# Performance Analysis of a WPCN-Based Underwater Acoustic Communication System
## 3 Performance Analysis
### Instantaneous SNR and Its PDF  
The instantaneous received SNR for the \(i^{th}\) subcarrier of the uplink from the terminal node to the H-AP is given as:  
\[\gamma_{u}(i)=\frac{P_{u}(i)|H_{u}(i)|^{2}}{N[f_{u}(i)]}. \tag{13}\]  
Since each subchannel is subjected to small-scale Rayleigh fading, the subchannel gain \(\left|H_{u}(i)\right|^{2}\) for the \(i^{th}\) subcarrier follows an exponential distribution, and the PDF of \(\left|H_{u}(i)\right|^{2}\) is given as [41]:  
\[f_{\left|H_{u}(i)\right|^{2}}(x)=\frac{1}{\Omega_{u}(i)}e^{-\frac{x}{\Omega_{ u}(i)}}U(x), \tag{14}\]  
where \(\Omega_{u}(i)=E[\left|H_{u}(i)\right|^{2}]=\lambda_{u}d^{-m}[\alpha(f_{i})]^{ -d}\) with \(\lambda_{u}\) is the power of Rayleigh fading and \(U(\cdot)\) is the unit step function.  
Since the subchannel gain \(\left|H_{d}(k)\right|^{2}\) follows an exponential distribution, according to Equation (11), \(P_{u}(i)\) follows a gamma distribution, and its PDF can be given as [42]:  
\[f_{P_{u}(i)}(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} U(x), \tag{15}\]  
where \(\alpha=\frac{\left[\underset{k=1}{\overset{K}{\sum}}(B\Omega_{d}(k)+C(k)) \right]^{2}}{\underset{k=1}{\overset{K}{\sum}}(B\Omega_{d}(k))^{2}},\beta= \frac{\underset{k=1}{\overset{K}{\sum}}(B\Omega_{d}(k)+C(k))}{\underset{k=1}{ \overset{K}{\sum}}(B\Omega_{d}(k))^{2}}\), and \(\Gamma(\cdot)\) is the gamma function.  
Let \(X=P_{u}(i)\) and \(Y=\frac{\left|H_{u}(i)\right|^{2}}{N[f_{u}(i)]}\). Then, from Equation (13), the instantaneous received SNR \(\gamma_{u}(i)\) can be expressed as \(\gamma_{u}(i)=XY\). For the two independent variables \(X_{1}\) and \(X_{2}\), the PDF of their product \(Z=X_{1}X_{2}\) can be expressed as:  
\[f_{Z}(z)=\int_{-\infty}^{+\infty}\frac{1}{|x|}f_{X_{1}}(x)f_{X_{2}}\left(\frac {z}{x}\right)dx, \tag{16}\]according to Equations (14) and (15), the PDF of \(\gamma_{u}(i)\) can be expressed as:  
\[f_{\gamma_{u}(i)}(z)=\int_{0}^{+\infty}\frac{\beta N[f_{u}(i)]}{\Gamma(\alpha) \Omega_{u}(i)}x^{\alpha-2}e^{-\beta x-\frac{zN[f_{u}(i)]}{x\Omega_{u}(i)}}\,dx. \tag{17}\]  
According to ([43], eq. (4.268)), the PDF of \(\gamma_{u}(i)\) can be transformed as:  
\[f_{\gamma_{u}(i)}(z)=\int_{0}^{+\infty}\frac{x^{\alpha-2}N[f_{u}(i)]\beta^{ \alpha}}{\Omega_{u}(i)\Gamma(\alpha)}G_{0,1}^{1,0}\bigg{(}\beta x\bigg{|}-0 \bigg{)}G_{0,0}^{1,0}\bigg{(}\frac{zN[f_{u}(i)]}{x\Omega_{u}(i)}\bigg{|}-0 \bigg{)}dx. \tag{18}\]  
With the aid of ([43], eq. (4.256)) and ([43], eq. (4.266)), the PDF of \(\gamma_{u}(i)\) can finally be derived as:  
\[f_{\gamma_{u}(i)}(z)=\frac{N[f_{u}(i)]\beta}{\Gamma(\alpha)\Omega_{u}(i)}G_{ 2,4}^{2,2}\bigg{(}\frac{N[f_{u}(i)]\beta z}{\Omega_{u}(i)}\bigg{|}\alpha-1,0,0,\alpha-1\bigg{)}, \tag{19}\]  
where \(G(\cdot)\) is the Meijer G function defined as ([43], eq. (4.254)):  
\[G_{p,q}^{m,n}\bigg{(}x\bigg{|}\begin{array}{c}a_{1},\ldots,a_{n},\ldots,a_{ p}\\ b_{1},\ldots,b_{m},\ldots,b_{q}\end{array}\bigg{)}=\frac{1}{2\pi i}\int\frac{ \prod\limits_{i=1}^{m}\Gamma(b_{i}-s)\prod\limits_{i=0}^{n}\Gamma(1-a_{i}+s) }{\prod\limits_{i=m+1}^{q}\Gamma(1-b_{i}+s)\prod\limits_{i=n+1}^{p}\Gamma(a_ {i}-s)}\,x^{s}\,\mathrm{d}s. \tag{20}\]