## 3 Our Contribution
### UWSN Channel Model
#### 3.1.2 Characterization of the Small Scale Acoustic Channel
However, it cannot be accepted universally with all conditions, especially when \(\sigma_{\delta_{\rm p}}\) remains in the order of \(1/f_{0}\) even with low Ricean factor and, therefore, low correlation between \(\gamma_{\rm p}({\rm f}_{0})\) and \(\gamma_{\rm p}({\rm f}_{0}+{\rm B})\) in a wideband communication. Path Correlation in the Time Domain  
Similar to the frequency correlation, the time correlation of the scattering coefficients is defined in terms of \(\Big{\{}\gamma_{\rm p}({\rm f},{\rm f}+\Delta{\rm t})\gamma_{\rm p}^{*}({\rm t })\Big{\}}\), that obtains the motion-affect in the communication field, which further influences the coefficients \(\gamma_{\rm p}({\rm f},{\rm t})\) due to the time-varying micro-path delays. To assess \({\rm E}\Big{\{}\gamma_{\rm p}({\rm f},{\rm t}+\Delta{\rm t})\gamma_{\rm p}^{* }({\rm t})\Big{\}}\), at first, we have estimated the PSD of \(\delta{\rm T}_{{\rm p},{\rm i}}({\rm t})\). Here, without losing the generality, we consider that \(\delta{\rm T}_{{\rm p},{\rm i}}({\rm t})\) follows the first-order autoregressive process, given by:  
\[\delta{\rm T}_{{\rm p},{\rm i}}({\rm t}+\Delta{\rm t})=\alpha_{\rm sp}\delta{ \rm T}_{{\rm p},{\rm i}}({\rm t})+\omega_{\rm sp,{\rm i}}({\rm t}) \tag{27}\]where  
\[\omega_{\rm SP,i}(t)\,\sim\mathcal{N}\Big{(}0,\sigma_{\rm SP_{p}}^{2}\Big{(}1- \alpha_{\rm SP_{p}}^{2}\Big{)}\Big{)} \tag{28}\]  
\[\alpha_{\rm SP}={\rm e}^{-\pi{\rm B}\delta_{\rm SP}{\rm At}} \tag{29}\]  
We selected \({\rm B}\delta_{\rm p}\) as the 3-dB width of the PSD of \(\delta{\rm T}_{\rm p,i}(t)\). The derived relationship states that the two delay parameters can be similar if they are narrowly spaced in the time domain. The time correlation function is (29):  
\[ \begin{split}&{\rm E}\Big{\{}\Big{[}\gamma_{\rm p}(f,t+\Delta t)- \overline{\gamma}_{\rm p}(f)\Big{]}\left[\gamma_{\rm p}(f,t)-\overline{\gamma} _{\rm p}(f)\right]^{*}\Big{\}}\\ &=\mu_{\rm P}^{2}S_{\rm p}e^{-(1-\alpha_{\rm SP})(2\pi f)^{2} \sigma_{\rm SP}^{2}}\left[1-{\rm e}^{-\alpha_{\rm SP}(2\pi f)^{2}\sigma_{\rm SP }^{2}}\right]\\ &\approx 2\sigma_{\rm p}^{2}(f){\rm e}^{-\pi{\rm B}_{\rm p}(f) \Delta t}\end{split}  \tag{30}\]  
With the approximation for \(\Delta t\ll 1/{\rm B}_{\rm SP}\), the parameter signifying the effective Doppler bandwidth of \(\gamma_{\rm p}(f,t)\) is defined as:  
\[{\rm B}_{\rm p}(f)=\big{(}2\pi{\rm f}\sigma_{\rm SP}\big{)}^{2}{\rm B}_{\rm SP} \tag{31}\]  
Considering (25) and (29), the generalized expression can be obtained as (31), where the variable \(f_{1,2}\) refers to either of the two distinct frequencies \(f_{1}\) or \(f_{2}\) in the considered signal bandwidth. Statistical Model for \(\gamma_{\rm p}(f,t)\)  
Consider a supplementary auto-regressive process  
\[ \begin{split}&\Delta\gamma_{\rm p}(f,t+\Delta t)=\alpha_{\rm p}(f) \Delta\gamma_{\rm p}(f,t)+\omega_{\rm p}(f,t)\\ &\gamma_{\rm p}(f,t+\Delta t)=\overline{\gamma}_{\rm p}(f)+\Delta \gamma_{\rm p}(f,t+\Delta t)\end{split}  \tag{32}\]  
where \(\omega_{\rm p}(t)\) represents the complex Gaussian random process  
\[\omega_{\rm p}(t)\,\sim\,\mathcal{CN}(0,2\sigma_{\rm p}^{2}(f)(1-\alpha_{\rm p }^{2}(f))) \tag{33}\]  
where  
\[\alpha_{\rm p}(f)={\rm e}^{-\pi{\rm B}_{\rm p}(f)\Delta t} \tag{34}\]  
The autoregressive process (29) can be presented in terms of a Gaussian probabilistic distribution function, given as:  
\[\gamma_{\rm p}^{\prime}(f,t)\,\sim\,\mathcal{CN}\Big{(}\overline{\gamma}_{\rm p }(f),2\sigma_{\rm p}^{2}(f)\Big{)} \tag{35}\]  
Similarly, the auto-covariance function is  
\[{\rm E}\Big{\{}\Delta\gamma_{\rm p}(f,t+\Delta t)\Delta\gamma_{\rm p}^{*}(f,t )\Big{\}}=2\sigma_{\rm p}^{2}(f)){\rm e}^{-\pi{\rm B}_{\rm p}(f)\Delta t} \tag{36}\]  
The statistical equivalent model states that the two processes are statistically equivalent. To derive a complete acoustic channel model, embedding the frequency correlation to the auto-regressive process \(\Delta\gamma_{\rm p}(f,t)\) is a must. In this paper, the frequency axis is split into multiple steps of \(\Delta t_{\rm s}\).