# Angular Circle Array Multiple Input Multiple Output Underwater Optical Wireless Communications +
## 2 System Model
### Receiver Rotation  
Similar considerations are made for the receivers by the scattering angle \(\theta_{r}\) and the azimuthal angle \(\left]\varphi_{r}\right.\). The receiver plane is shown in Figure 3. The plane can be expressed as follows: \(Ax+By+Cz+D=0\), where \(A=\sin\theta_{r}\cos\left]\varphi_{r}\right.\), \(B=\sin\theta_{r}\sin\left]\varphi_{r}\right.\), \(C=\cos\theta_{r}\), and \(D=-Ax_{r}-By_{r}-Cz_{r}\), and \(\overrightarrow{n}_{wc}=(A,B,C)\) is the normal vector of the plane.  
Figure 3: Schematic diagram of the receiver plane and photon scattering.  
Figure 2: Transceiver parameters in the angular circle array MIMO system.  
From Figure 3, we assume that a photon is scattered \(\text{n}-1\) times, and after the nth scattering event due to collision, the photon passes through the receiving plane. In this optical event, the position of the photon before the scattering has the coordinates of point M \((x_{n-1},y_{n-1},z_{n-1})\), and the coordinates of the photon after the scattering event are at point N \((x_{n},y_{n},z_{n})\). The photon coordinates before the nth scattering are above the receiving plane, and the photon coordinates after the nth scattering are below the receiving plane, expressed as follows: \(Ax_{n-1}+By_{n-1}+Cz_{n-1}+D\geq 0\), \(Ax_{n}+By_{n}+Cz_{n}+D\leq 0\). Photons within the receiver plane that pass through the reception plane will be intercepted at the reception plane, and their final path length \(s_{n}\) can be expressed as follows: \(s_{n}=equ\cdot s_{n}\), where  
\[equ=\frac{Ax_{n-1}+By_{n-1}+Cz_{n-1}+D}{|Ax_{n-1}+By_{n-1}+Cz_{n-1}+D|+|Ax_{n} +By_{n}+Cz_{n}+D|}, \tag{3}\]  
and \(s_{r_{n}}\) is the path length traveled by the photon that underwent the last scattering.  
According to the above derivation, the actual coordinates of the photon arriving at the receiving plane can be written as follows:  
\[(x_{t},y_{t},z_{t})=[(x_{n},y_{n},y_{n})-(x_{n-1},y_{n-1},z_{n-1})]\cdot equ +(x_{n},y_{n},y_{n}), \tag{4}\]  
and \(\alpha_{r}\) between the direction vector of a photon arriving at the receiving plane and the receiver normal vector can be expressed as follows: \(\alpha_{r}=\arccos\biggl{(}(\overrightarrow{n}_{\text{rec}}\cdot \overrightarrow{M})/(\left|\overrightarrow{n}_{\text{rec}}\right|\left| \overrightarrow{M}\right|)\biggr{)}\), where \(\overrightarrow{M}=(x_{n}-x_{n-1},y_{n}-y_{n-1},z_{n}-z_{n-1})\), \(\left|\overrightarrow{n}_{\text{rec}}\right|=\sqrt{A^{2}+B^{2}+C^{2}}\), \(\left|\overrightarrow{M}\right|=s_{r_{n}}\). Then, \(\alpha_{r}\) can also be written as follows:  
\[\alpha_{r}=\arccos\biggl{(}\frac{A(x_{n}-x_{n-1})+B(y_{n}-y_{n-1})+C(z_{n}-z_ {n-1})}{\sqrt{A^{2}+B^{2}+C^{2}}s_{r_{n}}}\biggr{)}. \tag{5}\]  
As the receiver plane rotates, the conditions for determining whether a photon is successfully received need to be adjusted. The receiver is defined by three key parameters: the coordinates of the receiver center position \((x_{r},y_{r},z_{r})\), the receiver plane diameter \(A_{r}\), and FOV. Initially, \(x_{r}\) and \(y_{r}\) are both 0, and \(z_{r}\) is the communication distance. As the receiver rotates, photons that satisfy [16]: \(\pi-\alpha_{r}\leq\text{FOV}/2\), and  
\[A_{r}\geq\sqrt{(x_{n_{r}}-x_{r})^{2}+(y_{n_{r}}-y_{r})^{2}+(z_{n_{r}}-z_{r})^{ 2}}, \tag{6}\]  
are considered to be received. \((x_{n},y_{n},z_{n_{r}})\) is the coordinates of the photon projected onto the receiver plane.