# Cooperative Opportunistic Pressure Based Routing for Underwater Wireless Sensor Networks
## 3 Our Proposed Work
### Channel Model  
We use binary phase shift keying (BPSK) as a modulation scheme because bandwidth in UWSNs is severely limited and BPSK allows data to be carried out efficiently on acoustic channel with given bandwidth. It is calculated as given in Equation (9) as follows:  
\[\begin{split} S_{0}=(\sqrt{2}E/T)(\cos(2\pi ft)\\ S_{1}=(\sqrt{2}E/T)(\cos(2\pi ft+\pi)\end{split} \tag{9}\]  
where \(S_{0}\) denotes when \(\theta\) is 0 and \(S_{1}\) shows \(\theta=180\). \(E\) is energy, \(T\) shows time period and \(f\) is the frequency of the signal which is transmitted over a path of length \(l\). Total attenuation due to spreading loss and absorption co-efficient on the basis of spreading is given as follows [25]:  
\[\alpha=\frac{10^{\alpha(f)}}{(10)} \tag{10}\]\[10\log(A(l,f))=k10\log(l)+l10\log(\alpha(f)) \tag{11}\]  
It is physically impossible to design a noise free channel because whenever signal is broadcasted in the channel, it is affected by different noises like turbulence \((N_{tu})\), wind \((N_{w})\), thermal \((N_{th})\), and shipping \((N_{s})\). The total noise (TN(f)) calculated as shown in Equation (12).  
\[TN(f)=N_{tu}(f)+N_{w}(f)+N_{th}(f)+N_{s}(f) \tag{12}\]  
where:  
\[10\log(N_{tu}(f))=17-30\log(f) \tag{13}\]  
\[10\log(N_{w}(f))=50+7.5w^{1/2}+20\log(f) \tag{14}\]  
\[10\log(N_{th}(f))=-15+20\log(f) \tag{15}\]  
\[10\log(N_{s}(f))=40+20(s-0.5)+26\log(f)-60\log(f+0.03) \tag{16}\]  
SNR of the received signals at the destination is calculated as follows in Equation (17):  
\[SNR=\frac{\mu}{\sigma} \tag{17}\]  
where \(\mu\) is the mean of the received signal with and without a noise \(S_{1}\) by Equation (9) and \(\sigma\) is the standard deviation of the signal. \(n_{01}\) denotes total number of received signals with and without noise at the destination. Mean is calculated as follows in Equation (18):  
\[\mu=\frac{(S_{0}+S_{1})}{n_{01}} \tag{18}\]  
where standard deviation is given below:  
\[\sigma=\sqrt{\frac{(\mu-S_{0})^{2}+(\mu-S_{1})^{2}}{n_{01}}} \tag{19}\]  
We assume the additive white gaussian noise (AWGN) model because it counters ambient noises in a better way. The channel capacity \(C\)[26] with bandwidth \(B\) at time \(T\) is given as follows:  
\[C=B\frac{1}{2T}\ln\left(1+\frac{\mu}{\sigma}\right) \tag{20}\]  
Figure 7 shows the basic cooperation technique procedure, S transmits its data packet to the destination node D directly and via a relay node R. Here, the noise and the fading are denoted by \(n\) and \(g\), respectively. The relationship between the transmitted and the received signals are given in Equations (21)-(23) [27]. The faded signal received at D directly from S is given as below:  
\[Y_{SD}=X_{S}g_{SD}+n_{SD} \tag{21}\]  
Signals received at R, and from R to D are given as:  
\[Y_{SR}=X_{S}g_{SR}+n_{SR} \tag{22}\]  
\[Y_{RD}=Y_{SR}g_{RD}+n_{RD} \tag{23}\]  
\(X_{S}\) is the original signal generated and transmitted by S, \(Y_{SD}\) is the signal received at D directly from S, \(Y_{SR}\) is the received signal at R and \(Y_{RD}\) is the signal received at D via R. \(g_{SD}\), \(g_{SR}\) and \(g_{RD}\) denote channel gains over source to destination, source to relay and relay to destination links, respectively.  
\(n_{SD}\) and \(n_{SR}\) denote channel noises over source to destination and to relay links, respectively. \(n_{RD}\) is the channel noise over relay to destination link.  
At D, MRC technique is used to combine two different copies of the same signal, one received directly from S and the other via R. Thus the signal at D after applying MRC is given as [28]:  
\[Y_{D}=Y_{SD}+\frac{Y_{SR}Y_{RD}}{1+Y_{SR}+Y_{RD}} \tag{24}\]