# Cross-Layer Protocol Based on Directional Reception in Underwater Acoustic Wireless Sensor Networks
## 3 System Model
### Communication Model  
In an omnidirectional network, nodes adopt omnidirectional transmission and omnidirectional reception (OO) communication mode. In this mode, packets will be heard by all nodes within communication range. Its advantage is that communication nodes are easier to coordinate, but its disadvantage is that it is easy to cause packet collisions and waste channel capacity. Vector hydrophones have been widely used in hydroacoustic sensor networks due to their ability to increase the spatial reuse rate of the network. In this paper, an omnidirectional transmit and directional receive (OD) communication mode is used. In the OD communication mode, packets can only be received by nodes whose beams are facing the sending node. This mode reduces communication interference between nodes, but in order to increase network throughput, reservation coordination between nodes is also required. The comparison of the two communication modes is shown in Figures 1 and 2.  
Figure 1: Communication mode of OO.  
Figure 2: Communication mode of OD.  
Figure 1 is a visualization of the OO communication mode, where packets are received by all nodes in communication range. Figure 2 is a visualization of the OD communication mode. Nodes \(A\) and \(B\) are within the beam of node \(D\), and node \(C\) is not, so packets sent by nodes \(A\) and \(B\) can be received by node \(D\), while packets sent by node \(C\) cannot be received by node \(D\).  
Assume that there is a pair of transmitter \(T_{0}\) and receiver \(R\) and that there are \(K\) interfering \(T_{i}\) (\(i=1,2,\ldots M\)) at \(R\).  
In the OO mode, ignoring the influence of noise, the received signal can be expressed as:  
\[y(t,f)=\sum_{i=0}^{M}H_{i}(d_{i},f)\text{Re}\{s(t)e^{i2\pi ft}\}, \tag{3}\]  
where \(f\) is the communication carrier frequency, and \(d_{i}\) is the distance from \(T_{i}\) to \(R\), \(H_{i}(d_{i},f)\) is the channel gain of the \(T_{i}\) signal reaching \(R\), and \(x_{i}\) denotes the signal sent by \(T_{i}\).  
Assuming that the angle between the interferer \(T_{i}\) and the receiver \(R\) receives the pointing direction \(\theta_{i}\), and that the pointing reception uses \(p+2\cdot v_{c}\) of the vector combination mode, then the output of the signal sent by \(T_{i}\) at \(R\) is:  
\[y_{i}(t,f)=H_{i}(d_{i},f)\text{Re}\{s(t)e^{i2\pi ft}\}[1+2\cos(\theta_{i})] \tag{4}\]  
If the distance from \(T_{i}\) to the receiver is \(d_{i}\) and the center frequency is \(f\), the channel gain \(H_{i}(d,f)\) at the receiver of the underwater acoustic signal sent by \(T_{i}\) follows the complex Gaussian distribution, and the variance of this distribution is:  
\[\sigma_{i}^{2}(d_{i},f)=d_{i}^{-k}k(f)^{-d_{i}}, \tag{5}\]  
where \(k\) is the propagation coefficient related to the marine geographical environment (\(k\) is usually taken as 1.5), and \(k(f)^{-d_{i}}\) is the absorption coefficient related to distance \(d_{i}\) and frequency \(f\).  
The signal-to-interference noise ratio (\(SINR\)) model at \(R\) is expressed as:  
\[SINR=\frac{P_{t}\cdot\left(H_{i}(d_{i},f)\right)^{2}\cdot G(0)}{\sum\limits_{ j=1}^{M}P_{t}\cdot\left(H_{j}(d_{j},f)\right)^{2}\cdot G(\theta_{j})+P_{n}}, \tag{6}\]  
where \(P_{t}\) is the transmission power of \(T_{i}\), and \(P_{n}\) is the noise power at \(R\). Assuming that the sound pressure noise power is \(\sigma_{n}^{2}\), the vibration speed noise power is \(\frac{\sigma_{n}^{2}}{2}\), and the noise power is \(3\sigma_{n}^{2}\) in the linear combination \(p+2\cdot v_{c}\), so the values of \(P_{n}\) under omnidirectional and directional reception models are \(\sigma_{n}^{2}\) and \(3\sigma_{n}^{2}\), respectively. \(\theta_{j}\) denotes the angle between the orientation of \(T_{j}\) to \(R\) and the \(R\) extreme pointing orientation.