrm{p}}(\mathrm{f})=\mu_{\mathrm{p0}}+\mu_{\mathrm{p}} \mathrm{S_{p}}\rho_{\mathrm{p}}(\mathrm{f}) \tag{21}\]\[2\sigma_{\rm p}^{2}({\rm f})=\mu_{\rm p}^{2}S_{\rm p}\Big{[}1-\rho_{\rm p}{}^{2}({ \rm f})\Big{]}+S_{\rm p}\
u_{\rm p}^{2}\approx\mu_{\rm p}^{2}S_{\rm p}\Big{[}1- \rho_{\rm p}{}^{2}({\rm f})\Big{]} \tag{22}\]  
where \(S_{\rm p}\) represents the number of intra-paths formed. Hence,  
\[\rho_{\rm p}({\rm f})={\rm E}\Big{\{}{\rm e}^{-{\rm i}2\pi{\rm f}S_{\rm p}{\rm i }}\Big{\}}={\rm e}^{-(2\pi{\rm f})^{2}\sigma_{\rm sp}^{2}}{}^{/2} \tag{23}\]  
Intra-Paths Correlation  
Scattering remains independent between paths stating that the reflection points of the paths remain noticeable. Therefore,  
\[{\rm E}\Big{\{}\gamma_{\rm p}({\rm f})\gamma_{\rm q}^{*}({\rm f})\Big{\}}= \overline{\gamma}_{\rm p}({\rm f})\overline{\gamma}_{\rm q}({\rm f})+\delta_{ \rm p,q}2\sigma_{\rm p}^{2}({\rm f}) \tag{24}\]  
Noticeably, though the propagation paths signify uncorrelated scattering,  
\[ \begin{split}{\rm E}\Big{\{}\Big{[}\gamma_{\rm p}({\rm f}_{1},{ \rm t}+\Delta{\rm t})-\overline{\gamma}_{\rm p}({\rm f}_{1})\Big{]}\Big{[} \gamma_{\rm p}({\rm f}_{2},{\rm t})-\overline{\gamma}_{\rm p}({\rm f}_{2}) \Big{]}^{*}\Big{\}}\\ =\mu_{\rm p}^{2}S_{\rm p}{\rm e}^{-(2\pi({\rm f}_{1}-{\rm f}_{2}) )^{2}\sigma_{\rm sp}^{2}/2}\times{\rm e}^{-(1-\alpha_{\rm sp})(2\pi{\rm f}_{1} )(2\pi{\rm f}_{2})\sigma_{\rm sp}^{2}[1-\rho_{\rm p}(\sqrt{2\alpha_{\rm sp}{ \rm f}_{1}{\rm f}_{2}})]}\\ \approx\rho_{\rm p}({\rm f}_{1}-{\rm f}_{2}){\rm e}^{-\pi{\rm B} _{\rm p}({\rm f}_{1})\Delta{\rm t}}2\sigma_{\rm p}({\rm f}_{1})\sigma_{\rm p}( {\rm f}_{2})\end{split}  \tag{25}\]  
\({\rm E}\Big{\{}\gamma_{\rm p}({\rm f})\gamma_{\rm q}^{*}({\rm f})\Big{\}}\) remains non-zero due to nonzero mean values. Path Correlation in Frequency Domain  
The frequency correlation of the small-scale path coefficients can be defined in terms of the function \({\rm E}\Big{\{}\gamma_{\rm p}({\rm f}+\Delta{\rm f},{\rm t})\gamma_{\rm q}^{* }({\rm f},{\rm t})\Big{\}}\)[24]. To estimate the frequency correlation, the PDF of the intra-path delays \(\delta{\rm T}_{{\rm p},{\rm i}}\) needs to be estimated and, for Gaussian-distributed delays along with zero mean and variance \(\sigma_{\delta_{\rm p}}^{2}\), the frequency correlation is (25). \[ \begin{split}{\rm E}\Big{\{}\Big{[}\gamma_{\rm p}({\rm f}+\Delta{ \rm f})-\overline{\gamma}_{\rm p}({\rm f}+\Delta{\rm f})\Big{]}\Big{[}\gamma_ {\rm p}({\rm f})-\overline{\gamma}_{\rm p}({\rm f})\Big{]}^{*}\Big{\}}\\ =\mu_{\rm p}^{2}S_{\rm p}\rho_{\rm p}(\Delta{\rm f})\Big{[}1- \rho_{\rm p}\Big{(}\sqrt{2\bar{\rm f}(\bar{\rm f}+\Delta{\rm f})}\Big{)}\Big{]} \\ \approx\rho_{\rm p}(\Delta{\rm f})2\sigma_{\rm p}({\rm f})\sigma_{ \rm p}({\rm f}+\Delta{\rm f})\end{split}  \tag{26}\]  
Here, it can be found that based on the standard deviation (i.e., \(\delta{\rm T}_{{\rm p},{\rm i}}\)), there can be a definite correlation between \(\gamma_{\rm p}\) coefficients within the signal bandwidth. However, the correlation might vary over the signal bandwidth. Since, in this paper, we intend to develop a multipath channel based data transmission, we assume the full correlation hypothesis across the signal bandwidth.