# Angular Circle Array Multiple Input Multiple Output Underwater Optical Wireless Communications +
## 3 Channel Capacity of Underwater Wireless Optical MIMO Communication System  
We used the Monte Carlo method to perform photon tracking for simulation modeling of the UWOC system. A Monte Carlo flow chart is shown in Figure 9. The random path length can be obtained by \(s=-\ln(\xi)/c\), where c is the attenuation coefficient of light underwater, which is equal to the sum of the absorption coefficient a and the scattering coefficient b, and \(\varepsilon_{s}\) obeys uniform distribution in the range (0,1). The azimuthal angle satisfies a uniform distribution of \((0,2\pi)\), and the scattering angle is generated from the scattering phase function. We have chosen turbid port water \((a=0.366m^{-1},\)\(c=2.19m^{-1})\) as our water medium, because if the proposed communication system model demonstrates an improvement in channel capacity under turbid water conditions, it will achieve even better communication performance in relatively clearer water environments. At the same time, the Fournier-Forand (FF) phase function is chosen because it fits [PERSON]'s measurements well and is the most accurate theoretical scattering phase function with high accuracy. We set \(w_{th}=10^{-4}\). With the recorded photon weights and propagation distances, we can use them to calculate the subsequent channel impulse response (CIR).  
For SISO systems, the channel impulse response (CIR) can be obtained by the following steps [17]: (1) By counting the total path length \(d\) of each photon to reach the receiver, we can calculate the propagation time of the received photons by \(t=d/v_{water}\), where \(v_{water}\) is the speed of light in water. (2) Taking the range from the shortest propagation time of the photon \(t_{\min}\) to the longest propagation time \(t_{\max}\), we can divide them into equal intervals as \(\{T_{m};m=1,2,\ldots\}\). (3) Calculating the power sum of the photons in each time interval and denoting it as \(\{P_{m};m=1,2,\ldots\}\), the photon intensity \(H_{m}\) can be obtained by: \(H_{m}=P_{m}/S_{area}\), where \(S_{area}\) is the receiver area. Generalizing this to MIMO, we get the channel matrix. Using the channel matrix, the channel capacity can be calculated, which is expressed as follows [13]:  
\[C=\sum_{i=1}^{q}\log_{2}(1+SNR\cdot\lambda_{i}), \tag{14}\]  
where \(q\) is the number of non-zero eigenvalues in the H matrix, SNR is the signal-to-noise ratio, and \(\lambda_{i}\) is the eigenvalue of H matrix. Following this, inverse extrapolation through the channel capacity is performed to obtain the channel matrix H. Based on the value of H, the outage probability of the communication system in this scenario can also be determined.  
Figure 8: Transceiver cross-section diagram of the angular circle array MIMO communication system: (**a**) transmitters and (**b**) receivers.  
In this paper, we fit CIR by using the Gaussian model, and its mathematical expression is as follows:  
\[h_{g}(t)=\sum\
olimits_{l=1}^{n_{l}}a_{1}e^{-(\frac{t-b_{1}}{c_{1}})^{2}}, \tag{15}\]  
where \(n_{l}\) is Gaussian function order, and \(a_{1}\),\(b_{1}\), and \(c_{1}\) are real constants.