## 3 Our Contribution
### UWSN Channel Model
#### 3.1.3 Channel Gain Characterization  
The time varying CTF is derived as in (50),  
\[\mathrm{H}(\mathrm{f},\mathrm{t})=\mathrm{\overline{H}_{0}}\mathrm{\sum_{\rm{ p}}}\mathrm{h}_{\rm{p}}\mathrm{\widetilde{\gamma}}_{\rm{p}}(\mathrm{f},\mathrm{t}) \mathrm{e}^{-\mathrm{i}2\pi\mathrm{f}\mathrm{T}_{\rm{p}}} \tag{50}\]  
where \(\mathrm{\widetilde{\gamma}}_{\rm{p}}(\mathrm{f},\mathrm{t})=\mathrm{\gamma}_{ \rm{p}}(\mathrm{f},\mathrm{t})\mathrm{e}^{2\pi\mathrm{a}_{\rm{rp}}\mathrm{f}}\). The CTF characterizes the channel for the specific set of large-scale parameters (\(\mathrm{h}_{\rm{p}}\) and \(\mathrm{T}_{\rm{p}}\)), path statistics \(\mathrm{\widetilde{\gamma}}_{\rm{p}}(\mathrm{f})\), \(\mathrm{\widetilde{o}}_{\rm{p}}^{2}(\mathrm{f})\) and the Doppler scaling factors \(\mathrm{a}_{\rm{p}}\). In practice, these channel parameters might change over time, with a personalized rate. The time range at which these channel parameters change results in the derivation of the \"figures of merit\" of the considered acoustic communication system design. Channel gain is considered as one of the key figures of merit characterizing the efficacy of a communication system. In our mode, the instantaneous channel gain of a communication system working in the frequency range \([\mathrm{f}_{0},\mathrm{f}_{0}+\mathrm{B}]\) is derived as:  
\[\mathrm{\widetilde{G}}(\mathrm{t})=\frac{1}{\mathrm{B}}\int_{\mathrm{f}_{0}}^{ \mathrm{f}_{0}+\mathrm{B}}\big{|}\mathrm{H}(\mathrm{f},\mathrm{t})\big{|}^{2 }\mathrm{d}\mathrm{f} \tag{51}\]Thus, the regionally averaged gain can be defined as (52):  
\[\mathrm{G}=\mathrm{E}_{\overline{\gamma}}\Big{\{}\widetilde{G}(\mathrm{t})\Big{\}}. \tag{52}\]  
In case the radio bandwidth is sufficient so that all multipaths are precisely retrieved, the gain by each multipath component is:  
\[\mathrm{G}=\sum_{\mathrm{P}}\mathrm{G}_{\mathrm{P}}\mathrm{h}_{\mathrm{P}}^{2} \tag{53}\]  
where  
\[\mathrm{G}_{\mathrm{P}}=\frac{1}{\mathrm{B}}\int_{\mathrm{f}_{0}}^{\mathrm{f}_ {0}+\mathrm{B}}\overline{H}_{0}^{2}(\mathrm{f})\Big{[}\overline{\gamma}_{ \mathrm{P}}^{2}(\mathrm{f})+2\sigma_{\mathrm{P}}^{2}(\mathrm{f})\Big{]} \mathrm{d}\mathrm{f} \tag{54}\]  
In our model, the gain parameter \(\mathrm{G}\) is stated as the large-scale gain that primarily relies on both path coefficients \(\mathrm{h}_{\mathrm{P}}\) and related statistics \(\overline{\gamma}_{\mathrm{P}}(\mathrm{f})\), \(\sigma_{\mathrm{P}}^{2}(\mathrm{f})\). With the known path gain, the gain distribution is estimated analytically. Considering (13), gain parameter can be derived as:  
\[\mathrm{G}(\mathrm{t})=\sum_{\mathrm{P}}\mathrm{G}_{\mathrm{P}}\overline{h}_{ \mathrm{P}}^{2}\mathrm{e}^{-\mathrm{E}_{\mathrm{P}}\mathrm{\Delta}\mathrm{l} _{\mathrm{P}}(\mathrm{t})} \tag{55}\]  
With the path lengths of Gaussian distributed with mean \(\overline{l}_{\mathrm{P}}\) and variance \(\sigma_{\mathrm{l}_{\mathrm{P}}}^{2}\), the gain parameters can be estimated as the sum of log-normally distributed random processes [51]. We used the Fenton-Wilkinson method described in [51] to estimate the parameters, \(\overline{l}_{\mathrm{P}}\) and \(\sigma_{\mathrm{l}_{\mathrm{P}}}^{2}\).