# Performance Analysis of a WPCN-Based Underwater Acoustic Communication System
## 2 System Model
### Channel Model  
We assume frequency-selective, sparse channels for underwater acoustic links. These channels are modeled by a tapped delay line (TDL) with the order of \(L\)[36]. Let \(h=[h(1),0,\ldots,0,h(2),0,\ldots,0,h(l)]^{T}_{(L\times 1)}\) represent the channel delay taps. The Rayleigh distribution is assumed to introduce randomness to the non-zero tap coefficient, which is denoted by \(h(l),l=1,2,\cdots,L_{p}\), where \(L_{p}\) is the number of non-zero taps. Due to the sparseness of typical UWA channels, we have \(L\gg L_{p}\). The total power is also normalized such that \(\sum\limits_{l=1}^{L_{p}}E\left[\left|h(l)\right|^{2}\right]=1\), where \(E[\cdot]\) is the expectation operation. The channel frequency  
Figure 3: The harvest-then-transmit protocol.  
Figure 2: Module components for a WPCN-based underwater acoustic terminal node.  
response corresponding to the \(n^{th}\) subcarrier between H-AP and the terminal node can be expressed as:  
\[H(n)=\sqrt{A[d,f(n)]}\sum_{l=1}^{L_{p}}h(l)e^{-\beta\pi nr(l)/T_{s}}e^{\beta 2\pi f_{d}(l)l_{m}}D[f_{d}(l)],\ \forall n\in\{1,2,\ldots,N\}, \tag{1}\]  
where \(T_{s}\) is the inverse of the subcarrier spacing and \(N\) denotes the number of subcarriers in one OFDM symbol starting at the time instant \(t_{m}\), \(d\) is the distance between the H-AP and terminal node, \(f(n)\) is the central frequency of the \(n^{th}\) subcarrier, and \(A[d,f(n)]\) is the path loss over the \(n^{th}\) subcarrier. In Equation (1), \(f_{d}(l)\) and \(\tau(l)\) denote the Doppler shift and the delay corresponding to the \(l^{th}\) path, respectively, and function \(D(\cdot)\) as reported in [37]:  
\[D(f)=\frac{1}{T_{s}}\int_{0}^{T_{s}}e^{\beta 2\pi f}dt=\begin{cases}1&f=0\\ \frac{e^{2\pi/T_{s}-1}}{2\pi f_{s}f_{s}}&f\
eq 0\end{cases}. \tag{2}\]  
According to [38], the path loss that occurs in an underwater acoustic channel over distance \(d\) for a signal with frequency \(f\) can be expressed as:  
\[A(d,f)=d^{-m}[\alpha(f)]^{-d}, \tag{3}\]  
where \(m\) is the spreading factor and its value is normally between 1 and 2 (for cylindrical and spherical spreading, respectively), \(\alpha(f)\) is the absorption loss that can be expressed by [PERSON]'s empirical formula as [39]:  
\[\alpha(f)=0.11\frac{f^{2}}{1+f^{2}}+44\frac{f^{2}}{4100+f^{2}}+2.75\times 10^{- 4}f^{2}+0.003. \tag{4}\]  
The ambient noise in the ocean includes the turbulence noise, the distant shipping noise, the wind-driven wave noise, and the thermal noise. The corresponding power spectral densities (PSDs) dB re \(\upmu\)Pa per Hz are, respectively, given by [40]:  
\[\begin{array}{l}10\log_{10}N_{t}(f)=17-30\log_{10}f,\\ 10\log_{10}N_{s}(f)=40+20(s-0.5)+26\log_{10}f-60\log_{10}(f+0.03),\\ 10\log_{10}N_{w}(f)=50+7.5w^{1/2}+20\log_{10}f-40\log_{10}(f+0.4),\\ 10\log_{10}N_{th}(f)=-15+20\log_{10}f,\end{array} \tag{5}\]  
where the shipping activity \(s\) ranges from 0 to 1 for low and high activity, respectively, and \(w\) corresponds to the wind speed measured in m/s. Then the PSD of the overall ambient noise can be expressed as:  
\[N(f)=N_{t}(f)+N_{s}(f)+N_{w}(f)+N_{th}(f). \tag{6}\]