# Performance Analysis of a WPCN-Based Underwater Acoustic Communication System
## 2 System Model
### Receiver  
For phase I transmission, the H-AP (e.g., the submarine or the AUVs in Figure 1) transmits the energy signal to the terminal node in the downlink. The baseband signal received at the terminal node can be given as:  
\[y_{d}=\sum_{k=1}^{K}\left[\sqrt{P_{d}(k)}H_{d}(k)x_{d}(k)+W_{d}(k)\right], \tag{7}\]  
where \(K\) denotes the number of subcarriers in one OFDM symbol for the downlink power transfer from the H-AP to the terminal node, \(P_{d}(k),\ \forall k\in\{1,2\cdots,K\}\) represents the adjustable transmit power for the \(k^{th}\) subcarrier at the H-AP, \(H_{d}(k)\) is the downlink channel coefficient given as Equation (1), \(x_{d}(k)\) denotes the transmitted baseband signal of the H-AP over the \(k^{th}\) subcarrier and satisfies \(E\left[|x_{d}(k)|^{2}\right]=1\), and \(W_{d}(k)\) is the fast Fouriertransformation (FFT) of the non-white Gaussian ambient noise terms with zero mean and variance of \(N[f_{d}(k)]\), where \(f_{d}(k)\) is the central frequency of the \(k^{th}\) subcarrier in the downlink. Therefore, the power received at the terminal node can readily be written as:  
\[P_{ER}=\sum_{k=1}^{\text{K}}\eta\Big{\{}P_{d}(k)|H_{d}(k)|^{2}+N[f_{d}(k)]\Big{\}}, \tag{8}\]  
where \(0<\eta<1\) is the energy harvesting efficiency at the terminal node. By substituting Equations (1) and (6) into Equation (8), the received SPL in the downlink energy transfer can be calculated, and the output power at the EH receiver of the terminal node can be obtained according to the SPL [7]. Table 1 illustrates the energy that can be harvested under various transmit power and transmission distances, where the harvested power is obtained according to the received SPL and can be referred to [7] for more details. Normally, the energy harvesting efficiency \(\eta\) is a non-linear function concerning the received SPL and can also be referred to [7]. Assuming unit block time, i.e., \(T=1\), the amount of energy harvested by the terminal node in the downlink can be expressed as:  
\[E_{u}=\tau_{0}TP_{ER}=\tau_{0}\eta\sum_{k=1}^{K}\Big{\{}P_{d}(k)|H_{d}(k)|^{2}+ N[f_{d}(k)]\Big{\}}. \tag{9}\]  
For phase II transmission, the terminal node uses the energy harvested in phase I to transfer information to the H-AP in the uplink. The received signal at H-AP is given by:  
\[y_{u}=\sum_{i=1}^{I}\biggl{[}\sqrt{P_{u}(i)}H_{u}(i)x_{u}(i)+W_{u}(i)\biggr{]}, \tag{10}\]  
where \(I\) denotes the number of subcarriers in one OFDM symbol for the uplink information transmission from the terminal node to the H-AP, \(P_{u}(i)\), \(\forall i\in\{1,2\cdots,I\}\) is the transmit power for the \(i^{th}\) subcarrier, \(H_{u}(i)\) is the uplink channel coefficient of the \(i^{th}\) subcarrier that can be obtained from Equation (1), \(x_{u}(i)\) is the BPSK-modulated signal transmitted from the terminal node over the \(i^{th}\) subcarrier and satisfies \(E\Bigl{[}\left|x_{u}(k)\right|^{2}\Bigr{]}=1\), and \(W_{u}(i)\) is the FFT of the non-white Gaussian ambient noise terms with zero mean and variance of \(N[f_{u}(i)]\), where \(f_{u}(i)\) is the central frequency of the \(i^{th}\) subcarrier in the uplink. For simplicity, we consider that the energy harvested at the terminal node in phase I is equally allocated to each subcarrier for the uplink information transmission in phase II. Then the transmit \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transmit** & **Transmission** & **Received SPL** & **Harvested** \\
**Power (W)** & **Distance (m)** & **(dB re \(\mu\)Pa)** & **Power (mW)** \\ \hline
20 & 100 & 163.1 & 36.3 \\
30 & 100 & 164.8 & 38.2 \\
40 & 100 & 166.1 & 39.8 \\
20 & 200 & 158.8 & 30.2 \\
30 & 200 & 160.6 & 32.9 \\
40 & 200 & 161.8 & 34.4 \\
20 & 300 & 155.9 & 27.9 \\
30 & 300 & 157.7 & 29.0 \\
40 & 300 & 158.9 & 30.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Harvested energy under various transmit power and transmission distances. (transducer directivity \(DI=10\) dB, signal center frequency \(f=15\) kHz, and spreading factor \(m=1.5\)).