# Angular Circle Array Multiple Input Multiple Output Underwater Optical Wireless Communications +
## 2 System Model
### Derivation of the Transceiver Structure  
Based on the above, the parameters required for designing the hemispherical transceivers can be determined. These parameters include the elevation angles \(\beta_{t}\) and \(\beta_{r}\), the azimuth angles \(\left|\varphi_{t}\right.\) and \(\left|\varphi_{r}\right.\), and the coordinate positions \(\left(x_{t}^{i},y_{t}^{j},z_{t}^{i}\right)\) and \(\left(x_{r}^{j},y_{r}^{j},z_{r}^{j}\right)\), where \(1\leq i=j\leq n_{r}+1\). \(n_{r}\) is the number of transceivers on the side. We design the transceiver positions as shown in Figure 4, aiming to achieve a \(360^{\circ}\) planar arrangement on the _xoy_ plane. Except for \(t_{1}\), all transmitters are tilted at specific angles, with their normal vectors pointing in the positive direction of the \(z\)-axis, to realize the three-dimensional structure of the transmitters. The structure of the receivers is similar to that of the transmitters. Next, the derivation of the angular parameters is performed. It should be noted that the transmitter \(t_{1}\) is located at the origin of the coordinate system and is similar to a SISO communication system, so there is no need to consider its azimuth and elevation parameterization. Based on the normal vector projection of the transmitter plane plotted in Figure 5a, the azimuthal angle can be expressed as follows:  
\[\varphi_{t}^{i}=(i-2)\frac{360^{\circ}}{n_{r}}, \tag{7}\]  
where \(2\leq i\leq n_{r}+1\). To ensure efficient use of each side transmitter, we define the transmitter plane inclination angle as \(\theta_{tin}\), such that the lower boundary of the beam divergence angle \(\theta_{div}\) is parallel to the transmitter plane, as shown in Figure 6a. The transmitter plane inclination can be expressed as follows: \(\theta_{tin}=\frac{180^{\circ}-\theta_{div}}{2}\). According to the definition of the elevation angle, it can be derived from \(\theta_{tin}\) as follows:  
\[\beta_{t}=90^{\circ}-\theta_{tin}. \tag{8}\]  
Figure 4: Projection diagram of the angular circle array MIMO communication system on the _xoy_ plane: (**a**) transmitters and (**b**) receivers. Figure 5: Projection diagram of the transceiver plane normal vectors of the angular circle array MIMO communication system on the _xoy_ plane: (**a**) transmitters and (**b**) receivers. Figure 6: Geometric structure of the angular circle array MIMO communication system in the _zox_ cross-section: (**a**) transmitters and (**b**) receivers. According to Figure 5b, it is evident that the receiver angle parameter derivation is similar to the transmitter angle parameter derivation. In both cases, the azimuth setting is the same, so the derivation process is not repeated here. Since the different side receivers are symmetrical about the center receiver, their elevation angles and FOVs are equal. To achieve omnidirectional coverage of the receiving plane, the plan view of the hemispherical receiver in the _voz_ plane needs to realize a \(360^{\circ}\) receiver plane arrangement. Therefore, the FOVs of each side receiver can be obtained as follows:  
\[\text{FOVs}=\frac{360^{\circ}}{n_{r}}. \tag{9}\]  
Here, the center receiver FOVc can be directly given by:  
\[\text{FOVs}=180^{\circ}-2\text{FOVs}. \tag{10}\]  
Similar to \(\theta_{lin}\), the plane inclination angle is given by the following equation: \(\theta_{rin}=\frac{180^{\circ}-\text{FOVs}}{2}\), and according to the definition of the elevation angle, it can be derived from \(\theta_{rin}\) as follows:  
\[\beta_{r}=90^{\circ}-\theta_{rin}. \tag{11}\]  
Based on the obtained transceiver angle parameters, the transceiver coordinate positions can be derived. As shown in Figure 7, \(r_{t}\), \(r_{r}\) are the radii of the projected circles of the transceiver center in the _xoy_ plane, respectively. In the following, the position of the transmitter coordinates is first deduced. According to the order of the transmitter serial number, the coordinates of transmitter \(t_{1}\left(x_{t}^{1},y_{t}^{1},z_{t}^{1}\right)\) are given as \(\left(0,0,z_{t}^{1}\right)\).