## 3 Our Contribution
### UWSN Channel Model
#### 3.1.2 Characterization of the Small Scale Acoustic Channel  
As discussed above, the CTF (14) can achieve only the large-scale effect and is unable to retrieve any small-scale phenomena such as scattering in acoustic medium. On the contrary, scattering plays a decisive role in signal strength over propagation. An acoustic signal with the frequency \(f\) suffers scattering on wavy surfaces and the objects having dimension of the order of a few signal wavelengths equal to c/f. For illustration, the wavelength associated with an acoustic frequency element of 15 kHz is 0.1 m, and hence the distance can be stated as \"small.\" To develop a scattering model in a UWSN channel, it is necessary to place emphasis on a single path, say, p. In the previous section, this path was designed by considering respective path gain \(\mathrm{h_{p}}\) and the propagation delay \(\mathrm{T_{p}}\). However, in practice, if scattering exists within the acoustic medium, particularly towards path p, it usually gets split into multiple micro-sized paths. Mathematically,  
\[\mathrm{H(f)}=\overline{H}_{\mathrm{o}}(\mathrm{f})\underset{\mathrm{P}}{ \sum}\underset{\mathrm{i}}{\sum}\mathrm{h_{p,i}}\;\mathrm{e}^{-\mathrm{j2\pi f }\mathrm{T_{p,i}}} \tag{16}\]  
where \(\mathrm{h_{p,i}}\) refers to the gain parameter of the intra-paths and \(\mathrm{T_{p,i}}=\mathrm{T_{p}}+\delta\mathrm{T_{p,i}}\) states the delay introduced across the intra-paths formed. Considering dynamicity in the acoustic network, \(\mathrm{h_{p,i}}\) and \(\delta\mathrm{T_{p,i}}\) are considered as random variables and, therefore, in our model, the small-scale fading coefficient is defined as (17):  
\[\gamma_{\mathrm{p}}(\mathrm{f})=\frac{1}{\mathrm{h_{p}}}\underset{\mathrm{i} \geq 0}{\sum}\mathrm{h_{p,i}}\;\mathrm{e}^{-\mathrm{j2\pi f}\delta\mathrm{T_{p,i}}} \tag{17}\]  
Thus, the eventual CTF can be derived as:  
\[\mathrm{H(f)}=\overline{H}_{\mathrm{o}}(\mathrm{f})\underset{\mathrm{P}}{ \sum}\mathrm{h_{p}}\gamma_{\mathrm{p}}(\mathrm{f})\mathrm{e}^{-\mathrm{j2\pi f }\mathrm{T_{p}}} \tag{18}\]  
Probability Density Function (PDF) for \(\gamma_{\mathrm{p}}(\mathrm{f})\)  
As the scattering locations can be far apart with the distance of \(\lambda\), \(\mathrm{h_{p,i}}\) is supposed to be equivalent for different intra-paths. On the contrary, the phases of \(\mathrm{2\pi f}\delta\mathrm{T_{p,i}}\) may differ between the different intra-paths, thus leading to variation in \(\gamma_{\mathrm{p}}(\mathrm{f})\). Assuming that the comprising terms of \(\gamma_{\mathrm{p}}(\mathrm{f})\) in (17) are diettically distributed, the central limit theorem (CLT) applies Gaussian distribution for \(\gamma_{\mathrm{p}}(\mathrm{f})\) with a higher number of micro-paths. The component with stable delay can be considered, and hence the distortion can be derived as  
\[\gamma_{\mathrm{p}}(\mathrm{f})=\gamma_{\mathrm{p,0}}+\underset{\mathrm{i} \geq 1}{\sum}\gamma_{\mathrm{p,i}}\mathrm{e}^{-\mathrm{j2\pi f}\delta\mathrm{T_{p,i}}} \tag{19}\]  
where \(\gamma_{\mathrm{p,0}}\) signifies the coefficient of the stable path with \(\delta\mathrm{T_{p,0}}=0\). Typically, \(\gamma_{\mathrm{p}}(\mathrm{f})\) is a complex Gaussian with the mean and standard deviation of \(\overline{\gamma}_{\mathrm{p}}(\mathrm{f})\) and \(2\sigma_{\mathrm{p}}^{2}(\mathrm{f})\), respectively. Such distributions can be applied for both the large-scale parameters \(\mathrm{h_{p}}\) and \(\mathrm{T_{p}}\), as well as small-scale parameters \(\overline{\gamma}_{\mathrm{p}}(\mathrm{f})\) and \(\sigma_{\mathrm{p}}^{2}(\mathrm{f})\). With the known variables of (17), the parameters \(\overline{\gamma}_{\mathrm{p}}(\mathrm{f})\) and \(\sigma_{\mathrm{p}}^{2}(\mathrm{f})\) can be estimated using an analytical or experimental process. To estimate the parameter using analytical measures, consider that the magnitude of the micro-path has the mean and variance of \(\mu_{\mathrm{p}}\) and \(\
u_{\mathrm{p}}^{2}\ll\mu_{\mathrm{p}}^{2}\), respectively.